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Integer Set Library: Manual
Version: isl-0.20

Sven Verdoolaege
July 28, 2018

Contents
1

User Manual
3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.1 Backward Incompatible Changes . . . . . . . . . . . . . . .
3
1.2 License . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3.1 Installation from the git repository . . . . . . . . . . . . . . .
7
1.3.2 Common installation instructions . . . . . . . . . . . . . . .
8
1.4 Integer Set Library . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.4.1 Memory Management . . . . . . . . . . . . . . . . . . . . .
9
1.4.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.3 Return Types . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.4 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.5 Sets and Relations . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.6 Error Handling . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.7 Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.8 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.9 Local Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.4.10 Creating New Sets and Relations . . . . . . . . . . . . . . . . 37
1.4.11 Inspecting Sets and Relations . . . . . . . . . . . . . . . . . 43
1.4.12 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.4.13 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.4.14 Input and Output . . . . . . . . . . . . . . . . . . . . . . . . 69
1.4.15 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
1.4.16 Unary Operations . . . . . . . . . . . . . . . . . . . . . . . . 89
1.4.17 Binary Operations . . . . . . . . . . . . . . . . . . . . . . . 122
1.4.18 Ternary Operations . . . . . . . . . . . . . . . . . . . . . . . 157
1.4.19 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
1.4.20 Associative arrays . . . . . . . . . . . . . . . . . . . . . . . 160
1.4.21 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
1.4.22 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
1.4.23 Bounds on Piecewise Quasipolynomials and Piecewise Quasipolynomial Reductions . . . . . . . . . . . . . . . . . . . . . . . 163
1.4.24 Parametric Vertex Enumeration . . . . . . . . . . . . . . . . 164
1.5 Polyhedral Compilation Library . . . . . . . . . . . . . . . . . . . . 165
1

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Implementation Details
2.1 Sets and Relations . . . . . . . . . . . . . . . . . . . . . . .
2.2 Simple Hull . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Parametric Integer Programming . . . . . . . . . . . . . . .
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2.3.2 The Dual Simplex Method . . . . . . . . . . . . . .
2.3.3 Gomory Cuts . . . . . . . . . . . . . . . . . . . . .
2.3.4 Negative Unknowns and Maximization . . . . . . .
2.3.5 Preprocessing . . . . . . . . . . . . . . . . . . . . .
2.3.6 Postprocessing . . . . . . . . . . . . . . . . . . . .
2.3.7 Context Tableau . . . . . . . . . . . . . . . . . . .
2.3.8 Experiments . . . . . . . . . . . . . . . . . . . . .
2.3.9 Online Symmetry Detection . . . . . . . . . . . . .
2.4 Coalescing . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Transitive Closure . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Computing an Approximation of Rk . . . . . . . . .
2.5.3 Checking Exactness . . . . . . . . . . . . . . . . .
2.5.4 Decomposing R into strongly connected components
2.5.5 Partitioning the domains and ranges of R . . . . . .
2.5.6 Incremental Computation . . . . . . . . . . . . . . .
2.5.7 An Omega-like implementation . . . . . . . . . . . .

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1.6

2

3

1.5.1 Schedule Trees . . . . . . . .
1.5.2 Dependence Analysis . . . . .
1.5.3 Scheduling . . . . . . . . . .
1.5.4 AST Generation . . . . . . .
Applications . . . . . . . . . . . . . .
1.6.1 isl_polyhedron_sample .
1.6.2 isl_pip . . . . . . . . . . .
1.6.3 isl_polyhedron_minimize
1.6.4 isl_polytope_scan . . . .
1.6.5 isl_flow . . . . . . . . . . .
1.6.6 isl_codegen . . . . . . . .
1.6.7 isl_schedule . . . . . . . .

Further Reading

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247

2

Chapter 1

User Manual
1.1

Introduction

isl is a thread-safe C library for manipulating sets and relations of integer points
bounded by affine constraints. The descriptions of the sets and relations may involve
both parameters and existentially quantified variables. All computations are performed
in exact integer arithmetic using GMP or imath. The isl library offers functionality
that is similar to that offered by the Omega and Omega+ libraries, but the underlying
algorithms are in most cases completely different.
The library is by no means complete and some fairly basic functionality is still
missing. Still, even in its current form, the library has been successfully used as a
backend polyhedral library for the polyhedral scanner CLooG and as part of an equivalence checker of static affine programs. For bug reports, feature requests and questions,
visit the discussion group at http://groups.google.com/group/isl-development.

1.1.1

Backward Incompatible Changes

Changes since isl-0.02
• The old printing functions have been deprecated and replaced by isl_printer
functions, see Input and Output.
• Most functions related to dependence analysis have acquired an extra must argument. To obtain the old behavior, this argument should be given the value 1.
See Dependence Analysis.
Changes since isl-0.03

• The function isl_pw_qpolynomial_fold_add has been renamed to isl_pw_qpolynomial_fold_fold.
Similarly, isl_union_pw_qpolynomial_fold_add has been renamed to isl_union_pw_qpolynomial_fold_

3

Changes since isl-0.04
• All header files have been renamed from isl_header.h to isl/header.h.
Changes since isl-0.05
• The functions isl_printer_print_basic_set and isl_printer_print_basic_map
no longer print a newline.
• The functions isl_flow_get_no_source and isl_union_map_compute_flow
now return the accesses for which no source could be found instead of the iterations where those accesses occur.
• The functions isl_basic_map_identity and isl_map_identity now take a
map space as input. An old call isl_map_identity(space) can be rewritten
to isl_map_identity(isl_space_map_from_set(space)).
• The function isl_map_power no longer takes a parameter position as input.
Instead, the exponent is now expressed as the domain of the resulting relation.
Changes since isl-0.06
• The format of isl_printer_print_qpolynomial’s ISL_FORMAT_ISL output
has changed. Use ISL_FORMAT_C to obtain the old output.
• The *_fast_* functions have been renamed to *_plain_*. Some of the old
names have been kept for backward compatibility, but they will be removed in
the future.
Changes since isl-0.07
• The function isl_pw_aff_max has been renamed to isl_pw_aff_union_max.
Similarly, the function isl_pw_aff_add has been renamed to isl_pw_aff_union_add.
• The isl_dim type has been renamed to isl_space along with the associated
functions. Some of the old names have been kept for backward compatibility,
but they will be removed in the future.
• Spaces of maps, sets and parameter domains are now treated differently. The
distinction between map spaces and set spaces has always been made on a conceptual level, but proper use of such spaces was never checked. Furthermore, up
until isl-0.07 there was no way of explicitly creating a parameter space. These
can now be created directly using isl_space_params_alloc or from other
spaces using isl_space_params.
• The space in which isl_aff, isl_pw_aff, isl_qpolynomial, isl_pw_qpolynomial,
isl_qpolynomial_fold and isl_pw_qpolynomial_fold objects live is now
a map space instead of a set space. This means, for example, that the dimensions
of the domain of an isl_aff are now considered to be of type isl_dim_in

4

instead of isl_dim_set. Extra functions have been added to obtain the domain
space. Some of the constructors still take a domain space and have therefore
been renamed.
• The functions isl_equality_alloc and isl_inequality_alloc now take
an isl_local_space instead of an isl_space. An isl_local_space can be
created from an isl_space using isl_local_space_from_space.
• The isl_div type has been removed. Functions that used to return an isl_div
now return an isl_aff. Note that the space of an isl_aff is that of relation.
When replacing a call to isl_div_get_coefficient by a call to isl_aff_get_coefficient
any isl_dim_set argument needs to be replaced by isl_dim_in. A call to
isl_aff_from_div can be replaced by a call to isl_aff_floor. A call to
isl_qpolynomial_div(div) call be replaced by the nested call
isl_qpolynomial_from_aff(isl_aff_floor(div))
The function isl_constraint_div has also been renamed to isl_constraint_get_div.
• The nparam argument has been removed from isl_map_read_from_str and
similar functions. When reading input in the original PolyLib format, the result
will have no parameters. If parameters are expected, the caller may want to
perform dimension manipulation on the result.
Changes since isl-0.09
• The schedule_split_parallel option has been replaced by the schedule_split_scaled
option.
• The first argument of isl_pw_aff_cond is now an isl_pw_aff instead of an
isl_set. A call isl_pw_aff_cond(a, b, c) can be replaced by
isl_pw_aff_cond(isl_set_indicator_function(a), b, c)
Changes since isl-0.10
• The functions isl_set_dim_has_lower_bound and isl_set_dim_has_upper_bound
have been renamed to isl_set_dim_has_any_lower_bound and isl_set_dim_has_any_upper_bound.
The new isl_set_dim_has_lower_bound and isl_set_dim_has_upper_bound
have slightly different meanings.
Changes since isl-0.12
• isl_int has been replaced by isl_val. Some of the old functions are still
available in isl/deprecated/*.h but they will be removed in the future.
• The functions isl_pw_qpolynomial_eval, isl_union_pw_qpolynomial_eval,
isl_pw_qpolynomial_fold_eval and isl_union_pw_qpolynomial_fold_eval
have been changed to return an isl_val instead of an isl_qpolynomial.
5

• The function isl_band_member_is_zero_distance has been removed. Essentially the same functionality is available through isl_band_member_is_coincident,
except that it requires setting up coincidence constraints. The option schedule_outer_zero_distance
has accordingly been replaced by the option schedule_outer_coincidence.
• The function isl_vertex_get_expr has been changed to return an isl_multi_aff
instead of a rational isl_basic_set. The function isl_vertex_get_domain
has been changed to return a regular basic set, rather than a rational basic set.
Changes since isl-0.14
• The function isl_union_pw_multi_aff_add now consistently computes the
sum on the shared definition domain. The function isl_union_pw_multi_aff_union_add
has been added to compute the sum on the union of definition domains. The
original behavior of isl_union_pw_multi_aff_add was confused and is no
longer available.
• Band forests have been replaced by schedule trees.
• The function isl_union_map_compute_flow has been replaced by the function isl_union_access_info_compute_flow. Note that the may dependence
relation returned by isl_union_flow_get_may_dependence is the union of
the two dependence relations returned by isl_union_map_compute_flow. Similarly for the no source relations. The function isl_union_map_compute_flow
is still available for backward compatibility, but it will be removed in the future.
• The function isl_basic_set_drop_constraint has been deprecated.

• The function isl_ast_build_ast_from_schedule has been renamed to isl_ast_build_node_from_schedu
The original name is still available for backward compatibility, but it will be removed in the future.
• The separation_class AST generation option has been deprecated.
• The functions isl_equality_alloc and isl_inequality_alloc have been
renamed to isl_constraint_alloc_equality and isl_constraint_alloc_inequality.
The original names have been kept for backward compatibility, but they will be
removed in the future.
• The schedule_fuse option has been replaced by the schedule_serialize_sccs
option. The effect of setting the schedule_fuse option to ISL_SCHEDULE_FUSE_MIN
is now obtained by turning on the schedule_serialize_sccs option.
Changes since isl-0.17
• The function isl_printer_print_ast_expr no longer prints in C format by
default. To print in C format, the output format of the printer needs to have been
explicitly set to ISL_FORMAT_C. As a result, the function isl_ast_expr_to_str
no longer prints the expression in C format. Use isl_ast_expr_to_C_str instead.
6

• The functions isl_set_align_divs and isl_map_align_divs have been deprecated. The function isl_set_lift has an effect that is similar to isl_set_align_divs
and could in some cases be used as an alternative.
Changes since isl-0.19
• Zero-dimensional objects of type isl_multi_pw_aff or isl_multi_union_pw_aff
can now keep track of an explicit domain. This explicit domain, if present, is
taken into account by various operations that take such objects as input.

1.2

License

isl is released under the MIT license.
Permission is hereby granted, free of charge, to any person obtaining a copy of this
software and associated documentation files (the "Software"), to deal in the Software
without restriction, including without limitation the rights to use, copy, modify, merge,
publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons
to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies
or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY
KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE
AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR
THE USE OR OTHER DEALINGS IN THE SOFTWARE.
Note that by default isl requires GMP, which is released under the GNU Lesser
General Public License (LGPL). This means that code linked against isl is also linked
against LGPL code.
When configuring with --with-int=imath or --with-int=imath-32, isl will
link against imath, a library for exact integer arithmetic released under the MIT license.

1.3

Installation

The source of isl can be obtained either as a tarball or from the git repository. Both
are available from http://isl.gforge.inria.fr/. The installation process depends on how
you obtained the source.

1.3.1

Installation from the git repository

1. Clone or update the repository

7

The first time the source is obtained, you need to clone the repository.
git clone git://repo.or.cz/isl.git
To obtain updates, you need to pull in the latest changes
git pull
2. Optionally get imath submodule
To build isl with imath, you need to obtain the imath submodule by running
in the git source tree of isl
git submodule init
git submodule update
This will fetch the required version of imath in a subdirectory of isl.
3. Generate configure
./autogen.sh
After performing the above steps, continue with the Common installation instructions.

1.3.2

Common installation instructions

1. Obtain GMP
By default, building isl requires GMP, including its headers files. Your distribution may not provide these header files by default and you may need to install a
package called gmp-devel or something similar. Alternatively, GMP can be built
from source, available from http://gmplib.org/. GMP is not needed if you build
isl with imath.
2. Configure
isl uses the standard autoconf configure script. To run it, just type
./configure
optionally followed by some configure options. A complete list of options can
be obtained by running
./configure --help
Below we discuss some of the more common options.
8

--prefix
Installation prefix for isl
--with-int=[gmp|imath|imath-32]
Select the integer library to be used by isl, the default is gmp. With
imath-32, isl will use 32 bit integers, but fall back to imath for values out of the 32 bit range. In most applications, isl will run fastest with
the imath-32 option, followed by gmp and imath, the slowest.
--with-gmp-prefix
Installation prefix for GMP (architecture-independent files).
--with-gmp-exec-prefix
Installation prefix for GMP (architecture-dependent files).
3. Compile
make
4. Install (optional)
make install

1.4

Integer Set Library

1.4.1

Memory Management

Since a high-level operation on isl objects usually involves several substeps and since
the user is usually not interested in the intermediate results, most functions that return
a new object will also release all the objects passed as arguments. If the user still wants
to use one or more of these arguments after the function call, she should pass along a
copy of the object rather than the object itself. The user is then responsible for making
sure that the original object gets used somewhere else or is explicitly freed.
The arguments and return values of all documented functions are annotated to make
clear which arguments are released and which arguments are preserved. In particular,
the following annotations are used
__isl_give
__isl_give means that a new object is returned. The user should make sure that
the returned pointer is used exactly once as a value for an __isl_take argument.
In between, it can be used as a value for as many __isl_keep arguments as the
user likes. There is one exception, and that is the case where the pointer returned
is NULL. Is this case, the user is free to use it as an __isl_take argument or not.
When applied to a char *, the returned pointer needs to be freed using free.
__isl_null
__isl_null means that a NULL value is returned.
9

__isl_take
__isl_take means that the object the argument points to is taken over by the
function and may no longer be used by the user as an argument to any other
function. The pointer value must be one returned by a function returning an
__isl_give pointer. If the user passes in a NULL value, then this will be treated
as an error in the sense that the function will not perform its usual operation.
However, it will still make sure that all the other __isl_take arguments are
released.
__isl_keep
__isl_keep means that the function will only use the object temporarily. After the function has finished, the user can still use it as an argument to other
functions. A NULL value will be treated in the same way as a NULL value for an
__isl_take argument. This annotation may also be used on return values of
type const char *, in which case the returned pointer should not be freed by
the user and is only valid until the object from which it was derived is updated
or freed.

1.4.2

Initialization

All manipulations of integer sets and relations occur within the context of an isl_ctx.
A given isl_ctx can only be used within a single thread. All arguments of a function
are required to have been allocated within the same context. There are currently no
functions available for moving an object from one isl_ctx to another isl_ctx. This
means that there is currently no way of safely moving an object from one thread to
another, unless the whole isl_ctx is moved.
An isl_ctx can be allocated using isl_ctx_alloc and freed using isl_ctx_free.
All objects allocated within an isl_ctx should be freed before the isl_ctx itself is
freed.
isl_ctx *isl_ctx_alloc();
void isl_ctx_free(isl_ctx *ctx);
The user can impose a bound on the number of low-level operations that can be
performed by an isl_ctx. This bound can be set and retrieved using the following
functions. A bound of zero means that no bound is imposed. The number of operations
performed can be reset using isl_ctx_reset_operations. Note that the number
of low-level operations needed to perform a high-level computation may differ significantly across different versions of isl, but it should be the same across different
platforms for the same version of isl.
Warning: This feature is experimental. isl has good support to abort and bail out
during the computation, but this feature may exercise error code paths that are normally
not used that much. Consequently, it is not unlikely that hidden bugs will be exposed.
void isl_ctx_set_max_operations(isl_ctx *ctx,
unsigned long max_operations);
10

unsigned long isl_ctx_get_max_operations(isl_ctx *ctx);
void isl_ctx_reset_operations(isl_ctx *ctx);
In order to be able to create an object in the same context as another object, most
object types (described later in this document) provide a function to obtain the context
in which the object was created.
#include 
isl_ctx *isl_val_get_ctx(__isl_keep isl_val *val);
isl_ctx *isl_multi_val_get_ctx(
__isl_keep isl_multi_val *mv);
#include 
isl_ctx *isl_id_get_ctx(__isl_keep isl_id *id);
#include 
isl_ctx *isl_local_space_get_ctx(
__isl_keep isl_local_space *ls);
#include 
isl_ctx *isl_set_list_get_ctx(
__isl_keep isl_set_list *list);
#include 
isl_ctx *isl_aff_get_ctx(__isl_keep isl_aff *aff);
isl_ctx *isl_multi_aff_get_ctx(
__isl_keep isl_multi_aff *maff);
isl_ctx *isl_pw_aff_get_ctx(__isl_keep isl_pw_aff *pa);
isl_ctx *isl_pw_multi_aff_get_ctx(
__isl_keep isl_pw_multi_aff *pma);
isl_ctx *isl_multi_pw_aff_get_ctx(
__isl_keep isl_multi_pw_aff *mpa);
isl_ctx *isl_union_pw_aff_get_ctx(
__isl_keep isl_union_pw_aff *upa);
isl_ctx *isl_union_pw_multi_aff_get_ctx(
__isl_keep isl_union_pw_multi_aff *upma);
isl_ctx *isl_multi_union_pw_aff_get_ctx(
__isl_keep isl_multi_union_pw_aff *mupa);
#include 
isl_ctx *isl_id_to_ast_expr_get_ctx(
__isl_keep isl_id_to_ast_expr *id2expr);
#include 
isl_ctx *isl_point_get_ctx(__isl_keep isl_point *pnt);
#include 
isl_ctx *isl_vec_get_ctx(__isl_keep isl_vec *vec);
11

#include 
isl_ctx *isl_mat_get_ctx(__isl_keep isl_mat *mat);
#include 
isl_ctx *isl_vertices_get_ctx(
__isl_keep isl_vertices *vertices);
isl_ctx *isl_vertex_get_ctx(__isl_keep isl_vertex *vertex);
isl_ctx *isl_cell_get_ctx(__isl_keep isl_cell *cell);
#include 
isl_ctx *isl_restriction_get_ctx(
__isl_keep isl_restriction *restr);
isl_ctx *isl_union_access_info_get_ctx(
__isl_keep isl_union_access_info *access);
isl_ctx *isl_union_flow_get_ctx(
__isl_keep isl_union_flow *flow);
#include 
isl_ctx *isl_schedule_get_ctx(
__isl_keep isl_schedule *sched);
isl_ctx *isl_schedule_constraints_get_ctx(
__isl_keep isl_schedule_constraints *sc);
#include 
isl_ctx *isl_schedule_node_get_ctx(
__isl_keep isl_schedule_node *node);
#include 
isl_ctx *isl_ast_build_get_ctx(
__isl_keep isl_ast_build *build);
#include 
isl_ctx *isl_ast_expr_get_ctx(
__isl_keep isl_ast_expr *expr);
isl_ctx *isl_ast_node_get_ctx(
__isl_keep isl_ast_node *node);
#include 
isl_ctx *isl_stride_info_get_ctx(
__isl_keep isl_stride_info *si);
#include 
isl_ctx *isl_fixed_box_get_ctx(
__isl_keep isl_fixed_box *box);

12

1.4.3

Return Types

isl uses two special return types for functions that either return a boolean or that in
principle do not return anything. In particular, the isl_bool type has three possible
values: isl_bool_true (a positive integer value), indicating true or yes; isl_bool_false
(the integer value zero), indicating false or no; and isl_bool_error (a negative integer value), indicating that something went wrong. The following function can be used
to negate an isl_bool, where the negation of isl_bool_error is isl_bool_error
again.
#include 
isl_bool isl_bool_not(isl_bool b);
The isl_stat type has two possible values: isl_stat_ok (the integer value
zero), indicating a successful operation; and isl_stat_error (a negative integer
value), indicating that something went wrong. See §1.4.6 for more information on
isl_bool_error and isl_stat_error.

1.4.4

Values

An isl_val represents an integer value, a rational value or one of three special values,
infinity, negative infinity and NaN. Some predefined values can be created using the
following functions.
#include 
__isl_give isl_val *isl_val_zero(isl_ctx *ctx);
__isl_give isl_val *isl_val_one(isl_ctx *ctx);
__isl_give isl_val *isl_val_negone(isl_ctx *ctx);
__isl_give isl_val *isl_val_nan(isl_ctx *ctx);
__isl_give isl_val *isl_val_infty(isl_ctx *ctx);
__isl_give isl_val *isl_val_neginfty(isl_ctx *ctx);
Specific integer values can be created using the following functions.
#include 
__isl_give isl_val *isl_val_int_from_si(isl_ctx *ctx,
long i);
__isl_give isl_val *isl_val_int_from_ui(isl_ctx *ctx,
unsigned long u);
__isl_give isl_val *isl_val_int_from_chunks(isl_ctx *ctx,
size_t n, size_t size, const void *chunks);
The function isl_val_int_from_chunks constructs an isl_val from the n digits, each consisting of size bytes, stored at chunks. The least significant digit is
assumed to be stored first.
Value objects can be copied and freed using the following functions.

13

#include 
__isl_give isl_val *isl_val_copy(__isl_keep isl_val *v);
__isl_null isl_val *isl_val_free(__isl_take isl_val *v);
They can be inspected using the following functions.
#include 
long isl_val_get_num_si(__isl_keep isl_val *v);
long isl_val_get_den_si(__isl_keep isl_val *v);
__isl_give isl_val *isl_val_get_den_val(
__isl_keep isl_val *v);
double isl_val_get_d(__isl_keep isl_val *v);
size_t isl_val_n_abs_num_chunks(__isl_keep isl_val *v,
size_t size);
int isl_val_get_abs_num_chunks(__isl_keep isl_val *v,
size_t size, void *chunks);
isl_val_n_abs_num_chunks returns the number of digits of size bytes needed
to store the absolute value of the numerator of v. isl_val_get_abs_num_chunks
stores these digits at chunks, which is assumed to have been preallocated by the
caller. The least significant digit is stored first. Note that isl_val_get_num_si,
isl_val_get_den_si, isl_val_get_d, isl_val_n_abs_num_chunks and isl_val_get_abs_num_chunks
can only be applied to rational values.
An isl_val can be modified using the following function.
#include 
__isl_give isl_val *isl_val_set_si(__isl_take isl_val *v,
long i);
The following unary properties are defined on isl_vals.
#include 
int isl_val_sgn(__isl_keep isl_val *v);
isl_bool isl_val_is_zero(__isl_keep isl_val *v);
isl_bool isl_val_is_one(__isl_keep isl_val *v);
isl_bool isl_val_is_negone(__isl_keep isl_val *v);
isl_bool isl_val_is_nonneg(__isl_keep isl_val *v);
isl_bool isl_val_is_nonpos(__isl_keep isl_val *v);
isl_bool isl_val_is_pos(__isl_keep isl_val *v);
isl_bool isl_val_is_neg(__isl_keep isl_val *v);
isl_bool isl_val_is_int(__isl_keep isl_val *v);
isl_bool isl_val_is_rat(__isl_keep isl_val *v);
isl_bool isl_val_is_nan(__isl_keep isl_val *v);
isl_bool isl_val_is_infty(__isl_keep isl_val *v);
isl_bool isl_val_is_neginfty(__isl_keep isl_val *v);
Note that the sign of NaN is undefined.
The following binary properties are defined on pairs of isl_vals.
14

#include 
isl_bool isl_val_lt(__isl_keep isl_val *v1,
__isl_keep isl_val *v2);
isl_bool isl_val_le(__isl_keep isl_val *v1,
__isl_keep isl_val *v2);
isl_bool isl_val_gt(__isl_keep isl_val *v1,
__isl_keep isl_val *v2);
isl_bool isl_val_ge(__isl_keep isl_val *v1,
__isl_keep isl_val *v2);
isl_bool isl_val_eq(__isl_keep isl_val *v1,
__isl_keep isl_val *v2);
isl_bool isl_val_ne(__isl_keep isl_val *v1,
__isl_keep isl_val *v2);
isl_bool isl_val_abs_eq(__isl_keep isl_val *v1,
__isl_keep isl_val *v2);
Comparisons to NaN always return false. That is, a NaN is not considered to hold
any relative position with respect to any value. In particular, a NaN is neither considered to be equal to nor to be different from any value (including another NaN). The
function isl_val_abs_eq checks whether its two arguments are equal in absolute
value.
For integer isl_vals we additionally have the following binary property.
#include 
isl_bool isl_val_is_divisible_by(__isl_keep isl_val *v1,
__isl_keep isl_val *v2);
An isl_val can also be compared to an integer using the following functions. The
result of isl_val_cmp_si undefined for NaN.
#include 
isl_bool isl_val_gt_si(__isl_keep isl_val *v, long i);
int isl_val_cmp_si(__isl_keep isl_val *v, long i);
The following unary operations are available on isl_vals.
#include 
__isl_give isl_val *isl_val_abs(__isl_take isl_val *v);
__isl_give isl_val *isl_val_neg(__isl_take isl_val *v);
__isl_give isl_val *isl_val_floor(__isl_take isl_val *v);
__isl_give isl_val *isl_val_ceil(__isl_take isl_val *v);
__isl_give isl_val *isl_val_trunc(__isl_take isl_val *v);
__isl_give isl_val *isl_val_inv(__isl_take isl_val *v);
__isl_give isl_val *isl_val_2exp(__isl_take isl_val *v);
The following binary operations are available on isl_vals.

15

#include 
__isl_give isl_val *isl_val_min(__isl_take isl_val *v1,
__isl_take isl_val *v2);
__isl_give isl_val *isl_val_max(__isl_take isl_val *v1,
__isl_take isl_val *v2);
__isl_give isl_val *isl_val_add(__isl_take isl_val *v1,
__isl_take isl_val *v2);
__isl_give isl_val *isl_val_add_ui(__isl_take isl_val *v1,
unsigned long v2);
__isl_give isl_val *isl_val_sub(__isl_take isl_val *v1,
__isl_take isl_val *v2);
__isl_give isl_val *isl_val_sub_ui(__isl_take isl_val *v1,
unsigned long v2);
__isl_give isl_val *isl_val_mul(__isl_take isl_val *v1,
__isl_take isl_val *v2);
__isl_give isl_val *isl_val_mul_ui(__isl_take isl_val *v1,
unsigned long v2);
__isl_give isl_val *isl_val_div(__isl_take isl_val *v1,
__isl_take isl_val *v2);
__isl_give isl_val *isl_val_div_ui(__isl_take isl_val *v1,
unsigned long v2);
On integer values, we additionally have the following operations.
#include 
__isl_give isl_val *isl_val_2exp(__isl_take isl_val *v);
__isl_give isl_val *isl_val_mod(__isl_take isl_val *v1,
__isl_take isl_val *v2);
__isl_give isl_val *isl_val_gcd(__isl_take isl_val *v1,
__isl_take isl_val *v2);
__isl_give isl_val *isl_val_gcdext(__isl_take isl_val *v1,
__isl_take isl_val *v2, __isl_give isl_val **x,
__isl_give isl_val **y);
The function isl_val_gcdext returns the greatest common divisor g of v1 and
v2 as well as two integers *x and *y such that *x * v1 + *y * v2 = g.
GMP specific functions
These functions are only available if isl has been compiled with GMP support.
Specific integer and rational values can be created from GMP values using the following functions.
#include 
__isl_give isl_val *isl_val_int_from_gmp(isl_ctx *ctx,
mpz_t z);
__isl_give isl_val *isl_val_from_gmp(isl_ctx *ctx,
const mpz_t n, const mpz_t d);
16

The numerator and denominator of a rational value can be extracted as GMP values
using the following functions.
#include 
int isl_val_get_num_gmp(__isl_keep isl_val *v, mpz_t z);
int isl_val_get_den_gmp(__isl_keep isl_val *v, mpz_t z);

1.4.5

Sets and Relations

isl uses six types of objects for representing sets and relations, isl_basic_set,
isl_basic_map, isl_set, isl_map, isl_union_set and isl_union_map. isl_basic_set
and isl_basic_map represent sets and relations that can be described as a conjunction
of affine constraints, while isl_set and isl_map represent unions of isl_basic_sets
and isl_basic_maps, respectively. However, all isl_basic_sets or isl_basic_maps
in the union need to live in the same space. isl_union_sets and isl_union_maps
represent unions of isl_sets or isl_maps in different spaces, where spaces are considered different if they have a different number of dimensions and/or different names
(see §1.4.8). The difference between sets and relations (maps) is that sets have one
set of variables, while relations have two sets of variables, input variables and output
variables.

1.4.6

Error Handling

isl supports different ways to react in case a runtime error is triggered. Runtime
errors arise, e.g., if a function such as isl_map_intersect is called with two maps
that have incompatible spaces. There are three possible ways to react on error: to warn,
to continue or to abort.
The default behavior is to warn. In this mode, isl prints a warning, stores the last
error in the corresponding isl_ctx and the function in which the error was triggered
returns a value indicating that some error has occurred. In case of functions returning a
pointer, this value is NULL. In case of functions returning an isl_bool or an isl_stat,
this value is isl_bool_error or isl_stat_error. An error does not corrupt internal
state, such that isl can continue to be used. isl also provides functions to read the last
error, including the specific error message, the isl source file where the error occurred
and the line number, and to reset all information about the last error. The last error
is only stored for information purposes. Its presence does not change the behavior of
isl. Hence, resetting an error is not required to continue to use isl, but only to observe
new errors.
#include 
enum isl_error isl_ctx_last_error(isl_ctx *ctx);
const char *isl_ctx_last_error_msg(isl_ctx *ctx);
const char *isl_ctx_last_error_file(isl_ctx *ctx);
int isl_ctx_last_error_line(isl_ctx *ctx);
void isl_ctx_reset_error(isl_ctx *ctx);

17

If no error has occurred since the last call to isl_ctx_reset_error, then the
functions isl_ctx_last_error_msg and isl_ctx_last_error_file return NULL.
Another option is to continue on error. This is similar to warn on error mode, except
that isl does not print any warning. This allows a program to implement its own error
reporting.
The last option is to directly abort the execution of the program from within the isl
library. This makes it obviously impossible to recover from an error, but it allows to
directly spot the error location. By aborting on error, debuggers break at the location
the error occurred and can provide a stack trace. Other tools that automatically provide stack traces on abort or that do not want to continue execution after an error was
triggered may also prefer to abort on error.
The on error behavior of isl can be specified by calling isl_options_set_on_error
or by setting the command line option --isl-on-error. Valid arguments for the function call are ISL_ON_ERROR_WARN, ISL_ON_ERROR_CONTINUE and ISL_ON_ERROR_ABORT.
The choices for the command line option are warn, continue and abort. It is also
possible to query the current error mode.
#include 
isl_stat isl_options_set_on_error(isl_ctx *ctx, int val);
int isl_options_get_on_error(isl_ctx *ctx);

1.4.7

Identifiers

Identifiers are used to identify both individual dimensions and tuples of dimensions.
They consist of an optional name and an optional user pointer. The name and the
user pointer cannot both be NULL, however. Identifiers with the same name but different pointer values are considered to be distinct. Similarly, identifiers with different
names but the same pointer value are also considered to be distinct. Equal identifiers
are represented using the same object. Pairs of identifiers can therefore be tested for
equality using the == operator. Identifiers can be constructed, copied, freed, inspected
and printed using the following functions.
#include 
__isl_give isl_id *isl_id_alloc(isl_ctx *ctx,
__isl_keep const char *name, void *user);
__isl_give isl_id *isl_id_set_free_user(
__isl_take isl_id *id,
void (*free_user)(void *user));
__isl_give isl_id *isl_id_copy(isl_id *id);
__isl_null isl_id *isl_id_free(__isl_take isl_id *id);
void *isl_id_get_user(__isl_keep isl_id *id);
__isl_keep const char *isl_id_get_name(__isl_keep isl_id *id);
__isl_give isl_printer *isl_printer_print_id(
__isl_take isl_printer *p, __isl_keep isl_id *id);

18

The callback set by isl_id_set_free_user is called on the user pointer when
the last reference to the isl_id is freed. Note that isl_id_get_name returns a pointer
to some internal data structure, so the result can only be used while the corresponding
isl_id is alive.

1.4.8

Spaces

Whenever a new set, relation or similar object is created from scratch, the space in
which it lives needs to be specified using an isl_space. Each space involves zero or
more parameters and zero, one or two tuples of set or input/output dimensions. The
parameters and dimensions are identified by an isl_dim_type and a position. The
type isl_dim_param refers to parameters, the type isl_dim_set refers to set dimensions (for spaces with a single tuple of dimensions) and the types isl_dim_in and
isl_dim_out refer to input and output dimensions (for spaces with two tuples of dimensions). Local spaces (see §1.4.9) also contain dimensions of type isl_dim_div.
Note that parameters are only identified by their position within a given object. Across
different objects, parameters are (usually) identified by their names or identifiers. Only
unnamed parameters are identified by their positions across objects. The use of unnamed parameters is discouraged.
#include 
__isl_give isl_space *isl_space_alloc(isl_ctx *ctx,
unsigned nparam, unsigned n_in, unsigned n_out);
__isl_give isl_space *isl_space_params_alloc(isl_ctx *ctx,
unsigned nparam);
__isl_give isl_space *isl_space_set_alloc(isl_ctx *ctx,
unsigned nparam, unsigned dim);
__isl_give isl_space *isl_space_copy(__isl_keep isl_space *space);
__isl_null isl_space *isl_space_free(__isl_take isl_space *space);
The space used for creating a parameter domain needs to be created using isl_space_params_alloc.
For other sets, the space needs to be created using isl_space_set_alloc, while for
a relation, the space needs to be created using isl_space_alloc.
To check whether a given space is that of a set or a map or whether it is a parameter
space, use these functions:
#include
isl_bool
isl_bool
isl_bool


isl_space_is_params(__isl_keep isl_space *space);
isl_space_is_set(__isl_keep isl_space *space);
isl_space_is_map(__isl_keep isl_space *space);

Spaces can be compared using the following functions:
#include 
isl_bool isl_space_is_equal(__isl_keep isl_space *space1,
__isl_keep isl_space *space2);
isl_bool isl_space_has_equal_params(
19

__isl_keep isl_space *space1,
__isl_keep isl_space *space2);
isl_bool isl_space_has_equal_tuples(
__isl_keep isl_space *space1,
__isl_keep isl_space *space2);
isl_bool isl_space_is_domain(__isl_keep isl_space *space1,
__isl_keep isl_space *space2);
isl_bool isl_space_is_range(__isl_keep isl_space *space1,
__isl_keep isl_space *space2);
isl_bool isl_space_tuple_is_equal(
__isl_keep isl_space *space1,
enum isl_dim_type type1,
__isl_keep isl_space *space2,
enum isl_dim_type type2);
isl_space_is_domain checks whether the first argument is equal to the domain
of the second argument. This requires in particular that the first argument is a set
space and that the second argument is a map space. isl_space_tuple_is_equal
checks whether the given tuples (isl_dim_in, isl_dim_out or isl_dim_set) of
the given spaces are the same. That is, it checks if they have the same identifier
(if any), the same dimension and the same internal structure (if any). The function
isl_space_has_equal_params checks whether two spaces have the same parameters in the same order. isl_space_has_equal_tuples check whether two spaces
have the same tuples. In contrast to isl_space_is_equal below, it does not check the
parameters. This is useful because many isl functions align the parameters before they
perform their operations, such that equivalence is not necessary. isl_space_is_equal
checks whether two spaces are identical, meaning that they have the same parameters
and the same tuples. That is, it checks whether both isl_space_has_equal_params
and isl_space_has_equal_tuples hold.
It is often useful to create objects that live in the same space as some other object.
This can be accomplished by creating the new objects (see §1.4.10 or §1.4.13) based
on the space of the original object.
#include 
__isl_give isl_space *isl_basic_set_get_space(
__isl_keep isl_basic_set *bset);
__isl_give isl_space *isl_set_get_space(__isl_keep isl_set *set);
#include 
__isl_give isl_space *isl_union_set_get_space(
__isl_keep isl_union_set *uset);
#include 
__isl_give isl_space *isl_basic_map_get_space(
__isl_keep isl_basic_map *bmap);
__isl_give isl_space *isl_map_get_space(__isl_keep isl_map *map);

20

#include 
__isl_give isl_space *isl_union_map_get_space(
__isl_keep isl_union_map *umap);
#include 
__isl_give isl_space *isl_constraint_get_space(
__isl_keep isl_constraint *constraint);
#include 
__isl_give isl_space *isl_qpolynomial_get_domain_space(
__isl_keep isl_qpolynomial *qp);
__isl_give isl_space *isl_qpolynomial_get_space(
__isl_keep isl_qpolynomial *qp);
__isl_give isl_space *
isl_qpolynomial_fold_get_domain_space(
__isl_keep isl_qpolynomial_fold *fold);
__isl_give isl_space *isl_qpolynomial_fold_get_space(
__isl_keep isl_qpolynomial_fold *fold);
__isl_give isl_space *isl_pw_qpolynomial_get_domain_space(
__isl_keep isl_pw_qpolynomial *pwqp);
__isl_give isl_space *isl_pw_qpolynomial_get_space(
__isl_keep isl_pw_qpolynomial *pwqp);
__isl_give isl_space *isl_pw_qpolynomial_fold_get_domain_space(
__isl_keep isl_pw_qpolynomial_fold *pwf);
__isl_give isl_space *isl_pw_qpolynomial_fold_get_space(
__isl_keep isl_pw_qpolynomial_fold *pwf);
__isl_give isl_space *isl_union_pw_qpolynomial_get_space(
__isl_keep isl_union_pw_qpolynomial *upwqp);
__isl_give isl_space *isl_union_pw_qpolynomial_fold_get_space(
__isl_keep isl_union_pw_qpolynomial_fold *upwf);
#include 
__isl_give isl_space *isl_multi_val_get_space(
__isl_keep isl_multi_val *mv);
#include 
__isl_give isl_space *isl_aff_get_domain_space(
__isl_keep isl_aff *aff);
__isl_give isl_space *isl_aff_get_space(
__isl_keep isl_aff *aff);
__isl_give isl_space *isl_pw_aff_get_domain_space(
__isl_keep isl_pw_aff *pwaff);
__isl_give isl_space *isl_pw_aff_get_space(
__isl_keep isl_pw_aff *pwaff);
__isl_give isl_space *isl_multi_aff_get_domain_space(
__isl_keep isl_multi_aff *maff);
__isl_give isl_space *isl_multi_aff_get_space(
21

__isl_keep isl_multi_aff *maff);
__isl_give isl_space *isl_pw_multi_aff_get_domain_space(
__isl_keep isl_pw_multi_aff *pma);
__isl_give isl_space *isl_pw_multi_aff_get_space(
__isl_keep isl_pw_multi_aff *pma);
__isl_give isl_space *isl_union_pw_aff_get_space(
__isl_keep isl_union_pw_aff *upa);
__isl_give isl_space *isl_union_pw_multi_aff_get_space(
__isl_keep isl_union_pw_multi_aff *upma);
__isl_give isl_space *isl_multi_pw_aff_get_domain_space(
__isl_keep isl_multi_pw_aff *mpa);
__isl_give isl_space *isl_multi_pw_aff_get_space(
__isl_keep isl_multi_pw_aff *mpa);
__isl_give isl_space *
isl_multi_union_pw_aff_get_domain_space(
__isl_keep isl_multi_union_pw_aff *mupa);
__isl_give isl_space *
isl_multi_union_pw_aff_get_space(
__isl_keep isl_multi_union_pw_aff *mupa);
#include 
__isl_give isl_space *isl_point_get_space(
__isl_keep isl_point *pnt);
#include 
__isl_give isl_space *isl_fixed_box_get_space(
__isl_keep isl_fixed_box *box);
The number of dimensions of a given type of space may be read off from a space or
an object that lives in a space using the following functions. In case of isl_space_dim,
type may be isl_dim_param, isl_dim_in (only for relations), isl_dim_out (only
for relations), isl_dim_set (only for sets) or isl_dim_all.
#include 
unsigned isl_space_dim(__isl_keep isl_space *space,
enum isl_dim_type type);
#include 
int isl_local_space_dim(__isl_keep isl_local_space *ls,
enum isl_dim_type type);
#include 
unsigned isl_basic_set_dim(__isl_keep isl_basic_set *bset,
enum isl_dim_type type);
unsigned isl_set_dim(__isl_keep isl_set *set,
enum isl_dim_type type);

22

#include 
unsigned isl_union_set_dim(__isl_keep isl_union_set *uset,
enum isl_dim_type type);
#include 
unsigned isl_basic_map_dim(__isl_keep isl_basic_map *bmap,
enum isl_dim_type type);
unsigned isl_map_dim(__isl_keep isl_map *map,
enum isl_dim_type type);
#include 
unsigned isl_union_map_dim(__isl_keep isl_union_map *umap,
enum isl_dim_type type);
#include 
unsigned isl_multi_val_dim(__isl_keep isl_multi_val *mv,
enum isl_dim_type type);
#include 
int isl_aff_dim(__isl_keep isl_aff *aff,
enum isl_dim_type type);
unsigned isl_multi_aff_dim(__isl_keep isl_multi_aff *maff,
enum isl_dim_type type);
unsigned isl_pw_aff_dim(__isl_keep isl_pw_aff *pwaff,
enum isl_dim_type type);
unsigned isl_pw_multi_aff_dim(
__isl_keep isl_pw_multi_aff *pma,
enum isl_dim_type type);
unsigned isl_multi_pw_aff_dim(
__isl_keep isl_multi_pw_aff *mpa,
enum isl_dim_type type);
unsigned isl_union_pw_aff_dim(
__isl_keep isl_union_pw_aff *upa,
enum isl_dim_type type);
unsigned isl_union_pw_multi_aff_dim(
__isl_keep isl_union_pw_multi_aff *upma,
enum isl_dim_type type);
unsigned isl_multi_union_pw_aff_dim(
__isl_keep isl_multi_union_pw_aff *mupa,
enum isl_dim_type type);
#include 
unsigned isl_union_pw_qpolynomial_dim(
__isl_keep isl_union_pw_qpolynomial *upwqp,
enum isl_dim_type type);
unsigned isl_union_pw_qpolynomial_fold_dim(
__isl_keep isl_union_pw_qpolynomial_fold *upwf,
enum isl_dim_type type);
23

Note that an isl_union_set, an isl_union_map, an isl_union_pw_multi_aff,
an isl_union_pw_qpolynomial and an isl_union_pw_qpolynomial_fold only
have parameters.
Additional parameters can be added to a space using the following function.
#include 
__isl_give isl_space *isl_space_add_param_id(
__isl_take isl_space *space,
__isl_take isl_id *id);
If a parameter with the given identifier already appears in the space, then it is not
added again.
The identifiers or names of the individual dimensions of spaces may be set or read
off using the following functions on spaces or objects that live in spaces. These functions are mostly useful to obtain the identifiers, positions or names of the parameters.
Identifiers of individual dimensions are essentially only useful for printing. They are
ignored by all other operations and may not be preserved across those operations.
#include 
__isl_give isl_space *isl_space_set_dim_id(
__isl_take isl_space *space,
enum isl_dim_type type, unsigned pos,
__isl_take isl_id *id);
isl_bool isl_space_has_dim_id(__isl_keep isl_space *space,
enum isl_dim_type type, unsigned pos);
__isl_give isl_id *isl_space_get_dim_id(
__isl_keep isl_space *space,
enum isl_dim_type type, unsigned pos);
__isl_give isl_space *isl_space_set_dim_name(
__isl_take isl_space *space,
enum isl_dim_type type, unsigned pos,
__isl_keep const char *name);
isl_bool isl_space_has_dim_name(__isl_keep isl_space *space,
enum isl_dim_type type, unsigned pos);
__isl_keep const char *isl_space_get_dim_name(
__isl_keep isl_space *space,
enum isl_dim_type type, unsigned pos);
#include 
__isl_give isl_local_space *isl_local_space_set_dim_id(
__isl_take isl_local_space *ls,
enum isl_dim_type type, unsigned pos,
__isl_take isl_id *id);
isl_bool isl_local_space_has_dim_id(
__isl_keep isl_local_space *ls,
enum isl_dim_type type, unsigned pos);
__isl_give isl_id *isl_local_space_get_dim_id(
24

__isl_keep isl_local_space *ls,
enum isl_dim_type type, unsigned pos);
__isl_give isl_local_space *isl_local_space_set_dim_name(
__isl_take isl_local_space *ls,
enum isl_dim_type type, unsigned pos, const char *s);
isl_bool isl_local_space_has_dim_name(
__isl_keep isl_local_space *ls,
enum isl_dim_type type, unsigned pos)
const char *isl_local_space_get_dim_name(
__isl_keep isl_local_space *ls,
enum isl_dim_type type, unsigned pos);
#include 
const char *isl_constraint_get_dim_name(
__isl_keep isl_constraint *constraint,
enum isl_dim_type type, unsigned pos);
#include 
__isl_give isl_id *isl_basic_set_get_dim_id(
__isl_keep isl_basic_set *bset,
enum isl_dim_type type, unsigned pos);
__isl_give isl_set *isl_set_set_dim_id(
__isl_take isl_set *set, enum isl_dim_type type,
unsigned pos, __isl_take isl_id *id);
isl_bool isl_set_has_dim_id(__isl_keep isl_set *set,
enum isl_dim_type type, unsigned pos);
__isl_give isl_id *isl_set_get_dim_id(
__isl_keep isl_set *set, enum isl_dim_type type,
unsigned pos);
const char *isl_basic_set_get_dim_name(
__isl_keep isl_basic_set *bset,
enum isl_dim_type type, unsigned pos);
isl_bool isl_set_has_dim_name(__isl_keep isl_set *set,
enum isl_dim_type type, unsigned pos);
const char *isl_set_get_dim_name(
__isl_keep isl_set *set,
enum isl_dim_type type, unsigned pos);
#include 
__isl_give isl_map *isl_map_set_dim_id(
__isl_take isl_map *map, enum isl_dim_type type,
unsigned pos, __isl_take isl_id *id);
isl_bool isl_basic_map_has_dim_id(
__isl_keep isl_basic_map *bmap,
enum isl_dim_type type, unsigned pos);
isl_bool isl_map_has_dim_id(__isl_keep isl_map *map,
enum isl_dim_type type, unsigned pos);
25

__isl_give isl_id *isl_map_get_dim_id(
__isl_keep isl_map *map, enum isl_dim_type type,
unsigned pos);
__isl_give isl_id *isl_union_map_get_dim_id(
__isl_keep isl_union_map *umap,
enum isl_dim_type type, unsigned pos);
const char *isl_basic_map_get_dim_name(
__isl_keep isl_basic_map *bmap,
enum isl_dim_type type, unsigned pos);
isl_bool isl_map_has_dim_name(__isl_keep isl_map *map,
enum isl_dim_type type, unsigned pos);
const char *isl_map_get_dim_name(
__isl_keep isl_map *map,
enum isl_dim_type type, unsigned pos);
#include 
__isl_give isl_multi_val *isl_multi_val_set_dim_id(
__isl_take isl_multi_val *mv,
enum isl_dim_type type, unsigned pos,
__isl_take isl_id *id);
__isl_give isl_id *isl_multi_val_get_dim_id(
__isl_keep isl_multi_val *mv,
enum isl_dim_type type, unsigned pos);
__isl_give isl_multi_val *isl_multi_val_set_dim_name(
__isl_take isl_multi_val *mv,
enum isl_dim_type type, unsigned pos, const char *s);
#include 
__isl_give isl_aff *isl_aff_set_dim_id(
__isl_take isl_aff *aff, enum isl_dim_type type,
unsigned pos, __isl_take isl_id *id);
__isl_give isl_multi_aff *isl_multi_aff_set_dim_id(
__isl_take isl_multi_aff *maff,
enum isl_dim_type type, unsigned pos,
__isl_take isl_id *id);
__isl_give isl_pw_aff *isl_pw_aff_set_dim_id(
__isl_take isl_pw_aff *pma,
enum isl_dim_type type, unsigned pos,
__isl_take isl_id *id);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_set_dim_id(
__isl_take isl_multi_pw_aff *mpa,
enum isl_dim_type type, unsigned pos,
__isl_take isl_id *id);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_set_dim_id(

26

__isl_take isl_multi_union_pw_aff *mupa,
enum isl_dim_type type, unsigned pos,
__isl_take isl_id *id);
__isl_give isl_id *isl_multi_aff_get_dim_id(
__isl_keep isl_multi_aff *ma,
enum isl_dim_type type, unsigned pos);
isl_bool isl_pw_aff_has_dim_id(__isl_keep isl_pw_aff *pa,
enum isl_dim_type type, unsigned pos);
__isl_give isl_id *isl_pw_aff_get_dim_id(
__isl_keep isl_pw_aff *pa,
enum isl_dim_type type, unsigned pos);
__isl_give isl_id *isl_pw_multi_aff_get_dim_id(
__isl_keep isl_pw_multi_aff *pma,
enum isl_dim_type type, unsigned pos);
__isl_give isl_id *isl_multi_pw_aff_get_dim_id(
__isl_keep isl_multi_pw_aff *mpa,
enum isl_dim_type type, unsigned pos);
__isl_give isl_id *isl_multi_union_pw_aff_get_dim_id(
__isl_keep isl_multi_union_pw_aff *mupa,
enum isl_dim_type type, unsigned pos);
__isl_give isl_aff *isl_aff_set_dim_name(
__isl_take isl_aff *aff, enum isl_dim_type type,
unsigned pos, const char *s);
__isl_give isl_multi_aff *isl_multi_aff_set_dim_name(
__isl_take isl_multi_aff *maff,
enum isl_dim_type type, unsigned pos, const char *s);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_set_dim_name(
__isl_take isl_multi_pw_aff *mpa,
enum isl_dim_type type, unsigned pos, const char *s);
__isl_give isl_union_pw_aff *
isl_union_pw_aff_set_dim_name(
__isl_take isl_union_pw_aff *upa,
enum isl_dim_type type, unsigned pos,
const char *s);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_set_dim_name(
__isl_take isl_union_pw_multi_aff *upma,
enum isl_dim_type type, unsigned pos,
const char *s);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_set_dim_name(
__isl_take isl_multi_union_pw_aff *mupa,
enum isl_dim_type type, unsigned pos,
const char *isl_aff_get_dim_name(__isl_keep isl_aff *aff,
enum isl_dim_type type, unsigned pos);
27

const char *isl_pw_aff_get_dim_name(
__isl_keep isl_pw_aff *pa,
enum isl_dim_type type, unsigned pos);
const char *isl_pw_multi_aff_get_dim_name(
__isl_keep isl_pw_multi_aff *pma,
enum isl_dim_type type, unsigned pos);
#include 
__isl_give isl_qpolynomial *isl_qpolynomial_set_dim_name(
__isl_take isl_qpolynomial *qp,
enum isl_dim_type type, unsigned pos,
const char *s);
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_set_dim_name(
__isl_take isl_pw_qpolynomial *pwqp,
enum isl_dim_type type, unsigned pos,
const char *s);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_set_dim_name(
__isl_take isl_pw_qpolynomial_fold *pwf,
enum isl_dim_type type, unsigned pos,
const char *s);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_set_dim_name(
__isl_take isl_union_pw_qpolynomial *upwqp,
enum isl_dim_type type, unsigned pos,
const char *s);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_set_dim_name(
__isl_take isl_union_pw_qpolynomial_fold *upwf,
enum isl_dim_type type, unsigned pos,
const char *s);
Note that isl_space_get_name returns a pointer to some internal data structure,
so the result can only be used while the corresponding isl_space is alive. Also note
that every function that operates on two sets or relations requires that both arguments
have the same parameters. This also means that if one of the arguments has named
parameters, then the other needs to have named parameters too and the names need to
match. Pairs of isl_set, isl_map, isl_union_set and/or isl_union_map arguments may have different parameters (as long as they are named), in which case the
result will have as parameters the union of the parameters of the arguments.
Given the identifier or name of a dimension (typically a parameter), its position can
be obtained from the following functions.
#include 
int isl_space_find_dim_by_id(__isl_keep isl_space *space,
enum isl_dim_type type, __isl_keep isl_id *id);
28

int isl_space_find_dim_by_name(__isl_keep isl_space *space,
enum isl_dim_type type, const char *name);
#include 
int isl_local_space_find_dim_by_name(
__isl_keep isl_local_space *ls,
enum isl_dim_type type, const char *name);
#include 
int isl_multi_val_find_dim_by_id(
__isl_keep isl_multi_val *mv,
enum isl_dim_type type, __isl_keep isl_id *id);
int isl_multi_val_find_dim_by_name(
__isl_keep isl_multi_val *mv,
enum isl_dim_type type, const char *name);
#include 
int isl_set_find_dim_by_id(__isl_keep isl_set *set,
enum isl_dim_type type, __isl_keep isl_id *id);
int isl_set_find_dim_by_name(__isl_keep isl_set *set,
enum isl_dim_type type, const char *name);
#include 
int isl_map_find_dim_by_id(__isl_keep isl_map *map,
enum isl_dim_type type, __isl_keep isl_id *id);
int isl_basic_map_find_dim_by_name(
__isl_keep isl_basic_map *bmap,
enum isl_dim_type type, const char *name);
int isl_map_find_dim_by_name(__isl_keep isl_map *map,
enum isl_dim_type type, const char *name);
int isl_union_map_find_dim_by_name(
__isl_keep isl_union_map *umap,
enum isl_dim_type type, const char *name);
#include 
int isl_multi_aff_find_dim_by_id(
__isl_keep isl_multi_aff *ma,
enum isl_dim_type type, __isl_keep isl_id *id);
int isl_multi_pw_aff_find_dim_by_id(
__isl_keep isl_multi_pw_aff *mpa,
enum isl_dim_type type, __isl_keep isl_id *id);
int isl_multi_union_pw_aff_find_dim_by_id(
__isl_keep isl_union_multi_pw_aff *mupa,
enum isl_dim_type type, __isl_keep isl_id *id);
int isl_aff_find_dim_by_name(__isl_keep isl_aff *aff,
enum isl_dim_type type, const char *name);
int isl_multi_aff_find_dim_by_name(
29

int
int

int

int

int

int

__isl_keep isl_multi_aff *ma,
enum isl_dim_type type, const char *name);
isl_pw_aff_find_dim_by_name(__isl_keep isl_pw_aff *pa,
enum isl_dim_type type, const char *name);
isl_multi_pw_aff_find_dim_by_name(
__isl_keep isl_multi_pw_aff *mpa,
enum isl_dim_type type, const char *name);
isl_pw_multi_aff_find_dim_by_name(
__isl_keep isl_pw_multi_aff *pma,
enum isl_dim_type type, const char *name);
isl_union_pw_aff_find_dim_by_name(
__isl_keep isl_union_pw_aff *upa,
enum isl_dim_type type, const char *name);
isl_union_pw_multi_aff_find_dim_by_name(
__isl_keep isl_union_pw_multi_aff *upma,
enum isl_dim_type type, const char *name);
isl_multi_union_pw_aff_find_dim_by_name(
__isl_keep isl_multi_union_pw_aff *mupa,
enum isl_dim_type type, const char *name);

#include 
int isl_pw_qpolynomial_find_dim_by_name(
__isl_keep isl_pw_qpolynomial *pwqp,
enum isl_dim_type type, const char *name);
int isl_pw_qpolynomial_fold_find_dim_by_name(
__isl_keep isl_pw_qpolynomial_fold *pwf,
enum isl_dim_type type, const char *name);
int isl_union_pw_qpolynomial_find_dim_by_name(
__isl_keep isl_union_pw_qpolynomial *upwqp,
enum isl_dim_type type, const char *name);
int isl_union_pw_qpolynomial_fold_find_dim_by_name(
__isl_keep isl_union_pw_qpolynomial_fold *upwf,
enum isl_dim_type type, const char *name);
The identifiers or names of entire spaces may be set or read off using the following
functions.
#include 
__isl_give isl_space *isl_space_set_tuple_id(
__isl_take isl_space *space,
enum isl_dim_type type, __isl_take isl_id *id);
__isl_give isl_space *isl_space_reset_tuple_id(
__isl_take isl_space *space, enum isl_dim_type type);
isl_bool isl_space_has_tuple_id(
__isl_keep isl_space *space,
enum isl_dim_type type);
__isl_give isl_id *isl_space_get_tuple_id(
30

__isl_keep isl_space *space, enum isl_dim_type type);
__isl_give isl_space *isl_space_set_tuple_name(
__isl_take isl_space *space,
enum isl_dim_type type, const char *s);
isl_bool isl_space_has_tuple_name(
__isl_keep isl_space *space,
enum isl_dim_type type);
__isl_keep const char *isl_space_get_tuple_name(
__isl_keep isl_space *space,
enum isl_dim_type type);
#include 
__isl_give isl_local_space *isl_local_space_set_tuple_id(
__isl_take isl_local_space *ls,
enum isl_dim_type type, __isl_take isl_id *id);
#include 
__isl_give isl_basic_set *isl_basic_set_set_tuple_id(
__isl_take isl_basic_set *bset,
__isl_take isl_id *id);
__isl_give isl_set *isl_set_set_tuple_id(
__isl_take isl_set *set, __isl_take isl_id *id);
__isl_give isl_set *isl_set_reset_tuple_id(
__isl_take isl_set *set);
isl_bool isl_set_has_tuple_id(__isl_keep isl_set *set);
__isl_give isl_id *isl_set_get_tuple_id(
__isl_keep isl_set *set);
__isl_give isl_basic_set *isl_basic_set_set_tuple_name(
__isl_take isl_basic_set *set, const char *s);
__isl_give isl_set *isl_set_set_tuple_name(
__isl_take isl_set *set, const char *s);
const char *isl_basic_set_get_tuple_name(
__isl_keep isl_basic_set *bset);
isl_bool isl_set_has_tuple_name(__isl_keep isl_set *set);
const char *isl_set_get_tuple_name(
__isl_keep isl_set *set);
#include 
__isl_give isl_basic_map *isl_basic_map_set_tuple_id(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type, __isl_take isl_id *id);
__isl_give isl_map *isl_map_set_tuple_id(
__isl_take isl_map *map, enum isl_dim_type type,
__isl_take isl_id *id);
__isl_give isl_map *isl_map_reset_tuple_id(
__isl_take isl_map *map, enum isl_dim_type type);
isl_bool isl_map_has_tuple_id(__isl_keep isl_map *map,
31

enum isl_dim_type type);
__isl_give isl_id *isl_map_get_tuple_id(
__isl_keep isl_map *map, enum isl_dim_type type);
__isl_give isl_map *isl_map_set_tuple_name(
__isl_take isl_map *map,
enum isl_dim_type type, const char *s);
const char *isl_basic_map_get_tuple_name(
__isl_keep isl_basic_map *bmap,
enum isl_dim_type type);
__isl_give isl_basic_map *isl_basic_map_set_tuple_name(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type, const char *s);
isl_bool isl_map_has_tuple_name(__isl_keep isl_map *map,
enum isl_dim_type type);
const char *isl_map_get_tuple_name(
__isl_keep isl_map *map,
enum isl_dim_type type);
#include 
__isl_give isl_multi_val *isl_multi_val_set_tuple_id(
__isl_take isl_multi_val *mv,
enum isl_dim_type type, __isl_take isl_id *id);
__isl_give isl_multi_val *isl_multi_val_reset_tuple_id(
__isl_take isl_multi_val *mv,
enum isl_dim_type type);
isl_bool isl_multi_val_has_tuple_id(
__isl_keep isl_multi_val *mv,
enum isl_dim_type type);
__isl_give isl_id *isl_multi_val_get_tuple_id(
__isl_keep isl_multi_val *mv,
enum isl_dim_type type);
__isl_give isl_multi_val *isl_multi_val_set_tuple_name(
__isl_take isl_multi_val *mv,
enum isl_dim_type type, const char *s);
const char *isl_multi_val_get_tuple_name(
__isl_keep isl_multi_val *mv,
enum isl_dim_type type);
#include 
__isl_give isl_aff *isl_aff_set_tuple_id(
__isl_take isl_aff *aff,
enum isl_dim_type type, __isl_take isl_id *id);
__isl_give isl_multi_aff *isl_multi_aff_set_tuple_id(
__isl_take isl_multi_aff *maff,
enum isl_dim_type type, __isl_take isl_id *id);
__isl_give isl_pw_aff *isl_pw_aff_set_tuple_id(

32

__isl_take isl_pw_aff *pwaff,
enum isl_dim_type type, __isl_take isl_id *id);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_tuple_id(
__isl_take isl_pw_multi_aff *pma,
enum isl_dim_type type, __isl_take isl_id *id);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_set_tuple_id(
__isl_take isl_multi_union_pw_aff *mupa,
enum isl_dim_type type, __isl_take isl_id *id);
__isl_give isl_multi_aff *isl_multi_aff_reset_tuple_id(
__isl_take isl_multi_aff *ma,
enum isl_dim_type type);
__isl_give isl_pw_aff *isl_pw_aff_reset_tuple_id(
__isl_take isl_pw_aff *pa,
enum isl_dim_type type);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_reset_tuple_id(
__isl_take isl_multi_pw_aff *mpa,
enum isl_dim_type type);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_reset_tuple_id(
__isl_take isl_pw_multi_aff *pma,
enum isl_dim_type type);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_reset_tuple_id(
__isl_take isl_multi_union_pw_aff *mupa,
enum isl_dim_type type);
isl_bool isl_multi_aff_has_tuple_id(
__isl_keep isl_multi_aff *ma,
enum isl_dim_type type);
__isl_give isl_id *isl_multi_aff_get_tuple_id(
__isl_keep isl_multi_aff *ma,
enum isl_dim_type type);
isl_bool isl_pw_aff_has_tuple_id(__isl_keep isl_pw_aff *pa,
enum isl_dim_type type);
__isl_give isl_id *isl_pw_aff_get_tuple_id(
__isl_keep isl_pw_aff *pa,
enum isl_dim_type type);
isl_bool isl_pw_multi_aff_has_tuple_id(
__isl_keep isl_pw_multi_aff *pma,
enum isl_dim_type type);
__isl_give isl_id *isl_pw_multi_aff_get_tuple_id(
__isl_keep isl_pw_multi_aff *pma,
enum isl_dim_type type);
isl_bool isl_multi_pw_aff_has_tuple_id(
__isl_keep isl_multi_pw_aff *mpa,
33

enum isl_dim_type type);
__isl_give isl_id *isl_multi_pw_aff_get_tuple_id(
__isl_keep isl_multi_pw_aff *mpa,
enum isl_dim_type type);
isl_bool isl_multi_union_pw_aff_has_tuple_id(
__isl_keep isl_multi_union_pw_aff *mupa,
enum isl_dim_type type);
__isl_give isl_id *isl_multi_union_pw_aff_get_tuple_id(
__isl_keep isl_multi_union_pw_aff *mupa,
enum isl_dim_type type);
__isl_give isl_multi_aff *isl_multi_aff_set_tuple_name(
__isl_take isl_multi_aff *maff,
enum isl_dim_type type, const char *s);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_set_tuple_name(
__isl_take isl_multi_pw_aff *mpa,
enum isl_dim_type type, const char *s);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_set_tuple_name(
__isl_take isl_multi_union_pw_aff *mupa,
enum isl_dim_type type, const char *s);
const char *isl_multi_aff_get_tuple_name(
__isl_keep isl_multi_aff *multi,
enum isl_dim_type type);
isl_bool isl_pw_multi_aff_has_tuple_name(
__isl_keep isl_pw_multi_aff *pma,
enum isl_dim_type type);
const char *isl_pw_multi_aff_get_tuple_name(
__isl_keep isl_pw_multi_aff *pma,
enum isl_dim_type type);
const char *isl_multi_union_pw_aff_get_tuple_name(
__isl_keep isl_multi_union_pw_aff *mupa,
enum isl_dim_type type);
The type argument needs to be one of isl_dim_in, isl_dim_out or isl_dim_set.
As with isl_space_get_name, the isl_space_get_tuple_name function returns
a pointer to some internal data structure. Binary operations require the corresponding
spaces of their arguments to have the same name.
To keep the names of all parameters and tuples, but reset the user pointers of all the
corresponding identifiers, use the following function.
#include 
__isl_give isl_space *isl_space_reset_user(
__isl_take isl_space *space);
#include 
__isl_give isl_set *isl_set_reset_user(
34

__isl_take isl_set *set);
#include 
__isl_give isl_map *isl_map_reset_user(
__isl_take isl_map *map);
#include 
__isl_give isl_union_set *isl_union_set_reset_user(
__isl_take isl_union_set *uset);
#include 
__isl_give isl_union_map *isl_union_map_reset_user(
__isl_take isl_union_map *umap);
#include 
__isl_give isl_multi_val *isl_multi_val_reset_user(
__isl_take isl_multi_val *mv);
#include 
__isl_give isl_multi_aff *isl_multi_aff_reset_user(
__isl_take isl_multi_aff *ma);
__isl_give isl_pw_aff *isl_pw_aff_reset_user(
__isl_take isl_pw_aff *pa);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_reset_user(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_reset_user(
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_union_pw_aff *isl_union_pw_aff_reset_user(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_reset_user(
__isl_take isl_multi_union_pw_aff *mupa);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_reset_user(
__isl_take isl_union_pw_multi_aff *upma);
#include 
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_reset_user(
__isl_take isl_pw_qpolynomial *pwqp);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_reset_user(
__isl_take isl_union_pw_qpolynomial *upwqp);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_reset_user(
__isl_take isl_pw_qpolynomial_fold *pwf);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_reset_user(
__isl_take isl_union_pw_qpolynomial_fold *upwf);
35

Spaces can be nested. In particular, the domain of a set or the domain or range of
a relation can be a nested relation. This process is also called wrapping. The functions
for detecting, constructing and deconstructing such nested spaces can be found in the
wrapping properties of §1.4.15, the wrapping operations of §1.4.16 and the Cartesian
product operations of §1.4.17.
Spaces can be created from other spaces using the functions described in §1.4.16
and §1.4.17.

1.4.9

Local Spaces

A local space is essentially a space with zero or more existentially quantified variables.
The local space of various objects can be obtained using the following functions.
#include 
__isl_give isl_local_space *isl_constraint_get_local_space(
__isl_keep isl_constraint *constraint);
#include 
__isl_give isl_local_space *isl_basic_set_get_local_space(
__isl_keep isl_basic_set *bset);
#include 
__isl_give isl_local_space *isl_basic_map_get_local_space(
__isl_keep isl_basic_map *bmap);
#include 
__isl_give isl_local_space
__isl_keep isl_aff
__isl_give isl_local_space
__isl_keep isl_aff

*isl_aff_get_domain_local_space(
*aff);
*isl_aff_get_local_space(
*aff);

A new local space can be created from a space using
#include 
__isl_give isl_local_space *isl_local_space_from_space(
__isl_take isl_space *space);
They can be inspected, modified, copied and freed using the following functions.
#include 
isl_bool isl_local_space_is_params(
__isl_keep isl_local_space *ls);
isl_bool isl_local_space_is_set(
__isl_keep isl_local_space *ls);
__isl_give isl_space *isl_local_space_get_space(
__isl_keep isl_local_space *ls);
__isl_give isl_aff *isl_local_space_get_div(

36

__isl_keep isl_local_space *ls, int pos);
__isl_give isl_local_space *isl_local_space_copy(
__isl_keep isl_local_space *ls);
__isl_null isl_local_space *isl_local_space_free(
__isl_take isl_local_space *ls);
Note that isl_local_space_get_div can only be used on local spaces of sets.
Two local spaces can be compared using
isl_bool isl_local_space_is_equal(
__isl_keep isl_local_space *ls1,
__isl_keep isl_local_space *ls2);
Local spaces can be created from other local spaces using the functions described
in §1.4.16 and §1.4.17.

1.4.10

Creating New Sets and Relations

isl has functions for creating some standard sets and relations.
• Empty sets and relations
__isl_give isl_basic_set *isl_basic_set_empty(
__isl_take isl_space *space);
__isl_give isl_basic_map *isl_basic_map_empty(
__isl_take isl_space *space);
__isl_give isl_set *isl_set_empty(
__isl_take isl_space *space);
__isl_give isl_map *isl_map_empty(
__isl_take isl_space *space);
__isl_give isl_union_set *isl_union_set_empty(
__isl_take isl_space *space);
__isl_give isl_union_map *isl_union_map_empty(
__isl_take isl_space *space);
For isl_union_sets and isl_union_maps, the space is only used to specify
the parameters.
• Universe sets and relations
__isl_give isl_basic_set *isl_basic_set_universe(
__isl_take isl_space *space);
__isl_give isl_basic_map *isl_basic_map_universe(
__isl_take isl_space *space);
__isl_give isl_set *isl_set_universe(
__isl_take isl_space *space);
__isl_give isl_map *isl_map_universe(
37

__isl_take isl_space *space);
__isl_give isl_union_set *isl_union_set_universe(
__isl_take isl_union_set *uset);
__isl_give isl_union_map *isl_union_map_universe(
__isl_take isl_union_map *umap);
The sets and relations constructed by the functions above contain all integer values, while those constructed by the functions below only contain non-negative
values.
__isl_give isl_basic_set *isl_basic_set_nat_universe(
__isl_take isl_space *space);
__isl_give isl_basic_map *isl_basic_map_nat_universe(
__isl_take isl_space *space);
__isl_give isl_set *isl_set_nat_universe(
__isl_take isl_space *space);
__isl_give isl_map *isl_map_nat_universe(
__isl_take isl_space *space);
• Identity relations
__isl_give isl_basic_map *isl_basic_map_identity(
__isl_take isl_space *space);
__isl_give isl_map *isl_map_identity(
__isl_take isl_space *space);
The number of input and output dimensions in space needs to be the same.
• Lexicographic order
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take

*isl_map_lex_lt(
isl_space *set_space);
*isl_map_lex_le(
isl_space *set_space);
*isl_map_lex_gt(
isl_space *set_space);
*isl_map_lex_ge(
isl_space *set_space);
*isl_map_lex_lt_first(
isl_space *space, unsigned
*isl_map_lex_le_first(
isl_space *space, unsigned
*isl_map_lex_gt_first(
isl_space *space, unsigned
*isl_map_lex_ge_first(
isl_space *space, unsigned
38

n);
n);
n);
n);

The first four functions take a space for a set and return relations that express
that the elements in the domain are lexicographically less (isl_map_lex_lt),
less or equal (isl_map_lex_le), greater (isl_map_lex_gt) or greater or equal
(isl_map_lex_ge) than the elements in the range. The last four functions take
a space for a map and return relations that express that the first n dimensions in
the domain are lexicographically less (isl_map_lex_lt_first), less or equal
(isl_map_lex_le_first), greater (isl_map_lex_gt_first) or greater or
equal (isl_map_lex_ge_first) than the first n dimensions in the range.
A basic set or relation can be converted to a set or relation using the following
functions.
__isl_give isl_set
__isl_take
__isl_give isl_map
__isl_take

*isl_set_from_basic_set(
isl_basic_set *bset);
*isl_map_from_basic_map(
isl_basic_map *bmap);

Sets and relations can be converted to union sets and relations using the following
functions.
__isl_give isl_union_set *isl_union_set_from_basic_set(
__isl_take isl_basic_set *bset);
__isl_give isl_union_map *isl_union_map_from_basic_map(
__isl_take isl_basic_map *bmap);
__isl_give isl_union_set *isl_union_set_from_set(
__isl_take isl_set *set);
__isl_give isl_union_map *isl_union_map_from_map(
__isl_take isl_map *map);
The inverse conversions below can only be used if the input union set or relation is
known to contain elements in exactly one space.
__isl_give isl_set
__isl_take
__isl_give isl_map
__isl_take

*isl_set_from_union_set(
isl_union_set *uset);
*isl_map_from_union_map(
isl_union_map *umap);

Sets and relations can be copied and freed again using the following functions.
__isl_give isl_basic_set *isl_basic_set_copy(
__isl_keep isl_basic_set *bset);
__isl_give isl_set *isl_set_copy(__isl_keep isl_set *set);
__isl_give isl_union_set *isl_union_set_copy(
__isl_keep isl_union_set *uset);
__isl_give isl_basic_map *isl_basic_map_copy(
__isl_keep isl_basic_map *bmap);
__isl_give isl_map *isl_map_copy(__isl_keep isl_map *map);

39

__isl_give isl_union_map *isl_union_map_copy(
__isl_keep isl_union_map *umap);
__isl_null isl_basic_set *isl_basic_set_free(
__isl_take isl_basic_set *bset);
__isl_null isl_set *isl_set_free(__isl_take isl_set *set);
__isl_null isl_union_set *isl_union_set_free(
__isl_take isl_union_set *uset);
__isl_null isl_basic_map *isl_basic_map_free(
__isl_take isl_basic_map *bmap);
__isl_null isl_map *isl_map_free(__isl_take isl_map *map);
__isl_null isl_union_map *isl_union_map_free(
__isl_take isl_union_map *umap);
Other sets and relations can be constructed by starting from a universe set or relation, adding equality and/or inequality constraints and then projecting out the existentially quantified variables, if any. Constraints can be constructed, manipulated and
added to (or removed from) (basic) sets and relations using the following functions.
#include 
__isl_give isl_constraint *isl_constraint_alloc_equality(
__isl_take isl_local_space *ls);
__isl_give isl_constraint *isl_constraint_alloc_inequality(
__isl_take isl_local_space *ls);
__isl_give isl_constraint *isl_constraint_set_constant_si(
__isl_take isl_constraint *constraint, int v);
__isl_give isl_constraint *isl_constraint_set_constant_val(
__isl_take isl_constraint *constraint,
__isl_take isl_val *v);
__isl_give isl_constraint *isl_constraint_set_coefficient_si(
__isl_take isl_constraint *constraint,
enum isl_dim_type type, int pos, int v);
__isl_give isl_constraint *
isl_constraint_set_coefficient_val(
__isl_take isl_constraint *constraint,
enum isl_dim_type type, int pos,
__isl_take isl_val *v);
__isl_give isl_basic_map *isl_basic_map_add_constraint(
__isl_take isl_basic_map *bmap,
__isl_take isl_constraint *constraint);
__isl_give isl_basic_set *isl_basic_set_add_constraint(
__isl_take isl_basic_set *bset,
__isl_take isl_constraint *constraint);
__isl_give isl_map *isl_map_add_constraint(
__isl_take isl_map *map,
__isl_take isl_constraint *constraint);
__isl_give isl_set *isl_set_add_constraint(

40

__isl_take isl_set *set,
__isl_take isl_constraint *constraint);
For example, to create a set containing the even integers between 10 and 42, you
would use the following code.
isl_space *space;
isl_local_space *ls;
isl_constraint *c;
isl_basic_set *bset;
space = isl_space_set_alloc(ctx, 0, 2);
bset = isl_basic_set_universe(isl_space_copy(space));
ls = isl_local_space_from_space(space);
c = isl_constraint_alloc_equality(isl_local_space_copy(ls));
c = isl_constraint_set_coefficient_si(c, isl_dim_set, 0, -1);
c = isl_constraint_set_coefficient_si(c, isl_dim_set, 1, 2);
bset = isl_basic_set_add_constraint(bset, c);
c = isl_constraint_alloc_inequality(isl_local_space_copy(ls));
c = isl_constraint_set_constant_si(c, -10);
c = isl_constraint_set_coefficient_si(c, isl_dim_set, 0, 1);
bset = isl_basic_set_add_constraint(bset, c);
c = isl_constraint_alloc_inequality(ls);
c = isl_constraint_set_constant_si(c, 42);
c = isl_constraint_set_coefficient_si(c, isl_dim_set, 0, -1);
bset = isl_basic_set_add_constraint(bset, c);
bset = isl_basic_set_project_out(bset, isl_dim_set, 1, 1);
Or, alternatively,
isl_basic_set *bset;
bset = isl_basic_set_read_from_str(ctx,
"{[i] : exists (a : i = 2a and i >= 10 and i <= 42)}");
A basic set or relation can also be constructed from two matrices describing the
equalities and the inequalities.
__isl_give isl_basic_set *isl_basic_set_from_constraint_matrices(
__isl_take isl_space *space,
__isl_take isl_mat *eq, __isl_take isl_mat *ineq,
enum isl_dim_type c1,
enum isl_dim_type c2, enum isl_dim_type c3,
enum isl_dim_type c4);
__isl_give isl_basic_map *isl_basic_map_from_constraint_matrices(
41

__isl_take isl_space *space,
__isl_take isl_mat *eq, __isl_take isl_mat *ineq,
enum isl_dim_type c1,
enum isl_dim_type c2, enum isl_dim_type c3,
enum isl_dim_type c4, enum isl_dim_type c5);
The isl_dim_type arguments indicate the order in which different kinds of variables appear in the input matrices and should be a permutation of isl_dim_cst,
isl_dim_param, isl_dim_set and isl_dim_div for sets and of isl_dim_cst,
isl_dim_param, isl_dim_in, isl_dim_out and isl_dim_div for relations.
A (basic or union) set or relation can also be constructed from a (union) (piecewise)
(multiple) affine expression or a list of affine expressions (See §1.4.13), provided these
affine expressions do not involve any NaN.
#include 
__isl_give isl_basic_set *isl_basic_set_from_multi_aff(
__isl_take isl_multi_aff *ma);
__isl_give isl_set *isl_set_from_multi_aff(
__isl_take isl_multi_aff *ma);
#include 
__isl_give isl_basic_map *isl_basic_map_from_aff(
__isl_take isl_aff *aff);
__isl_give isl_map *isl_map_from_aff(
__isl_take isl_aff *aff);
__isl_give isl_basic_map *isl_basic_map_from_aff_list(
__isl_take isl_space *domain_space,
__isl_take isl_aff_list *list);
__isl_give isl_basic_map *isl_basic_map_from_multi_aff(
__isl_take isl_multi_aff *maff)
__isl_give isl_map *isl_map_from_multi_aff(
__isl_take isl_multi_aff *maff)
#include 
__isl_give isl_set *isl_set_from_pw_aff(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_map *isl_map_from_pw_aff(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_set *isl_set_from_pw_multi_aff(
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_map *isl_map_from_pw_multi_aff(
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_set *isl_set_from_multi_pw_aff(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_map *isl_map_from_multi_pw_aff(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_union_map *isl_union_map_from_union_pw_aff(
42

__isl_take isl_union_pw_aff *upa);
__isl_give isl_union_map *
isl_union_map_from_union_pw_multi_aff(
__isl_take isl_union_pw_multi_aff *upma);
__isl_give isl_union_map *
isl_union_map_from_multi_union_pw_aff(
__isl_take isl_multi_union_pw_aff *mupa);
The domain_space argument describes the domain of the resulting basic relation.
It is required because the list may consist of zero affine expressions. The mupa
passed to isl_union_map_from_multi_union_pw_aff is not allowed to be zerodimensional. The domain of the result is the shared domain of the union piecewise
affine elements.

1.4.11

Inspecting Sets and Relations

Usually, the user should not have to care about the actual constraints of the sets and
maps, but should instead apply the abstract operations explained in the following sections. Occasionally, however, it may be required to inspect the individual coefficients
of the constraints. This section explains how to do so. In these cases, it may also be
useful to have isl compute an explicit representation of the existentially quantified
variables.
__isl_give isl_set *isl_set_compute_divs(
__isl_take isl_set *set);
__isl_give isl_map *isl_map_compute_divs(
__isl_take isl_map *map);
__isl_give isl_union_set *isl_union_set_compute_divs(
__isl_take isl_union_set *uset);
__isl_give isl_union_map *isl_union_map_compute_divs(
__isl_take isl_union_map *umap);
This explicit representation defines the existentially quantified variables as integer
divisions of the other variables, possibly including earlier existentially quantified variables. An explicitly represented existentially quantified variable therefore has a unique
value when the values of the other variables are known.
Alternatively, the existentially quantified variables can be removed using the following functions, which compute an overapproximation.
#include 
__isl_give isl_basic_set *isl_basic_set_remove_divs(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_set_remove_divs(
__isl_take isl_set *set);
#include 
__isl_give isl_basic_map *isl_basic_map_remove_divs(
43

__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_remove_divs(
__isl_take isl_map *map);
#include 
__isl_give isl_union_set *isl_union_set_remove_divs(
__isl_take isl_union_set *bset);
#include 
__isl_give isl_union_map *isl_union_map_remove_divs(
__isl_take isl_union_map *bmap);
It is also possible to only remove those divs that are defined in terms of a given
range of dimensions or only those for which no explicit representation is known.
__isl_give isl_basic_set *
isl_basic_set_remove_divs_involving_dims(
__isl_take isl_basic_set *bset,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_basic_map *
isl_basic_map_remove_divs_involving_dims(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_set *isl_set_remove_divs_involving_dims(
__isl_take isl_set *set, enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_map *isl_map_remove_divs_involving_dims(
__isl_take isl_map *map, enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_basic_set *
isl_basic_set_remove_unknown_divs(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_set_remove_unknown_divs(
__isl_take isl_set *set);
__isl_give isl_map *isl_map_remove_unknown_divs(
__isl_take isl_map *map);
To iterate over all the sets or maps in a union set or map, use
#include 
isl_stat isl_union_set_foreach_set(
__isl_keep isl_union_set *uset,
isl_stat (*fn)(__isl_take isl_set *set, void *user),
void *user);
44

#include 
isl_stat isl_union_map_foreach_map(
__isl_keep isl_union_map *umap,
isl_stat (*fn)(__isl_take isl_map *map, void *user),
void *user);
isl_bool isl_union_map_every_map(
__isl_keep isl_union_map *umap,
isl_bool (*test)(__isl_keep isl_map *map,
void *user),
void *user);
These functions call the callback function once for each (pair of) space(s) for which
there are elements in the input. The argument to the callback contains all elements in
the input with that (pair of) space(s). The isl_union_map_every_map variant check
whether each call to the callback returns true and stops checking as soon as one of these
calls returns false.
The number of sets or maps in a union set or map can be obtained from
int isl_union_set_n_set(__isl_keep isl_union_set *uset);
int isl_union_map_n_map(__isl_keep isl_union_map *umap);
To extract the set or map in a given space from a union, use
__isl_give isl_set
__isl_keep
__isl_take
__isl_give isl_map
__isl_keep
__isl_take

*isl_union_set_extract_set(
isl_union_set *uset,
isl_space *space);
*isl_union_map_extract_map(
isl_union_map *umap,
isl_space *space);

To iterate over all the basic sets or maps in a set or map, use
isl_stat isl_set_foreach_basic_set(__isl_keep isl_set *set,
isl_stat (*fn)(__isl_take isl_basic_set *bset,
void *user),
void *user);
isl_stat isl_map_foreach_basic_map(__isl_keep isl_map *map,
isl_stat (*fn)(__isl_take isl_basic_map *bmap,
void *user),
void *user);
The callback function fn should return isl_stat_ok if successful and isl_stat_error
if an error occurs. In the latter case, or if any other error occurs, the above functions
will return isl_stat_error.
It should be noted that isl does not guarantee that the basic sets or maps passed
to fn are disjoint. If this is required, then the user should call one of the following
functions first.
45

__isl_give isl_set
__isl_take
__isl_give isl_map
__isl_take

*isl_set_make_disjoint(
isl_set *set);
*isl_map_make_disjoint(
isl_map *map);

The number of basic sets in a set can be obtained or the number of basic maps in a
map can be obtained from
#include 
int isl_set_n_basic_set(__isl_keep isl_set *set);
#include 
int isl_map_n_basic_map(__isl_keep isl_map *map);
It is also possible to obtain a list of (basic) sets from a set or union set, a list of
basic maps from a map and a list of maps from a union map.
#include 
__isl_give isl_basic_set_list *isl_set_get_basic_set_list(
__isl_keep isl_set *set);
#include 
__isl_give isl_basic_set_list *
isl_union_set_get_basic_set_list(
__isl_keep isl_union_set *uset);
__isl_give isl_set_list *isl_union_set_get_set_list(
__isl_keep isl_union_set *uset);
#include 
__isl_give isl_basic_map_list *isl_map_get_basic_map_list(
__isl_keep isl_map *map);
#include 
__isl_give isl_map_list *isl_union_map_get_map_list(
__isl_keep isl_union_map *umap);
The returned list can be manipulated using the functions in §1.4.19.
To iterate over the constraints of a basic set or map, use
#include 
int isl_basic_set_n_constraint(
__isl_keep isl_basic_set *bset);
isl_stat isl_basic_set_foreach_constraint(
__isl_keep isl_basic_set *bset,
isl_stat (*fn)(__isl_take isl_constraint *c,
void *user),
void *user);
46

int isl_basic_map_n_constraint(
__isl_keep isl_basic_map *bmap);
isl_stat isl_basic_map_foreach_constraint(
__isl_keep isl_basic_map *bmap,
isl_stat (*fn)(__isl_take isl_constraint *c,
void *user),
void *user);
__isl_null isl_constraint *isl_constraint_free(
__isl_take isl_constraint *c);
Again, the callback function fn should return isl_stat_ok if successful and
isl_stat_error if an error occurs. In the latter case, or if any other error occurs,
the above functions will return isl_stat_error. The constraint c represents either
an equality or an inequality. Use the following function to find out whether a constraint
represents an equality. If not, it represents an inequality.
isl_bool isl_constraint_is_equality(
__isl_keep isl_constraint *constraint);
It is also possible to obtain a list of constraints from a basic map or set
#include 
__isl_give isl_constraint_list *
isl_basic_map_get_constraint_list(
__isl_keep isl_basic_map *bmap);
__isl_give isl_constraint_list *
isl_basic_set_get_constraint_list(
__isl_keep isl_basic_set *bset);
These functions require that all existentially quantified variables have an explicit
representation. The returned list can be manipulated using the functions in §1.4.19.
The coefficients of the constraints can be inspected using the following functions.
isl_bool isl_constraint_is_lower_bound(
__isl_keep isl_constraint *constraint,
enum isl_dim_type type, unsigned pos);
isl_bool isl_constraint_is_upper_bound(
__isl_keep isl_constraint *constraint,
enum isl_dim_type type, unsigned pos);
__isl_give isl_val *isl_constraint_get_constant_val(
__isl_keep isl_constraint *constraint);
__isl_give isl_val *isl_constraint_get_coefficient_val(
__isl_keep isl_constraint *constraint,
enum isl_dim_type type, int pos);
The explicit representations of the existentially quantified variables can be inspected
using the following function. Note that the user is only allowed to use this function if the inspected set or map is the result of a call to isl_set_compute_divs or
47

isl_map_compute_divs. The existentially quantified variable is equal to the floor of
the returned affine expression. The affine expression itself can be inspected using the
functions in §1.4.13.
__isl_give isl_aff *isl_constraint_get_div(
__isl_keep isl_constraint *constraint, int pos);
To obtain the constraints of a basic set or map in matrix form, use the following
functions.
__isl_give isl_mat *isl_basic_set_equalities_matrix(
__isl_keep isl_basic_set *bset,
enum isl_dim_type c1, enum isl_dim_type c2,
enum isl_dim_type c3, enum isl_dim_type c4);
__isl_give isl_mat *isl_basic_set_inequalities_matrix(
__isl_keep isl_basic_set *bset,
enum isl_dim_type c1, enum isl_dim_type c2,
enum isl_dim_type c3, enum isl_dim_type c4);
__isl_give isl_mat *isl_basic_map_equalities_matrix(
__isl_keep isl_basic_map *bmap,
enum isl_dim_type c1,
enum isl_dim_type c2, enum isl_dim_type c3,
enum isl_dim_type c4, enum isl_dim_type c5);
__isl_give isl_mat *isl_basic_map_inequalities_matrix(
__isl_keep isl_basic_map *bmap,
enum isl_dim_type c1,
enum isl_dim_type c2, enum isl_dim_type c3,
enum isl_dim_type c4, enum isl_dim_type c5);
The isl_dim_type arguments dictate the order in which different kinds of variables appear in the resulting matrix. For set inputs, they should be a permutation
of isl_dim_cst, isl_dim_param, isl_dim_set and isl_dim_div. For map inputs, they should be a permutation of isl_dim_cst, isl_dim_param, isl_dim_in,
isl_dim_out and isl_dim_div.

1.4.12

Points

Points are elements of a set. They can be used to construct simple sets (boxes) or they
can be used to represent the individual elements of a set. The zero point (the origin)
can be created using
__isl_give isl_point *isl_point_zero(__isl_take isl_space *space);
The coordinates of a point can be inspected, set and changed using
__isl_give isl_val *isl_point_get_coordinate_val(
__isl_keep isl_point *pnt,
48

enum isl_dim_type type, int pos);
__isl_give isl_point *isl_point_set_coordinate_val(
__isl_take isl_point *pnt,
enum isl_dim_type type, int pos,
__isl_take isl_val *v);
__isl_give isl_point *isl_point_add_ui(
__isl_take isl_point *pnt,
enum isl_dim_type type, int pos, unsigned val);
__isl_give isl_point *isl_point_sub_ui(
__isl_take isl_point *pnt,
enum isl_dim_type type, int pos, unsigned val);
Points can be copied or freed using
__isl_give isl_point *isl_point_copy(
__isl_keep isl_point *pnt);
__isl_null isl_point *isl_point_free(
__isl_take isl_point *pnt);
A singleton set can be created from a point using
__isl_give isl_basic_set *isl_basic_set_from_point(
__isl_take isl_point *pnt);
__isl_give isl_set *isl_set_from_point(
__isl_take isl_point *pnt);
__isl_give isl_union_set *isl_union_set_from_point(
__isl_take isl_point *pnt);
and a box can be created from two opposite extremal points using
__isl_give isl_basic_set *isl_basic_set_box_from_points(
__isl_take isl_point *pnt1,
__isl_take isl_point *pnt2);
__isl_give isl_set *isl_set_box_from_points(
__isl_take isl_point *pnt1,
__isl_take isl_point *pnt2);
All elements of a bounded (union) set can be enumerated using the following functions.
isl_stat isl_set_foreach_point(__isl_keep isl_set *set,
isl_stat (*fn)(__isl_take isl_point *pnt,
void *user),
void *user);
isl_stat isl_union_set_foreach_point(
__isl_keep isl_union_set *uset,
isl_stat (*fn)(__isl_take isl_point *pnt,
void *user),
void *user);
49

The function fn is called for each integer point in set with as second argument
the last argument of the isl_set_foreach_point call. The function fn should return isl_stat_ok on success and isl_stat_error on failure. In the latter case,
isl_set_foreach_point will stop enumerating and return isl_stat_error as well.
If the enumeration is performed successfully and to completion, then isl_set_foreach_point
returns isl_stat_ok.
To obtain a single point of a (basic or union) set, use
__isl_give isl_point *isl_basic_set_sample_point(
__isl_take isl_basic_set *bset);
__isl_give isl_point *isl_set_sample_point(
__isl_take isl_set *set);
__isl_give isl_point *isl_union_set_sample_point(
__isl_take isl_union_set *uset);
If set does not contain any (integer) points, then the resulting point will be “void”,
a property that can be tested using
isl_bool isl_point_is_void(__isl_keep isl_point *pnt);

1.4.13

Functions

Besides sets and relation, isl also supports various types of functions. Each of these
types is derived from the value type (see §1.4.4) or from one of two primitive function
types through the application of zero or more type constructors. We first describe the
primitive type and then we describe the types derived from these primitive types.
Primitive Functions
isl support two primitive function types, quasi-affine expressions and quasipolynomials. A quasi-affine expression is defined either over a parameter space or over a set and
is composed of integer constants, parameters and set variables, addition, subtraction
and integer division by an integer constant. For example, the quasi-affine expression
[n] -> { [x] -> [2*floor((4 n + x)/9)] }
maps x to 2*floor((4 n + x)/9. A quasipolynomial is a polynomial expression in quasi-affine expression. That is, it additionally allows for multiplication. Note,
though, that it is not allowed to construct an integer division of an expression involving multiplications. Here is an example of a quasipolynomial that is not quasi-affine
expression
[n] -> { [x] -> (n*floor((4 n + x)/9)) }
Note that the external representations of quasi-affine expressions and quasipolynomials are different. Quasi-affine expressions use a notation with square brackets just
like binary relations, while quasipolynomials do not. This might change at some point.

50

If a primitive function is defined over a parameter space, then the space of the function itself is that of a set. If it is defined over a set, then the space of the function is that
of a relation. In both cases, the set space (or the output space) is single-dimensional,
anonymous and unstructured. To create functions with multiple dimensions or with
other kinds of set or output spaces, use multiple expressions (see §1.4.13).
• Quasi-affine Expressions
Besides the expressions described above, a quasi-affine expression can also be
set to NaN. Such expressions typically represent a failure to represent a result as
a quasi-affine expression.
The zero quasi affine expression or the quasi affine expression that is equal to a
given value, parameter or a specified dimension on a given domain can be created
using
#include 
__isl_give isl_aff *isl_aff_zero_on_domain(
__isl_take isl_local_space *ls);
__isl_give isl_aff *isl_aff_val_on_domain(
__isl_take isl_local_space *ls,
__isl_take isl_val *val);
__isl_give isl_aff *isl_aff_param_on_domain_space_id(
__isl_take isl_space *space,
__isl_take isl_id *id);
__isl_give isl_aff *isl_aff_var_on_domain(
__isl_take isl_local_space *ls,
enum isl_dim_type type, unsigned pos);
__isl_give isl_aff *isl_aff_nan_on_domain(
__isl_take isl_local_space *ls);
The space passed to isl_aff_param_on_domain_space_id is required to have
a parameter with the given identifier.
Quasi affine expressions can be copied and freed using
#include 
__isl_give isl_aff *isl_aff_copy(
__isl_keep isl_aff *aff);
__isl_null isl_aff *isl_aff_free(
__isl_take isl_aff *aff);
A (rational) bound on a dimension can be extracted from an isl_constraint
using the following function. The constraint is required to have a non-zero coefficient for the specified dimension.
#include 

51

__isl_give isl_aff *isl_constraint_get_bound(
__isl_keep isl_constraint *constraint,
enum isl_dim_type type, int pos);
The entire affine expression of the constraint can also be extracted using the
following function.
#include 
__isl_give isl_aff *isl_constraint_get_aff(
__isl_keep isl_constraint *constraint);
Conversely, an equality constraint equating the affine expression to zero or an
inequality constraint enforcing the affine expression to be non-negative, can be
constructed using
__isl_give isl_constraint *isl_equality_from_aff(
__isl_take isl_aff *aff);
__isl_give isl_constraint *isl_inequality_from_aff(
__isl_take isl_aff *aff);
The coefficients and the integer divisions of an affine expression can be inspected
using the following functions.
#include 
__isl_give isl_val *isl_aff_get_constant_val(
__isl_keep isl_aff *aff);
__isl_give isl_val *isl_aff_get_coefficient_val(
__isl_keep isl_aff *aff,
enum isl_dim_type type, int pos);
int isl_aff_coefficient_sgn(__isl_keep isl_aff *aff,
enum isl_dim_type type, int pos);
__isl_give isl_val *isl_aff_get_denominator_val(
__isl_keep isl_aff *aff);
__isl_give isl_aff *isl_aff_get_div(
__isl_keep isl_aff *aff, int pos);
They can be modified using the following functions.
#include 
__isl_give isl_aff *isl_aff_set_constant_si(
__isl_take isl_aff *aff, int v);
__isl_give isl_aff *isl_aff_set_constant_val(
__isl_take isl_aff *aff, __isl_take isl_val *v);
__isl_give isl_aff *isl_aff_set_coefficient_si(
__isl_take isl_aff *aff,
52

enum isl_dim_type type, int pos, int v);
__isl_give isl_aff *isl_aff_set_coefficient_val(
__isl_take isl_aff *aff,
enum isl_dim_type type, int pos,
__isl_take isl_val *v);
__isl_give isl_aff *isl_aff_add_constant_si(
__isl_take isl_aff *aff, int v);
__isl_give isl_aff *isl_aff_add_constant_val(
__isl_take isl_aff *aff, __isl_take isl_val *v);
__isl_give isl_aff *isl_aff_add_constant_num_si(
__isl_take isl_aff *aff, int v);
__isl_give isl_aff *isl_aff_add_coefficient_si(
__isl_take isl_aff *aff,
enum isl_dim_type type, int pos, int v);
__isl_give isl_aff *isl_aff_add_coefficient_val(
__isl_take isl_aff *aff,
enum isl_dim_type type, int pos,
__isl_take isl_val *v);
Note that isl_aff_set_constant_si and isl_aff_set_coefficient_si
set the numerator of the constant or coefficient, while isl_aff_set_constant_val
and isl_aff_set_coefficient_val set the constant or coefficient as a whole.
The add_constant and add_coefficient functions add an integer or rational
value to the possibly rational constant or coefficient. The add_constant_num
functions add an integer value to the numerator.
• Quasipolynomials
Some simple quasipolynomials can be created using the following functions.
#include 
__isl_give isl_qpolynomial *isl_qpolynomial_zero_on_domain(
__isl_take isl_space *domain);
__isl_give isl_qpolynomial *isl_qpolynomial_one_on_domain(
__isl_take isl_space *domain);
__isl_give isl_qpolynomial *isl_qpolynomial_infty_on_domain(
__isl_take isl_space *domain);
__isl_give isl_qpolynomial *isl_qpolynomial_neginfty_on_domain(
__isl_take isl_space *domain);
__isl_give isl_qpolynomial *isl_qpolynomial_nan_on_domain(
__isl_take isl_space *domain);
__isl_give isl_qpolynomial *isl_qpolynomial_val_on_domain(
__isl_take isl_space *domain,
__isl_take isl_val *val);
__isl_give isl_qpolynomial *isl_qpolynomial_var_on_domain(
__isl_take isl_space *domain,
53

enum isl_dim_type type, unsigned pos);
__isl_give isl_qpolynomial *isl_qpolynomial_from_aff(
__isl_take isl_aff *aff);
Recall that the space in which a quasipolynomial lives is a map space with a
one-dimensional range. The domain argument in some of the functions above
corresponds to the domain of this map space.
Quasipolynomials can be copied and freed again using the following functions.
#include 
__isl_give isl_qpolynomial *isl_qpolynomial_copy(
__isl_keep isl_qpolynomial *qp);
__isl_null isl_qpolynomial *isl_qpolynomial_free(
__isl_take isl_qpolynomial *qp);
The constant term of a quasipolynomial can be extracted using
__isl_give isl_val *isl_qpolynomial_get_constant_val(
__isl_keep isl_qpolynomial *qp);
To iterate over all terms in a quasipolynomial, use
isl_stat isl_qpolynomial_foreach_term(
__isl_keep isl_qpolynomial *qp,
isl_stat (*fn)(__isl_take isl_term *term,
void *user), void *user);
The terms themselves can be inspected and freed using these functions
unsigned isl_term_dim(__isl_keep isl_term *term,
enum isl_dim_type type);
__isl_give isl_val *isl_term_get_coefficient_val(
__isl_keep isl_term *term);
int isl_term_get_exp(__isl_keep isl_term *term,
enum isl_dim_type type, unsigned pos);
__isl_give isl_aff *isl_term_get_div(
__isl_keep isl_term *term, unsigned pos);
void isl_term_free(__isl_take isl_term *term);
Each term is a product of parameters, set variables and integer divisions. The
function isl_term_get_exp returns the exponent of a given dimensions in the
given term.

54

Reductions
A reduction represents a maximum or a minimum of its base expressions. The only
reduction type defined by isl is isl_qpolynomial_fold.
There are currently no functions to directly create such objects, but they do appear
in the piecewise quasipolynomial reductions returned by the isl_pw_qpolynomial_bound
function. See §1.4.23.
Reductions can be copied and freed using the following functions.
#include 
__isl_give isl_qpolynomial_fold *
isl_qpolynomial_fold_copy(
__isl_keep isl_qpolynomial_fold *fold);
void isl_qpolynomial_fold_free(
__isl_take isl_qpolynomial_fold *fold);
To iterate over all quasipolynomials in a reduction, use
isl_stat isl_qpolynomial_fold_foreach_qpolynomial(
__isl_keep isl_qpolynomial_fold *fold,
isl_stat (*fn)(__isl_take isl_qpolynomial *qp,
void *user), void *user);
Multiple Expressions
A multiple expression represents a sequence of zero or more base expressions, all defined on the same domain space. The domain space of the multiple expression is the
same as that of the base expressions, but the range space can be any space. In case
the base expressions have a set space, the corresponding multiple expression also has
a set space. Objects of the value type do not have an associated space. The space
of a multiple value is therefore always a set space. Similarly, the space of a multiple
union piecewise affine expression is always a set space. If the base expressions are not
total, then a corresponding zero-dimensional multiple expression may have an explicit
domain that keeps track of the domain outside of any base expressions.
The multiple expression types defined by isl are isl_multi_val, isl_multi_aff,
isl_multi_pw_aff, isl_multi_union_pw_aff.
A multiple expression with the value zero for each output (or set) dimension can be
created using the following functions.
#include 
__isl_give isl_multi_val *isl_multi_val_zero(
__isl_take isl_space *space);
#include 
__isl_give isl_multi_aff *isl_multi_aff_zero(
__isl_take isl_space *space);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_zero(

55

__isl_take isl_space *space);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_zero(
__isl_take isl_space *space);
Since there is no canonical way of representing a zero value of type isl_union_pw_aff,
the space passed to isl_multi_union_pw_aff_zero needs to be zero-dimensional.
An identity function can be created using the following functions. The space needs
to be that of a relation with the same number of input and output dimensions.
#include 
__isl_give isl_multi_aff *isl_multi_aff_identity(
__isl_take isl_space *space);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_identity(
__isl_take isl_space *space);
A function that performs a projection on a universe relation or set can be created
using the following functions. See also the corresponding projection operations in
§1.4.16.
#include 
__isl_give isl_multi_aff *isl_multi_aff_domain_map(
__isl_take isl_space *space);
__isl_give isl_multi_aff *isl_multi_aff_range_map(
__isl_take isl_space *space);
__isl_give isl_multi_aff *isl_multi_aff_project_out_map(
__isl_take isl_space *space,
enum isl_dim_type type,
unsigned first, unsigned n);
A multiple expression can be created from a single base expression using the following functions. The space of the created multiple expression is the same as that
of the base expression, except for isl_multi_union_pw_aff_from_union_pw_aff
where the input lives in a parameter space and the output lives in a single-dimensional
set space.
#include 
__isl_give isl_multi_aff *isl_multi_aff_from_aff(
__isl_take isl_aff *aff);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_pw_aff(
__isl_take isl_pw_aff *pa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_from_union_pw_aff(
__isl_take isl_union_pw_aff *upa);
A multiple expression can be created from a list of base expression in a specified
space. The domain of this space needs to be the same as the domains of the base
expressions in the list. If the base expressions have a set space (or no associated space),
then this space also needs to be a set space.
56

#include 
__isl_give isl_multi_val *isl_multi_val_from_val_list(
__isl_take isl_space *space,
__isl_take isl_val_list *list);
#include 
__isl_give isl_multi_aff *isl_multi_aff_from_aff_list(
__isl_take isl_space *space,
__isl_take isl_aff_list *list);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_from_pw_aff_list(
__isl_take isl_space *space,
__isl_take isl_pw_aff_list *list);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_from_union_pw_aff_list(
__isl_take isl_space *space,
__isl_take isl_union_pw_aff_list *list);
As a convenience, a multiple piecewise expression can also be created from a multiple expression. Each piecewise expression in the result has a single universe cell.
#include 
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_from_multi_aff(
__isl_take isl_multi_aff *ma);
Similarly, a multiple union expression can be created from a multiple expression.
#include 
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_from_multi_aff(
__isl_take isl_multi_aff *ma);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_from_multi_pw_aff(
__isl_take isl_multi_pw_aff *mpa);
A multiple quasi-affine expression can be created from a multiple value with a given
domain space using the following function.
#include 
__isl_give isl_multi_aff *
isl_multi_aff_multi_val_on_space(
__isl_take isl_space *space,
__isl_take isl_multi_val *mv);
Similarly, a multiple union piecewise affine expression can be created from a multiple value with a given domain or a (piecewise) multiple affine expression with a given
domain using the following functions.
57

#include 
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_multi_val_on_domain(
__isl_take isl_union_set *domain,
__isl_take isl_multi_val *mv);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_multi_aff_on_domain(
__isl_take isl_union_set *domain,
__isl_take isl_multi_aff *ma);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_pw_multi_aff_on_domain(
__isl_take isl_union_set *domain,
__isl_take isl_pw_multi_aff *pma);
Multiple expressions can be copied and freed using the following functions.
#include 
__isl_give isl_multi_val *isl_multi_val_copy(
__isl_keep isl_multi_val *mv);
__isl_null isl_multi_val *isl_multi_val_free(
__isl_take isl_multi_val *mv);
#include 
__isl_give isl_multi_aff *isl_multi_aff_copy(
__isl_keep isl_multi_aff *maff);
__isl_null isl_multi_aff *isl_multi_aff_free(
__isl_take isl_multi_aff *maff);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_copy(
__isl_keep isl_multi_pw_aff *mpa);
__isl_null isl_multi_pw_aff *isl_multi_pw_aff_free(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_copy(
__isl_keep isl_multi_union_pw_aff *mupa);
__isl_null isl_multi_union_pw_aff *
isl_multi_union_pw_aff_free(
__isl_take isl_multi_union_pw_aff *mupa);
The base expression at a given position of a multiple expression can be extracted
using the following functions.
#include 
__isl_give isl_val *isl_multi_val_get_val(
__isl_keep isl_multi_val *mv, int pos);
#include 
__isl_give isl_aff *isl_multi_aff_get_aff(
58

__isl_keep isl_multi_aff *multi, int pos);
__isl_give isl_pw_aff *isl_multi_pw_aff_get_pw_aff(
__isl_keep isl_multi_pw_aff *mpa, int pos);
__isl_give isl_union_pw_aff *
isl_multi_union_pw_aff_get_union_pw_aff(
__isl_keep isl_multi_union_pw_aff *mupa, int pos);
It can be replaced using the following functions.
#include 
__isl_give isl_multi_val *isl_multi_val_set_val(
__isl_take isl_multi_val *mv, int pos,
__isl_take isl_val *val);
#include 
__isl_give isl_multi_aff *isl_multi_aff_set_aff(
__isl_take isl_multi_aff *multi, int pos,
__isl_take isl_aff *aff);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_set_union_pw_aff(
__isl_take isl_multi_union_pw_aff *mupa, int pos,
__isl_take isl_union_pw_aff *upa);
As a convenience, a sequence of base expressions that have their domains in a
given space can be extracted from a sequence of union expressions using the following
function.
#include 
__isl_give isl_multi_pw_aff *
isl_multi_union_pw_aff_extract_multi_pw_aff(
__isl_keep isl_multi_union_pw_aff *mupa,
__isl_take isl_space *space);
Note that there is a difference between isl_multi_union_pw_aff and isl_union_pw_multi_aff
objects. The first is a sequence of unions of piecewise expressions, while the second is a union of piecewise sequences. In particular, multiple affine expressions in
an isl_union_pw_multi_aff may live in different spaces, while there is only a
single multiple expression in an isl_multi_union_pw_aff, which can therefore
only live in a single space. This means that not every isl_union_pw_multi_aff
can be converted to an isl_multi_union_pw_aff. Conversely, the elements of an
isl_multi_union_pw_aff may be defined over different domains, while each multiple expression inside an isl_union_pw_multi_aff has a single domain. The conversion of an isl_union_pw_multi_aff of dimension greater than one may therefore
not be exact. The following functions can be used to perform these conversions when
they are possible.
#include 
59

__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_from_union_pw_multi_aff(
__isl_take isl_union_pw_multi_aff *upma);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_from_multi_union_pw_aff(
__isl_take isl_multi_union_pw_aff *mupa);
Piecewise Expressions
A piecewise expression is an expression that is described using zero or more base
expression defined over the same number of cells in the domain space of the base expressions. All base expressions are defined over the same domain space and the cells
are disjoint. The space of a piecewise expression is the same as that of the base expressions. If the union of the cells is a strict subset of the domain space, then the value of the
piecewise expression outside this union is different for types derived from quasi-affine
expressions and those derived from quasipolynomials. Piecewise expressions derived
from quasi-affine expressions are considered to be undefined outside the union of their
cells. Piecewise expressions derived from quasipolynomials are considered to be zero
outside the union of their cells.
Piecewise quasipolynomials are mainly used by the barvinok library for representing the number of elements in a parametric set or map. For example, the piecewise
quasipolynomial
[n] -> { [x] -> ((1 + n) - x) : x <= n and x >= 0 }
represents the number of points in the map
[n] -> { [x] -> [y] : x,y >= 0 and 0 <= x + y <= n }
The piecewise expression types defined by isl are isl_pw_aff, isl_pw_multi_aff,
isl_pw_qpolynomial and isl_pw_qpolynomial_fold.
A piecewise expression with no cells can be created using the following functions.
#include 
__isl_give isl_pw_aff *isl_pw_aff_empty(
__isl_take isl_space *space);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_empty(
__isl_take isl_space *space);
A piecewise expression with a single universe cell can be created using the following functions.
#include 
__isl_give isl_pw_aff *isl_pw_aff_from_aff(
__isl_take isl_aff *aff);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_from_multi_aff(
__isl_take isl_multi_aff *ma);
60

#include 
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_from_qpolynomial(
__isl_take isl_qpolynomial *qp);
A piecewise expression with a single specified cell can be created using the following functions.
#include 
__isl_give isl_pw_aff *isl_pw_aff_alloc(
__isl_take isl_set *set, __isl_take isl_aff *aff);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_alloc(
__isl_take isl_set *set,
__isl_take isl_multi_aff *maff);
#include 
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_alloc(
__isl_take isl_set *set,
__isl_take isl_qpolynomial *qp);
The following convenience functions first create a base expression and then create
a piecewise expression over a universe domain.
#include 
__isl_give isl_pw_aff *isl_pw_aff_zero_on_domain(
__isl_take isl_local_space *ls);
__isl_give isl_pw_aff *isl_pw_aff_var_on_domain(
__isl_take isl_local_space *ls,
enum isl_dim_type type, unsigned pos);
__isl_give isl_pw_aff *isl_pw_aff_nan_on_domain(
__isl_take isl_local_space *ls);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_zero(
__isl_take isl_space *space);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_identity(
__isl_take isl_space *space);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_range_map(
__isl_take isl_space *space);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_project_out_map(
__isl_take isl_space *space,
enum isl_dim_type type,
unsigned first, unsigned n);
#include 
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_zero(
__isl_take isl_space *space);

61

The following convenience functions first create a base expression and then create
a piecewise expression over a given domain.
#include 
__isl_give isl_pw_aff *isl_pw_aff_val_on_domain(
__isl_take isl_set *domain,
__isl_take isl_val *v);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_multi_val_on_domain(
__isl_take isl_set *domain,
__isl_take isl_multi_val *mv);
As a convenience, a piecewise multiple expression can also be created from a piecewise expression. Each multiple expression in the result is derived from the corresponding base expression.
#include 
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_pw_aff(
__isl_take isl_pw_aff *pa);
Similarly, a piecewise quasipolynomial can be created from a piecewise quasiaffine expression using the following function.
#include 
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_from_pw_aff(
__isl_take isl_pw_aff *pwaff);
Piecewise expressions can be copied and freed using the following functions.
#include 
__isl_give isl_pw_aff *isl_pw_aff_copy(
__isl_keep isl_pw_aff *pwaff);
__isl_null isl_pw_aff *isl_pw_aff_free(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_copy(
__isl_keep isl_pw_multi_aff *pma);
__isl_null isl_pw_multi_aff *isl_pw_multi_aff_free(
__isl_take isl_pw_multi_aff *pma);
#include 
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_copy(
__isl_keep isl_pw_qpolynomial *pwqp);
__isl_null isl_pw_qpolynomial *isl_pw_qpolynomial_free(
__isl_take isl_pw_qpolynomial *pwqp);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_copy(

62

__isl_keep isl_pw_qpolynomial_fold *pwf);
__isl_null isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_free(
__isl_take isl_pw_qpolynomial_fold *pwf);
To iterate over the different cells of a piecewise expression, use the following functions.
#include 
isl_bool isl_pw_aff_is_empty(__isl_keep isl_pw_aff *pwaff);
int isl_pw_aff_n_piece(__isl_keep isl_pw_aff *pwaff);
isl_stat isl_pw_aff_foreach_piece(
__isl_keep isl_pw_aff *pwaff,
isl_stat (*fn)(__isl_take isl_set *set,
__isl_take isl_aff *aff,
void *user), void *user);
int isl_pw_multi_aff_n_piece(
__isl_keep isl_pw_multi_aff *pma);
isl_stat isl_pw_multi_aff_foreach_piece(
__isl_keep isl_pw_multi_aff *pma,
isl_stat (*fn)(__isl_take isl_set *set,
__isl_take isl_multi_aff *maff,
void *user), void *user);
#include 
int isl_pw_qpolynomial_n_piece(
__isl_keep isl_pw_qpolynomial *pwqp);
isl_stat isl_pw_qpolynomial_foreach_piece(
__isl_keep isl_pw_qpolynomial *pwqp,
isl_stat (*fn)(__isl_take isl_set *set,
__isl_take isl_qpolynomial *qp,
void *user), void *user);
isl_stat isl_pw_qpolynomial_foreach_lifted_piece(
__isl_keep isl_pw_qpolynomial *pwqp,
isl_stat (*fn)(__isl_take isl_set *set,
__isl_take isl_qpolynomial *qp,
void *user), void *user);
int isl_pw_qpolynomial_fold_n_piece(
__isl_keep isl_pw_qpolynomial_fold *pwf);
isl_stat isl_pw_qpolynomial_fold_foreach_piece(
__isl_keep isl_pw_qpolynomial_fold *pwf,
isl_stat (*fn)(__isl_take isl_set *set,
__isl_take isl_qpolynomial_fold *fold,
void *user), void *user);
isl_stat isl_pw_qpolynomial_fold_foreach_lifted_piece(
__isl_keep isl_pw_qpolynomial_fold *pwf,
isl_stat (*fn)(__isl_take isl_set *set,
63

__isl_take isl_qpolynomial_fold *fold,
void *user), void *user);

As usual, the function fn should return isl_stat_ok on success and isl_stat_error
on failure. The difference between isl_pw_qpolynomial_foreach_piece and isl_pw_qpolynomial_foreach_lif
is that isl_pw_qpolynomial_foreach_lifted_piece will first compute unique
representations for all existentially quantified variables and then turn these existentially quantified variables into extra set variables, adapting the associated quasipolynomial accordingly. This means that the set passed to fn will not have any existentially
quantified variables, but that the dimensions of the sets may be different for different
invocations of fn. Similarly for isl_pw_qpolynomial_fold_foreach_piece and
isl_pw_qpolynomial_fold_foreach_lifted_piece.
A piecewise expression consisting of the expressions at a given position of a piecewise multiple expression can be extracted using the following function.
#include 
__isl_give isl_pw_aff *isl_pw_multi_aff_get_pw_aff(
__isl_keep isl_pw_multi_aff *pma, int pos);
These expressions can be replaced using the following function.
#include 
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_pw_aff(
__isl_take isl_pw_multi_aff *pma, unsigned pos,
__isl_take isl_pw_aff *pa);
Note that there is a difference between isl_multi_pw_aff and isl_pw_multi_aff
objects. The first is a sequence of piecewise affine expressions, while the second is a
piecewise sequence of affine expressions. In particular, each of the piecewise affine
expressions in an isl_multi_pw_aff may have a different domain, while all multiple
expressions associated to a cell in an isl_pw_multi_aff have the same domain. It is
possible to convert between the two, but when converting an isl_multi_pw_aff to
an isl_pw_multi_aff, the domain of the result is the intersection of the domains of
the input. The reverse conversion is exact.
#include 
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_from_multi_pw_aff(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_from_pw_multi_aff(
__isl_take isl_pw_multi_aff *pma);
Union Expressions
A union expression collects base expressions defined over different domains. The space
of a union expression is that of the shared parameter space.
64

The union expression types defined by isl are isl_union_pw_aff, isl_union_pw_multi_aff,
isl_union_pw_qpolynomial and isl_union_pw_qpolynomial_fold. In case of
isl_union_pw_aff, isl_union_pw_qpolynomial and isl_union_pw_qpolynomial_fold,
there can be at most one base expression for a given domain space. In case of isl_union_pw_multi_aff,
there can be multiple such expressions for a given domain space, but the domains of
these expressions need to be disjoint.
An empty union expression can be created using the following functions.
#include 
__isl_give isl_union_pw_aff *isl_union_pw_aff_empty(
__isl_take isl_space *space);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_empty(
__isl_take isl_space *space);
#include 
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_zero(
__isl_take isl_space *space);
A union expression containing a single base expression can be created using the
following functions.
#include 
__isl_give isl_union_pw_aff *
isl_union_pw_aff_from_pw_aff(
__isl_take isl_pw_aff *pa);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_from_aff(
__isl_take isl_aff *aff);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_from_pw_multi_aff(
__isl_take isl_pw_multi_aff *pma);
#include 
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_from_pw_qpolynomial(
__isl_take isl_pw_qpolynomial *pwqp);
The following functions create a base expression on each of the sets in the union
set and collect the results.
#include 
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_from_union_pw_aff(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_union_pw_aff *
65

isl_union_pw_multi_aff_get_union_pw_aff(
__isl_keep isl_union_pw_multi_aff *upma, int pos);
__isl_give isl_union_pw_aff *
isl_union_pw_aff_val_on_domain(
__isl_take isl_union_set *domain,
__isl_take isl_val *v);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_multi_val_on_domain(
__isl_take isl_union_set *domain,
__isl_take isl_multi_val *mv);
__isl_give isl_union_pw_aff *
isl_union_pw_aff_param_on_domain_id(
__isl_take isl_union_set *domain,
__isl_take isl_id *id);
The id argument of isl_union_pw_aff_param_on_domain_id is the identifier
of a parameter that may or may not already be present in domain.
An isl_union_pw_aff that is equal to a (parametric) affine or piecewise affine
expression on a given domain can be created using the following functions.
#include 
__isl_give isl_union_pw_aff *
isl_union_pw_aff_aff_on_domain(
__isl_take isl_union_set *domain,
__isl_take isl_aff *aff);
__isl_give isl_union_pw_aff *
isl_union_pw_aff_pw_aff_on_domain(
__isl_take isl_union_set *domain,
__isl_take isl_pw_aff *pa);
A base expression can be added to a union expression using the following functions.
#include 
__isl_give isl_union_pw_aff *
isl_union_pw_aff_add_pw_aff(
__isl_take isl_union_pw_aff *upa,
__isl_take isl_pw_aff *pa);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_add_pw_multi_aff(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_pw_multi_aff *pma);
#include 
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_add_pw_qpolynomial(
__isl_take isl_union_pw_qpolynomial *upwqp,
__isl_take isl_pw_qpolynomial *pwqp);
66

Union expressions can be copied and freed using the following functions.
#include 
__isl_give isl_union_pw_aff *isl_union_pw_aff_copy(
__isl_keep isl_union_pw_aff *upa);
__isl_null isl_union_pw_aff *isl_union_pw_aff_free(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_copy(
__isl_keep isl_union_pw_multi_aff *upma);
__isl_null isl_union_pw_multi_aff *
isl_union_pw_multi_aff_free(
__isl_take isl_union_pw_multi_aff *upma);
#include 
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_copy(
__isl_keep isl_union_pw_qpolynomial *upwqp);
__isl_null isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_free(
__isl_take isl_union_pw_qpolynomial *upwqp);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_copy(
__isl_keep isl_union_pw_qpolynomial_fold *upwf);
__isl_null isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_free(
__isl_take isl_union_pw_qpolynomial_fold *upwf);
To iterate over the base expressions in a union expression, use the following functions.
#include 
int isl_union_pw_aff_n_pw_aff(
__isl_keep isl_union_pw_aff *upa);
isl_stat isl_union_pw_aff_foreach_pw_aff(
__isl_keep isl_union_pw_aff *upa,
isl_stat (*fn)(__isl_take isl_pw_aff *pa,
void *user), void *user);
int isl_union_pw_multi_aff_n_pw_multi_aff(
__isl_keep isl_union_pw_multi_aff *upma);
isl_stat isl_union_pw_multi_aff_foreach_pw_multi_aff(
__isl_keep isl_union_pw_multi_aff *upma,
isl_stat (*fn)(__isl_take isl_pw_multi_aff *pma,
void *user), void *user);
#include 
int isl_union_pw_qpolynomial_n_pw_qpolynomial(
67

__isl_keep isl_union_pw_qpolynomial *upwqp);
isl_stat isl_union_pw_qpolynomial_foreach_pw_qpolynomial(
__isl_keep isl_union_pw_qpolynomial *upwqp,
isl_stat (*fn)(__isl_take isl_pw_qpolynomial *pwqp,
void *user), void *user);
int isl_union_pw_qpolynomial_fold_n_pw_qpolynomial_fold(
__isl_keep isl_union_pw_qpolynomial_fold *upwf);
isl_stat isl_union_pw_qpolynomial_fold_foreach_pw_qpolynomial_fold(
__isl_keep isl_union_pw_qpolynomial_fold *upwf,
isl_stat (*fn)(__isl_take isl_pw_qpolynomial_fold *pwf,
void *user), void *user);
To extract the base expression in a given space from a union, use the following
functions.
#include 
__isl_give isl_pw_aff *isl_union_pw_aff_extract_pw_aff(
__isl_keep isl_union_pw_aff *upa,
__isl_take isl_space *space);
__isl_give isl_pw_multi_aff *
isl_union_pw_multi_aff_extract_pw_multi_aff(
__isl_keep isl_union_pw_multi_aff *upma,
__isl_take isl_space *space);
#include 
__isl_give isl_pw_qpolynomial *
isl_union_pw_qpolynomial_extract_pw_qpolynomial(
__isl_keep isl_union_pw_qpolynomial *upwqp,
__isl_take isl_space *space);
It is also possible to obtain a list of the base expressions using the following functions.
#include 
__isl_give isl_pw_aff_list *
isl_union_pw_aff_get_pw_aff_list(
__isl_keep isl_union_pw_aff *upa);
__isl_give isl_pw_multi_aff_list *
isl_union_pw_multi_aff_get_pw_multi_aff_list(
__isl_keep isl_union_pw_multi_aff *upma);
#include 
__isl_give isl_pw_qpolynomial_list *
isl_union_pw_qpolynomial_get_pw_qpolynomial_list(
__isl_keep isl_union_pw_qpolynomial *upwqp);
__isl_give isl_pw_qpolynomial_fold_list *
isl_union_pw_qpolynomial_fold_get_pw_qpolynomial_fold_list(
__isl_keep isl_union_pw_qpolynomial_fold *upwf);
The returned list can be manipulated using the functions in §1.4.19.
68

1.4.14

Input and Output

For set and relation, isl supports its own input/output format, which is similar to the
Omega format, but also supports the PolyLib format in some cases. For other object
types, typically only an isl format is supported.
isl format
The isl format is similar to that of Omega, but has a different syntax for describing the
parameters and allows for the definition of an existentially quantified variable as the
integer division of an affine expression. For example, the set of integers i between 0
and n such that i % 10 <= 6 can be described as
[n] -> { [i] : exists (a = [i/10] : 0 <= i and i <= n and
i - 10 a <= 6) }
A set or relation can have several disjuncts, separated by the keyword or. Each
disjunct is either a conjunction of constraints or a projection (exists) of a conjunction
of constraints. The constraints are separated by the keyword and.
PolyLib format
If the represented set is a union, then the first line contains a single number representing
the number of disjuncts. Otherwise, a line containing the number 1 is optional.
Each disjunct is represented by a matrix of constraints. The first line contains two
numbers representing the number of rows and columns, where the number of rows is
equal to the number of constraints and the number of columns is equal to two plus
the number of variables. The following lines contain the actual rows of the constraint
matrix. In each row, the first column indicates whether the constraint is an equality (0)
or inequality (1). The final column corresponds to the constant term.
If the set is parametric, then the coefficients of the parameters appear in the last
columns before the constant column. The coefficients of any existentially quantified
variables appear between those of the set variables and those of the parameters.
Extended PolyLib format
The extended PolyLib format is nearly identical to the PolyLib format. The only
difference is that the line containing the number of rows and columns of a constraint
matrix also contains four additional numbers: the number of output dimensions, the
number of input dimensions, the number of local dimensions (i.e., the number of existentially quantified variables) and the number of parameters. For sets, the number
of “output” dimensions is equal to the number of set dimensions, while the number of
“input” dimensions is zero.
Input
Objects can be read from input using the following functions.

69

#include 
__isl_give isl_val *isl_val_read_from_str(isl_ctx *ctx,
const char *str);
__isl_give isl_multi_val *isl_multi_val_read_from_str(
isl_ctx *ctx, const char *str);
#include 
__isl_give isl_basic_set *isl_basic_set_read_from_file(
isl_ctx *ctx, FILE *input);
__isl_give isl_basic_set *isl_basic_set_read_from_str(
isl_ctx *ctx, const char *str);
__isl_give isl_set *isl_set_read_from_file(isl_ctx *ctx,
FILE *input);
__isl_give isl_set *isl_set_read_from_str(isl_ctx *ctx,
const char *str);
#include 
__isl_give isl_basic_map *isl_basic_map_read_from_file(
isl_ctx *ctx, FILE *input);
__isl_give isl_basic_map *isl_basic_map_read_from_str(
isl_ctx *ctx, const char *str);
__isl_give isl_map *isl_map_read_from_file(
isl_ctx *ctx, FILE *input);
__isl_give isl_map *isl_map_read_from_str(isl_ctx *ctx,
const char *str);
#include 
__isl_give isl_union_set *isl_union_set_read_from_file(
isl_ctx *ctx, FILE *input);
__isl_give isl_union_set *isl_union_set_read_from_str(
isl_ctx *ctx, const char *str);
#include 
__isl_give isl_union_map *isl_union_map_read_from_file(
isl_ctx *ctx, FILE *input);
__isl_give isl_union_map *isl_union_map_read_from_str(
isl_ctx *ctx, const char *str);
#include 
__isl_give isl_aff *isl_aff_read_from_str(
isl_ctx *ctx, const char *str);
__isl_give isl_multi_aff *isl_multi_aff_read_from_str(
isl_ctx *ctx, const char *str);
__isl_give isl_pw_aff *isl_pw_aff_read_from_str(
isl_ctx *ctx, const char *str);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_read_from_str(
isl_ctx *ctx, const char *str);
70

__isl_give isl_multi_pw_aff *isl_multi_pw_aff_read_from_str(
isl_ctx *ctx, const char *str);
__isl_give isl_union_pw_aff *
isl_union_pw_aff_read_from_str(
isl_ctx *ctx, const char *str);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_read_from_str(
isl_ctx *ctx, const char *str);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_read_from_str(
isl_ctx *ctx, const char *str);
#include 
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_read_from_str(
isl_ctx *ctx, const char *str);
For sets and relations, the input format is autodetected and may be either the
PolyLib format or the isl format.
Output
Before anything can be printed, an isl_printer needs to be created.
__isl_give isl_printer *isl_printer_to_file(isl_ctx *ctx,
FILE *file);
__isl_give isl_printer *isl_printer_to_str(isl_ctx *ctx);
__isl_null isl_printer *isl_printer_free(
__isl_take isl_printer *printer);
isl_printer_to_file prints to the given file, while isl_printer_to_str prints
to a string that can be extracted using the following function.
#include 
__isl_give char *isl_printer_get_str(
__isl_keep isl_printer *printer);
The printer can be inspected using the following functions.
FILE *isl_printer_get_file(
__isl_keep isl_printer *printer);
int isl_printer_get_output_format(
__isl_keep isl_printer *p);
int isl_printer_get_yaml_style(__isl_keep isl_printer *p);
The behavior of the printer can be modified in various ways

71

__isl_give isl_printer *isl_printer_set_output_format(
__isl_take isl_printer *p, int output_format);
__isl_give isl_printer *isl_printer_set_indent(
__isl_take isl_printer *p, int indent);
__isl_give isl_printer *isl_printer_set_indent_prefix(
__isl_take isl_printer *p, const char *prefix);
__isl_give isl_printer *isl_printer_indent(
__isl_take isl_printer *p, int indent);
__isl_give isl_printer *isl_printer_set_prefix(
__isl_take isl_printer *p, const char *prefix);
__isl_give isl_printer *isl_printer_set_suffix(
__isl_take isl_printer *p, const char *suffix);
__isl_give isl_printer *isl_printer_set_yaml_style(
__isl_take isl_printer *p, int yaml_style);
The output_format may be either ISL_FORMAT_ISL, ISL_FORMAT_OMEGA, ISL_FORMAT_POLYLIB,
ISL_FORMAT_EXT_POLYLIB or ISL_FORMAT_LATEX and defaults to ISL_FORMAT_ISL.
Each line in the output is prefixed by indent_prefix, indented by indent (set by
isl_printer_set_indent) spaces (default: 0), prefixed by prefix and suffixed by
suffix. In the PolyLib format output, the coefficients of the existentially quantified variables appear between those of the set variables and those of the parameters.
The function isl_printer_indent increases the indentation by the specified amount
(which may be negative). The YAML style may be either ISL_YAML_STYLE_BLOCK or
ISL_YAML_STYLE_FLOW and when we are printing something in YAML format.
To actually print something, use
#include 
__isl_give isl_printer *isl_printer_print_double(
__isl_take isl_printer *p, double d);
#include 
__isl_give isl_printer *isl_printer_print_val(
__isl_take isl_printer *p, __isl_keep isl_val *v);
#include 
__isl_give isl_printer *isl_printer_print_basic_set(
__isl_take isl_printer *printer,
__isl_keep isl_basic_set *bset);
__isl_give isl_printer *isl_printer_print_set(
__isl_take isl_printer *printer,
__isl_keep isl_set *set);
#include 
__isl_give isl_printer *isl_printer_print_basic_map(
__isl_take isl_printer *printer,
__isl_keep isl_basic_map *bmap);
__isl_give isl_printer *isl_printer_print_map(
72

__isl_take isl_printer *printer,
__isl_keep isl_map *map);
#include 
__isl_give isl_printer *isl_printer_print_union_set(
__isl_take isl_printer *p,
__isl_keep isl_union_set *uset);
#include 
__isl_give isl_printer *isl_printer_print_union_map(
__isl_take isl_printer *p,
__isl_keep isl_union_map *umap);
#include 
__isl_give isl_printer *isl_printer_print_multi_val(
__isl_take isl_printer *p,
__isl_keep isl_multi_val *mv);
#include 
__isl_give isl_printer *isl_printer_print_aff(
__isl_take isl_printer *p, __isl_keep isl_aff *aff);
__isl_give isl_printer *isl_printer_print_multi_aff(
__isl_take isl_printer *p,
__isl_keep isl_multi_aff *maff);
__isl_give isl_printer *isl_printer_print_pw_aff(
__isl_take isl_printer *p,
__isl_keep isl_pw_aff *pwaff);
__isl_give isl_printer *isl_printer_print_pw_multi_aff(
__isl_take isl_printer *p,
__isl_keep isl_pw_multi_aff *pma);
__isl_give isl_printer *isl_printer_print_multi_pw_aff(
__isl_take isl_printer *p,
__isl_keep isl_multi_pw_aff *mpa);
__isl_give isl_printer *isl_printer_print_union_pw_aff(
__isl_take isl_printer *p,
__isl_keep isl_union_pw_aff *upa);
__isl_give isl_printer *isl_printer_print_union_pw_multi_aff(
__isl_take isl_printer *p,
__isl_keep isl_union_pw_multi_aff *upma);
__isl_give isl_printer *
isl_printer_print_multi_union_pw_aff(
__isl_take isl_printer *p,
__isl_keep isl_multi_union_pw_aff *mupa);
#include 
__isl_give isl_printer *isl_printer_print_qpolynomial(
__isl_take isl_printer *p,
73

__isl_keep isl_qpolynomial *qp);
__isl_give isl_printer *isl_printer_print_pw_qpolynomial(
__isl_take isl_printer *p,
__isl_keep isl_pw_qpolynomial *pwqp);
__isl_give isl_printer *isl_printer_print_union_pw_qpolynomial(
__isl_take isl_printer *p,
__isl_keep isl_union_pw_qpolynomial *upwqp);
__isl_give isl_printer *
isl_printer_print_pw_qpolynomial_fold(
__isl_take isl_printer *p,
__isl_keep isl_pw_qpolynomial_fold *pwf);
__isl_give isl_printer *
isl_printer_print_union_pw_qpolynomial_fold(
__isl_take isl_printer *p,
__isl_keep isl_union_pw_qpolynomial_fold *upwf);
For isl_printer_print_qpolynomial, isl_printer_print_pw_qpolynomial
and isl_printer_print_pw_qpolynomial_fold, the output format of the printer
needs to be set to either ISL_FORMAT_ISL or ISL_FORMAT_C. For isl_printer_print_union_pw_qpolynomial
and isl_printer_print_union_pw_qpolynomial_fold, only ISL_FORMAT_ISL
is supported. In case of printing in ISL_FORMAT_C, the user may want to set the names
of all dimensions first.
isl also provides limited support for printing YAML documents, just enough for
the internal use for printing such documents.
#include 
__isl_give isl_printer *isl_printer_yaml_start_mapping(
__isl_take isl_printer *p);
__isl_give isl_printer *isl_printer_yaml_end_mapping(
__isl_take isl_printer *p);
__isl_give isl_printer *isl_printer_yaml_start_sequence(
__isl_take isl_printer *p);
__isl_give isl_printer *isl_printer_yaml_end_sequence(
__isl_take isl_printer *p);
__isl_give isl_printer *isl_printer_yaml_next(
__isl_take isl_printer *p);
A document is started by a call to either isl_printer_yaml_start_mapping or
isl_printer_yaml_start_sequence. Anything printed to the printer after such a
call belong to the first key of the mapping or the first element in the sequence. The
function isl_printer_yaml_next moves to the value if we are currently printing a
mapping key, the next key if we are printing a value or the next element if we are printing an element in a sequence. Nested mappings and sequences are initiated by the same
isl_printer_yaml_start_mapping or isl_printer_yaml_start_sequence. Each
call to these functions needs to have a corresponding call to isl_printer_yaml_end_mapping
or isl_printer_yaml_end_sequence.
74

When called on a file printer, the following function flushes the file. When called
on a string printer, the buffer is cleared.
__isl_give isl_printer *isl_printer_flush(
__isl_take isl_printer *p);
The following functions allow the user to attach notes to a printer in order to keep
track of additional state.
#include 
isl_bool isl_printer_has_note(__isl_keep isl_printer *p,
__isl_keep isl_id *id);
__isl_give isl_id *isl_printer_get_note(
__isl_keep isl_printer *p, __isl_take isl_id *id);
__isl_give isl_printer *isl_printer_set_note(
__isl_take isl_printer *p,
__isl_take isl_id *id, __isl_take isl_id *note);
isl_printer_set_note associates the given note to the given identifier in the
printer. isl_printer_get_note retrieves a note associated to an identifier, while
isl_printer_has_note checks if there is such a note. isl_printer_get_note
fails if the requested note does not exist.
Alternatively, a string representation can be obtained directly using the following
functions, which always print in isl format.
#include 
__isl_give char *isl_id_to_str(
__isl_keep isl_id *id);
#include 
__isl_give char *isl_space_to_str(
__isl_keep isl_space *space);
#include 
__isl_give char *isl_val_to_str(__isl_keep isl_val *v);
__isl_give char *isl_multi_val_to_str(
__isl_keep isl_multi_val *mv);
#include 
__isl_give char *isl_basic_set_to_str(
__isl_keep isl_basic_set *bset);
__isl_give char *isl_set_to_str(
__isl_keep isl_set *set);
#include 
__isl_give char *isl_union_set_to_str(
__isl_keep isl_union_set *uset);

75

#include 
__isl_give char *isl_basic_map_to_str(
__isl_keep isl_basic_map *bmap);
__isl_give char *isl_map_to_str(
__isl_keep isl_map *map);
#include 
__isl_give char *isl_union_map_to_str(
__isl_keep isl_union_map *umap);
#include 
__isl_give char *isl_aff_to_str(__isl_keep isl_aff *aff);
__isl_give char *isl_pw_aff_to_str(
__isl_keep isl_pw_aff *pa);
__isl_give char *isl_multi_aff_to_str(
__isl_keep isl_multi_aff *ma);
__isl_give char *isl_pw_multi_aff_to_str(
__isl_keep isl_pw_multi_aff *pma);
__isl_give char *isl_multi_pw_aff_to_str(
__isl_keep isl_multi_pw_aff *mpa);
__isl_give char *isl_union_pw_aff_to_str(
__isl_keep isl_union_pw_aff *upa);
__isl_give char *isl_union_pw_multi_aff_to_str(
__isl_keep isl_union_pw_multi_aff *upma);
__isl_give char *isl_multi_union_pw_aff_to_str(
__isl_keep isl_multi_union_pw_aff *mupa);
#include 
__isl_give char *isl_point_to_str(
__isl_keep isl_point *pnt);
#include 
__isl_give char *isl_pw_qpolynomial_to_str(
__isl_keep isl_pw_qpolynomial *pwqp);
__isl_give char *isl_union_pw_qpolynomial_to_str(
__isl_keep isl_union_pw_qpolynomial *upwqp);

1.4.15

Properties

Unary Properties
• Emptiness
The following functions test whether the given set or relation contains any integer
points. The “plain” variants do not perform any computations, but simply check
if the given set or relation is already known to be empty.
#include 
76

isl_bool isl_basic_set_plain_is_empty(
__isl_keep isl_basic_set *bset);
isl_bool isl_basic_set_is_empty(
__isl_keep isl_basic_set *bset);
isl_bool isl_set_plain_is_empty(
__isl_keep isl_set *set);
isl_bool isl_set_is_empty(__isl_keep isl_set *set);
#include 
isl_bool isl_union_set_is_empty(
__isl_keep isl_union_set *uset);
#include 
isl_bool isl_basic_map_plain_is_empty(
__isl_keep isl_basic_map *bmap);
isl_bool isl_basic_map_is_empty(
__isl_keep isl_basic_map *bmap);
isl_bool isl_map_plain_is_empty(
__isl_keep isl_map *map);
isl_bool isl_map_is_empty(__isl_keep isl_map *map);
#include 
isl_bool isl_union_map_plain_is_empty(
__isl_keep isl_union_map *umap);
isl_bool isl_union_map_is_empty(
__isl_keep isl_union_map *umap);
• Universality
isl_bool isl_basic_set_plain_is_universe(
__isl_keep isl_basic_set *bset);
isl_bool isl_basic_set_is_universe(
__isl_keep isl_basic_set *bset);
isl_bool isl_basic_map_plain_is_universe(
__isl_keep isl_basic_map *bmap);
isl_bool isl_basic_map_is_universe(
__isl_keep isl_basic_map *bmap);
isl_bool isl_set_plain_is_universe(
__isl_keep isl_set *set);
isl_bool isl_map_plain_is_universe(
__isl_keep isl_map *map);
• Single-valuedness
#include 
isl_bool isl_set_is_singleton(__isl_keep isl_set *set);
77

#include 
isl_bool isl_basic_map_is_single_valued(
__isl_keep isl_basic_map *bmap);
isl_bool isl_map_plain_is_single_valued(
__isl_keep isl_map *map);
isl_bool isl_map_is_single_valued(__isl_keep isl_map *map);
#include 
isl_bool isl_union_map_is_single_valued(
__isl_keep isl_union_map *umap);
• Injectivity
isl_bool isl_map_plain_is_injective(
__isl_keep isl_map *map);
isl_bool isl_map_is_injective(
__isl_keep isl_map *map);
isl_bool isl_union_map_plain_is_injective(
__isl_keep isl_union_map *umap);
isl_bool isl_union_map_is_injective(
__isl_keep isl_union_map *umap);
• Bijectivity
isl_bool isl_map_is_bijective(
__isl_keep isl_map *map);
isl_bool isl_union_map_is_bijective(
__isl_keep isl_union_map *umap);
• Identity
The following functions test whether the given relation only maps elements to
themselves.
#include 
isl_bool isl_map_is_identity(
__isl_keep isl_map *map);
#include 
isl_bool isl_union_map_is_identity(
__isl_keep isl_union_map *umap);
• Position

78

__isl_give isl_val *
isl_basic_map_plain_get_val_if_fixed(
__isl_keep isl_basic_map *bmap,
enum isl_dim_type type, unsigned pos);
__isl_give isl_val *isl_set_plain_get_val_if_fixed(
__isl_keep isl_set *set,
enum isl_dim_type type, unsigned pos);
__isl_give isl_val *isl_map_plain_get_val_if_fixed(
__isl_keep isl_map *map,
enum isl_dim_type type, unsigned pos);
If the set or relation obviously lies on a hyperplane where the given dimension
has a fixed value, then return that value. Otherwise return NaN.
• Stride
isl_stat isl_set_dim_residue_class_val(
__isl_keep isl_set *set,
int pos, __isl_give isl_val **modulo,
__isl_give isl_val **residue);
Check if the values of the given set dimension are equal to a fixed value modulo
some integer value. If so, assign the modulo to *modulo and the fixed value to
*residue. If the given dimension attains only a single value, then assign 0 to
*modulo and the fixed value to *residue. If the dimension does not attain only
a single value and if no modulo can be found then assign 1 to *modulo and 1 to
*residue.
#include 
__isl_give isl_stride_info *isl_set_get_stride_info(
__isl_keep isl_set *set, int pos);
__isl_give isl_val *isl_set_get_stride(
__isl_keep isl_set *set, int pos);
#include 
__isl_give isl_stride_info *
isl_map_get_range_stride_info(
__isl_keep isl_map *map, int pos);
Check if the values of the given set dimension are equal to some affine expression
of the other dimensions (the offset) modulo some integer stride or check if the
values of the given output dimensions are equal to some affine expression of
the input dimensions (the offset) modulo some integer stride. If no more specific
information can be found, then the stride is taken to be one and the offset is taken
to be the zero expression. The function isl_set_get_stride performs the
same computation as isl_set_get_stride_info but only returns the stride.
For the other functions, the stride and offset can be extracted from the returned
object using the following functions.
79

#include 
__isl_give isl_val *isl_stride_info_get_stride(
__isl_keep isl_stride_info *si);
__isl_give isl_aff *isl_stride_info_get_offset(
__isl_keep isl_stride_info *si);
The stride info object can be copied and released using the following functions.
#include 
__isl_give isl_stride_info *isl_stride_info_copy(
__isl_keep isl_stride_info *si);
__isl_null isl_stride_info *isl_stride_info_free(
__isl_take isl_stride_info *si);
• Dependence
To check whether the description of a set, relation or function depends on one or
more given dimensions, the following functions can be used.
#include 
isl_bool isl_constraint_involves_dims(
__isl_keep isl_constraint *constraint,
enum isl_dim_type type, unsigned first, unsigned n);
#include 
isl_bool isl_basic_set_involves_dims(
__isl_keep isl_basic_set *bset,
enum isl_dim_type type, unsigned first, unsigned n);
isl_bool isl_set_involves_dims(__isl_keep isl_set *set,
enum isl_dim_type type, unsigned first, unsigned n);
#include 
isl_bool isl_basic_map_involves_dims(
__isl_keep isl_basic_map *bmap,
enum isl_dim_type type, unsigned first, unsigned n);
isl_bool isl_map_involves_dims(__isl_keep isl_map *map,
enum isl_dim_type type, unsigned first, unsigned n);
#include 
isl_bool isl_union_map_involves_dims(
__isl_keep isl_union_map *umap,
enum isl_dim_type type, unsigned first, unsigned n);
#include 
isl_bool isl_aff_involves_dims(__isl_keep isl_aff *aff,
enum isl_dim_type type, unsigned first, unsigned n);
80

isl_bool isl_pw_aff_involves_dims(
__isl_keep isl_pw_aff *pwaff,
enum isl_dim_type type, unsigned first,
isl_bool isl_multi_aff_involves_dims(
__isl_keep isl_multi_aff *ma,
enum isl_dim_type type, unsigned first,
isl_bool isl_pw_multi_aff_involves_dims(
__isl_keep isl_pw_multi_aff *pma,
enum isl_dim_type type, unsigned first,
isl_bool isl_multi_pw_aff_involves_dims(
__isl_keep isl_multi_pw_aff *mpa,
enum isl_dim_type type, unsigned first,

unsigned n);

unsigned n);

unsigned n);

unsigned n);

#include 
isl_bool isl_qpolynomial_involves_dims(
__isl_keep isl_qpolynomial *qp,
enum isl_dim_type type, unsigned first, unsigned n);
Similarly, the following functions can be used to check whether a given dimension is involved in any lower or upper bound.
#include 
isl_bool isl_set_dim_has_any_lower_bound(
__isl_keep isl_set *set,
enum isl_dim_type type, unsigned pos);
isl_bool isl_set_dim_has_any_upper_bound(
__isl_keep isl_set *set,
enum isl_dim_type type, unsigned pos);
Note that these functions return true even if there is a bound on the dimension on
only some of the basic sets of set. To check if they have a bound for all of the
basic sets in set, use the following functions instead.
#include 
isl_bool isl_set_dim_has_lower_bound(
__isl_keep isl_set *set,
enum isl_dim_type type, unsigned pos);
isl_bool isl_set_dim_has_upper_bound(
__isl_keep isl_set *set,
enum isl_dim_type type, unsigned pos);
• Space
To check whether a set is a parameter domain, use this function:

81

isl_bool isl_set_is_params(__isl_keep isl_set *set);
isl_bool isl_union_set_is_params(
__isl_keep isl_union_set *uset);
• Wrapping
The following functions check whether the space of the given (basic) set or relation domain and/or range is a wrapped relation.
#include 
isl_bool isl_space_is_wrapping(
__isl_keep isl_space *space);
isl_bool isl_space_domain_is_wrapping(
__isl_keep isl_space *space);
isl_bool isl_space_range_is_wrapping(
__isl_keep isl_space *space);
isl_bool isl_space_is_product(
__isl_keep isl_space *space);
#include 
isl_bool isl_basic_set_is_wrapping(
__isl_keep isl_basic_set *bset);
isl_bool isl_set_is_wrapping(__isl_keep isl_set *set);
#include 
isl_bool isl_map_domain_is_wrapping(
__isl_keep isl_map *map);
isl_bool isl_map_range_is_wrapping(
__isl_keep isl_map *map);
isl_bool isl_map_is_product(__isl_keep isl_map *map);
#include 
isl_bool isl_multi_val_range_is_wrapping(
__isl_keep isl_multi_val *mv);
#include 
isl_bool isl_multi_aff_range_is_wrapping(
__isl_keep isl_multi_aff *ma);
isl_bool isl_multi_pw_aff_range_is_wrapping(
__isl_keep isl_multi_pw_aff *mpa);
isl_bool isl_multi_union_pw_aff_range_is_wrapping(
__isl_keep isl_multi_union_pw_aff *mupa);
The input to isl_space_is_wrapping should be the space of a set, while that
of isl_space_domain_is_wrapping and isl_space_range_is_wrapping
should be the space of a relation. The input to isl_space_is_product can be
either the space of a set or that of a binary relation. In case the input is the space
of a binary relation, it checks whether both domain and range are wrapping.
82

• Internal Product
isl_bool isl_basic_map_can_zip(
__isl_keep isl_basic_map *bmap);
isl_bool isl_map_can_zip(__isl_keep isl_map *map);
Check whether the product of domain and range of the given relation can be
computed, i.e., whether both domain and range are nested relations.
• Currying
#include 
isl_bool isl_space_can_curry(
__isl_keep isl_space *space);
#include 
isl_bool isl_basic_map_can_curry(
__isl_keep isl_basic_map *bmap);
isl_bool isl_map_can_curry(__isl_keep isl_map *map);
Check whether the domain of the (basic) relation is a wrapped relation.
#include 
__isl_give isl_space *isl_space_uncurry(
__isl_take isl_space *space);
#include 
isl_bool isl_basic_map_can_uncurry(
__isl_keep isl_basic_map *bmap);
isl_bool isl_map_can_uncurry(__isl_keep isl_map *map);
Check whether the range of the (basic) relation is a wrapped relation.
#include 
isl_bool isl_space_can_range_curry(
__isl_keep isl_space *space);
#include 
isl_bool isl_map_can_range_curry(
__isl_keep isl_map *map);
Check whether the domain of the relation wrapped in the range of the input is
itself a wrapped relation.
• Special Values

83

#include
isl_bool
isl_bool
isl_bool


isl_aff_is_cst(__isl_keep isl_aff *aff);
isl_pw_aff_is_cst(__isl_keep isl_pw_aff *pwaff);
isl_multi_pw_aff_is_cst(
__isl_keep isl_multi_pw_aff *mpa);

Check whether the given expression is a constant.
#include 
isl_bool isl_multi_val_involves_nan(
__isl_keep isl_multi_val *mv);
#include 
isl_bool isl_aff_is_nan(__isl_keep isl_aff *aff);
isl_bool isl_multi_aff_involves_nan(
__isl_keep isl_multi_aff *ma);
isl_bool isl_pw_aff_involves_nan(
__isl_keep isl_pw_aff *pa);
isl_bool isl_pw_multi_aff_involves_nan(
__isl_keep isl_pw_multi_aff *pma);
isl_bool isl_multi_pw_aff_involves_nan(
__isl_keep isl_multi_pw_aff *mpa);
isl_bool isl_union_pw_aff_involves_nan(
__isl_keep isl_union_pw_aff *upa);
isl_bool isl_union_pw_multi_aff_involves_nan(
__isl_keep isl_union_pw_multi_aff *upma);
isl_bool isl_multi_union_pw_aff_involves_nan(
__isl_keep isl_multi_union_pw_aff *mupa);
#include 
isl_bool isl_qpolynomial_is_nan(
__isl_keep isl_qpolynomial *qp);
isl_bool isl_qpolynomial_fold_is_nan(
__isl_keep isl_qpolynomial_fold *fold);
isl_bool isl_pw_qpolynomial_involves_nan(
__isl_keep isl_pw_qpolynomial *pwqp);
isl_bool isl_pw_qpolynomial_fold_involves_nan(
__isl_keep isl_pw_qpolynomial_fold *pwf);
isl_bool isl_union_pw_qpolynomial_involves_nan(
__isl_keep isl_union_pw_qpolynomial *upwqp);
isl_bool isl_union_pw_qpolynomial_fold_involves_nan(
__isl_keep isl_union_pw_qpolynomial_fold *upwf);
Check whether the given expression is equal to or involves NaN.

84

#include 
isl_bool isl_aff_plain_is_zero(
__isl_keep isl_aff *aff);
Check whether the affine expression is obviously zero.
Binary Properties
• Equality
The following functions check whether two objects represent the same set, relation or function. The plain variants only return true if the objects are obviously
the same. That is, they may return false even if the objects are the same, but they
will never return true if the objects are not the same.
#include 
isl_bool isl_basic_set_plain_is_equal(
__isl_keep isl_basic_set *bset1,
__isl_keep isl_basic_set *bset2);
isl_bool isl_basic_set_is_equal(
__isl_keep isl_basic_set *bset1,
__isl_keep isl_basic_set *bset2);
isl_bool isl_set_plain_is_equal(
__isl_keep isl_set *set1,
__isl_keep isl_set *set2);
isl_bool isl_set_is_equal(__isl_keep isl_set *set1,
__isl_keep isl_set *set2);
#include 
isl_bool isl_basic_map_is_equal(
__isl_keep isl_basic_map *bmap1,
__isl_keep isl_basic_map *bmap2);
isl_bool isl_map_is_equal(__isl_keep isl_map *map1,
__isl_keep isl_map *map2);
isl_bool isl_map_plain_is_equal(
__isl_keep isl_map *map1,
__isl_keep isl_map *map2);
#include 
isl_bool isl_union_set_is_equal(
__isl_keep isl_union_set *uset1,
__isl_keep isl_union_set *uset2);
#include 
isl_bool isl_union_map_is_equal(
__isl_keep isl_union_map *umap1,
__isl_keep isl_union_map *umap2);
85

#include 
isl_bool isl_multi_val_plain_is_equal(
__isl_keep isl_multi_val *mv1,
__isl_keep isl_multi_val *mv2);
#include 
isl_bool isl_aff_plain_is_equal(
__isl_keep isl_aff *aff1,
__isl_keep isl_aff *aff2);
isl_bool isl_multi_aff_plain_is_equal(
__isl_keep isl_multi_aff *maff1,
__isl_keep isl_multi_aff *maff2);
isl_bool isl_pw_aff_plain_is_equal(
__isl_keep isl_pw_aff *pwaff1,
__isl_keep isl_pw_aff *pwaff2);
isl_bool isl_pw_aff_is_equal(
__isl_keep isl_pw_aff *pa1,
__isl_keep isl_pw_aff *pa2);
isl_bool isl_pw_multi_aff_plain_is_equal(
__isl_keep isl_pw_multi_aff *pma1,
__isl_keep isl_pw_multi_aff *pma2);
isl_bool isl_pw_multi_aff_is_equal(
__isl_keep isl_pw_multi_aff *pma1,
__isl_keep isl_pw_multi_aff *pma2);
isl_bool isl_multi_pw_aff_plain_is_equal(
__isl_keep isl_multi_pw_aff *mpa1,
__isl_keep isl_multi_pw_aff *mpa2);
isl_bool isl_multi_pw_aff_is_equal(
__isl_keep isl_multi_pw_aff *mpa1,
__isl_keep isl_multi_pw_aff *mpa2);
isl_bool isl_union_pw_aff_plain_is_equal(
__isl_keep isl_union_pw_aff *upa1,
__isl_keep isl_union_pw_aff *upa2);
isl_bool isl_union_pw_multi_aff_plain_is_equal(
__isl_keep isl_union_pw_multi_aff *upma1,
__isl_keep isl_union_pw_multi_aff *upma2);
isl_bool isl_multi_union_pw_aff_plain_is_equal(
__isl_keep isl_multi_union_pw_aff *mupa1,
__isl_keep isl_multi_union_pw_aff *mupa2);
#include 
isl_bool isl_union_pw_qpolynomial_plain_is_equal(
__isl_keep isl_union_pw_qpolynomial *upwqp1,
__isl_keep isl_union_pw_qpolynomial *upwqp2);
isl_bool isl_union_pw_qpolynomial_fold_plain_is_equal(
__isl_keep isl_union_pw_qpolynomial_fold *upwf1,
86

__isl_keep isl_union_pw_qpolynomial_fold *upwf2);
• Disjointness
#include 
isl_bool isl_basic_set_is_disjoint(
__isl_keep isl_basic_set *bset1,
__isl_keep isl_basic_set *bset2);
isl_bool isl_set_plain_is_disjoint(
__isl_keep isl_set *set1,
__isl_keep isl_set *set2);
isl_bool isl_set_is_disjoint(__isl_keep isl_set *set1,
__isl_keep isl_set *set2);
#include 
isl_bool isl_basic_map_is_disjoint(
__isl_keep isl_basic_map *bmap1,
__isl_keep isl_basic_map *bmap2);
isl_bool isl_map_is_disjoint(__isl_keep isl_map *map1,
__isl_keep isl_map *map2);
#include 
isl_bool isl_union_set_is_disjoint(
__isl_keep isl_union_set *uset1,
__isl_keep isl_union_set *uset2);
#include 
isl_bool isl_union_map_is_disjoint(
__isl_keep isl_union_map *umap1,
__isl_keep isl_union_map *umap2);
• Subset
isl_bool isl_basic_set_is_subset(
__isl_keep isl_basic_set *bset1,
__isl_keep isl_basic_set *bset2);
isl_bool isl_set_is_subset(__isl_keep isl_set *set1,
__isl_keep isl_set *set2);
isl_bool isl_set_is_strict_subset(
__isl_keep isl_set *set1,
__isl_keep isl_set *set2);
isl_bool isl_union_set_is_subset(
__isl_keep isl_union_set *uset1,
__isl_keep isl_union_set *uset2);
isl_bool isl_union_set_is_strict_subset(
87

__isl_keep isl_union_set *uset1,
__isl_keep isl_union_set *uset2);
isl_bool isl_basic_map_is_subset(
__isl_keep isl_basic_map *bmap1,
__isl_keep isl_basic_map *bmap2);
isl_bool isl_basic_map_is_strict_subset(
__isl_keep isl_basic_map *bmap1,
__isl_keep isl_basic_map *bmap2);
isl_bool isl_map_is_subset(
__isl_keep isl_map *map1,
__isl_keep isl_map *map2);
isl_bool isl_map_is_strict_subset(
__isl_keep isl_map *map1,
__isl_keep isl_map *map2);
isl_bool isl_union_map_is_subset(
__isl_keep isl_union_map *umap1,
__isl_keep isl_union_map *umap2);
isl_bool isl_union_map_is_strict_subset(
__isl_keep isl_union_map *umap1,
__isl_keep isl_union_map *umap2);
Check whether the first argument is a (strict) subset of the second argument.
• Order
Every comparison function returns a negative value if the first argument is considered smaller than the second, a positive value if the first argument is considered greater and zero if the two constraints are considered the same by the
comparison criterion.
#include 
int isl_constraint_plain_cmp(
__isl_keep isl_constraint *c1,
__isl_keep isl_constraint *c2);
This function is useful for sorting isl_constraints. The order depends on the
internal representation of the inputs. The order is fixed over different calls to the
function (assuming the internal representation of the inputs has not changed), but
may change over different versions of isl.
#include 
int isl_constraint_cmp_last_non_zero(
__isl_keep isl_constraint *c1,
__isl_keep isl_constraint *c2);
This function can be used to sort constraints that live in the same local space.
Constraints that involve “earlier” dimensions or that have a smaller coefficient
88

for the shared latest dimension are considered smaller than other constraints.
This function only defines a partial order.
#include 
int isl_set_plain_cmp(__isl_keep isl_set *set1,
__isl_keep isl_set *set2);
This function is useful for sorting isl_sets. The order depends on the internal
representation of the inputs. The order is fixed over different calls to the function
(assuming the internal representation of the inputs has not changed), but may
change over different versions of isl.
#include 
int isl_multi_aff_plain_cmp(
__isl_keep isl_multi_aff *ma1,
__isl_keep isl_multi_aff *ma2);
int isl_pw_aff_plain_cmp(__isl_keep isl_pw_aff *pa1,
__isl_keep isl_pw_aff *pa2);
The functions isl_multi_aff_plain_cmp and isl_pw_aff_plain_cmp can
be used to sort isl_multi_affs and isl_pw_affs. The order is not strictly
defined. The current order sorts expressions that only involve earlier dimensions
before those that involve later dimensions.

1.4.16

Unary Operations

• Complement
__isl_give isl_set
__isl_take
__isl_give isl_map
__isl_take

*isl_set_complement(
isl_set *set);
*isl_map_complement(
isl_map *map);

• Inverse map
#include 
__isl_give isl_space *isl_space_reverse(
__isl_take isl_space *space);
#include 
__isl_give isl_basic_map *isl_basic_map_reverse(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_reverse(
__isl_take isl_map *map);

89

#include 
__isl_give isl_union_map *isl_union_map_reverse(
__isl_take isl_union_map *umap);
• Projection
#include 
__isl_give isl_space *isl_space_domain(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_range(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_params(
__isl_take isl_space *space);
#include 
__isl_give isl_local_space *isl_local_space_domain(
__isl_take isl_local_space *ls);
__isl_give isl_local_space *isl_local_space_range(
__isl_take isl_local_space *ls);
#include 
__isl_give isl_basic_set *isl_basic_set_project_out(
__isl_take isl_basic_set *bset,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_set *isl_set_project_out(__isl_take isl_set *set,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_map *isl_set_project_onto_map(
__isl_take isl_set *set,
enum isl_dim_type type, unsigned first,
unsigned n);
__isl_give isl_basic_set *isl_basic_set_params(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_set_params(__isl_take isl_set *set);
The function isl_set_project_onto_map returns a relation that projects the
input set onto the given set dimensions.
#include 
__isl_give isl_basic_map *isl_basic_map_project_out(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_map *isl_map_project_out(__isl_take isl_map *map,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_basic_set *isl_basic_map_domain(
__isl_take isl_basic_map *bmap);
90

__isl_give isl_basic_set *isl_basic_map_range(
__isl_take isl_basic_map *bmap);
__isl_give isl_set *isl_map_params(__isl_take isl_map *map);
__isl_give isl_set *isl_map_domain(
__isl_take isl_map *bmap);
__isl_give isl_set *isl_map_range(
__isl_take isl_map *map);
#include 
__isl_give isl_union_set *isl_union_set_project_out(
__isl_take isl_union_set *uset,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_set *isl_union_set_params(
__isl_take isl_union_set *uset);
The function isl_union_set_project_out can only project out parameters.
#include 
__isl_give isl_union_map *isl_union_map_project_out(
__isl_take isl_union_map *umap,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_union_map *
isl_union_map_project_out_all_params(
__isl_take isl_union_map *umap);
__isl_give isl_set *isl_union_map_params(
__isl_take isl_union_map *umap);
__isl_give isl_union_set *isl_union_map_domain(
__isl_take isl_union_map *umap);
__isl_give isl_union_set *isl_union_map_range(
__isl_take isl_union_map *umap);
The function isl_union_map_project_out can only project out parameters.
#include 
__isl_give isl_aff *isl_aff_project_domain_on_params(
__isl_take isl_aff *aff);
__isl_give isl_multi_aff *
isl_multi_aff_project_domain_on_params(
__isl_take isl_multi_aff *ma);
__isl_give isl_pw_aff *
isl_pw_aff_project_domain_on_params(
__isl_take isl_pw_aff *pa);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_project_domain_on_params(
91

__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_project_domain_on_params(
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_set *isl_pw_aff_domain(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_set *isl_pw_multi_aff_domain(
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_set *isl_multi_pw_aff_domain(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_union_set *isl_union_pw_aff_domain(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_union_set *isl_union_pw_multi_aff_domain(
__isl_take isl_union_pw_multi_aff *upma);
__isl_give isl_union_set *
isl_multi_union_pw_aff_domain(
__isl_take isl_multi_union_pw_aff *mupa);
__isl_give isl_set *isl_pw_aff_params(
__isl_take isl_pw_aff *pwa);
If no explicit domain was set on a zero-dimensional input to isl_multi_union_pw_aff_domain,
then this function will return a parameter set.
#include 
__isl_give isl_qpolynomial *
isl_qpolynomial_project_domain_on_params(
__isl_take isl_qpolynomial *qp);
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_project_domain_on_params(
__isl_take isl_pw_qpolynomial *pwqp);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_project_domain_on_params(
__isl_take isl_pw_qpolynomial_fold *pwf);
__isl_give isl_set *isl_pw_qpolynomial_domain(
__isl_take isl_pw_qpolynomial *pwqp);
__isl_give isl_union_set *isl_union_pw_qpolynomial_fold_domain(
__isl_take isl_union_pw_qpolynomial_fold *upwf);
__isl_give isl_union_set *isl_union_pw_qpolynomial_domain(
__isl_take isl_union_pw_qpolynomial *upwqp);
#include 
__isl_give isl_space *isl_space_domain_map(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_range_map(
__isl_take isl_space *space);

92

#include 
__isl_give isl_map *isl_set_wrapped_domain_map(
__isl_take isl_set *set);
__isl_give isl_basic_map *isl_basic_map_domain_map(
__isl_take isl_basic_map *bmap);
__isl_give isl_basic_map *isl_basic_map_range_map(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_domain_map(__isl_take isl_map *map);
__isl_give isl_map *isl_map_range_map(__isl_take isl_map *map);
#include 
__isl_give isl_union_map *isl_union_map_domain_map(
__isl_take isl_union_map *umap);
__isl_give isl_union_pw_multi_aff *
isl_union_map_domain_map_union_pw_multi_aff(
__isl_take isl_union_map *umap);
__isl_give isl_union_map *isl_union_map_range_map(
__isl_take isl_union_map *umap);
__isl_give isl_union_map *
isl_union_set_wrapped_domain_map(
__isl_take isl_union_set *uset);
The functions above construct a (basic, regular or union) relation that maps (a
wrapped version of) the input relation to its domain or range. isl_set_wrapped_domain_map
maps the input set to the domain of its wrapped relation.
• Elimination
__isl_give isl_basic_set *isl_basic_set_eliminate(
__isl_take isl_basic_set *bset,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_set *isl_set_eliminate(
__isl_take isl_set *set, enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_basic_map *isl_basic_map_eliminate(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_map *isl_map_eliminate(
__isl_take isl_map *map, enum isl_dim_type type,
unsigned first, unsigned n);
Eliminate the coefficients for the given dimensions from the constraints, without
removing the dimensions.

93

• Constructing a set from a parameter domain
A zero-dimensional (local) space or (basic) set can be constructed on a given
parameter domain using the following functions.
#include 
__isl_give isl_space *isl_space_set_from_params(
__isl_take isl_space *space);
#include 
__isl_give isl_local_space *
isl_local_space_set_from_params(
__isl_take isl_local_space *ls);
#include 
__isl_give isl_basic_set *isl_basic_set_from_params(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_set_from_params(
__isl_take isl_set *set);
• Constructing a relation from one or two sets
Create a relation with the given set(s) as domain and/or range. If only the domain
or the range is specified, then the range or domain of the created relation is a
zero-dimensional flat anonymous space.
#include 
__isl_give isl_space *isl_space_from_domain(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_from_range(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_map_from_set(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_map_from_domain_and_range(
__isl_take isl_space *domain,
__isl_take isl_space *range);
#include 
__isl_give isl_local_space *isl_local_space_from_domain(
__isl_take isl_local_space *ls);
#include 
__isl_give isl_map *isl_map_from_domain(
__isl_take isl_set *set);
__isl_give isl_map *isl_map_from_range(
__isl_take isl_set *set);
94

#include 
__isl_give isl_union_map *isl_union_map_from_domain(
__isl_take isl_union_set *uset);
__isl_give isl_union_map *isl_union_map_from_range(
__isl_take isl_union_set *uset);
__isl_give isl_union_map *
isl_union_map_from_domain_and_range(
__isl_take isl_union_set *domain,
__isl_take isl_union_set *range);
#include 
__isl_give isl_multi_val *isl_multi_val_from_range(
__isl_take isl_multi_val *mv);
#include 
__isl_give isl_aff *isl_aff_from_range(
__isl_take isl_aff *aff);
__isl_give isl_multi_aff *isl_multi_aff_from_range(
__isl_take isl_multi_aff *ma);
__isl_give isl_pw_aff *isl_pw_aff_from_range(
__isl_take isl_pw_aff *pwa);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_range(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_from_range(
__isl_take isl_multi_union_pw_aff *mupa);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_domain(
__isl_take isl_set *set);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_from_domain(
__isl_take isl_union_set *uset);
#include 
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_from_range(
__isl_take isl_pw_qpolynomial *pwqp);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_from_range(
__isl_take isl_pw_qpolynomial_fold *pwf);
• Slicing
#include 
__isl_give isl_basic_set *isl_basic_set_fix_si(
__isl_take isl_basic_set *bset,

95

enum isl_dim_type type, unsigned pos, int value);
__isl_give isl_basic_set *isl_basic_set_fix_val(
__isl_take isl_basic_set *bset,
enum isl_dim_type type, unsigned pos,
__isl_take isl_val *v);
__isl_give isl_set *isl_set_fix_si(__isl_take isl_set *set,
enum isl_dim_type type, unsigned pos, int value);
__isl_give isl_set *isl_set_fix_val(
__isl_take isl_set *set,
enum isl_dim_type type, unsigned pos,
__isl_take isl_val *v);
#include 
__isl_give isl_basic_map *isl_basic_map_fix_si(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type, unsigned pos, int value);
__isl_give isl_basic_map *isl_basic_map_fix_val(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type, unsigned pos,
__isl_take isl_val *v);
__isl_give isl_map *isl_map_fix_si(__isl_take isl_map *map,
enum isl_dim_type type, unsigned pos, int value);
__isl_give isl_map *isl_map_fix_val(
__isl_take isl_map *map,
enum isl_dim_type type, unsigned pos,
__isl_take isl_val *v);
#include 
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_fix_si(
__isl_take isl_pw_multi_aff *pma,
enum isl_dim_type type, unsigned pos, int value);
#include 
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_fix_val(
__isl_take isl_pw_qpolynomial *pwqp,
enum isl_dim_type type, unsigned n,
__isl_take isl_val *v);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_fix_val(
__isl_take isl_pw_qpolynomial_fold *pwf,
enum isl_dim_type type, unsigned n,
__isl_take isl_val *v);
Intersect the set, relation or function domain with the hyperplane where the given
dimension has the fixed given value.

96

#include 
__isl_give isl_basic_set *
isl_basic_set_lower_bound_val(
__isl_take isl_basic_set *bset,
enum isl_dim_type type, unsigned pos,
__isl_take isl_val *value);
__isl_give isl_basic_set *
isl_basic_set_upper_bound_val(
__isl_take isl_basic_set *bset,
enum isl_dim_type type, unsigned pos,
__isl_take isl_val *value);
__isl_give isl_set *isl_set_lower_bound_si(
__isl_take isl_set *set,
enum isl_dim_type type, unsigned pos, int value);
__isl_give isl_set *isl_set_lower_bound_val(
__isl_take isl_set *set,
enum isl_dim_type type, unsigned pos,
__isl_take isl_val *value);
__isl_give isl_set *isl_set_upper_bound_si(
__isl_take isl_set *set,
enum isl_dim_type type, unsigned pos, int value);
__isl_give isl_set *isl_set_upper_bound_val(
__isl_take isl_set *set,
enum isl_dim_type type, unsigned pos,
__isl_take isl_val *value);
#include 
__isl_give isl_basic_map *isl_basic_map_lower_bound_si(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type, unsigned pos, int value);
__isl_give isl_basic_map *isl_basic_map_upper_bound_si(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type, unsigned pos, int value);
__isl_give isl_map *isl_map_lower_bound_si(
__isl_take isl_map *map,
enum isl_dim_type type, unsigned pos, int value);
__isl_give isl_map *isl_map_upper_bound_si(
__isl_take isl_map *map,
enum isl_dim_type type, unsigned pos, int value);
Intersect the set or relation with the half-space where the given dimension has a
value bounded by the fixed given integer value.
__isl_give isl_set *isl_set_equate(__isl_take isl_set *set,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
97

__isl_give isl_basic_map *isl_basic_map_equate(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
__isl_give isl_map *isl_map_equate(__isl_take isl_map *map,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
Intersect the set or relation with the hyperplane where the given dimensions are
equal to each other.
__isl_give isl_map *isl_map_oppose(__isl_take isl_map *map,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
Intersect the relation with the hyperplane where the given dimensions have opposite values.
__isl_give isl_map *isl_map_order_le(
__isl_take isl_map *map,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
__isl_give isl_basic_map *isl_basic_map_order_ge(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
__isl_give isl_map *isl_map_order_ge(
__isl_take isl_map *map,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
__isl_give isl_map *isl_map_order_lt(__isl_take isl_map *map,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
__isl_give isl_basic_map *isl_basic_map_order_gt(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
__isl_give isl_map *isl_map_order_gt(__isl_take isl_map *map,
enum isl_dim_type type1, int pos1,
enum isl_dim_type type2, int pos2);
Intersect the relation with the half-space where the given dimensions satisfy the
given ordering.
#include 
98

__isl_give isl_union_map *isl_union_map_remove_map_if(
__isl_take isl_union_map *umap,
isl_bool (*fn)(__isl_keep isl_map *map,
void *user), void *user);
This function calls the callback function once for each pair of spaces for which
there are elements in the input. If the callback returns isl_bool_true, then all
those elements are removed from the result. The only remaining elements in the
output are then those for which the callback returns isl_bool_false.
• Locus
#include 
__isl_give isl_basic_set *isl_aff_zero_basic_set(
__isl_take isl_aff *aff);
__isl_give isl_basic_set *isl_aff_neg_basic_set(
__isl_take isl_aff *aff);
__isl_give isl_set *isl_pw_aff_pos_set(
__isl_take isl_pw_aff *pa);
__isl_give isl_set *isl_pw_aff_nonneg_set(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_set *isl_pw_aff_zero_set(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_set *isl_pw_aff_non_zero_set(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_union_set *
isl_union_pw_aff_zero_union_set(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_union_set *
isl_multi_union_pw_aff_zero_union_set(
__isl_take isl_multi_union_pw_aff *mupa);
The function isl_aff_neg_basic_set returns a basic set containing those
elements in the domain space of aff where aff is negative. The function
isl_pw_aff_nonneg_set returns a set containing those elements in the domain
of pwaff where pwaff is non-negative. The function isl_multi_union_pw_aff_zero_union_set
returns a union set containing those elements in the domains of its elements
where they are all zero.
• Identity
__isl_give isl_map *isl_set_identity(
__isl_take isl_set *set);
__isl_give isl_union_map *isl_union_set_identity(
__isl_take isl_union_set *uset);
__isl_give isl_union_pw_multi_aff *
isl_union_set_identity_union_pw_multi_aff(
__isl_take isl_union_set *uset);
99

Construct an identity relation on the given (union) set.
• Function Extraction
A piecewise quasi affine expression that is equal to 1 on a set and 0 outside the
set can be created using the following function.
#include 
__isl_give isl_pw_aff *isl_set_indicator_function(
__isl_take isl_set *set);
A piecewise multiple quasi affine expression can be extracted from an isl_set
or isl_map, provided the isl_set is a singleton and the isl_map is singlevalued. In case of a conversion from an isl_union_map to an isl_union_pw_multi_aff,
these properties need to hold in each domain space. A conversion to a isl_multi_union_pw_aff
additionally requires that the input is non-empty and involves only a single range
space.
#include 
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_set(
__isl_take isl_set *set);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_map(
__isl_take isl_map *map);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_from_union_set(
__isl_take isl_union_set *uset);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_from_union_map(
__isl_take isl_union_map *umap);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_from_union_map(
__isl_take isl_union_map *umap);
• Deltas
__isl_give isl_basic_set *isl_basic_map_deltas(
__isl_take isl_basic_map *bmap);
__isl_give isl_set *isl_map_deltas(__isl_take isl_map *map);
__isl_give isl_union_set *isl_union_map_deltas(
__isl_take isl_union_map *umap);
These functions return a (basic) set containing the differences between image
elements and corresponding domain elements in the input.

100

__isl_give isl_basic_map *isl_basic_map_deltas_map(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_deltas_map(
__isl_take isl_map *map);
__isl_give isl_union_map *isl_union_map_deltas_map(
__isl_take isl_union_map *umap);
The functions above construct a (basic, regular or union) relation that maps (a
wrapped version of) the input relation to its delta set.
• Coalescing
Simplify the representation of a set, relation or functions by trying to combine
pairs of basic sets or relations into a single basic set or relation.
#include 
__isl_give isl_set *isl_set_coalesce(__isl_take isl_set *set);
#include 
__isl_give isl_map *isl_map_coalesce(__isl_take isl_map *map);
#include 
__isl_give isl_union_set *isl_union_set_coalesce(
__isl_take isl_union_set *uset);
#include 
__isl_give isl_union_map *isl_union_map_coalesce(
__isl_take isl_union_map *umap);
#include 
__isl_give isl_pw_aff *isl_pw_aff_coalesce(
__isl_take isl_pw_aff *pwqp);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_coalesce(
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_coalesce(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_union_pw_aff *isl_union_pw_aff_coalesce(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_coalesce(
__isl_take isl_union_pw_multi_aff *upma);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_coalesce(
__isl_take isl_multi_union_pw_aff *aff);

101

#include 
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_coalesce(
__isl_take isl_pw_qpolynomial_fold *pwf);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_coalesce(
__isl_take isl_union_pw_qpolynomial *upwqp);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_coalesce(
__isl_take isl_union_pw_qpolynomial_fold *upwf);
One of the methods for combining pairs of basic sets or relations can result in
coefficients that are much larger than those that appear in the constraints of the
input. By default, the coefficients are not allowed to grow larger, but this can be
changed by unsetting the following option.
isl_stat isl_options_set_coalesce_bounded_wrapping(
isl_ctx *ctx, int val);
int isl_options_get_coalesce_bounded_wrapping(
isl_ctx *ctx);
• Detecting equalities
__isl_give isl_basic_set *isl_basic_set_detect_equalities(
__isl_take isl_basic_set *bset);
__isl_give isl_basic_map *isl_basic_map_detect_equalities(
__isl_take isl_basic_map *bmap);
__isl_give isl_set *isl_set_detect_equalities(
__isl_take isl_set *set);
__isl_give isl_map *isl_map_detect_equalities(
__isl_take isl_map *map);
__isl_give isl_union_set *isl_union_set_detect_equalities(
__isl_take isl_union_set *uset);
__isl_give isl_union_map *isl_union_map_detect_equalities(
__isl_take isl_union_map *umap);
Simplify the representation of a set or relation by detecting implicit equalities.
• Removing redundant constraints
#include 
__isl_give isl_basic_set *isl_basic_set_remove_redundancies(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_set_remove_redundancies(
__isl_take isl_set *set);

102

#include 
__isl_give isl_union_set *
isl_union_set_remove_redundancies(
__isl_take isl_union_set *uset);
#include 
__isl_give isl_basic_map *isl_basic_map_remove_redundancies(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_remove_redundancies(
__isl_take isl_map *map);
#include 
__isl_give isl_union_map *
isl_union_map_remove_redundancies(
__isl_take isl_union_map *umap);
• Convex hull
__isl_give isl_basic_set *isl_set_convex_hull(
__isl_take isl_set *set);
__isl_give isl_basic_map *isl_map_convex_hull(
__isl_take isl_map *map);
If the input set or relation has any existentially quantified variables, then the
result of these operations is currently undefined.
• Simple hull
#include 
__isl_give isl_basic_set *
isl_set_unshifted_simple_hull(
__isl_take isl_set *set);
__isl_give isl_basic_set *isl_set_simple_hull(
__isl_take isl_set *set);
__isl_give isl_basic_set *
isl_set_plain_unshifted_simple_hull(
__isl_take isl_set *set);
__isl_give isl_basic_set *
isl_set_unshifted_simple_hull_from_set_list(
__isl_take isl_set *set,
__isl_take isl_set_list *list);
#include 
__isl_give isl_basic_map *
isl_map_unshifted_simple_hull(
__isl_take isl_map *map);
103

__isl_give isl_basic_map *isl_map_simple_hull(
__isl_take isl_map *map);
__isl_give isl_basic_map *
isl_map_plain_unshifted_simple_hull(
__isl_take isl_map *map);
__isl_give isl_basic_map *
isl_map_unshifted_simple_hull_from_map_list(
__isl_take isl_map *map,
__isl_take isl_map_list *list);
#include 
__isl_give isl_union_map *isl_union_map_simple_hull(
__isl_take isl_union_map *umap);

These functions compute a single basic set or relation that contains the whole
input set or relation. In particular, the output is described by translates of the constraints describing the basic sets or relations in the input. In case of isl_set_unshifted_simple_hull,
only the original constraints are used, without any translation. In case of isl_set_plain_unshifted_simple_hu
and isl_map_plain_unshifted_simple_hull, the result is described by original constraints that are obviously satisfied by the entire input set or relation. In
case of isl_set_unshifted_simple_hull_from_set_list and isl_map_unshifted_simple_hull_from_
the constraints are taken from the elements of the second argument.
(See Section 2.2.)
• Affine hull
__isl_give isl_basic_set *isl_basic_set_affine_hull(
__isl_take isl_basic_set *bset);
__isl_give isl_basic_set *isl_set_affine_hull(
__isl_take isl_set *set);
__isl_give isl_union_set *isl_union_set_affine_hull(
__isl_take isl_union_set *uset);
__isl_give isl_basic_map *isl_basic_map_affine_hull(
__isl_take isl_basic_map *bmap);
__isl_give isl_basic_map *isl_map_affine_hull(
__isl_take isl_map *map);
__isl_give isl_union_map *isl_union_map_affine_hull(
__isl_take isl_union_map *umap);
In case of union sets and relations, the affine hull is computed per space.
• Polyhedral hull
__isl_give isl_basic_set *isl_set_polyhedral_hull(
__isl_take isl_set *set);
__isl_give isl_basic_map *isl_map_polyhedral_hull(
104

__isl_take isl_map *map);
__isl_give isl_union_set *isl_union_set_polyhedral_hull(
__isl_take isl_union_set *uset);
__isl_give isl_union_map *isl_union_map_polyhedral_hull(
__isl_take isl_union_map *umap);
These functions compute a single basic set or relation not involving any existentially quantified variables that contains the whole input set or relation. In case of
union sets and relations, the polyhedral hull is computed per space.
• Box hull
#include 
__isl_give isl_fixed_box *
isl_map_get_range_simple_fixed_box_hull(
__isl_keep isl_map *map);
This function tries to approximate the range of the map by a box of fixed size.
The box is described in terms of an offset living in the same space as the input
map and a size living in the range space. For any element in the input map, the
range value is greater than or equal to the offset applied to the domain value and
the difference with this offset is strictly smaller than the size. If no fixed-size
approximation of the range can be found, an invalid box is returned, i.e., one for
which isl_fixed_box_is_valid below returns false.
The validity, the offset and the size of the box can be obtained using the following
functions.
#include 
isl_bool isl_fixed_box_is_valid(
__isl_keep isl_fixed_box *box);
__isl_give isl_multi_aff *isl_fixed_box_get_offset(
__isl_keep isl_fixed_box *box);
__isl_give isl_multi_val *isl_fixed_box_get_size(
__isl_keep isl_fixed_box *box);
The box can be copied and freed using the following functions.
#include 
__isl_give isl_fixed_box *isl_fixed_box_copy(
__isl_keep isl_fixed_box *box);
__isl_null isl_fixed_box *isl_fixed_box_free(
__isl_take isl_fixed_box *box);
• Other approximations

105

#include 
__isl_give isl_basic_set *
isl_basic_set_drop_constraints_involving_dims(
__isl_take isl_basic_set *bset,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_basic_set *
isl_basic_set_drop_constraints_not_involving_dims(
__isl_take isl_basic_set *bset,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_set *
isl_set_drop_constraints_involving_dims(
__isl_take isl_set *set,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_set *
isl_set_drop_constraints_not_involving_dims(
__isl_take isl_set *set,
enum isl_dim_type type,
unsigned first, unsigned n);
#include 
__isl_give isl_basic_map *
isl_basic_map_drop_constraints_involving_dims(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_basic_map *
isl_basic_map_drop_constraints_not_involving_dims(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_map *
isl_map_drop_constraints_involving_dims(
__isl_take isl_map *map,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_map *
isl_map_drop_constraints_not_involving_dims(
__isl_take isl_map *map,
enum isl_dim_type type,
unsigned first, unsigned n);
These functions drop any constraints (not) involving the specified dimensions.
Note that the result depends on the representation of the input.
106

#include 
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_to_polynomial(
__isl_take isl_pw_qpolynomial *pwqp, int sign);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_to_polynomial(
__isl_take isl_union_pw_qpolynomial *upwqp, int sign);
Approximate each quasipolynomial by a polynomial. If sign is positive, the
polynomial will be an overapproximation. If sign is negative, it will be an
underapproximation. If sign is zero, the approximation will lie somewhere in
between.
• Feasibility
__isl_give isl_basic_set *isl_basic_set_sample(
__isl_take isl_basic_set *bset);
__isl_give isl_basic_set *isl_set_sample(
__isl_take isl_set *set);
__isl_give isl_basic_map *isl_basic_map_sample(
__isl_take isl_basic_map *bmap);
__isl_give isl_basic_map *isl_map_sample(
__isl_take isl_map *map);
If the input (basic) set or relation is non-empty, then return a singleton subset of
the input. Otherwise, return an empty set.
• Optimization
#include 
__isl_give isl_val *isl_basic_set_max_val(
__isl_keep isl_basic_set *bset,
__isl_keep isl_aff *obj);
__isl_give isl_val *isl_set_min_val(
__isl_keep isl_set *set,
__isl_keep isl_aff *obj);
__isl_give isl_val *isl_set_max_val(
__isl_keep isl_set *set,
__isl_keep isl_aff *obj);
__isl_give isl_multi_val *
isl_union_set_min_multi_union_pw_aff(
__isl_keep isl_union_set *uset,
__isl_keep isl_multi_union_pw_aff *obj);
Compute the minimum or maximum of the integer affine expression obj over
the points in set. The result is NULL in case of an error, the optimal value in case
there is one, negative infinity or infinity if the problem is unbounded and NaN if
the problem is empty.
107

#include 
__isl_give isl_val *isl_union_pw_aff_min_val(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_val *isl_union_pw_aff_max_val(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_multi_val *
isl_multi_union_pw_aff_min_multi_val(
__isl_take isl_multi_union_pw_aff *mupa);
__isl_give isl_multi_val *
isl_multi_union_pw_aff_max_multi_val(
__isl_take isl_multi_union_pw_aff *mupa);
Compute the minimum or maximum of the integer affine expression over its
definition domain. The result is NULL in case of an error, the optimal value in
case there is one, negative infinity or infinity if the problem is unbounded and
NaN if the problem is empty.
#include 
__isl_give isl_val *isl_basic_set_dim_max_val(
__isl_take isl_basic_set *bset, int pos);
Return the maximal value attained by the given set dimension, independently of
the parameter values and of any other dimensions. The result is NULL in case
of an error, the optimal value in case there is one, infinity if the problem is
unbounded and NaN if the input is empty.
• Parametric optimization
__isl_give isl_pw_aff *isl_set_dim_min(
__isl_take isl_set *set, int pos);
__isl_give isl_pw_aff *isl_set_dim_max(
__isl_take isl_set *set, int pos);
__isl_give isl_pw_aff *isl_map_dim_min(
__isl_take isl_map *map, int pos);
__isl_give isl_pw_aff *isl_map_dim_max(
__isl_take isl_map *map, int pos);
Compute the minimum or maximum of the given set or output dimension as a
function of the parameters (and input dimensions), but independently of the other
set or output dimensions. For lexicographic optimization, see §1.4.17.
• Dual
The following functions compute either the set of (rational) coefficient values
of valid constraints for the given set or the set of (rational) values satisfying the
constraints with coefficients from the given set. Internally, these two sets of functions perform essentially the same operations, except that the set of coefficients
108

is assumed to be a cone, while the set of values may be any polyhedron. The
current implementation is based on the Farkas lemma and Fourier-Motzkin elimination, but this may change or be made optional in future. In particular, future
implementations may use different dualization algorithms or skip the elimination
step.
#include 
__isl_give isl_basic_set *isl_basic_set_coefficients(
__isl_take isl_basic_set *bset);
__isl_give isl_basic_set_list *
isl_basic_set_list_coefficients(
__isl_take isl_basic_set_list *list);
__isl_give isl_basic_set *isl_set_coefficients(
__isl_take isl_set *set);
__isl_give isl_union_set *isl_union_set_coefficients(
__isl_take isl_union_set *bset);
__isl_give isl_basic_set *isl_basic_set_solutions(
__isl_take isl_basic_set *bset);
__isl_give isl_basic_set *isl_set_solutions(
__isl_take isl_set *set);
__isl_give isl_union_set *isl_union_set_solutions(
__isl_take isl_union_set *bset);
• Power
__isl_give isl_map *isl_map_fixed_power_val(
__isl_take isl_map *map,
__isl_take isl_val *exp);
__isl_give isl_union_map *
isl_union_map_fixed_power_val(
__isl_take isl_union_map *umap,
__isl_take isl_val *exp);
Compute the given power of map, where exp is assumed to be non-zero. If
the exponent exp is negative, then the -exp th power of the inverse of map is
computed.
__isl_give isl_map *isl_map_power(__isl_take isl_map *map,
int *exact);
__isl_give isl_union_map *isl_union_map_power(
__isl_take isl_union_map *umap, int *exact);
Compute a parametric representation for all positive powers k of map. The result
maps k to a nested relation corresponding to the kth power of map. The result
may be an overapproximation. If the result is known to be exact, then *exact is
set to 1.
109

• Transitive closure
__isl_give isl_map *isl_map_transitive_closure(
__isl_take isl_map *map, int *exact);
__isl_give isl_union_map *isl_union_map_transitive_closure(
__isl_take isl_union_map *umap, int *exact);
Compute the transitive closure of map. The result may be an overapproximation.
If the result is known to be exact, then *exact is set to 1.
• Reaching path lengths
__isl_give isl_map *isl_map_reaching_path_lengths(
__isl_take isl_map *map, int *exact);
Compute a relation that maps each element in the range of map to the lengths of
all paths composed of edges in map that end up in the given element. The result
may be an overapproximation. If the result is known to be exact, then *exact
is set to 1. To compute the maximal path length, the resulting relation should
be postprocessed by isl_map_lexmax. In particular, if the input relation is a
dependence relation (mapping sources to sinks), then the maximal path length
corresponds to the free schedule. Note, however, that isl_map_lexmax expects
the maximum to be finite, so if the path lengths are unbounded (possibly due to
the overapproximation), then you will get an error message.
• Wrapping
#include 
__isl_give isl_space *isl_space_wrap(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_unwrap(
__isl_take isl_space *space);
#include 
__isl_give isl_local_space *isl_local_space_wrap(
__isl_take isl_local_space *ls);
#include 
__isl_give isl_basic_map *isl_basic_set_unwrap(
__isl_take isl_basic_set *bset);
__isl_give isl_map *isl_set_unwrap(
__isl_take isl_set *set);
#include 
__isl_give isl_basic_set *isl_basic_map_wrap(
__isl_take isl_basic_map *bmap);
__isl_give isl_set *isl_map_wrap(
__isl_take isl_map *map);
110

#include 
__isl_give isl_union_map *isl_union_set_unwrap(
__isl_take isl_union_set *uset);
#include 
__isl_give isl_union_set *isl_union_map_wrap(
__isl_take isl_union_map *umap);
The input to isl_space_unwrap should be the space of a set, while that of
isl_space_wrap should be the space of a relation. Conversely, the output of
isl_space_unwrap is the space of a relation, while that of isl_space_wrap
is the space of a set.
• Flattening
Remove any internal structure of domain (and range) of the given set or relation.
If there is any such internal structure in the input, then the name of the space is
also removed.
#include 
__isl_give isl_space *isl_space_flatten_domain(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_flatten_range(
__isl_take isl_space *space);
#include 
__isl_give isl_local_space *
isl_local_space_flatten_domain(
__isl_take isl_local_space *ls);
__isl_give isl_local_space *
isl_local_space_flatten_range(
__isl_take isl_local_space *ls);
#include 
__isl_give isl_basic_set *isl_basic_set_flatten(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_set_flatten(
__isl_take isl_set *set);
#include 
__isl_give isl_basic_map *isl_basic_map_flatten_domain(
__isl_take isl_basic_map *bmap);
__isl_give isl_basic_map *isl_basic_map_flatten_range(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_flatten_range(
__isl_take isl_map *map);
111

__isl_give isl_map *isl_map_flatten_domain(
__isl_take isl_map *map);
__isl_give isl_basic_map *isl_basic_map_flatten(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_flatten(
__isl_take isl_map *map);
#include 
__isl_give isl_multi_val *isl_multi_val_flatten_range(
__isl_take isl_multi_val *mv);
#include 
__isl_give isl_multi_aff *isl_multi_aff_flatten_domain(
__isl_take isl_multi_aff *ma);
__isl_give isl_multi_aff *isl_multi_aff_flatten_range(
__isl_take isl_multi_aff *ma);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_flatten_range(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_flatten_range(
__isl_take isl_multi_union_pw_aff *mupa);
#include 
__isl_give isl_map *isl_set_flatten_map(
__isl_take isl_set *set);
The function above constructs a relation that maps the input set to a flattened
version of the set.
• Lifting
Lift the input set to a space with extra dimensions corresponding to the existentially quantified variables in the input. In particular, the result lives in a wrapped
map where the domain is the original space and the range corresponds to the
original existentially quantified variables.
#include 
__isl_give isl_basic_set *isl_basic_set_lift(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_set_lift(
__isl_take isl_set *set);
__isl_give isl_union_set *isl_union_set_lift(
__isl_take isl_union_set *uset);
Given a local space that contains the existentially quantified variables of a set, a
basic relation that, when applied to a basic set, has essentially the same effect as
isl_basic_set_lift, can be constructed using the following function.
112

#include 
__isl_give isl_basic_map *isl_local_space_lifting(
__isl_take isl_local_space *ls);
#include 
__isl_give isl_multi_aff *isl_multi_aff_lift(
__isl_take isl_multi_aff *maff,
__isl_give isl_local_space **ls);
If the ls argument of isl_multi_aff_lift is not NULL, then it is assigned the
local space that lies at the basis of the lifting applied.
• Internal Product
#include 
__isl_give isl_space *isl_space_zip(
__isl_take isl_space *space);
#include 
__isl_give isl_basic_map *isl_basic_map_zip(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_zip(
__isl_take isl_map *map);
#include 
__isl_give isl_union_map *isl_union_map_zip(
__isl_take isl_union_map *umap);
Given a relation with nested relations for domain and range, interchange the
range of the domain with the domain of the range.
• Currying
#include 
__isl_give isl_space *isl_space_curry(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_uncurry(
__isl_take isl_space *space);
#include 
__isl_give isl_basic_map *isl_basic_map_curry(
__isl_take isl_basic_map *bmap);
__isl_give isl_basic_map *isl_basic_map_uncurry(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_curry(
__isl_take isl_map *map);
__isl_give isl_map *isl_map_uncurry(
__isl_take isl_map *map);
113

#include 
__isl_give isl_union_map *isl_union_map_curry(
__isl_take isl_union_map *umap);
__isl_give isl_union_map *isl_union_map_uncurry(
__isl_take isl_union_map *umap);
Given a relation with a nested relation for domain, the curry functions move the
range of the nested relation out of the domain and use it as the domain of a nested
relation in the range, with the original range as range of this nested relation. The
uncurry functions perform the inverse operation.
#include 
__isl_give isl_space *isl_space_range_curry(
__isl_take isl_space *space);
#include 
__isl_give isl_map *isl_map_range_curry(
__isl_take isl_map *map);
#include 
__isl_give isl_union_map *isl_union_map_range_curry(
__isl_take isl_union_map *umap);
These functions apply the currying to the relation that is nested inside the range
of the input.
• Aligning parameters
Change the order of the parameters of the given set, relation or function such that
the first parameters match those of model. This may involve the introduction of
extra parameters. All parameters need to be named.
#include 
__isl_give isl_space *isl_space_align_params(
__isl_take isl_space *space1,
__isl_take isl_space *space2)
#include 
__isl_give isl_basic_set *isl_basic_set_align_params(
__isl_take isl_basic_set *bset,
__isl_take isl_space *model);
__isl_give isl_set *isl_set_align_params(
__isl_take isl_set *set,
__isl_take isl_space *model);

114

#include 
__isl_give isl_basic_map *isl_basic_map_align_params(
__isl_take isl_basic_map *bmap,
__isl_take isl_space *model);
__isl_give isl_map *isl_map_align_params(
__isl_take isl_map *map,
__isl_take isl_space *model);
#include 
__isl_give isl_multi_val *isl_multi_val_align_params(
__isl_take isl_multi_val *mv,
__isl_take isl_space *model);
#include 
__isl_give isl_aff *isl_aff_align_params(
__isl_take isl_aff *aff,
__isl_take isl_space *model);
__isl_give isl_multi_aff *isl_multi_aff_align_params(
__isl_take isl_multi_aff *multi,
__isl_take isl_space *model);
__isl_give isl_pw_aff *isl_pw_aff_align_params(
__isl_take isl_pw_aff *pwaff,
__isl_take isl_space *model);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_align_params(
__isl_take isl_pw_multi_aff *pma,
__isl_take isl_space *model);
__isl_give isl_union_pw_aff *
isl_union_pw_aff_align_params(
__isl_take isl_union_pw_aff *upa,
__isl_take isl_space *model);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_align_params(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_space *model);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_align_params(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_space *model);
#include 
__isl_give isl_qpolynomial *isl_qpolynomial_align_params(
__isl_take isl_qpolynomial *qp,
__isl_take isl_space *model);
• Drop unused parameters

115

Drop parameters that are not referenced by the isl object. All parameters need to
be named.
#include 
__isl_give isl_basic_set *
isl_basic_set_drop_unused_params(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_set_drop_unused_params(
__isl_take isl_set *set);
#include 
__isl_give isl_basic_map *
isl_basic_map_drop_unused_params(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_drop_unused_params(
__isl_take isl_map *map);
#include 
__isl_give isl_pw_aff *isl_pw_aff_drop_unused_params(
__isl_take isl_pw_aff *pa);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_drop_unused_params(
__isl_take isl_pw_multi_aff *pma);
#include 
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_drop_unused_params(
__isl_take isl_pw_qpolynomial *pwqp);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_drop_unused_params(
__isl_take isl_pw_qpolynomial_fold *pwf);
• Unary Arithmetic Operations
#include 
__isl_give isl_set *isl_set_neg(
__isl_take isl_set *set);
#include 
__isl_give isl_map *isl_map_neg(
__isl_take isl_map *map);
isl_set_neg constructs a set containing the opposites of the elements in its argument. The domain of the result of isl_map_neg is the same as the domain of
its argument. The corresponding range elements are the opposites of the corresponding range elements in the argument.
116

#include 
__isl_give isl_multi_val *isl_multi_val_neg(
__isl_take isl_multi_val *mv);
#include 
__isl_give isl_aff *isl_aff_neg(
__isl_take isl_aff *aff);
__isl_give isl_multi_aff *isl_multi_aff_neg(
__isl_take isl_multi_aff *ma);
__isl_give isl_pw_aff *isl_pw_aff_neg(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_neg(
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_neg(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_union_pw_aff *isl_union_pw_aff_neg(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_neg(
__isl_take isl_union_pw_multi_aff *upma);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_neg(
__isl_take isl_multi_union_pw_aff *mupa);
__isl_give isl_aff *isl_aff_ceil(
__isl_take isl_aff *aff);
__isl_give isl_pw_aff *isl_pw_aff_ceil(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_aff *isl_aff_floor(
__isl_take isl_aff *aff);
__isl_give isl_multi_aff *isl_multi_aff_floor(
__isl_take isl_multi_aff *ma);
__isl_give isl_pw_aff *isl_pw_aff_floor(
__isl_take isl_pw_aff *pwaff);
__isl_give isl_union_pw_aff *isl_union_pw_aff_floor(
__isl_take isl_union_pw_aff *upa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_floor(
__isl_take isl_multi_union_pw_aff *mupa);
#include 
__isl_give isl_pw_aff *isl_pw_aff_list_min(
__isl_take isl_pw_aff_list *list);
__isl_give isl_pw_aff *isl_pw_aff_list_max(
__isl_take isl_pw_aff_list *list);
#include 
117

__isl_give isl_qpolynomial *isl_qpolynomial_neg(
__isl_take isl_qpolynomial *qp);
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_neg(
__isl_take isl_pw_qpolynomial *pwqp);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_neg(
__isl_take isl_union_pw_qpolynomial *upwqp);
__isl_give isl_qpolynomial *isl_qpolynomial_pow(
__isl_take isl_qpolynomial *qp,
unsigned exponent);
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_pow(
__isl_take isl_pw_qpolynomial *pwqp,
unsigned exponent);
• Evaluation
The following functions evaluate a function in a point.
#include 
__isl_give isl_val *isl_aff_eval(
__isl_take isl_aff *aff,
__isl_take isl_point *pnt);
__isl_give isl_val *isl_pw_aff_eval(
__isl_take isl_pw_aff *pa,
__isl_take isl_point *pnt);
#include 
__isl_give isl_val *isl_pw_qpolynomial_eval(
__isl_take isl_pw_qpolynomial *pwqp,
__isl_take isl_point *pnt);
__isl_give isl_val *isl_pw_qpolynomial_fold_eval(
__isl_take isl_pw_qpolynomial_fold *pwf,
__isl_take isl_point *pnt);
__isl_give isl_val *isl_union_pw_qpolynomial_eval(
__isl_take isl_union_pw_qpolynomial *upwqp,
__isl_take isl_point *pnt);
__isl_give isl_val *isl_union_pw_qpolynomial_fold_eval(
__isl_take isl_union_pw_qpolynomial_fold *upwf,
__isl_take isl_point *pnt);
These functions return NaN when evaluated at a void point. Note that isl_pw_aff_eval
returns NaN when the function is evaluated outside its definition domain, while
isl_pw_qpolynomial_eval returns zero when the function is evaluated outside its explicit domain.
• Dimension manipulation

118

It is usually not advisable to directly change the (input or output) space of a set
or a relation as this removes the name and the internal structure of the space.
However, the functions below can be useful to add new parameters, assuming
isl_set_align_params and isl_map_align_params are not sufficient.
#include 
__isl_give isl_space *isl_space_add_dims(
__isl_take isl_space *space,
enum isl_dim_type type, unsigned n);
__isl_give isl_space *isl_space_insert_dims(
__isl_take isl_space *space,
enum isl_dim_type type, unsigned pos, unsigned n);
__isl_give isl_space *isl_space_drop_dims(
__isl_take isl_space *space,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_space *isl_space_move_dims(
__isl_take isl_space *space,
enum isl_dim_type dst_type, unsigned dst_pos,
enum isl_dim_type src_type, unsigned src_pos,
unsigned n);
#include 
__isl_give isl_local_space *isl_local_space_add_dims(
__isl_take isl_local_space *ls,
enum isl_dim_type type, unsigned n);
__isl_give isl_local_space *isl_local_space_insert_dims(
__isl_take isl_local_space *ls,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_local_space *isl_local_space_drop_dims(
__isl_take isl_local_space *ls,
enum isl_dim_type type, unsigned first, unsigned n);
#include 
__isl_give isl_basic_set *isl_basic_set_add_dims(
__isl_take isl_basic_set *bset,
enum isl_dim_type type, unsigned n);
__isl_give isl_set *isl_set_add_dims(
__isl_take isl_set *set,
enum isl_dim_type type, unsigned n);
__isl_give isl_basic_set *isl_basic_set_insert_dims(
__isl_take isl_basic_set *bset,
enum isl_dim_type type, unsigned pos,
unsigned n);
__isl_give isl_set *isl_set_insert_dims(
__isl_take isl_set *set,
enum isl_dim_type type, unsigned pos, unsigned n);
119

__isl_give isl_basic_set *isl_basic_set_move_dims(
__isl_take isl_basic_set *bset,
enum isl_dim_type dst_type, unsigned dst_pos,
enum isl_dim_type src_type, unsigned src_pos,
unsigned n);
__isl_give isl_set *isl_set_move_dims(
__isl_take isl_set *set,
enum isl_dim_type dst_type, unsigned dst_pos,
enum isl_dim_type src_type, unsigned src_pos,
unsigned n);
#include 
__isl_give isl_basic_map *isl_basic_map_add_dims(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type, unsigned n);
__isl_give isl_map *isl_map_add_dims(
__isl_take isl_map *map,
enum isl_dim_type type, unsigned n);
__isl_give isl_basic_map *isl_basic_map_insert_dims(
__isl_take isl_basic_map *bmap,
enum isl_dim_type type, unsigned pos,
unsigned n);
__isl_give isl_map *isl_map_insert_dims(
__isl_take isl_map *map,
enum isl_dim_type type, unsigned pos, unsigned n);
__isl_give isl_basic_map *isl_basic_map_move_dims(
__isl_take isl_basic_map *bmap,
enum isl_dim_type dst_type, unsigned dst_pos,
enum isl_dim_type src_type, unsigned src_pos,
unsigned n);
__isl_give isl_map *isl_map_move_dims(
__isl_take isl_map *map,
enum isl_dim_type dst_type, unsigned dst_pos,
enum isl_dim_type src_type, unsigned src_pos,
unsigned n);
#include 
__isl_give isl_multi_val *isl_multi_val_insert_dims(
__isl_take isl_multi_val *mv,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_multi_val *isl_multi_val_add_dims(
__isl_take isl_multi_val *mv,
enum isl_dim_type type, unsigned n);
__isl_give isl_multi_val *isl_multi_val_drop_dims(
__isl_take isl_multi_val *mv,
enum isl_dim_type type, unsigned first, unsigned n);
120

#include 
__isl_give isl_aff *isl_aff_insert_dims(
__isl_take isl_aff *aff,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_multi_aff *isl_multi_aff_insert_dims(
__isl_take isl_multi_aff *ma,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_pw_aff *isl_pw_aff_insert_dims(
__isl_take isl_pw_aff *pwaff,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_insert_dims(
__isl_take isl_multi_pw_aff *mpa,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_aff *isl_aff_add_dims(
__isl_take isl_aff *aff,
enum isl_dim_type type, unsigned n);
__isl_give isl_multi_aff *isl_multi_aff_add_dims(
__isl_take isl_multi_aff *ma,
enum isl_dim_type type, unsigned n);
__isl_give isl_pw_aff *isl_pw_aff_add_dims(
__isl_take isl_pw_aff *pwaff,
enum isl_dim_type type, unsigned n);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_add_dims(
__isl_take isl_multi_pw_aff *mpa,
enum isl_dim_type type, unsigned n);
__isl_give isl_aff *isl_aff_drop_dims(
__isl_take isl_aff *aff,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_multi_aff *isl_multi_aff_drop_dims(
__isl_take isl_multi_aff *maff,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_pw_aff *isl_pw_aff_drop_dims(
__isl_take isl_pw_aff *pwaff,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_drop_dims(
__isl_take isl_pw_multi_aff *pma,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_union_pw_aff *isl_union_pw_aff_drop_dims(
__isl_take isl_union_pw_aff *upa,
enum isl_dim_type type, unsigned first, unsigned n);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_drop_dims(
__isl_take isl_union_pw_multi_aff *upma,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_multi_union_pw_aff *
121

isl_multi_union_pw_aff_drop_dims(
__isl_take isl_multi_union_pw_aff *mupa,
enum isl_dim_type type, unsigned first,
unsigned n);
__isl_give isl_aff *isl_aff_move_dims(
__isl_take isl_aff *aff,
enum isl_dim_type dst_type, unsigned dst_pos,
enum isl_dim_type src_type, unsigned src_pos,
unsigned n);
__isl_give isl_multi_aff *isl_multi_aff_move_dims(
__isl_take isl_multi_aff *ma,
enum isl_dim_type dst_type, unsigned dst_pos,
enum isl_dim_type src_type, unsigned src_pos,
unsigned n);
__isl_give isl_pw_aff *isl_pw_aff_move_dims(
__isl_take isl_pw_aff *pa,
enum isl_dim_type dst_type, unsigned dst_pos,
enum isl_dim_type src_type, unsigned src_pos,
unsigned n);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_move_dims(
__isl_take isl_multi_pw_aff *pma,
enum isl_dim_type dst_type, unsigned dst_pos,
enum isl_dim_type src_type, unsigned src_pos,
unsigned n);
#include 
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_drop_dims(
__isl_take isl_union_pw_qpolynomial *upwqp,
enum isl_dim_type type,
unsigned first, unsigned n);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_drop_dims(
__isl_take isl_union_pw_qpolynomial_fold *upwf,
enum isl_dim_type type,
unsigned first, unsigned n);
The operations on union expressions can only manipulate parameters.

1.4.17

Binary Operations

The two arguments of a binary operation not only need to live in the same isl_ctx,
they currently also need to have the same (number of) parameters.
Basic Operations
• Intersection
122

#include 
__isl_give isl_local_space *isl_local_space_intersect(
__isl_take isl_local_space *ls1,
__isl_take isl_local_space *ls2);
#include 
__isl_give isl_basic_set *isl_basic_set_intersect_params(
__isl_take isl_basic_set *bset1,
__isl_take isl_basic_set *bset2);
__isl_give isl_basic_set *isl_basic_set_intersect(
__isl_take isl_basic_set *bset1,
__isl_take isl_basic_set *bset2);
__isl_give isl_basic_set *isl_basic_set_list_intersect(
__isl_take struct isl_basic_set_list *list);
__isl_give isl_set *isl_set_intersect_params(
__isl_take isl_set *set,
__isl_take isl_set *params);
__isl_give isl_set *isl_set_intersect(
__isl_take isl_set *set1,
__isl_take isl_set *set2);
#include 
__isl_give isl_basic_map *isl_basic_map_intersect_domain(
__isl_take isl_basic_map *bmap,
__isl_take isl_basic_set *bset);
__isl_give isl_basic_map *isl_basic_map_intersect_range(
__isl_take isl_basic_map *bmap,
__isl_take isl_basic_set *bset);
__isl_give isl_basic_map *isl_basic_map_intersect(
__isl_take isl_basic_map *bmap1,
__isl_take isl_basic_map *bmap2);
__isl_give isl_basic_map *isl_basic_map_list_intersect(
__isl_take isl_basic_map_list *list);
__isl_give isl_map *isl_map_intersect_params(
__isl_take isl_map *map,
__isl_take isl_set *params);
__isl_give isl_map *isl_map_intersect_domain(
__isl_take isl_map *map,
__isl_take isl_set *set);
__isl_give isl_map *isl_map_intersect_range(
__isl_take isl_map *map,
__isl_take isl_set *set);
__isl_give isl_map *isl_map_intersect(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
__isl_give isl_map *
123

isl_map_intersect_domain_factor_range(
__isl_take isl_map *map,
__isl_take isl_map *factor);
__isl_give isl_map *
isl_map_intersect_range_factor_range(
__isl_take isl_map *map,
__isl_take isl_map *factor);
#include 
__isl_give isl_union_set *isl_union_set_intersect_params(
__isl_take isl_union_set *uset,
__isl_take isl_set *set);
__isl_give isl_union_set *isl_union_set_intersect(
__isl_take isl_union_set *uset1,
__isl_take isl_union_set *uset2);
#include 
__isl_give isl_union_map *isl_union_map_intersect_params(
__isl_take isl_union_map *umap,
__isl_take isl_set *set);
__isl_give isl_union_map *isl_union_map_intersect_domain(
__isl_take isl_union_map *umap,
__isl_take isl_union_set *uset);
__isl_give isl_union_map *isl_union_map_intersect_range(
__isl_take isl_union_map *umap,
__isl_take isl_union_set *uset);
__isl_give isl_union_map *isl_union_map_intersect(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
__isl_give isl_union_map *
isl_union_map_intersect_range_factor_range(
__isl_take isl_union_map *umap,
__isl_take isl_union_map *factor);
#include 
__isl_give isl_pw_aff *isl_pw_aff_intersect_domain(
__isl_take isl_pw_aff *pa,
__isl_take isl_set *set);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_intersect_domain(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_set *domain);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_intersect_domain(
__isl_take isl_pw_multi_aff *pma,
__isl_take isl_set *set);
__isl_give isl_union_pw_aff *isl_union_pw_aff_intersect_domain(
124

__isl_take isl_union_pw_aff *upa,
__isl_take isl_union_set *uset);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_intersect_domain(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_union_set *uset);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_intersect_domain(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_union_set *uset);
__isl_give isl_pw_aff *isl_pw_aff_intersect_params(
__isl_take isl_pw_aff *pa,
__isl_take isl_set *set);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_intersect_params(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_set *set);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_intersect_params(
__isl_take isl_pw_multi_aff *pma,
__isl_take isl_set *set);
__isl_give isl_union_pw_aff *
isl_union_pw_aff_intersect_params(
__isl_take isl_union_pw_aff *upa,
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_intersect_params(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_set *set);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_intersect_params(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_set *params);
isl_multi_union_pw_aff_intersect_range(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_set *set);
#include 
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_intersect_domain(
__isl_take isl_pw_qpolynomial *pwpq,
__isl_take isl_set *set);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_intersect_domain(
__isl_take isl_union_pw_qpolynomial *upwpq,
__isl_take isl_union_set *uset);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_intersect_domain(
125

__isl_take isl_union_pw_qpolynomial_fold *upwf,
__isl_take isl_union_set *uset);
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_intersect_params(
__isl_take isl_pw_qpolynomial *pwpq,
__isl_take isl_set *set);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_intersect_params(
__isl_take isl_pw_qpolynomial_fold *pwf,
__isl_take isl_set *set);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_intersect_params(
__isl_take isl_union_pw_qpolynomial *upwpq,
__isl_take isl_set *set);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_intersect_params(
__isl_take isl_union_pw_qpolynomial_fold *upwf,
__isl_take isl_set *set);
The second argument to the _params functions needs to be a parametric (basic) set. For the other functions, a parametric set for either argument is only
allowed if the other argument is a parametric set as well. The list passed to
isl_basic_set_list_intersect needs to have at least one element and all
elements need to live in the same space. The function isl_multi_union_pw_aff_intersect_range
restricts the input function to those shared domain elements that map to the specified range.
• Union
#include 
__isl_give isl_set *isl_basic_set_union(
__isl_take isl_basic_set *bset1,
__isl_take isl_basic_set *bset2);
__isl_give isl_set *isl_set_union(
__isl_take isl_set *set1,
__isl_take isl_set *set2);
__isl_give isl_set *isl_set_list_union(
__isl_take isl_set_list *list);
#include 
__isl_give isl_map *isl_basic_map_union(
__isl_take isl_basic_map *bmap1,
__isl_take isl_basic_map *bmap2);
__isl_give isl_map *isl_map_union(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
126

#include 
__isl_give isl_union_set *isl_union_set_union(
__isl_take isl_union_set *uset1,
__isl_take isl_union_set *uset2);
__isl_give isl_union_set *isl_union_set_list_union(
__isl_take isl_union_set_list *list);
#include 
__isl_give isl_union_map *isl_union_map_union(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
The list passed to isl_set_list_union needs to have at least one element and
all elements need to live in the same space.
• Set difference
#include 
__isl_give isl_set *isl_set_subtract(
__isl_take isl_set *set1,
__isl_take isl_set *set2);
#include 
__isl_give isl_map *isl_map_subtract(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
__isl_give isl_map *isl_map_subtract_domain(
__isl_take isl_map *map,
__isl_take isl_set *dom);
__isl_give isl_map *isl_map_subtract_range(
__isl_take isl_map *map,
__isl_take isl_set *dom);
#include 
__isl_give isl_union_set *isl_union_set_subtract(
__isl_take isl_union_set *uset1,
__isl_take isl_union_set *uset2);
#include 
__isl_give isl_union_map *isl_union_map_subtract(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
__isl_give isl_union_map *isl_union_map_subtract_domain(
__isl_take isl_union_map *umap,
__isl_take isl_union_set *dom);
__isl_give isl_union_map *isl_union_map_subtract_range(
127

__isl_take isl_union_map *umap,
__isl_take isl_union_set *dom);
#include 
__isl_give isl_pw_aff *isl_pw_aff_subtract_domain(
__isl_take isl_pw_aff *pa,
__isl_take isl_set *set);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_subtract_domain(
__isl_take isl_pw_multi_aff *pma,
__isl_take isl_set *set);
__isl_give isl_union_pw_aff *
isl_union_pw_aff_subtract_domain(
__isl_take isl_union_pw_aff *upa,
__isl_take isl_union_set *uset);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_subtract_domain(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_set *set);
#include 
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_subtract_domain(
__isl_take isl_pw_qpolynomial *pwpq,
__isl_take isl_set *set);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_subtract_domain(
__isl_take isl_pw_qpolynomial_fold *pwf,
__isl_take isl_set *set);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_subtract_domain(
__isl_take isl_union_pw_qpolynomial *upwpq,
__isl_take isl_union_set *uset);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_subtract_domain(
__isl_take isl_union_pw_qpolynomial_fold *upwf,
__isl_take isl_union_set *uset);
• Application
#include 
__isl_give isl_space *isl_space_join(
__isl_take isl_space *left,
__isl_take isl_space *right);
#include 
128

__isl_give isl_basic_set *isl_basic_set_apply(
__isl_take isl_basic_set *bset,
__isl_take isl_basic_map *bmap);
__isl_give isl_set *isl_set_apply(
__isl_take isl_set *set,
__isl_take isl_map *map);
__isl_give isl_union_set *isl_union_set_apply(
__isl_take isl_union_set *uset,
__isl_take isl_union_map *umap);
__isl_give isl_basic_map *isl_basic_map_apply_domain(
__isl_take isl_basic_map *bmap1,
__isl_take isl_basic_map *bmap2);
__isl_give isl_basic_map *isl_basic_map_apply_range(
__isl_take isl_basic_map *bmap1,
__isl_take isl_basic_map *bmap2);
__isl_give isl_map *isl_map_apply_domain(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
__isl_give isl_map *isl_map_apply_range(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
#include 
__isl_give isl_union_map *isl_union_map_apply_domain(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
__isl_give isl_union_map *isl_union_map_apply_range(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
#include 
__isl_give isl_union_pw_aff *
isl_multi_union_pw_aff_apply_aff(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_aff *aff);
__isl_give isl_union_pw_aff *
isl_multi_union_pw_aff_apply_pw_aff(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_pw_aff *pa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_apply_multi_aff(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_multi_aff *ma);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_apply_pw_multi_aff(
__isl_take isl_multi_union_pw_aff *mupa,
129

__isl_take isl_pw_multi_aff *pma);
The result of isl_multi_union_pw_aff_apply_aff is defined over the shared
domain of the elements of the input. The dimension is required to be greater than
zero. The isl_multi_union_pw_aff argument of isl_multi_union_pw_aff_apply_multi_aff
is allowed to be zero-dimensional, but only if the range of the isl_multi_aff
argument is also zero-dimensional. Similarly for isl_multi_union_pw_aff_apply_pw_multi_aff.
#include 
__isl_give isl_pw_qpolynomial_fold *
isl_set_apply_pw_qpolynomial_fold(
__isl_take isl_set *set,
__isl_take isl_pw_qpolynomial_fold *pwf,
int *tight);
__isl_give isl_pw_qpolynomial_fold *
isl_map_apply_pw_qpolynomial_fold(
__isl_take isl_map *map,
__isl_take isl_pw_qpolynomial_fold *pwf,
int *tight);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_set_apply_union_pw_qpolynomial_fold(
__isl_take isl_union_set *uset,
__isl_take isl_union_pw_qpolynomial_fold *upwf,
int *tight);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_map_apply_union_pw_qpolynomial_fold(
__isl_take isl_union_map *umap,
__isl_take isl_union_pw_qpolynomial_fold *upwf,
int *tight);
The functions taking a map compose the given map with the given piecewise
quasipolynomial reduction. That is, compute a bound (of the same type as pwf
or upwf itself) over all elements in the intersection of the range of the map and
the domain of the piecewise quasipolynomial reduction as a function of an element in the domain of the map. The functions taking a set compute a bound
over all elements in the intersection of the set and the domain of the piecewise
quasipolynomial reduction.
• Preimage
#include 
__isl_give isl_basic_set *
isl_basic_set_preimage_multi_aff(
__isl_take isl_basic_set *bset,
__isl_take isl_multi_aff *ma);
__isl_give isl_set *isl_set_preimage_multi_aff(
130

__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take

isl_set *set,
isl_multi_aff *ma);
*isl_set_preimage_pw_multi_aff(
isl_set *set,
isl_pw_multi_aff *pma);
*isl_set_preimage_multi_pw_aff(
isl_set *set,
isl_multi_pw_aff *mpa);

#include 
__isl_give isl_union_set *
isl_union_set_preimage_multi_aff(
__isl_take isl_union_set *uset,
__isl_take isl_multi_aff *ma);
__isl_give isl_union_set *
isl_union_set_preimage_pw_multi_aff(
__isl_take isl_union_set *uset,
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_union_set *
isl_union_set_preimage_union_pw_multi_aff(
__isl_take isl_union_set *uset,
__isl_take isl_union_pw_multi_aff *upma);
#include 
__isl_give isl_basic_map *
isl_basic_map_preimage_domain_multi_aff(
__isl_take isl_basic_map *bmap,
__isl_take isl_multi_aff *ma);
__isl_give isl_map *isl_map_preimage_domain_multi_aff(
__isl_take isl_map *map,
__isl_take isl_multi_aff *ma);
__isl_give isl_map *isl_map_preimage_range_multi_aff(
__isl_take isl_map *map,
__isl_take isl_multi_aff *ma);
__isl_give isl_map *
isl_map_preimage_domain_pw_multi_aff(
__isl_take isl_map *map,
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_map *
isl_map_preimage_range_pw_multi_aff(
__isl_take isl_map *map,
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_map *
isl_map_preimage_domain_multi_pw_aff(
__isl_take isl_map *map,
__isl_take isl_multi_pw_aff *mpa);
131

__isl_give isl_basic_map *
isl_basic_map_preimage_range_multi_aff(
__isl_take isl_basic_map *bmap,
__isl_take isl_multi_aff *ma);
#include 
__isl_give isl_union_map *
isl_union_map_preimage_domain_multi_aff(
__isl_take isl_union_map *umap,
__isl_take isl_multi_aff *ma);
__isl_give isl_union_map *
isl_union_map_preimage_range_multi_aff(
__isl_take isl_union_map *umap,
__isl_take isl_multi_aff *ma);
__isl_give isl_union_map *
isl_union_map_preimage_domain_pw_multi_aff(
__isl_take isl_union_map *umap,
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_union_map *
isl_union_map_preimage_range_pw_multi_aff(
__isl_take isl_union_map *umap,
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_union_map *
isl_union_map_preimage_domain_union_pw_multi_aff(
__isl_take isl_union_map *umap,
__isl_take isl_union_pw_multi_aff *upma);
__isl_give isl_union_map *
isl_union_map_preimage_range_union_pw_multi_aff(
__isl_take isl_union_map *umap,
__isl_take isl_union_pw_multi_aff *upma);
These functions compute the preimage of the given set or map domain/range
under the given function. In other words, the expression is plugged into the set
description or into the domain/range of the map.
• Pullback
#include 
__isl_give isl_aff *isl_aff_pullback_aff(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_aff *isl_aff_pullback_multi_aff(
__isl_take isl_aff *aff,
__isl_take isl_multi_aff *ma);
__isl_give isl_pw_aff *isl_pw_aff_pullback_multi_aff(
__isl_take isl_pw_aff *pa,
132

__isl_take isl_multi_aff *ma);
__isl_give isl_pw_aff *isl_pw_aff_pullback_pw_multi_aff(
__isl_take isl_pw_aff *pa,
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_pw_aff *isl_pw_aff_pullback_multi_pw_aff(
__isl_take isl_pw_aff *pa,
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_multi_aff *isl_multi_aff_pullback_multi_aff(
__isl_take isl_multi_aff *ma1,
__isl_take isl_multi_aff *ma2);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_pullback_multi_aff(
__isl_take isl_pw_multi_aff *pma,
__isl_take isl_multi_aff *ma);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_pullback_multi_aff(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_multi_aff *ma);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_pullback_pw_multi_aff(
__isl_take isl_pw_multi_aff *pma1,
__isl_take isl_pw_multi_aff *pma2);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_pullback_pw_multi_aff(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_pullback_multi_pw_aff(
__isl_take isl_multi_pw_aff *mpa1,
__isl_take isl_multi_pw_aff *mpa2);
__isl_give isl_union_pw_aff *
isl_union_pw_aff_pullback_union_pw_multi_aff(
__isl_take isl_union_pw_aff *upa,
__isl_take isl_union_pw_multi_aff *upma);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_pullback_union_pw_multi_aff(
__isl_take isl_union_pw_multi_aff *upma1,
__isl_take isl_union_pw_multi_aff *upma2);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_pullback_union_pw_multi_aff(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_union_pw_multi_aff *upma);
These functions precompose the first expression by the second function. In other
words, the second function is plugged into the first expression.

133

• Locus
#include 
__isl_give isl_basic_set *isl_aff_eq_basic_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_set *isl_aff_eq_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_set *isl_aff_ne_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_basic_set *isl_aff_le_basic_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_set *isl_aff_le_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_basic_set *isl_aff_lt_basic_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_set *isl_aff_lt_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_basic_set *isl_aff_ge_basic_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_set *isl_aff_ge_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_basic_set *isl_aff_gt_basic_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_set *isl_aff_gt_set(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_set *isl_pw_aff_eq_set(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_set *isl_pw_aff_ne_set(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_set *isl_pw_aff_le_set(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_set *isl_pw_aff_lt_set(
134

__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take

isl_pw_aff *pwaff1,
isl_pw_aff *pwaff2);
*isl_pw_aff_ge_set(
isl_pw_aff *pwaff1,
isl_pw_aff *pwaff2);
*isl_pw_aff_gt_set(
isl_pw_aff *pwaff1,
isl_pw_aff *pwaff2);

__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take

*isl_multi_aff_lex_le_set(
isl_multi_aff *ma1,
isl_multi_aff *ma2);
*isl_multi_aff_lex_lt_set(
isl_multi_aff *ma1,
isl_multi_aff *ma2);
*isl_multi_aff_lex_ge_set(
isl_multi_aff *ma1,
isl_multi_aff *ma2);
*isl_multi_aff_lex_gt_set(
isl_multi_aff *ma1,
isl_multi_aff *ma2);

__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take
__isl_give isl_set
__isl_take
__isl_take

*isl_pw_aff_list_eq_set(
isl_pw_aff_list *list1,
isl_pw_aff_list *list2);
*isl_pw_aff_list_ne_set(
isl_pw_aff_list *list1,
isl_pw_aff_list *list2);
*isl_pw_aff_list_le_set(
isl_pw_aff_list *list1,
isl_pw_aff_list *list2);
*isl_pw_aff_list_lt_set(
isl_pw_aff_list *list1,
isl_pw_aff_list *list2);
*isl_pw_aff_list_ge_set(
isl_pw_aff_list *list1,
isl_pw_aff_list *list2);
*isl_pw_aff_list_gt_set(
isl_pw_aff_list *list1,
isl_pw_aff_list *list2);

The function isl_aff_ge_basic_set returns a basic set containing those elements in the shared space of aff1 and aff2 where aff1 is greater than or
equal to aff2. The function isl_pw_aff_ge_set returns a set containing
those elements in the shared domain of pwaff1 and pwaff2 where pwaff1 is
greater than or equal to pwaff2. The function isl_multi_aff_lex_le_set
135

returns a set containing those elements in the shared domain space where ma1
is lexicographically smaller than or equal to ma2. The functions operating on
isl_pw_aff_list apply the corresponding isl_pw_aff function to each pair
of elements in the two lists.
#include 
__isl_give isl_map *isl_pw_aff_eq_map(
__isl_take isl_pw_aff *pa1,
__isl_take isl_pw_aff *pa2);
__isl_give isl_map *isl_pw_aff_lt_map(
__isl_take isl_pw_aff *pa1,
__isl_take isl_pw_aff *pa2);
__isl_give isl_map *isl_pw_aff_gt_map(
__isl_take isl_pw_aff *pa1,
__isl_take isl_pw_aff *pa2);
__isl_give isl_map
__isl_take
__isl_take
__isl_give isl_map
__isl_take
__isl_take
__isl_give isl_map
__isl_take
__isl_take

*isl_multi_pw_aff_eq_map(
isl_multi_pw_aff *mpa1,
isl_multi_pw_aff *mpa2);
*isl_multi_pw_aff_lex_lt_map(
isl_multi_pw_aff *mpa1,
isl_multi_pw_aff *mpa2);
*isl_multi_pw_aff_lex_gt_map(
isl_multi_pw_aff *mpa1,
isl_multi_pw_aff *mpa2);

These functions return a map between domain elements of the arguments where
the function values satisfy the given relation.
#include 
__isl_give isl_union_map *
isl_union_map_eq_at_multi_union_pw_aff(
__isl_take isl_union_map *umap,
__isl_take isl_multi_union_pw_aff *mupa);
__isl_give isl_union_map *
isl_union_map_lex_lt_at_multi_union_pw_aff(
__isl_take isl_union_map *umap,
__isl_take isl_multi_union_pw_aff *mupa);
__isl_give isl_union_map *
isl_union_map_lex_gt_at_multi_union_pw_aff(
__isl_take isl_union_map *umap,
__isl_take isl_multi_union_pw_aff *mupa);
These functions select the subset of elements in the union map that have an equal
or lexicographically smaller function value.
136

• Cartesian Product
#include 
__isl_give isl_space *isl_space_product(
__isl_take isl_space *space1,
__isl_take isl_space *space2);
__isl_give isl_space *isl_space_domain_product(
__isl_take isl_space *space1,
__isl_take isl_space *space2);
__isl_give isl_space *isl_space_range_product(
__isl_take isl_space *space1,
__isl_take isl_space *space2);
The functions isl_space_product, isl_space_domain_product and isl_space_range_product
take pairs or relation spaces and produce a single relations space, where either
the domain, the range or both domain and range are wrapped spaces of relations
between the domains and/or ranges of the input spaces. If the product is only
constructed over the domain or the range then the ranges or the domains of the
inputs should be the same. The function isl_space_product also accepts a
pair of set spaces, in which case it returns a wrapped space of a relation between
the two input spaces.
#include 
__isl_give isl_set *isl_set_product(
__isl_take isl_set *set1,
__isl_take isl_set *set2);
#include 
__isl_give isl_basic_map *isl_basic_map_domain_product(
__isl_take isl_basic_map *bmap1,
__isl_take isl_basic_map *bmap2);
__isl_give isl_basic_map *isl_basic_map_range_product(
__isl_take isl_basic_map *bmap1,
__isl_take isl_basic_map *bmap2);
__isl_give isl_basic_map *isl_basic_map_product(
__isl_take isl_basic_map *bmap1,
__isl_take isl_basic_map *bmap2);
__isl_give isl_map *isl_map_domain_product(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
__isl_give isl_map *isl_map_range_product(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
__isl_give isl_map *isl_map_product(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
137

#include 
__isl_give isl_union_set *isl_union_set_product(
__isl_take isl_union_set *uset1,
__isl_take isl_union_set *uset2);
#include 
__isl_give isl_union_map *isl_union_map_domain_product(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
__isl_give isl_union_map *isl_union_map_range_product(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
__isl_give isl_union_map *isl_union_map_product(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
#include 
__isl_give isl_multi_val *isl_multi_val_range_product(
__isl_take isl_multi_val *mv1,
__isl_take isl_multi_val *mv2);
__isl_give isl_multi_val *isl_multi_val_product(
__isl_take isl_multi_val *mv1,
__isl_take isl_multi_val *mv2);
#include 
__isl_give isl_multi_aff *isl_multi_aff_range_product(
__isl_take isl_multi_aff *ma1,
__isl_take isl_multi_aff *ma2);
__isl_give isl_multi_aff *isl_multi_aff_product(
__isl_take isl_multi_aff *ma1,
__isl_take isl_multi_aff *ma2);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_range_product(
__isl_take isl_multi_pw_aff *mpa1,
__isl_take isl_multi_pw_aff *mpa2);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_product(
__isl_take isl_multi_pw_aff *mpa1,
__isl_take isl_multi_pw_aff *mpa2);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_range_product(
__isl_take isl_pw_multi_aff *pma1,
__isl_take isl_pw_multi_aff *pma2);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_product(
__isl_take isl_pw_multi_aff *pma1,
__isl_take isl_pw_multi_aff *pma2);
__isl_give isl_multi_union_pw_aff *
138

isl_multi_union_pw_aff_range_product(
__isl_take isl_multi_union_pw_aff *mupa1,
__isl_take isl_multi_union_pw_aff *mupa2);
The above functions compute the cross product of the given sets, relations or
functions. The domains and ranges of the results are wrapped maps between
domains and ranges of the inputs. To obtain a “flat” product, use the following
functions instead.
#include 
__isl_give isl_basic_set *isl_basic_set_flat_product(
__isl_take isl_basic_set *bset1,
__isl_take isl_basic_set *bset2);
__isl_give isl_set *isl_set_flat_product(
__isl_take isl_set *set1,
__isl_take isl_set *set2);
#include 
__isl_give isl_basic_map *isl_basic_map_flat_range_product(
__isl_take isl_basic_map *bmap1,
__isl_take isl_basic_map *bmap2);
__isl_give isl_map *isl_map_flat_domain_product(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
__isl_give isl_map *isl_map_flat_range_product(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
__isl_give isl_basic_map *isl_basic_map_flat_product(
__isl_take isl_basic_map *bmap1,
__isl_take isl_basic_map *bmap2);
__isl_give isl_map *isl_map_flat_product(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
#include 
__isl_give isl_union_map *
isl_union_map_flat_domain_product(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
__isl_give isl_union_map *
isl_union_map_flat_range_product(
__isl_take isl_union_map *umap1,
__isl_take isl_union_map *umap2);
#include 
139

__isl_give isl_multi_val *isl_multi_val_flat_range_product(
__isl_take isl_multi_val *mv1,
__isl_take isl_multi_val *mv2);
#include 
__isl_give isl_multi_aff *isl_multi_aff_flat_range_product(
__isl_take isl_multi_aff *ma1,
__isl_take isl_multi_aff *ma2);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_flat_range_product(
__isl_take isl_pw_multi_aff *pma1,
__isl_take isl_pw_multi_aff *pma2);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_flat_range_product(
__isl_take isl_multi_pw_aff *mpa1,
__isl_take isl_multi_pw_aff *mpa2);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_flat_range_product(
__isl_take isl_union_pw_multi_aff *upma1,
__isl_take isl_union_pw_multi_aff *upma2);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_flat_range_product(
__isl_take isl_multi_union_pw_aff *mupa1,
__isl_take isl_multi_union_pw_aff *mupa2);
#include 
__isl_give isl_space *isl_space_factor_domain(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_factor_range(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_domain_factor_domain(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_domain_factor_range(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_range_factor_domain(
__isl_take isl_space *space);
__isl_give isl_space *isl_space_range_factor_range(
__isl_take isl_space *space);
The functions isl_space_range_factor_domain and isl_space_range_factor_range
extract the two arguments from the result of a call to isl_space_range_product.
The arguments of a call to a product can be extracted from the result using the
following functions.
#include 
140

__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take
__isl_give isl_map
__isl_take

*isl_map_factor_domain(
isl_map *map);
*isl_map_factor_range(
isl_map *map);
*isl_map_domain_factor_domain(
isl_map *map);
*isl_map_domain_factor_range(
isl_map *map);
*isl_map_range_factor_domain(
isl_map *map);
*isl_map_range_factor_range(
isl_map *map);

#include 
__isl_give isl_union_map *isl_union_map_factor_domain(
__isl_take isl_union_map *umap);
__isl_give isl_union_map *isl_union_map_factor_range(
__isl_take isl_union_map *umap);
__isl_give isl_union_map *
isl_union_map_domain_factor_domain(
__isl_take isl_union_map *umap);
__isl_give isl_union_map *
isl_union_map_domain_factor_range(
__isl_take isl_union_map *umap);
__isl_give isl_union_map *
isl_union_map_range_factor_domain(
__isl_take isl_union_map *umap);
__isl_give isl_union_map *
isl_union_map_range_factor_range(
__isl_take isl_union_map *umap);
#include 
__isl_give isl_multi_val *isl_multi_val_factor_range(
__isl_take isl_multi_val *mv);
__isl_give isl_multi_val *
isl_multi_val_range_factor_domain(
__isl_take isl_multi_val *mv);
__isl_give isl_multi_val *
isl_multi_val_range_factor_range(
__isl_take isl_multi_val *mv);
#include 
__isl_give isl_multi_aff *isl_multi_aff_factor_range(
__isl_take isl_multi_aff *ma);
__isl_give isl_multi_aff *
isl_multi_aff_range_factor_domain(
141

__isl_take isl_multi_aff *ma);
__isl_give isl_multi_aff *
isl_multi_aff_range_factor_range(
__isl_take isl_multi_aff *ma);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_factor_range(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_range_factor_domain(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_range_factor_range(
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_factor_range(
__isl_take isl_multi_union_pw_aff *mupa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_range_factor_domain(
__isl_take isl_multi_union_pw_aff *mupa);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_range_factor_range(
__isl_take isl_multi_union_pw_aff *mupa);

The splice functions are a generalization of the flat product functions, where the
second argument may be inserted at any position inside the first argument rather
than being placed at the end. The functions isl_multi_val_factor_range,
isl_multi_aff_factor_range, isl_multi_pw_aff_factor_range and isl_multi_union_pw_aff_facto
take functions that live in a set space.
#include 
__isl_give isl_multi_val *isl_multi_val_range_splice(
__isl_take isl_multi_val *mv1, unsigned pos,
__isl_take isl_multi_val *mv2);
#include 
__isl_give isl_multi_aff *isl_multi_aff_range_splice(
__isl_take isl_multi_aff *ma1, unsigned pos,
__isl_take isl_multi_aff *ma2);
__isl_give isl_multi_aff *isl_multi_aff_splice(
__isl_take isl_multi_aff *ma1,
unsigned in_pos, unsigned out_pos,
__isl_take isl_multi_aff *ma2);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_range_splice(
__isl_take isl_multi_pw_aff *mpa1, unsigned pos,
__isl_take isl_multi_pw_aff *mpa2);
142

__isl_give isl_multi_pw_aff *isl_multi_pw_aff_splice(
__isl_take isl_multi_pw_aff *mpa1,
unsigned in_pos, unsigned out_pos,
__isl_take isl_multi_pw_aff *mpa2);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_range_splice(
__isl_take isl_multi_union_pw_aff *mupa1,
unsigned pos,
__isl_take isl_multi_union_pw_aff *mupa2);
• Simplification
When applied to a set or relation, the gist operation returns a set or relation that
has the same intersection with the context as the input set or relation. Any implicit equality in the intersection is made explicit in the result, while all inequalities that are redundant with respect to the intersection are removed. In case of
union sets and relations, the gist operation is performed per space.
When applied to a function, the gist operation applies the set gist operation to
each of the cells in the domain of the input piecewise expression. The context is
also exploited to simplify the expression associated to each cell.
#include 
__isl_give isl_basic_set *isl_basic_set_gist(
__isl_take isl_basic_set *bset,
__isl_take isl_basic_set *context);
__isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
__isl_take isl_set *context);
__isl_give isl_set *isl_set_gist_params(
__isl_take isl_set *set,
__isl_take isl_set *context);
#include 
__isl_give isl_basic_map *isl_basic_map_gist(
__isl_take isl_basic_map *bmap,
__isl_take isl_basic_map *context);
__isl_give isl_basic_map *isl_basic_map_gist_domain(
__isl_take isl_basic_map *bmap,
__isl_take isl_basic_set *context);
__isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
__isl_take isl_map *context);
__isl_give isl_map *isl_map_gist_params(
__isl_take isl_map *map,
__isl_take isl_set *context);
__isl_give isl_map *isl_map_gist_domain(
__isl_take isl_map *map,
__isl_take isl_set *context);
143

__isl_give isl_map *isl_map_gist_range(
__isl_take isl_map *map,
__isl_take isl_set *context);
#include 
__isl_give isl_union_set *isl_union_set_gist(
__isl_take isl_union_set *uset,
__isl_take isl_union_set *context);
__isl_give isl_union_set *isl_union_set_gist_params(
__isl_take isl_union_set *uset,
__isl_take isl_set *set);
#include 
__isl_give isl_union_map *isl_union_map_gist(
__isl_take isl_union_map *umap,
__isl_take isl_union_map *context);
__isl_give isl_union_map *isl_union_map_gist_params(
__isl_take isl_union_map *umap,
__isl_take isl_set *set);
__isl_give isl_union_map *isl_union_map_gist_domain(
__isl_take isl_union_map *umap,
__isl_take isl_union_set *uset);
__isl_give isl_union_map *isl_union_map_gist_range(
__isl_take isl_union_map *umap,
__isl_take isl_union_set *uset);
#include 
__isl_give isl_aff *isl_aff_gist_params(
__isl_take isl_aff *aff,
__isl_take isl_set *context);
__isl_give isl_aff *isl_aff_gist(__isl_take isl_aff *aff,
__isl_take isl_set *context);
__isl_give isl_multi_aff *isl_multi_aff_gist_params(
__isl_take isl_multi_aff *maff,
__isl_take isl_set *context);
__isl_give isl_multi_aff *isl_multi_aff_gist(
__isl_take isl_multi_aff *maff,
__isl_take isl_set *context);
__isl_give isl_pw_aff *isl_pw_aff_gist_params(
__isl_take isl_pw_aff *pwaff,
__isl_take isl_set *context);
__isl_give isl_pw_aff *isl_pw_aff_gist(
__isl_take isl_pw_aff *pwaff,
__isl_take isl_set *context);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_gist_params(
__isl_take isl_pw_multi_aff *pma,
144

__isl_take isl_set *set);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_gist(
__isl_take isl_pw_multi_aff *pma,
__isl_take isl_set *set);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_gist_params(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_set *set);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_gist(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_set *set);
__isl_give isl_union_pw_aff *isl_union_pw_aff_gist(
__isl_take isl_union_pw_aff *upa,
__isl_take isl_union_set *context);
__isl_give isl_union_pw_aff *isl_union_pw_aff_gist_params(
__isl_take isl_union_pw_aff *upa,
__isl_take isl_set *context);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_gist_params(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_set *context);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_gist(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_union_set *context);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_gist_params(
__isl_take isl_multi_union_pw_aff *aff,
__isl_take isl_set *context);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_gist(
__isl_take isl_multi_union_pw_aff *aff,
__isl_take isl_union_set *context);
#include 
__isl_give isl_qpolynomial *isl_qpolynomial_gist_params(
__isl_take isl_qpolynomial *qp,
__isl_take isl_set *context);
__isl_give isl_qpolynomial *isl_qpolynomial_gist(
__isl_take isl_qpolynomial *qp,
__isl_take isl_set *context);
__isl_give isl_qpolynomial_fold *
isl_qpolynomial_fold_gist_params(
__isl_take isl_qpolynomial_fold *fold,
__isl_take isl_set *context);
__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_gist(
__isl_take isl_qpolynomial_fold *fold,
145

__isl_take isl_set *context);
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_gist_params(
__isl_take isl_pw_qpolynomial *pwqp,
__isl_take isl_set *context);
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_gist(
__isl_take isl_pw_qpolynomial *pwqp,
__isl_take isl_set *context);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_gist(
__isl_take isl_pw_qpolynomial_fold *pwf,
__isl_take isl_set *context);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_gist_params(
__isl_take isl_pw_qpolynomial_fold *pwf,
__isl_take isl_set *context);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_gist_params(
__isl_take isl_union_pw_qpolynomial *upwqp,
__isl_take isl_set *context);
__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_gist(
__isl_take isl_union_pw_qpolynomial *upwqp,
__isl_take isl_union_set *context);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_gist(
__isl_take isl_union_pw_qpolynomial_fold *upwf,
__isl_take isl_union_set *context);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_gist_params(
__isl_take isl_union_pw_qpolynomial_fold *upwf,
__isl_take isl_set *context);
• Binary Arithmetic Operations
#include 
__isl_give isl_set *isl_set_sum(
__isl_take isl_set *set1,
__isl_take isl_set *set2);
#include 
__isl_give isl_map *isl_map_sum(
__isl_take isl_map *map1,
__isl_take isl_map *map2);
isl_set_sum computes the Minkowski sum of its two arguments, i.e., the set
containing the sums of pairs of elements from set1 and set2. The domain of
the result of isl_map_sum is the intersection of the domains of its two arguments. The corresponding range elements are the sums of the corresponding
range elements in the two arguments.
146

#include 
__isl_give isl_multi_val *isl_multi_val_add(
__isl_take isl_multi_val *mv1,
__isl_take isl_multi_val *mv2);
__isl_give isl_multi_val *isl_multi_val_sub(
__isl_take isl_multi_val *mv1,
__isl_take isl_multi_val *mv2);
#include 
__isl_give isl_aff *isl_aff_add(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_multi_aff *isl_multi_aff_add(
__isl_take isl_multi_aff *maff1,
__isl_take isl_multi_aff *maff2);
__isl_give isl_pw_aff *isl_pw_aff_add(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_add(
__isl_take isl_multi_pw_aff *mpa1,
__isl_take isl_multi_pw_aff *mpa2);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_add(
__isl_take isl_pw_multi_aff *pma1,
__isl_take isl_pw_multi_aff *pma2);
__isl_give isl_union_pw_aff *isl_union_pw_aff_add(
__isl_take isl_union_pw_aff *upa1,
__isl_take isl_union_pw_aff *upa2);
__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_add(
__isl_take isl_union_pw_multi_aff *upma1,
__isl_take isl_union_pw_multi_aff *upma2);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_add(
__isl_take isl_multi_union_pw_aff *mupa1,
__isl_take isl_multi_union_pw_aff *mupa2);
__isl_give isl_pw_aff *isl_pw_aff_min(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_pw_aff *isl_pw_aff_max(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_aff *isl_aff_sub(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_multi_aff *isl_multi_aff_sub(
__isl_take isl_multi_aff *ma1,
__isl_take isl_multi_aff *ma2);
147

__isl_give isl_pw_aff *isl_pw_aff_sub(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_sub(
__isl_take isl_multi_pw_aff *mpa1,
__isl_take isl_multi_pw_aff *mpa2);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_sub(
__isl_take isl_pw_multi_aff *pma1,
__isl_take isl_pw_multi_aff *pma2);
__isl_give isl_union_pw_aff *isl_union_pw_aff_sub(
__isl_take isl_union_pw_aff *upa1,
__isl_take isl_union_pw_aff *upa2);
__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_sub(
__isl_take isl_union_pw_multi_aff *upma1,
__isl_take isl_union_pw_multi_aff *upma2);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_sub(
__isl_take isl_multi_union_pw_aff *mupa1,
__isl_take isl_multi_union_pw_aff *mupa2);
isl_aff_sub subtracts the second argument from the first.
#include 
__isl_give isl_qpolynomial *isl_qpolynomial_add(
__isl_take isl_qpolynomial *qp1,
__isl_take isl_qpolynomial *qp2);
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add(
__isl_take isl_pw_qpolynomial *pwqp1,
__isl_take isl_pw_qpolynomial *pwqp2);
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add_disjoint(
__isl_take isl_pw_qpolynomial *pwqp1,
__isl_take isl_pw_qpolynomial *pwqp2);
__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_add(
__isl_take isl_pw_qpolynomial_fold *pwf1,
__isl_take isl_pw_qpolynomial_fold *pwf2);
__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_add(
__isl_take isl_union_pw_qpolynomial *upwqp1,
__isl_take isl_union_pw_qpolynomial *upwqp2);
__isl_give isl_qpolynomial *isl_qpolynomial_sub(
__isl_take isl_qpolynomial *qp1,
__isl_take isl_qpolynomial *qp2);
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_sub(
__isl_take isl_pw_qpolynomial *pwqp1,
__isl_take isl_pw_qpolynomial *pwqp2);
__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_sub(
__isl_take isl_union_pw_qpolynomial *upwqp1,
148

__isl_take isl_union_pw_qpolynomial *upwqp2);
__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_fold(
__isl_take isl_pw_qpolynomial_fold *pwf1,
__isl_take isl_pw_qpolynomial_fold *pwf2);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_fold(
__isl_take isl_union_pw_qpolynomial_fold *upwf1,
__isl_take isl_union_pw_qpolynomial_fold *upwf2);
#include 
__isl_give isl_pw_aff *isl_pw_aff_union_add(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_add(
__isl_take isl_pw_multi_aff *pma1,
__isl_take isl_pw_multi_aff *pma2);
__isl_give isl_union_pw_aff *isl_union_pw_aff_union_add(
__isl_take isl_union_pw_aff *upa1,
__isl_take isl_union_pw_aff *upa2);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_union_add(
__isl_take isl_union_pw_multi_aff *upma1,
__isl_take isl_union_pw_multi_aff *upma2);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_union_add(
__isl_take isl_multi_union_pw_aff *mupa1,
__isl_take isl_multi_union_pw_aff *mupa2);
__isl_give isl_pw_aff *isl_pw_aff_union_min(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_pw_aff *isl_pw_aff_union_max(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
The function isl_pw_aff_union_max computes a piecewise quasi-affine expression with a domain that is the union of those of pwaff1 and pwaff2 and
such that on each cell, the quasi-affine expression is the maximum of those of
pwaff1 and pwaff2. If only one of pwaff1 or pwaff2 is defined on a given
cell, then the associated expression is the defined one. This in contrast to the
isl_pw_aff_max function, which is only defined on the shared definition domain of the arguments.
#include 
__isl_give isl_multi_val *isl_multi_val_add_val(
__isl_take isl_multi_val *mv,
__isl_take isl_val *v);
149

__isl_give isl_multi_val *isl_multi_val_mod_val(
__isl_take isl_multi_val *mv,
__isl_take isl_val *v);
__isl_give isl_multi_val *isl_multi_val_scale_val(
__isl_take isl_multi_val *mv,
__isl_take isl_val *v);
__isl_give isl_multi_val *isl_multi_val_scale_down_val(
__isl_take isl_multi_val *mv,
__isl_take isl_val *v);
#include 
__isl_give isl_aff *isl_aff_mod_val(__isl_take isl_aff *aff,
__isl_take isl_val *mod);
__isl_give isl_pw_aff *isl_pw_aff_mod_val(
__isl_take isl_pw_aff *pa,
__isl_take isl_val *mod);
__isl_give isl_union_pw_aff *isl_union_pw_aff_mod_val(
__isl_take isl_union_pw_aff *upa,
__isl_take isl_val *f);
__isl_give isl_aff *isl_aff_scale_val(__isl_take isl_aff *aff,
__isl_take isl_val *v);
__isl_give isl_multi_aff *isl_multi_aff_scale_val(
__isl_take isl_multi_aff *ma,
__isl_take isl_val *v);
__isl_give isl_pw_aff *isl_pw_aff_scale_val(
__isl_take isl_pw_aff *pa, __isl_take isl_val *v);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_scale_val(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_val *v);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_val(
__isl_take isl_pw_multi_aff *pma,
__isl_take isl_val *v);
__isl_give isl_union_pw_multi_aff *
__isl_give isl_union_pw_aff *isl_union_pw_aff_scale_val(
__isl_take isl_union_pw_aff *upa,
__isl_take isl_val *f);
isl_union_pw_multi_aff_scale_val(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_val *val);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_scale_val(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_val *v);
__isl_give isl_aff *isl_aff_scale_down_ui(
__isl_take isl_aff *aff, unsigned f);
__isl_give isl_aff *isl_aff_scale_down_val(
150

__isl_take isl_aff *aff, __isl_take isl_val *v);
__isl_give isl_multi_aff *isl_multi_aff_scale_down_val(
__isl_take isl_multi_aff *ma,
__isl_take isl_val *v);
__isl_give isl_pw_aff *isl_pw_aff_scale_down_val(
__isl_take isl_pw_aff *pa,
__isl_take isl_val *f);
__isl_give isl_multi_pw_aff *isl_multi_pw_aff_scale_down_val(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_val *v);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_down_val(
__isl_take isl_pw_multi_aff *pma,
__isl_take isl_val *v);
__isl_give isl_union_pw_aff *isl_union_pw_aff_scale_down_val(
__isl_take isl_union_pw_aff *upa,
__isl_take isl_val *v);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_scale_down_val(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_val *val);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_scale_down_val(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_val *v);
#include 
__isl_give isl_qpolynomial *isl_qpolynomial_scale_val(
__isl_take isl_qpolynomial *qp,
__isl_take isl_val *v);
__isl_give isl_qpolynomial_fold *
isl_qpolynomial_fold_scale_val(
__isl_take isl_qpolynomial_fold *fold,
__isl_take isl_val *v);
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_scale_val(
__isl_take isl_pw_qpolynomial *pwqp,
__isl_take isl_val *v);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_scale_val(
__isl_take isl_pw_qpolynomial_fold *pwf,
__isl_take isl_val *v);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_scale_val(
__isl_take isl_union_pw_qpolynomial *upwqp,
__isl_take isl_val *v);
__isl_give isl_union_pw_qpolynomial_fold *
151

isl_union_pw_qpolynomial_fold_scale_val(
__isl_take isl_union_pw_qpolynomial_fold *upwf,
__isl_take isl_val *v);
__isl_give isl_qpolynomial *
isl_qpolynomial_scale_down_val(
__isl_take isl_qpolynomial *qp,
__isl_take isl_val *v);
__isl_give isl_qpolynomial_fold *
isl_qpolynomial_fold_scale_down_val(
__isl_take isl_qpolynomial_fold *fold,
__isl_take isl_val *v);
__isl_give isl_pw_qpolynomial *
isl_pw_qpolynomial_scale_down_val(
__isl_take isl_pw_qpolynomial *pwqp,
__isl_take isl_val *v);
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_fold_scale_down_val(
__isl_take isl_pw_qpolynomial_fold *pwf,
__isl_take isl_val *v);
__isl_give isl_union_pw_qpolynomial *
isl_union_pw_qpolynomial_scale_down_val(
__isl_take isl_union_pw_qpolynomial *upwqp,
__isl_take isl_val *v);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_fold_scale_down_val(
__isl_take isl_union_pw_qpolynomial_fold *upwf,
__isl_take isl_val *v);
#include 
__isl_give isl_multi_val *isl_multi_val_mod_multi_val(
__isl_take isl_multi_val *mv1,
__isl_take isl_multi_val *mv2);
__isl_give isl_multi_val *isl_multi_val_scale_multi_val(
__isl_take isl_multi_val *mv1,
__isl_take isl_multi_val *mv2);
__isl_give isl_multi_val *
isl_multi_val_scale_down_multi_val(
__isl_take isl_multi_val *mv1,
__isl_take isl_multi_val *mv2);
#include 
__isl_give isl_multi_aff *isl_multi_aff_mod_multi_val(
__isl_take isl_multi_aff *ma,
__isl_take isl_multi_val *mv);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_mod_multi_val(
152

__isl_take isl_multi_union_pw_aff *upma,
__isl_take isl_multi_val *mv);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_mod_multi_val(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_multi_val *mv);
__isl_give isl_multi_aff *isl_multi_aff_scale_multi_val(
__isl_take isl_multi_aff *ma,
__isl_take isl_multi_val *mv);
__isl_give isl_pw_multi_aff *
isl_pw_multi_aff_scale_multi_val(
__isl_take isl_pw_multi_aff *pma,
__isl_take isl_multi_val *mv);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_scale_multi_val(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_multi_val *mv);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_scale_multi_val(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_multi_val *mv);
__isl_give isl_union_pw_multi_aff *
isl_union_pw_multi_aff_scale_multi_val(
__isl_take isl_union_pw_multi_aff *upma,
__isl_take isl_multi_val *mv);
__isl_give isl_multi_aff *
isl_multi_aff_scale_down_multi_val(
__isl_take isl_multi_aff *ma,
__isl_take isl_multi_val *mv);
__isl_give isl_multi_pw_aff *
isl_multi_pw_aff_scale_down_multi_val(
__isl_take isl_multi_pw_aff *mpa,
__isl_take isl_multi_val *mv);
__isl_give isl_multi_union_pw_aff *
isl_multi_union_pw_aff_scale_down_multi_val(
__isl_take isl_multi_union_pw_aff *mupa,
__isl_take isl_multi_val *mv);
isl_multi_aff_scale_multi_val scales the elements of ma by the corresponding elements of mv.
#include 
__isl_give isl_aff *isl_aff_mul(
__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_aff *isl_aff_div(
153

__isl_take isl_aff *aff1,
__isl_take isl_aff *aff2);
__isl_give isl_pw_aff *isl_pw_aff_mul(
__isl_take isl_pw_aff *pwaff1,
__isl_take isl_pw_aff *pwaff2);
__isl_give isl_pw_aff *isl_pw_aff_div(
__isl_take isl_pw_aff *pa1,
__isl_take isl_pw_aff *pa2);
__isl_give isl_pw_aff *isl_pw_aff_tdiv_q(
__isl_take isl_pw_aff *pa1,
__isl_take isl_pw_aff *pa2);
__isl_give isl_pw_aff *isl_pw_aff_tdiv_r(
__isl_take isl_pw_aff *pa1,
__isl_take isl_pw_aff *pa2);
When multiplying two affine expressions, at least one of the two needs to be a
constant. Similarly, when dividing an affine expression by another, the second
expression needs to be a constant. isl_pw_aff_tdiv_q computes the quotient
of an integer division with rounding towards zero. isl_pw_aff_tdiv_r computes the corresponding remainder.
#include 
__isl_give isl_qpolynomial *isl_qpolynomial_mul(
__isl_take isl_qpolynomial *qp1,
__isl_take isl_qpolynomial *qp2);
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_mul(
__isl_take isl_pw_qpolynomial *pwqp1,
__isl_take isl_pw_qpolynomial *pwqp2);
__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_mul(
__isl_take isl_union_pw_qpolynomial *upwqp1,
__isl_take isl_union_pw_qpolynomial *upwqp2);
Lexicographic Optimization
Given a (basic) set set (or bset) and a zero-dimensional domain dom, the following
functions compute a set that contains the lexicographic minimum or maximum of the
elements in set (or bset) for those values of the parameters that satisfy dom. If empty
is not NULL, then *empty is assigned a set that contains the parameter values in dom
for which set (or bset) has no elements. In other words, the union of the parameter
values for which the result is non-empty and of *empty is equal to dom.
#include 
__isl_give isl_set *isl_basic_set_partial_lexmin(
__isl_take isl_basic_set *bset,
__isl_take isl_basic_set *dom,
__isl_give isl_set **empty);
154

__isl_give isl_set
__isl_take
__isl_take
__isl_give
__isl_give isl_set
__isl_take
__isl_give
__isl_give isl_set
__isl_take
__isl_give

*isl_basic_set_partial_lexmax(
isl_basic_set *bset,
isl_basic_set *dom,
isl_set **empty);
*isl_set_partial_lexmin(
isl_set *set, __isl_take isl_set *dom,
isl_set **empty);
*isl_set_partial_lexmax(
isl_set *set, __isl_take isl_set *dom,
isl_set **empty);

Given a (basic) set set (or bset), the following functions simply return a set containing the lexicographic minimum or maximum of the elements in set (or bset). In
case of union sets, the optimum is computed per space.
#include 
__isl_give isl_set *isl_basic_set_lexmin(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_basic_set_lexmax(
__isl_take isl_basic_set *bset);
__isl_give isl_set *isl_set_lexmin(
__isl_take isl_set *set);
__isl_give isl_set *isl_set_lexmax(
__isl_take isl_set *set);
__isl_give isl_union_set *isl_union_set_lexmin(
__isl_take isl_union_set *uset);
__isl_give isl_union_set *isl_union_set_lexmax(
__isl_take isl_union_set *uset);
Given a (basic) relation map (or bmap) and a domain dom, the following functions
compute a relation that maps each element of dom to the single lexicographic minimum
or maximum of the elements that are associated to that same element in map (or bmap).
If empty is not NULL, then *empty is assigned a set that contains the elements in dom
that do not map to any elements in map (or bmap). In other words, the union of the
domain of the result and of *empty is equal to dom.
#include 
__isl_give isl_map *isl_basic_map_partial_lexmax(
__isl_take isl_basic_map *bmap,
__isl_take isl_basic_set *dom,
__isl_give isl_set **empty);
__isl_give isl_map *isl_basic_map_partial_lexmin(
__isl_take isl_basic_map *bmap,
__isl_take isl_basic_set *dom,
__isl_give isl_set **empty);
__isl_give isl_map *isl_map_partial_lexmax(
__isl_take isl_map *map, __isl_take isl_set *dom,
155

__isl_give
__isl_give isl_map
__isl_take
__isl_give

isl_set **empty);
*isl_map_partial_lexmin(
isl_map *map, __isl_take isl_set *dom,
isl_set **empty);

Given a (basic) map map (or bmap), the following functions simply return a map
mapping each element in the domain of map (or bmap) to the lexicographic minimum
or maximum of all elements associated to that element. In case of union relations, the
optimum is computed per space.
#include 
__isl_give isl_map *isl_basic_map_lexmin(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_basic_map_lexmax(
__isl_take isl_basic_map *bmap);
__isl_give isl_map *isl_map_lexmin(
__isl_take isl_map *map);
__isl_give isl_map *isl_map_lexmax(
__isl_take isl_map *map);
__isl_give isl_union_map *isl_union_map_lexmin(
__isl_take isl_union_map *umap);
__isl_give isl_union_map *isl_union_map_lexmax(
__isl_take isl_union_map *umap);
The following functions return their result in the form of a piecewise multi-affine
expression, but are otherwise equivalent to the corresponding functions returning a
basic set or relation.
#include 
__isl_give isl_pw_multi_aff *
isl_basic_set_partial_lexmin_pw_multi_aff(
__isl_take isl_basic_set *bset,
__isl_take isl_basic_set *dom,
__isl_give isl_set **empty);
__isl_give isl_pw_multi_aff *
isl_basic_set_partial_lexmax_pw_multi_aff(
__isl_take isl_basic_set *bset,
__isl_take isl_basic_set *dom,
__isl_give isl_set **empty);
__isl_give isl_pw_multi_aff *isl_set_lexmin_pw_multi_aff(
__isl_take isl_set *set);
__isl_give isl_pw_multi_aff *isl_set_lexmax_pw_multi_aff(
__isl_take isl_set *set);
#include 
__isl_give isl_pw_multi_aff *
isl_basic_map_lexmin_pw_multi_aff(
156

__isl_take isl_basic_map *bmap);
__isl_give isl_pw_multi_aff *
isl_basic_map_partial_lexmin_pw_multi_aff(
__isl_take isl_basic_map *bmap,
__isl_take isl_basic_set *dom,
__isl_give isl_set **empty);
__isl_give isl_pw_multi_aff *
isl_basic_map_partial_lexmax_pw_multi_aff(
__isl_take isl_basic_map *bmap,
__isl_take isl_basic_set *dom,
__isl_give isl_set **empty);
__isl_give isl_pw_multi_aff *isl_map_lexmin_pw_multi_aff(
__isl_take isl_map *map);
__isl_give isl_pw_multi_aff *isl_map_lexmax_pw_multi_aff(
__isl_take isl_map *map);
The following functions return the lexicographic minimum or maximum on the
shared domain of the inputs and the single defined function on those parts of the domain
where only a single function is defined.
#include 
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmin(
__isl_take isl_pw_multi_aff *pma1,
__isl_take isl_pw_multi_aff *pma2);
__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmax(
__isl_take isl_pw_multi_aff *pma1,
__isl_take isl_pw_multi_aff *pma2);
If the input to a lexicographic optimization problem has multiple constraints with
the same coefficients for the optimized variables, then, by default, this symmetry is
exploited by replacing those constraints by a single constraint with an abstract bound,
which is in turn bounded by the corresponding terms in the original constraints. Without this optimization, the solver would typically consider all possible orderings of those
original bounds, resulting in a needless decomposition of the domain. However, the
optimization can also result in slowdowns since an extra parameter is introduced that
may get used in additional integer divisions. The following option determines whether
symmetry detection is applied during lexicographic optimization.
#include 
isl_stat isl_options_set_pip_symmetry(isl_ctx *ctx,
int val);
int isl_options_get_pip_symmetry(isl_ctx *ctx);
See also Section 2.3.5.

1.4.18

Ternary Operations
#include 
157

__isl_give isl_pw_aff *isl_pw_aff_cond(
__isl_take isl_pw_aff *cond,
__isl_take isl_pw_aff *pwaff_true,
__isl_take isl_pw_aff *pwaff_false);
The function isl_pw_aff_cond performs a conditional operator and returns an
expression that is equal to pwaff_true for elements where cond is non-zero and equal
to pwaff_false for elements where cond is zero.

1.4.19

Lists

Lists are defined over several element types, including isl_val, isl_id, isl_aff,
isl_pw_aff, isl_pw_multi_aff, isl_union_pw_aff, isl_union_pw_multi_aff,
isl_pw_qpolynomial, isl_pw_qpolynomial_fold, isl_constraint, isl_basic_set,
isl_set, isl_basic_map, isl_map, isl_union_set, isl_union_map, isl_ast_expr
and isl_ast_node. Here we take lists of isl_sets as an example. Lists can be created, copied, modified and freed using the following functions.
#include 
__isl_give isl_set_list *isl_set_list_from_set(
__isl_take isl_set *el);
__isl_give isl_set_list *isl_set_list_alloc(
isl_ctx *ctx, int n);
__isl_give isl_set_list *isl_set_list_copy(
__isl_keep isl_set_list *list);
__isl_give isl_set_list *isl_set_list_insert(
__isl_take isl_set_list *list, unsigned pos,
__isl_take isl_set *el);
__isl_give isl_set_list *isl_set_list_add(
__isl_take isl_set_list *list,
__isl_take isl_set *el);
__isl_give isl_set_list *isl_set_list_drop(
__isl_take isl_set_list *list,
unsigned first, unsigned n);
__isl_give isl_set_list *isl_set_list_swap(
__isl_take isl_set_list *list,
unsigned pos1, unsigned pos2);
__isl_give isl_set_list *isl_set_list_reverse(
__isl_take isl_set_list *list);
__isl_give isl_set_list *isl_set_list_set_set(
__isl_take isl_set_list *list, int index,
__isl_take isl_set *set);
__isl_give isl_set_list *isl_set_list_concat(
__isl_take isl_set_list *list1,
__isl_take isl_set_list *list2);
__isl_give isl_set_list *isl_set_list_map(

158

__isl_take isl_set_list *list,
__isl_give isl_set *(*fn)(__isl_take isl_set *el,
void *user),
void *user);
__isl_give isl_set_list *isl_set_list_sort(
__isl_take isl_set_list *list,
int (*cmp)(__isl_keep isl_set *a,
__isl_keep isl_set *b, void *user),
void *user);
__isl_null isl_set_list *isl_set_list_free(
__isl_take isl_set_list *list);
isl_set_list_alloc creates an empty list with an initial capacity for n elements. isl_set_list_insert and isl_set_list_add add elements to a list, increasing its capacity as needed. isl_set_list_from_set creates a list with a single element. isl_set_list_swap swaps the elements at the specified locations.
isl_set_list_reverse reverses the elements in the list.
Lists can be inspected using the following functions.
#include 
int isl_set_list_size(__isl_keep isl_set_list *list);
int isl_set_list_n_set(__isl_keep isl_set_list *list);
__isl_give isl_set *isl_set_list_get_at(
__isl_keep isl_set_list *list, int index);
__isl_give isl_set *isl_set_list_get_set(
__isl_keep isl_set_list *list, int index);
isl_stat isl_set_list_foreach(__isl_keep isl_set_list *list,
isl_stat (*fn)(__isl_take isl_set *el, void *user),
void *user);
isl_stat isl_set_list_foreach_scc(
__isl_keep isl_set_list *list,
isl_bool (*follows)(__isl_keep isl_set *a,
__isl_keep isl_set *b, void *user),
void *follows_user,
isl_stat (*fn)(__isl_take isl_set *el, void *user),
void *fn_user);
isl_set_list_n_set is an alternative name for isl_set_list_size. Similarly, isl_set_list_get_set is an alternative name for isl_set_list_get_at.
The function isl_set_list_foreach_scc calls fn on each of the strongly connected
components of the graph with as vertices the elements of list and a directed edge
from vertex b to vertex a iff follows(a, b) returns isl_bool_true. The callbacks
follows and fn should return isl_bool_error or isl_stat_error on error.
Lists can be printed using
#include 
__isl_give isl_printer *isl_printer_print_set_list(
159

__isl_take isl_printer *p,
__isl_keep isl_set_list *list);

1.4.20

Associative arrays

Associative arrays map isl objects of a specific type to isl objects of some (other) specific type. They are defined for several pairs of types, including (isl_map, isl_basic_set),
(isl_id, isl_ast_expr), (isl_id, isl_id) and (isl_id, isl_pw_aff). Here, we
take associative arrays that map isl_ids to isl_ast_exprs as an example.
Associative arrays can be created, copied and freed using the following functions.
#include 
__isl_give isl_id_to_ast_expr *isl_id_to_ast_expr_alloc(
isl_ctx *ctx, int min_size);
__isl_give isl_id_to_ast_expr *isl_id_to_ast_expr_copy(
__isl_keep isl_id_to_ast_expr *id2expr);
__isl_null isl_id_to_ast_expr *isl_id_to_ast_expr_free(
__isl_take isl_id_to_ast_expr *id2expr);
The min_size argument to isl_id_to_ast_expr_alloc can be used to specify
the expected size of the associative array. The associative array will be grown automatically as needed.
Associative arrays can be inspected using the following functions.
#include 
__isl_give isl_maybe_isl_ast_expr
isl_id_to_ast_expr_try_get(
__isl_keep isl_id_to_ast_expr *id2expr,
__isl_keep isl_id *key);
isl_bool isl_id_to_ast_expr_has(
__isl_keep isl_id_to_ast_expr *id2expr,
__isl_keep isl_id *key);
__isl_give isl_ast_expr *isl_id_to_ast_expr_get(
__isl_keep isl_id_to_ast_expr *id2expr,
__isl_take isl_id *key);
isl_stat isl_id_to_ast_expr_foreach(
__isl_keep isl_id_to_ast_expr *id2expr,
isl_stat (*fn)(__isl_take isl_id *key,
__isl_take isl_ast_expr *val, void *user),
void *user);
The function isl_id_to_ast_expr_try_get returns a structure containing two
elements, valid and value. If there is a value associated to the key, then valid is
set to isl_bool_true and value contains a copy of the associated value. Otherwise
value is NULL and valid may be isl_bool_error or isl_bool_false depending
on whether some error has occurred or there simply is no associated value. The function

160

isl_id_to_ast_expr_has returns the valid field in the structure and the function
isl_id_to_ast_expr_get returns the value field.
Associative arrays can be modified using the following functions.
#include 
__isl_give isl_id_to_ast_expr *isl_id_to_ast_expr_set(
__isl_take isl_id_to_ast_expr *id2expr,
__isl_take isl_id *key,
__isl_take isl_ast_expr *val);
__isl_give isl_id_to_ast_expr *isl_id_to_ast_expr_drop(
__isl_take isl_id_to_ast_expr *id2expr,
__isl_take isl_id *key);
Associative arrays can be printed using the following function.
#include 
__isl_give isl_printer *isl_printer_print_id_to_ast_expr(
__isl_take isl_printer *p,
__isl_keep isl_id_to_ast_expr *id2expr);

1.4.21

Vectors

Vectors can be created, copied and freed using the following functions.
#include 
__isl_give isl_vec *isl_vec_alloc(isl_ctx *ctx,
unsigned size);
__isl_give isl_vec *isl_vec_zero(isl_ctx *ctx,
unsigned size);
__isl_give isl_vec *isl_vec_copy(__isl_keep isl_vec *vec);
__isl_null isl_vec *isl_vec_free(__isl_take isl_vec *vec);
Note that the elements of a vector created by isl_vec_alloc may have arbitrary
values. A vector created by isl_vec_zero has elements with value zero. The elements can be changed and inspected using the following functions.
int isl_vec_size(__isl_keep isl_vec *vec);
__isl_give isl_val *isl_vec_get_element_val(
__isl_keep isl_vec *vec, int pos);
__isl_give isl_vec *isl_vec_set_element_si(
__isl_take isl_vec *vec, int pos, int v);
__isl_give isl_vec *isl_vec_set_element_val(
__isl_take isl_vec *vec, int pos,
__isl_take isl_val *v);
__isl_give isl_vec *isl_vec_set_si(__isl_take isl_vec *vec,
int v);
__isl_give isl_vec *isl_vec_set_val(
161

__isl_take isl_vec *vec, __isl_take isl_val *v);
int isl_vec_cmp_element(__isl_keep isl_vec *vec1,
__isl_keep isl_vec *vec2, int pos);
isl_vec_get_element will return a negative value if anything went wrong. In
that case, the value of *v is undefined.
The following function can be used to concatenate two vectors.
__isl_give isl_vec *isl_vec_concat(__isl_take isl_vec *vec1,
__isl_take isl_vec *vec2);

1.4.22

Matrices

Matrices can be created, copied and freed using the following functions.
#include 
__isl_give isl_mat *isl_mat_alloc(isl_ctx *ctx,
unsigned n_row, unsigned n_col);
__isl_give isl_mat *isl_mat_copy(__isl_keep isl_mat *mat);
__isl_null isl_mat *isl_mat_free(__isl_take isl_mat *mat);
Note that the elements of a newly created matrix may have arbitrary values. The
elements can be changed and inspected using the following functions.
int isl_mat_rows(__isl_keep isl_mat *mat);
int isl_mat_cols(__isl_keep isl_mat *mat);
__isl_give isl_val *isl_mat_get_element_val(
__isl_keep isl_mat *mat, int row, int col);
__isl_give isl_mat *isl_mat_set_element_si(__isl_take isl_mat *mat,
int row, int col, int v);
__isl_give isl_mat *isl_mat_set_element_val(
__isl_take isl_mat *mat, int row, int col,
__isl_take isl_val *v);
The following function computes the rank of a matrix. The return value may be -1
if some error occurred.
#include 
int isl_mat_rank(__isl_keep isl_mat *mat);
The following function can be used to compute the (right) inverse of a matrix, i.e.,
a matrix such that the product of the original and the inverse (in that order) is a multiple
of the identity matrix. The input matrix is assumed to be of full row-rank.
__isl_give isl_mat *isl_mat_right_inverse(__isl_take isl_mat *mat);
The following function can be used to compute the (right) kernel (or null space) of
a matrix, i.e., a matrix such that the product of the original and the kernel (in that order)
is the zero matrix.
162

__isl_give isl_mat *isl_mat_right_kernel(__isl_take isl_mat *mat);
The following function computes a basis for the space spanned by the rows of a
matrix.
__isl_give isl_mat *isl_mat_row_basis(
__isl_take isl_mat *mat);
The following function computes rows that extend a basis of mat1 to a basis that
also covers mat2.
__isl_give isl_mat *isl_mat_row_basis_extension(
__isl_take isl_mat *mat1,
__isl_take isl_mat *mat2);
The following function checks whether there is no linear dependence among the
combined rows of "mat1" and "mat2" that is not already present in "mat1" or "mat2"
individually. If "mat1" and "mat2" have linearly independent rows by themselves, then
this means that there is no linear dependence among all rows together.
isl_bool isl_mat_has_linearly_independent_rows(
__isl_keep isl_mat *mat1,
__isl_keep isl_mat *mat2);

1.4.23

Bounds on Piecewise Quasipolynomials and Piecewise Quasipolynomial Reductions

The following functions determine an upper or lower bound on a quasipolynomial over
its domain.
__isl_give isl_pw_qpolynomial_fold *
isl_pw_qpolynomial_bound(
__isl_take isl_pw_qpolynomial *pwqp,
enum isl_fold type, int *tight);
__isl_give isl_union_pw_qpolynomial_fold *
isl_union_pw_qpolynomial_bound(
__isl_take isl_union_pw_qpolynomial *upwqp,
enum isl_fold type, int *tight);
The type argument may be either isl_fold_min or isl_fold_max. If tight
is not NULL, then *tight is set to 1 is the returned bound is known be tight, i.e., for
each value of the parameters there is at least one element in the domain that reaches
the bound. If the domain of pwqp is not wrapping, then the bound is computed over all
elements in that domain and the result has a purely parametric domain. If the domain of
pwqp is wrapping, then the bound is computed over the range of the wrapped relation.
The domain of the wrapped relation becomes the domain of the result.

163

1.4.24

Parametric Vertex Enumeration

The parametric vertex enumeration described in this section is mainly intended to be
used internally and by the barvinok library.
#include 
__isl_give isl_vertices *isl_basic_set_compute_vertices(
__isl_keep isl_basic_set *bset);
The function isl_basic_set_compute_vertices performs the actual computation of the parametric vertices and the chamber decomposition and stores the result in
an isl_vertices object. This information can be queried by either iterating over all
the vertices or iterating over all the chambers or cells and then iterating over all vertices
that are active on the chamber.
isl_stat isl_vertices_foreach_vertex(
__isl_keep isl_vertices *vertices,
isl_stat (*fn)(__isl_take isl_vertex *vertex,
void *user), void *user);
isl_stat isl_vertices_foreach_cell(
__isl_keep isl_vertices *vertices,
isl_stat (*fn)(__isl_take isl_cell *cell,
void *user), void *user);
isl_stat isl_cell_foreach_vertex(__isl_keep isl_cell *cell,
isl_stat (*fn)(__isl_take isl_vertex *vertex,
void *user), void *user);
Other operations that can be performed on an isl_vertices object are the following.
int isl_vertices_get_n_vertices(
__isl_keep isl_vertices *vertices);
__isl_null isl_vertices *isl_vertices_free(
__isl_take isl_vertices *vertices);
Vertices can be inspected and destroyed using the following functions.
int isl_vertex_get_id(__isl_keep isl_vertex *vertex);
__isl_give isl_basic_set *isl_vertex_get_domain(
__isl_keep isl_vertex *vertex);
__isl_give isl_multi_aff *isl_vertex_get_expr(
__isl_keep isl_vertex *vertex);
void isl_vertex_free(__isl_take isl_vertex *vertex);
isl_vertex_get_expr returns a multiple quasi-affine expression describing the
vertex in terms of the parameters, while isl_vertex_get_domain returns the activity
domain of the vertex.
Chambers can be inspected and destroyed using the following functions.
164

__isl_give isl_basic_set *isl_cell_get_domain(
__isl_keep isl_cell *cell);
void isl_cell_free(__isl_take isl_cell *cell);

1.5

Polyhedral Compilation Library

This section collects functionality in isl that has been specifically designed for use
during polyhedral compilation.

1.5.1

Schedule Trees

A schedule tree is a structured representation of a schedule, assigning a relative order to a set of domain elements. The relative order expressed by the schedule tree
is defined recursively. In particular, the order between two domain elements is determined by the node that is closest to the root that refers to both elements and that orders
them apart. Each node in the tree is of one of several types. The root node is always
of type isl_schedule_node_domain (or isl_schedule_node_extension) and it
describes the (extra) domain elements to which the schedule applies. The other types
of nodes are as follows.
isl_schedule_node_band
A band of schedule dimensions. Each schedule dimension is represented by a
union piecewise quasi-affine expression. If this expression assigns a different
value to two domain elements, while all previous schedule dimensions in the
same band assign them the same value, then the two domain elements are ordered
according to these two different values. Each expression is required to be total in
the domain elements that reach the band node.
isl_schedule_node_expansion
An expansion node maps each of the domain elements that reach the node to one
or more domain elements. The image of this mapping forms the set of domain
elements that reach the child of the expansion node. The function that maps each
of the expanded domain elements to the original domain element from which it
was expanded is called the contraction.
isl_schedule_node_filter
A filter node does not impose any ordering, but rather intersects the set of domain
elements that the current subtree refers to with a given union set. The subtree of
the filter node only refers to domain elements in the intersection. A filter node is
typically only used as a child of a sequence or set node.
isl_schedule_node_leaf
A leaf of the schedule tree. Leaf nodes do not impose any ordering.

165

isl_schedule_node_mark
A mark node can be used to attach any kind of information to a subtree of the
schedule tree.
isl_schedule_node_sequence
A sequence node has one or more children, each of which is a filter node. The
filters on these filter nodes form a partition of the domain elements that the current subtree refers to. If two domain elements appear in distinct filters then the
sequence node orders them according to the child positions of the corresponding
filter nodes.
isl_schedule_node_set
A set node is similar to a sequence node, except that it expresses that domain elements appearing in distinct filters may have any order. The order of the children
of a set node is therefore also immaterial.
The following node types are only supported by the AST generator.
isl_schedule_node_context
The context describes constraints on the parameters and the schedule dimensions
of outer bands that the AST generator may assume to hold. It is also the only
kind of node that may introduce additional parameters. The space of the context
is that of the flat product of the outer band nodes. In particular, if there are no
outer band nodes, then this space is the unnamed zero-dimensional space. Since
a context node references the outer band nodes, any tree containing a context
node is considered to be anchored.
isl_schedule_node_extension
An extension node instructs the AST generator to add additional domain elements that need to be scheduled. The additional domain elements are described
by the range of the extension map in terms of the outer schedule dimensions, i.e.,
the flat product of the outer band nodes. Note that domain elements are added
whenever the AST generator reaches the extension node, meaning that there are
still some active domain elements for which an AST needs to be generated. The
conditions under which some domain elements are still active may however not
be completely described by the outer AST nodes generated at that point. Since an
extension node references the outer band nodes, any tree containing an extension
node is considered to be anchored.
An extension node may also appear as the root of a schedule tree, when it is intended to be inserted into another tree using isl_schedule_node_graft_before
or isl_schedule_node_graft_after. In this case, the domain of the extension node should correspond to the flat product of the outer band nodes in this
other schedule tree at the point where the extension tree will be inserted.
isl_schedule_node_guard

166

The guard describes constraints on the parameters and the schedule dimensions
of outer bands that need to be enforced by the outer nodes in the generated AST.
That is, the part of the AST that is generated from descendants of the guard node
can assume that these constraints are satisfied. The space of the guard is that
of the flat product of the outer band nodes. In particular, if there are no outer
band nodes, then this space is the unnamed zero-dimensional space. Since a
guard node references the outer band nodes, any tree containing a guard node is
considered to be anchored.
Except for the isl_schedule_node_context nodes, none of the nodes may introduce any parameters that were not already present in the root domain node.
A schedule tree is encapsulated in an isl_schedule object. The simplest such
objects, those with a tree consisting of single domain node, can be created using the
following functions with either an empty domain or a given domain.
#include 
__isl_give isl_schedule *isl_schedule_empty(
__isl_take isl_space *space);
__isl_give isl_schedule *isl_schedule_from_domain(
__isl_take isl_union_set *domain);
The function isl_schedule_constraints_compute_schedule described in §1.5.3
can also be used to construct schedules.
isl_schedule objects may be copied and freed using the following functions.
#include 
__isl_give isl_schedule *isl_schedule_copy(
__isl_keep isl_schedule *sched);
__isl_null isl_schedule *isl_schedule_free(
__isl_take isl_schedule *sched);
The following functions checks whether two isl_schedule objects are obviously
the same.
#include 
isl_bool isl_schedule_plain_is_equal(
__isl_keep isl_schedule *schedule1,
__isl_keep isl_schedule *schedule2);
The domain of the schedule, i.e., the domain described by the root node, can be
obtained using the following function.
#include 
__isl_give isl_union_set *isl_schedule_get_domain(
__isl_keep isl_schedule *schedule);
An extra top-level band node (right underneath the domain node) can be introduced
into the schedule using the following function. The schedule tree is assumed not to have
any anchored nodes.
167

#include 
__isl_give isl_schedule *
isl_schedule_insert_partial_schedule(
__isl_take isl_schedule *schedule,
__isl_take isl_multi_union_pw_aff *partial);
A top-level context node (right underneath the domain node) can be introduced into
the schedule using the following function.
#include 
__isl_give isl_schedule *isl_schedule_insert_context(
__isl_take isl_schedule *schedule,
__isl_take isl_set *context)
A top-level guard node (right underneath the domain node) can be introduced into
the schedule using the following function.
#include 
__isl_give isl_schedule *isl_schedule_insert_guard(
__isl_take isl_schedule *schedule,
__isl_take isl_set *guard)
A schedule that combines two schedules either in the given order or in an arbitrary
order, i.e., with an isl_schedule_node_sequence or an isl_schedule_node_set
node, can be created using the following functions.
#include 
__isl_give isl_schedule *isl_schedule_sequence(
__isl_take isl_schedule *schedule1,
__isl_take isl_schedule *schedule2);
__isl_give isl_schedule *isl_schedule_set(
__isl_take isl_schedule *schedule1,
__isl_take isl_schedule *schedule2);
The domains of the two input schedules need to be disjoint.
The following function can be used to restrict the domain of a schedule with a
domain node as root to be a subset of the given union set. This operation may remove
nodes in the tree that have become redundant.
#include 
__isl_give isl_schedule *isl_schedule_intersect_domain(
__isl_take isl_schedule *schedule,
__isl_take isl_union_set *domain);
The following function can be used to simplify the domain of a schedule with a
domain node as root with respect to the given parameter domain.

168

#include 
__isl_give isl_schedule *isl_schedule_gist_domain_params(
__isl_take isl_schedule *schedule,
__isl_take isl_set *context);
The following function resets the user pointers on all parameter and tuple identifiers
referenced by the nodes of the given schedule.
#include 
__isl_give isl_schedule *isl_schedule_reset_user(
__isl_take isl_schedule *schedule);
The following function aligns the parameters of all nodes in the given schedule to
the given space.
#include 
__isl_give isl_schedule *isl_schedule_align_params(
__isl_take isl_schedule *schedule,
__isl_take isl_space *space);
The following function allows the user to plug in a given function in the iteration
domains. The input schedule is not allowed to contain any expansion nodes.
#include 
__isl_give isl_schedule *
isl_schedule_pullback_union_pw_multi_aff(
__isl_take isl_schedule *schedule,
__isl_take isl_union_pw_multi_aff *upma);
The following function can be used to plug in the schedule expansion in the leaves
of schedule, where contraction describes how the domain elements of expansion
map to the domain elements at the original leaves of schedule. The resulting schedule
will contain expansion nodes, unless contraction is an identity function.
#include 
__isl_give isl_schedule *isl_schedule_expand(
__isl_take isl_schedule *schedule,
__isl_take isl_union_pw_multi_aff *contraction,
__isl_take isl_schedule *expansion);
An isl_union_map representation of the schedule can be obtained from an isl_schedule
using the following function.
#include 
__isl_give isl_union_map *isl_schedule_get_map(
__isl_keep isl_schedule *sched);

169

The resulting relation encodes the same relative ordering as the schedule by mapping the domain elements to a common schedule space. If the schedule_separate_components
option is set, then the order of the children of a set node is explicitly encoded in the
result. If the tree contains any expansion nodes, then the relation is formulated in terms
of the expanded domain elements.
Schedules can be read from input using the following functions.
#include 
__isl_give isl_schedule *isl_schedule_read_from_file(
isl_ctx *ctx, FILE *input);
__isl_give isl_schedule *isl_schedule_read_from_str(
isl_ctx *ctx, const char *str);
A representation of the schedule can be printed using
#include 
__isl_give isl_printer *isl_printer_print_schedule(
__isl_take isl_printer *p,
__isl_keep isl_schedule *schedule);
__isl_give char *isl_schedule_to_str(
__isl_keep isl_schedule *schedule);
isl_schedule_to_str prints the schedule in flow format.
The schedule tree can be traversed through the use of isl_schedule_node objects
that point to a particular position in the schedule tree. Whenever a isl_schedule_node
is used to modify a node in the schedule tree, the original schedule tree is left untouched and the modifications are performed to a copy of the tree. The returned
isl_schedule_node then points to this modified copy of the tree.
The root of the schedule tree can be obtained using the following function.
#include 
__isl_give isl_schedule_node *isl_schedule_get_root(
__isl_keep isl_schedule *schedule);
A pointer to a newly created schedule tree with a single domain node can be created
using the following functions.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_from_domain(
__isl_take isl_union_set *domain);
__isl_give isl_schedule_node *
isl_schedule_node_from_extension(
__isl_take isl_union_map *extension);
isl_schedule_node_from_extension creates a tree with an extension node as
root.
Schedule nodes can be copied and freed using the following functions.
170

#include 
__isl_give isl_schedule_node *isl_schedule_node_copy(
__isl_keep isl_schedule_node *node);
__isl_null isl_schedule_node *isl_schedule_node_free(
__isl_take isl_schedule_node *node);
The following functions can be used to check if two schedule nodes point to the
same position in the same schedule.
#include 
isl_bool isl_schedule_node_is_equal(
__isl_keep isl_schedule_node *node1,
__isl_keep isl_schedule_node *node2);
The following properties can be obtained from a schedule node.
#include 
enum isl_schedule_node_type isl_schedule_node_get_type(
__isl_keep isl_schedule_node *node);
enum isl_schedule_node_type
isl_schedule_node_get_parent_type(
__isl_keep isl_schedule_node *node);
__isl_give isl_schedule *isl_schedule_node_get_schedule(
__isl_keep isl_schedule_node *node);
The function isl_schedule_node_get_type returns the type of the node, while
isl_schedule_node_get_parent_type returns type of the parent of the node, which
is required to exist. The function isl_schedule_node_get_schedule returns a
copy to the schedule to which the node belongs.
The following functions can be used to move the schedule node to a different position in the tree or to check if such a position exists.
#include 
isl_bool isl_schedule_node_has_parent(
__isl_keep isl_schedule_node *node);
__isl_give isl_schedule_node *isl_schedule_node_parent(
__isl_take isl_schedule_node *node);
__isl_give isl_schedule_node *isl_schedule_node_root(
__isl_take isl_schedule_node *node);
__isl_give isl_schedule_node *isl_schedule_node_ancestor(
__isl_take isl_schedule_node *node,
int generation);
int isl_schedule_node_n_children(
__isl_keep isl_schedule_node *node);
__isl_give isl_schedule_node *isl_schedule_node_child(
__isl_take isl_schedule_node *node, int pos);
isl_bool isl_schedule_node_has_children(
171

__isl_keep isl_schedule_node *node);
__isl_give isl_schedule_node *isl_schedule_node_first_child(
__isl_take isl_schedule_node *node);
isl_bool isl_schedule_node_has_previous_sibling(
__isl_keep isl_schedule_node *node);
__isl_give isl_schedule_node *
isl_schedule_node_previous_sibling(
__isl_take isl_schedule_node *node);
isl_bool isl_schedule_node_has_next_sibling(
__isl_keep isl_schedule_node *node);
__isl_give isl_schedule_node *
isl_schedule_node_next_sibling(
__isl_take isl_schedule_node *node);
For isl_schedule_node_ancestor, the ancestor of generation 0 is the node itself, the ancestor of generation 1 is its parent and so on.
It is also possible to query the number of ancestors of a node, the position of the current node within the children of its parent, the position of the subtree containing a node
within the children of an ancestor or to obtain a copy of a given child without destroying
the current node. Given two nodes that point to the same schedule, their closest shared
ancestor can be obtained using isl_schedule_node_get_shared_ancestor.
#include 
int isl_schedule_node_get_tree_depth(
__isl_keep isl_schedule_node *node);
int isl_schedule_node_get_child_position(
__isl_keep isl_schedule_node *node);
int isl_schedule_node_get_ancestor_child_position(
__isl_keep isl_schedule_node *node,
__isl_keep isl_schedule_node *ancestor);
__isl_give isl_schedule_node *isl_schedule_node_get_child(
__isl_keep isl_schedule_node *node, int pos);
__isl_give isl_schedule_node *
isl_schedule_node_get_shared_ancestor(
__isl_keep isl_schedule_node *node1,
__isl_keep isl_schedule_node *node2);
All nodes in a schedule tree or all descendants of a specific node (including the
node) can be visited in depth-first pre-order using the following functions.
#include 
isl_stat isl_schedule_foreach_schedule_node_top_down(
__isl_keep isl_schedule *sched,
isl_bool (*fn)(__isl_keep isl_schedule_node *node,
void *user), void *user);
#include 
172

isl_stat isl_schedule_node_foreach_descendant_top_down(
__isl_keep isl_schedule_node *node,
isl_bool (*fn)(__isl_keep isl_schedule_node *node,
void *user), void *user);
The callback function is slightly different from the usual callbacks in that it not only
indicates success (non-negative result) or failure (negative result), but also indicates
whether the children of the given node should be visited. In particular, if the callback
returns a positive value, then the children are visited, but if the callback returns zero,
then the children are not visited.
The following functions checks whether all descendants of a specific node (including the node itself) satisfy a user-specified test.
#include 
isl_bool isl_schedule_node_every_descendant(
__isl_keep isl_schedule_node *node,
isl_bool (*test)(__isl_keep isl_schedule_node *node,
void *user), void *user)
The ancestors of a node in a schedule tree can be visited from the root down to and
including the parent of the node using the following function.
#include 
isl_stat isl_schedule_node_foreach_ancestor_top_down(
__isl_keep isl_schedule_node *node,
isl_stat (*fn)(__isl_keep isl_schedule_node *node,
void *user), void *user);
The following functions allows for a depth-first post-order traversal of the nodes
in a schedule tree or of the descendants of a specific node (including the node itself),
where the user callback is allowed to modify the visited node.
#include 
__isl_give isl_schedule *
isl_schedule_map_schedule_node_bottom_up(
__isl_take isl_schedule *schedule,
__isl_give isl_schedule_node *(*fn)(
__isl_take isl_schedule_node *node,
void *user), void *user);
#include 
__isl_give isl_schedule_node *
isl_schedule_node_map_descendant_bottom_up(
__isl_take isl_schedule_node *node,
__isl_give isl_schedule_node *(*fn)(
__isl_take isl_schedule_node *node,
void *user), void *user);
173

The traversal continues from the node returned by the callback function. It is the
responsibility of the user to ensure that this does not lead to an infinite loop. It is safest
to always return a pointer to the same position (same ancestors and child positions) as
the input node.
The following function removes a node (along with its descendants) from a schedule tree and returns a pointer to the leaf at the same position in the updated tree. It is
not allowed to remove the root of a schedule tree or a child of a set or sequence node.
#include 
__isl_give isl_schedule_node *isl_schedule_node_cut(
__isl_take isl_schedule_node *node);
The following function removes a single node from a schedule tree and returns a
pointer to the child of the node, now located at the position of the original node or to a
leaf node at that position if there was no child. It is not allowed to remove the root of a
schedule tree, a set or sequence node, a child of a set or sequence node or a band node
with an anchored subtree.
#include 
__isl_give isl_schedule_node *isl_schedule_node_delete(
__isl_take isl_schedule_node *node);
Most nodes in a schedule tree only contain local information. In some cases, however, a node may also refer to the schedule dimensions of its outer band nodes. This
means that the position of the node within the tree should not be changed, or at least
that no changes are performed to the outer band nodes. The following function can be
used to test whether the subtree rooted at a given node contains any such nodes.
#include 
isl_bool isl_schedule_node_is_subtree_anchored(
__isl_keep isl_schedule_node *node);
The following function resets the user pointers on all parameter and tuple identifiers
referenced by the given schedule node.
#include 
__isl_give isl_schedule_node *isl_schedule_node_reset_user(
__isl_take isl_schedule_node *node);
The following function aligns the parameters of the given schedule node to the
given space.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_align_params(
__isl_take isl_schedule_node *node,
__isl_take isl_space *space);

174

Several node types have their own functions for querying (and in some cases setting) some node type specific properties.
#include 
__isl_give isl_space *isl_schedule_node_band_get_space(
__isl_keep isl_schedule_node *node);
__isl_give isl_multi_union_pw_aff *
isl_schedule_node_band_get_partial_schedule(
__isl_keep isl_schedule_node *node);
__isl_give isl_union_map *
isl_schedule_node_band_get_partial_schedule_union_map(
__isl_keep isl_schedule_node *node);
unsigned isl_schedule_node_band_n_member(
__isl_keep isl_schedule_node *node);
isl_bool isl_schedule_node_band_member_get_coincident(
__isl_keep isl_schedule_node *node, int pos);
__isl_give isl_schedule_node *
isl_schedule_node_band_member_set_coincident(
__isl_take isl_schedule_node *node, int pos,
int coincident);
isl_bool isl_schedule_node_band_get_permutable(
__isl_keep isl_schedule_node *node);
__isl_give isl_schedule_node *
isl_schedule_node_band_set_permutable(
__isl_take isl_schedule_node *node, int permutable);
enum isl_ast_loop_type
isl_schedule_node_band_member_get_ast_loop_type(
__isl_keep isl_schedule_node *node, int pos);
__isl_give isl_schedule_node *
isl_schedule_node_band_member_set_ast_loop_type(
__isl_take isl_schedule_node *node, int pos,
enum isl_ast_loop_type type);
__isl_give isl_union_set *
enum isl_ast_loop_type
isl_schedule_node_band_member_get_isolate_ast_loop_type(
__isl_keep isl_schedule_node *node, int pos);
__isl_give isl_schedule_node *
isl_schedule_node_band_member_set_isolate_ast_loop_type(
__isl_take isl_schedule_node *node, int pos,
enum isl_ast_loop_type type);
isl_schedule_node_band_get_ast_build_options(
__isl_keep isl_schedule_node *node);
__isl_give isl_schedule_node *
isl_schedule_node_band_set_ast_build_options(
__isl_take isl_schedule_node *node,
__isl_take isl_union_set *options);
175

__isl_give isl_set *
isl_schedule_node_band_get_ast_isolate_option(
__isl_keep isl_schedule_node *node);

The function isl_schedule_node_band_get_space returns the space of the
partial schedule of the band. The function isl_schedule_node_band_get_partial_schedule_union_map
returns a representation of the partial schedule of the band node in the form of an
isl_union_map. The coincident and permutable properties are set by isl_schedule_constraints_compute_schedu
on the schedule tree it produces. A scheduling dimension is considered to be “coincident” if it satisfies the coincidence constraints within its band. That is, if the dependence distances of the coincidence constraints are all zero in that direction (for fixed
iterations of outer bands). A band is marked permutable if it was produced using the
Pluto-like scheduler. Note that the scheduler may have to resort to a Feautrier style
scheduling step even if the default scheduler is used. An isl_ast_loop_type is one
of isl_ast_loop_default, isl_ast_loop_atomic, isl_ast_loop_unroll or
isl_ast_loop_separate. For the meaning of these loop AST generation types and
the difference between the regular loop AST generation type and the isolate loop AST
generation type, see §1.5.4. The functions isl_schedule_node_band_member_get_ast_loop_type
and isl_schedule_node_band_member_get_isolate_ast_loop_type may return
isl_ast_loop_error if an error occurs. The AST build options govern how an AST
is generated for the individual schedule dimensions during AST generation. See §1.5.4.
The isolate option for the given node can be extracted from these AST build options
using the function isl_schedule_node_band_get_ast_isolate_option.
#include 
__isl_give isl_set *
isl_schedule_node_context_get_context(
__isl_keep isl_schedule_node *node);
#include 
__isl_give isl_union_set *
isl_schedule_node_domain_get_domain(
__isl_keep isl_schedule_node *node);
#include 
__isl_give isl_union_map *
isl_schedule_node_expansion_get_expansion(
__isl_keep isl_schedule_node *node);
__isl_give isl_union_pw_multi_aff *
isl_schedule_node_expansion_get_contraction(
__isl_keep isl_schedule_node *node);
#include 
__isl_give isl_union_map *
isl_schedule_node_extension_get_extension(
__isl_keep isl_schedule_node *node);

176

#include 
__isl_give isl_union_set *
isl_schedule_node_filter_get_filter(
__isl_keep isl_schedule_node *node);
#include 
__isl_give isl_set *isl_schedule_node_guard_get_guard(
__isl_keep isl_schedule_node *node);
#include 
__isl_give isl_id *isl_schedule_node_mark_get_id(
__isl_keep isl_schedule_node *node);
The following functions can be used to obtain an isl_multi_union_pw_aff, an
isl_union_pw_multi_aff or isl_union_map representation of partial schedules
related to the node.
#include 
__isl_give isl_multi_union_pw_aff *
isl_schedule_node_get_prefix_schedule_multi_union_pw_aff(
__isl_keep isl_schedule_node *node);
__isl_give isl_union_pw_multi_aff *
isl_schedule_node_get_prefix_schedule_union_pw_multi_aff(
__isl_keep isl_schedule_node *node);
__isl_give isl_union_map *
isl_schedule_node_get_prefix_schedule_union_map(
__isl_keep isl_schedule_node *node);
__isl_give isl_union_map *
isl_schedule_node_get_prefix_schedule_relation(
__isl_keep isl_schedule_node *node);
__isl_give isl_union_map *
isl_schedule_node_get_subtree_schedule_union_map(
__isl_keep isl_schedule_node *node);

In particular, the functions isl_schedule_node_get_prefix_schedule_multi_union_pw_aff,
isl_schedule_node_get_prefix_schedule_union_pw_multi_aff and isl_schedule_node_get_prefix_sch
return a relative ordering on the domain elements that reach the given node determined
by its ancestors. The function isl_schedule_node_get_prefix_schedule_relation
additionally includes the domain constraints in the result. The function isl_schedule_node_get_subtree_schedule_
returns a representation of the partial schedule defined by the subtree rooted at the given
node. If the tree contains any expansion nodes, then the subtree schedule is formulated
in terms of the expanded domain elements. The tree passed to functions returning a
prefix schedule may only contain extension nodes if these would not affect the result
of these functions. That is, if one of the ancestors is an extension node, then all of
the domain elements that were added by the extension node need to have been filtered
out by filter nodes between the extension node and the input node. The tree passed

177

to isl_schedule_node_get_subtree_schedule_union_map may not contain in
extension nodes in the selected subtree.
The expansion/contraction defined by an entire subtree, combining the expansions/contractions on the expansion nodes in the subtree, can be obtained using the following
functions.
#include 
__isl_give isl_union_map *
isl_schedule_node_get_subtree_expansion(
__isl_keep isl_schedule_node *node);
__isl_give isl_union_pw_multi_aff *
isl_schedule_node_get_subtree_contraction(
__isl_keep isl_schedule_node *node);
The total number of outer band members of given node, i.e., the shared output dimension of the maps in the result of isl_schedule_node_get_prefix_schedule_union_map
can be obtained using the following function.
#include 
int isl_schedule_node_get_schedule_depth(
__isl_keep isl_schedule_node *node);
The following functions return the elements that reach the given node or the union
of universes in the spaces that contain these elements.
#include 
__isl_give isl_union_set *
isl_schedule_node_get_domain(
__isl_keep isl_schedule_node *node);
__isl_give isl_union_set *
isl_schedule_node_get_universe_domain(
__isl_keep isl_schedule_node *node);
The input tree of isl_schedule_node_get_domain may only contain extension
nodes if these would not affect the result of this function. That is, if one of the ancestors
is an extension node, then all of the domain elements that were added by the extension
node need to have been filtered out by filter nodes between the extension node and the
input node.
The following functions can be used to introduce additional nodes in the schedule
tree. The new node is introduced at the point in the tree where the isl_schedule_node
points to and the results points to the new node.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_insert_partial_schedule(
__isl_take isl_schedule_node *node,
__isl_take isl_multi_union_pw_aff *schedule);
178

This function inserts a new band node with (the greatest integer part of) the given
partial schedule. The subtree rooted at the given node is assumed not to have any
anchored nodes.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_insert_context(
__isl_take isl_schedule_node *node,
__isl_take isl_set *context);
This function inserts a new context node with the given context constraints.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_insert_filter(
__isl_take isl_schedule_node *node,
__isl_take isl_union_set *filter);
This function inserts a new filter node with the given filter. If the original node
already pointed to a filter node, then the two filter nodes are merged into one.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_insert_guard(
__isl_take isl_schedule_node *node,
__isl_take isl_set *guard);
This function inserts a new guard node with the given guard constraints.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_insert_mark(
__isl_take isl_schedule_node *node,
__isl_take isl_id *mark);
This function inserts a new mark node with the give mark identifier.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_insert_sequence(
__isl_take isl_schedule_node *node,
__isl_take isl_union_set_list *filters);
__isl_give isl_schedule_node *
isl_schedule_node_insert_set(
__isl_take isl_schedule_node *node,
__isl_take isl_union_set_list *filters);
These functions insert a new sequence or set node with the given filters as children.
179

#include 
__isl_give isl_schedule_node *isl_schedule_node_group(
__isl_take isl_schedule_node *node,
__isl_take isl_id *group_id);
This function introduces an expansion node in between the current node and its
parent that expands instances of a space with tuple identifier group_id to the original
domain elements that reach the node. The group instances are identified by the prefix
schedule of those domain elements. The ancestors of the node are adjusted to refer to
the group instances instead of the original domain elements. The return value points to
the same node in the updated schedule tree as the input node, i.e., to the child of the
newly introduced expansion node. Grouping instances of different statements ensures
that they will be treated as a single statement by the AST generator up to the point of
the expansion node.
The following function can be used to flatten a nested sequence.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_sequence_splice_child(
__isl_take isl_schedule_node *node, int pos);
That is, given a sequence node node that has another sequence node in its child at
position pos (in particular, the child of that filter node is a sequence node), attach the
children of that other sequence node as children of node, replacing the original child
at position pos.
The partial schedule of a band node can be scaled (down) or reduced using the
following functions.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_band_scale(
__isl_take isl_schedule_node *node,
__isl_take isl_multi_val *mv);
__isl_give isl_schedule_node *
isl_schedule_node_band_scale_down(
__isl_take isl_schedule_node *node,
__isl_take isl_multi_val *mv);
__isl_give isl_schedule_node *
isl_schedule_node_band_mod(
__isl_take isl_schedule_node *node,
__isl_take isl_multi_val *mv);
The spaces of the two arguments need to match. After scaling, the partial schedule
is replaced by its greatest integer part to ensure that the schedule remains integral.
The partial schedule of a band node can be shifted by an isl_multi_union_pw_aff
with a domain that is a superset of the domain of the partial schedule using the following function.
180

#include 
__isl_give isl_schedule_node *
isl_schedule_node_band_shift(
__isl_take isl_schedule_node *node,
__isl_take isl_multi_union_pw_aff *shift);
A band node can be tiled using the following function.
#include 
__isl_give isl_schedule_node *isl_schedule_node_band_tile(
__isl_take isl_schedule_node *node,
__isl_take isl_multi_val *sizes);
isl_stat isl_options_set_tile_scale_tile_loops(isl_ctx *ctx,
int val);
int isl_options_get_tile_scale_tile_loops(isl_ctx *ctx);
isl_stat isl_options_set_tile_shift_point_loops(isl_ctx *ctx,
int val);
int isl_options_get_tile_shift_point_loops(isl_ctx *ctx);
The isl_schedule_node_band_tile function tiles the band using the given tile
sizes inside its schedule. A new child band node is created to represent the point loops
and it is inserted between the modified band and its children. The subtree rooted at the
given node is assumed not to have any anchored nodes. The tile_scale_tile_loops
option specifies whether the tile loops iterators should be scaled by the tile sizes. If the
tile_shift_point_loops option is set, then the point loops are shifted to start at
zero.
A band node can be split into two nested band nodes using the following function.
#include 
__isl_give isl_schedule_node *isl_schedule_node_band_split(
__isl_take isl_schedule_node *node, int pos);
The resulting outer band node contains the first pos dimensions of the schedule of
node while the inner band contains the remaining dimensions. The schedules of the
two band nodes live in anonymous spaces. The loop AST generation type options and
the isolate option are split over the two band nodes.
A band node can be moved down to the leaves of the subtree rooted at the band
node using the following function.
#include 
__isl_give isl_schedule_node *isl_schedule_node_band_sink(
__isl_take isl_schedule_node *node);
The subtree rooted at the given node is assumed not to have any anchored nodes.
The result points to the node in the resulting tree that is in the same position as the node
pointed to by node in the original tree.

181

#include 
__isl_give isl_schedule_node *
isl_schedule_node_order_before(
__isl_take isl_schedule_node *node,
__isl_take isl_union_set *filter);
__isl_give isl_schedule_node *
isl_schedule_node_order_after(
__isl_take isl_schedule_node *node,
__isl_take isl_union_set *filter);
These functions split the domain elements that reach node into those that satisfy
filter and those that do not and arranges for the elements that do satisfy the filter to
be executed before (in case of isl_schedule_node_order_before) or after (in case
of isl_schedule_node_order_after) those that do not. The order is imposed by a
sequence node, possibly reusing the grandparent of node on two copies of the subtree
attached to the original node. Both copies are simplified with respect to their filter.
Return a pointer to the copy of the subtree that does not satisfy filter. If there is
no such copy (because all reaching domain elements satisfy the filter), then return the
original pointer.
#include 
__isl_give isl_schedule_node *
isl_schedule_node_graft_before(
__isl_take isl_schedule_node
__isl_take isl_schedule_node
__isl_give isl_schedule_node *
isl_schedule_node_graft_after(
__isl_take isl_schedule_node
__isl_take isl_schedule_node

*node,
*graft);

*node,
*graft);

This function inserts the graft tree into the tree containing node such that it is
executed before (in case of isl_schedule_node_graft_before) or after (in case
of isl_schedule_node_graft_after) node. The root node of graft should be an
extension node where the domain of the extension is the flat product of all outer band
nodes of node. The root node may also be a domain node. The elements of the domain
or the range of the extension may not intersect with the domain elements that reach
"node". The schedule tree of graft may not be anchored.
The schedule tree of node is modified to include an extension node corresponding
to the root node of graft as a child of the original parent of node. The original
node that node points to and the child of the root node of graft are attached to this
extension node through a sequence, with appropriate filters and with the child of graft
appearing before or after the original node.
If node already appears inside a sequence that is the child of an extension node
and if the spaces of the new domain elements do not overlap with those of the original
domain elements, then that extension node is extended with the new extension rather
than introducing a new segment of extension and sequence nodes.

182

Return a pointer to the same node in the modified tree that node pointed to in the
original tree.
A representation of the schedule node can be printed using
#include 
__isl_give isl_printer *isl_printer_print_schedule_node(
__isl_take isl_printer *p,
__isl_keep isl_schedule_node *node);
__isl_give char *isl_schedule_node_to_str(
__isl_keep isl_schedule_node *node);
isl_schedule_node_to_str prints the schedule node in block format.

1.5.2

Dependence Analysis

isl contains specialized functionality for performing array dataflow analysis. That is,
given a sink access relation, a collection of possible source accesses and a collection
of kill accesses, isl can compute relations that describe for each iteration of the sink
access, which iterations of which of the source access relations may have accessed the
same data element before the given iteration of the sink access without any intermediate
kill of that data element. The resulting dependence relations map source iterations to
either the corresponding sink iterations or pairs of corresponding sink iterations and
accessed data elements. To compute standard flow dependences, the sink should be a
read, while the sources should be writes. If no kills are specified, then memory based
dependence analysis is performed. If, on the other hand, all sources are also kills,
then value based dependence analysis is performed. If any of the source accesses are
marked as being must accesses, then they are also treated as kills. Furthermore, the
specification of must-sources results in the computation of must-dependences. Only
dependences originating in a must access not coscheduled with any other access to
the same element and without any may accesses between the must access and the sink
access are considered to be must dependences.
High-level Interface
A high-level interface to dependence analysis is provided by the following function.
#include 
__isl_give isl_union_flow *
isl_union_access_info_compute_flow(
__isl_take isl_union_access_info *access);
The input isl_union_access_info object describes the sink access relations,
the source access relations and a schedule, while the output isl_union_flow object
describes the resulting dependence relations and the subsets of the sink relations for
which no source was found.
An isl_union_access_info is created, modified, copied and freed using the
following functions.
183

#include 
__isl_give isl_union_access_info *
isl_union_access_info_from_sink(
__isl_take isl_union_map *sink);
__isl_give isl_union_access_info *
isl_union_access_info_set_kill(
__isl_take isl_union_access_info *access,
__isl_take isl_union_map *kill);
__isl_give isl_union_access_info *
isl_union_access_info_set_may_source(
__isl_take isl_union_access_info *access,
__isl_take isl_union_map *may_source);
__isl_give isl_union_access_info *
isl_union_access_info_set_must_source(
__isl_take isl_union_access_info *access,
__isl_take isl_union_map *must_source);
__isl_give isl_union_access_info *
isl_union_access_info_set_schedule(
__isl_take isl_union_access_info *access,
__isl_take isl_schedule *schedule);
__isl_give isl_union_access_info *
isl_union_access_info_set_schedule_map(
__isl_take isl_union_access_info *access,
__isl_take isl_union_map *schedule_map);
__isl_give isl_union_access_info *
isl_union_access_info_copy(
__isl_keep isl_union_access_info *access);
__isl_null isl_union_access_info *
isl_union_access_info_free(
__isl_take isl_union_access_info *access);
The may sources set by isl_union_access_info_set_may_source do not need
to include the must sources set by isl_union_access_info_set_must_source as
a subset. The kills set by isl_union_access_info_set_kill may overlap with the
may-sources and/or must-sources. The user is free not to call one (or more) of these
functions, in which case the corresponding set is kept to its empty default. Similarly,
the default schedule initialized by isl_union_access_info_from_sink is empty.
The current schedule is determined by the last call to either isl_union_access_info_set_schedule
or isl_union_access_info_set_schedule_map. The domain of the schedule corresponds to the domains of the access relations. In particular, the domains of the access
relations are effectively intersected with the domain of the schedule and only the resulting accesses are considered by the dependence analysis.
An isl_union_access_info object can be read from input using the following
function.
#include 

184

__isl_give isl_union_access_info *
isl_union_access_info_read_from_file(isl_ctx *ctx,
FILE *input);
A representation of the information contained in an object of type isl_union_access_info
can be obtained using
#include 
__isl_give isl_printer *
isl_printer_print_union_access_info(
__isl_take isl_printer *p,
__isl_keep isl_union_access_info *access);
__isl_give char *isl_union_access_info_to_str(
__isl_keep isl_union_access_info *access);
isl_union_access_info_to_str prints the information in flow format.
The output of isl_union_access_info_compute_flow can be examined, copied,
and freed using the following functions.
#include 
__isl_give isl_union_map *isl_union_flow_get_must_dependence(
__isl_keep isl_union_flow *flow);
__isl_give isl_union_map *isl_union_flow_get_may_dependence(
__isl_keep isl_union_flow *flow);
__isl_give isl_union_map *
isl_union_flow_get_full_must_dependence(
__isl_keep isl_union_flow *flow);
__isl_give isl_union_map *
isl_union_flow_get_full_may_dependence(
__isl_keep isl_union_flow *flow);
__isl_give isl_union_map *isl_union_flow_get_must_no_source(
__isl_keep isl_union_flow *flow);
__isl_give isl_union_map *isl_union_flow_get_may_no_source(
__isl_keep isl_union_flow *flow);
__isl_give isl_union_flow *isl_union_flow_copy(
__isl_keep isl_union_flow *flow);
__isl_null isl_union_flow *isl_union_flow_free(
__isl_take isl_union_flow *flow);
The relation returned by isl_union_flow_get_must_dependence relates domain elements of must sources to domain elements of the sink. The relation returned
by isl_union_flow_get_may_dependence relates domain elements of must or may
sources to domain elements of the sink and includes the previous relation as a subset.
The relation returned by isl_union_flow_get_full_must_dependence relates domain elements of must sources to pairs of domain elements of the sink and accessed
data elements. The relation returned by isl_union_flow_get_full_may_dependence
relates domain elements of must or may sources to pairs of domain elements of the
185

sink and accessed data elements. This relation includes the previous relation as a subset. The relation returned by isl_union_flow_get_must_no_source is the subset of the sink relation for which no dependences have been found. The relation
returned by isl_union_flow_get_may_no_source is the subset of the sink relation for which no definite dependences have been found. That is, it contains those
sink access that do not contribute to any of the elements in the relation returned by
isl_union_flow_get_must_dependence.
A representation of the information contained in an object of type isl_union_flow
can be obtained using
#include 
__isl_give isl_printer *isl_printer_print_union_flow(
__isl_take isl_printer *p,
__isl_keep isl_union_flow *flow);
__isl_give char *isl_union_flow_to_str(
__isl_keep isl_union_flow *flow);
isl_union_flow_to_str prints the information in flow format.
Low-level Interface
A lower-level interface is provided by the following functions.
#include 
typedef int (*isl_access_level_before)(void *first, void *second);
__isl_give isl_access_info *isl_access_info_alloc(
__isl_take isl_map *sink,
void *sink_user, isl_access_level_before fn,
int max_source);
__isl_give isl_access_info *isl_access_info_add_source(
__isl_take isl_access_info *acc,
__isl_take isl_map *source, int must,
void *source_user);
__isl_null isl_access_info *isl_access_info_free(
__isl_take isl_access_info *acc);
__isl_give isl_flow *isl_access_info_compute_flow(
__isl_take isl_access_info *acc);
isl_stat isl_flow_foreach(__isl_keep isl_flow *deps,
isl_stat (*fn)(__isl_take isl_map *dep, int must,
void *dep_user, void *user),
void *user);
__isl_give isl_map *isl_flow_get_no_source(
__isl_keep isl_flow *deps, int must);
void isl_flow_free(__isl_take isl_flow *deps);
186

The function isl_access_info_compute_flow performs the actual dependence
analysis. The other functions are used to construct the input for this function or to read
off the output.
The input is collected in an isl_access_info, which can be created through a
call to isl_access_info_alloc. The arguments to this functions are the sink access
relation sink, a token sink_user used to identify the sink access to the user, a callback
function for specifying the relative order of source and sink accesses, and the number
of source access relations that will be added.
The callback function has type int (*)(void *first, void *second). The
function is called with two user supplied tokens identifying either a source or the sink
and it should return the shared nesting level and the relative order of the two accesses.
In particular, let n be the number of loops shared by the two accesses. If first precedes second textually, then the function should return 2 * n + 1; otherwise, it should
return 2 * n. The low-level interface assumes that no sources are coscheduled. If the
information returned by the callback does not allow the relative order to be determined,
then one of the sources is arbitrarily taken to be executed after the other(s).
The sources can be added to the isl_access_info object by performing (at most)
max_source calls to isl_access_info_add_source. must indicates whether the
source is a must access or a may access. Note that a multi-valued access relation
should only be marked must if every iteration in the domain of the relation accesses
all elements in its image. The source_user token is again used to identify the source
access. The range of the source access relation source should have the same dimension as the range of the sink access relation. The isl_access_info_free function should usually not be called explicitly, because it is already called implicitly by
isl_access_info_compute_flow.
The result of the dependence analysis is collected in an isl_flow. There may be
elements of the sink access for which no preceding source access could be found or for
which all preceding sources are may accesses. The relations containing these elements
can be obtained through calls to isl_flow_get_no_source, the first with must set
and the second with must unset. In the case of standard flow dependence analysis,
with the sink a read and the sources must writes, the first relation corresponds to the
reads from uninitialized array elements and the second relation is empty. The actual
flow dependences can be extracted using isl_flow_foreach. This function will call
the user-specified callback function fn for each non-empty dependence between a
source and the sink. The callback function is called with four arguments, the actual
flow dependence relation mapping source iterations to sink iterations, a boolean that
indicates whether it is a must or may dependence, a token identifying the source and an
additional void * with value equal to the third argument of the isl_flow_foreach
call. A dependence is marked must if it originates from a must source and if it is not
followed by any may sources.
After finishing with an isl_flow, the user should call isl_flow_free to free all
associated memory.

187

Interaction with the Low-level Interface
During the dependence analysis, we frequently need to perform the following operation. Given a relation between sink iterations and potential source iterations from a
particular source domain, what is the last potential source iteration corresponding to
each sink iteration. It can sometimes be convenient to adjust the set of potential source
iterations before or after each such operation. The prototypical example is fuzzy array
dataflow analysis, where we need to analyze if, based on data-dependent constraints,
the sink iteration can ever be executed without one or more of the corresponding potential source iterations being executed. If so, we can introduce extra parameters and
select an unknown but fixed source iteration from the potential source iterations. To be
able to perform such manipulations, isl provides the following function.
#include 
typedef __isl_give isl_restriction *(*isl_access_restrict)(
__isl_keep isl_map *source_map,
__isl_keep isl_set *sink, void *source_user,
void *user);
__isl_give isl_access_info *isl_access_info_set_restrict(
__isl_take isl_access_info *acc,
isl_access_restrict fn, void *user);
The function isl_access_info_set_restrict should be called before calling
isl_access_info_compute_flow and registers a callback function that will be called
any time isl is about to compute the last potential source. The first argument is the
(reverse) proto-dependence, mapping sink iterations to potential source iterations. The
second argument represents the sink iterations for which we want to compute the last
source iteration. The third argument is the token corresponding to the source and the
final argument is the token passed to isl_access_info_set_restrict. The callback is expected to return a restriction on either the input or the output of the operation
computing the last potential source. If the input needs to be restricted then restrictions
are needed for both the source and the sink iterations. The sink iterations and the potential source iterations will be intersected with these sets. If the output needs to be
restricted then only a restriction on the source iterations is required. If any error occurs, the callback should return NULL. An isl_restriction object can be created,
freed and inspected using the following functions.
#include 
__isl_give isl_restriction
__isl_take isl_set
__isl_take isl_set
__isl_give isl_restriction
__isl_take isl_set
__isl_give isl_restriction
__isl_take isl_map
188

*isl_restriction_input(
*source_restr,
*sink_restr);
*isl_restriction_output(
*source_restr);
*isl_restriction_none(
*source_map);

__isl_give isl_restriction *isl_restriction_empty(
__isl_take isl_map *source_map);
__isl_null isl_restriction *isl_restriction_free(
__isl_take isl_restriction *restr);
isl_restriction_none and isl_restriction_empty are special cases of isl_restriction_input.
isl_restriction_none is essentially equivalent to
isl_restriction_input(isl_set_universe(
isl_space_range(isl_map_get_space(source_map))),
isl_set_universe(
isl_space_domain(isl_map_get_space(source_map))));
whereas isl_restriction_empty is essentially equivalent to
isl_restriction_input(isl_set_empty(
isl_space_range(isl_map_get_space(source_map))),
isl_set_universe(
isl_space_domain(isl_map_get_space(source_map))));

1.5.3

Scheduling
#include 
__isl_give isl_schedule *
isl_schedule_constraints_compute_schedule(
__isl_take isl_schedule_constraints *sc);

The function isl_schedule_constraints_compute_schedule can be used to
compute a schedule that satisfies the given schedule constraints. These schedule constraints include the iteration domain for which a schedule should be computed and dependences between pairs of iterations. In particular, these dependences include validity
dependences and proximity dependences. By default, the algorithm used to construct
the schedule is similar to that of Pluto. Alternatively, Feautrier’s multi-dimensional
scheduling algorithm can be selected. The generated schedule respects all validity dependences. That is, all dependence distances over these dependences in the scheduled
space are lexicographically positive.
The default algorithm tries to ensure that the dependence distances over coincidence constraints are zero and to minimize the dependence distances over proximity
dependences. Moreover, it tries to obtain sequences (bands) of schedule dimensions
for groups of domains where the dependence distances over validity dependences have
only non-negative values. Note that when minimizing the maximal dependence distance over proximity dependences, a single affine expression in the parameters is constructed that bounds all dependence distances. If no such expression exists, then the
algorithm will fail and resort to an alternative scheduling algorithm. In particular, this
means that adding proximity dependences may eliminate valid solutions. A typical example where this phenomenon may occur is when some subset of the proximity dependences has no restriction on some parameter, forcing the coefficient of that parameter
189

to be zero, while some other subset forces the dependence distance to depend on that
parameter, requiring the same coefficient to be non-zero. When using Feautrier’s algorithm, the coincidence and proximity constraints are only taken into account during the
extension to a full-dimensional schedule.
An isl_schedule_constraints object can be constructed and manipulated using the following functions.
#include 
__isl_give isl_schedule_constraints *
isl_schedule_constraints_copy(
__isl_keep isl_schedule_constraints *sc);
__isl_give isl_schedule_constraints *
isl_schedule_constraints_on_domain(
__isl_take isl_union_set *domain);
__isl_give isl_schedule_constraints *
isl_schedule_constraints_set_context(
__isl_take isl_schedule_constraints *sc,
__isl_take isl_set *context);
__isl_give isl_schedule_constraints *
isl_schedule_constraints_set_validity(
__isl_take isl_schedule_constraints *sc,
__isl_take isl_union_map *validity);
__isl_give isl_schedule_constraints *
isl_schedule_constraints_set_coincidence(
__isl_take isl_schedule_constraints *sc,
__isl_take isl_union_map *coincidence);
__isl_give isl_schedule_constraints *
isl_schedule_constraints_set_proximity(
__isl_take isl_schedule_constraints *sc,
__isl_take isl_union_map *proximity);
__isl_give isl_schedule_constraints *
isl_schedule_constraints_set_conditional_validity(
__isl_take isl_schedule_constraints *sc,
__isl_take isl_union_map *condition,
__isl_take isl_union_map *validity);
__isl_give isl_schedule_constraints *
isl_schedule_constraints_apply(
__isl_take isl_schedule_constraints *sc,
__isl_take isl_union_map *umap);
__isl_null isl_schedule_constraints *
isl_schedule_constraints_free(
__isl_take isl_schedule_constraints *sc);
The initial isl_schedule_constraints object created by isl_schedule_constraints_on_domain
does not impose any constraints. That is, it has an empty set of dependences. The
function isl_schedule_constraints_set_context allows the user to specify ad-

190

ditional constraints on the parameters that may be assumed to hold during the construction of the schedule. The function isl_schedule_constraints_set_validity replaces the validity dependences, mapping domain elements i to domain elements that
should be scheduled after i. The function isl_schedule_constraints_set_coincidence
replaces the coincidence dependences, mapping domain elements i to domain elements
that should be scheduled together with I, if possible. The function isl_schedule_constraints_set_proximity
replaces the proximity dependences, mapping domain elements i to domain elements
that should be scheduled either before I or as early as possible after i.
The function isl_schedule_constraints_set_conditional_validity replaces the conditional validity constraints. A conditional validity constraint is only
imposed when any of the corresponding conditions is satisfied, i.e., when any of them
is non-zero. That is, the scheduler ensures that within each band if the dependence distances over the condition constraints are not all zero then all corresponding conditional
validity constraints are respected. A conditional validity constraint corresponds to a
condition if the two are adjacent, i.e., if the domain of one relation intersect the range
of the other relation. The typical use case of conditional validity constraints is to allow
order constraints between live ranges to be violated as long as the live ranges themselves are local to the band. To allow more fine-grained control over which conditions
correspond to which conditional validity constraints, the domains and ranges of these
relations may include tags. That is, the domains and ranges of those relation may themselves be wrapped relations where the iteration domain appears in the domain of those
wrapped relations and the range of the wrapped relations can be arbitrarily chosen by
the user. Conditions and conditional validity constraints are only considered adjacent
to each other if the entire wrapped relation matches. In particular, a relation with a tag
will never be considered adjacent to a relation without a tag.
The function isl_schedule_constraints_apply takes schedule constraints that
are defined on some set of domain elements and transforms them to schedule constraints on the elements to which these domain elements are mapped by the given
transformation.
An isl_schedule_constraints object can be inspected using the following
functions.
#include 
__isl_give isl_union_set *
isl_schedule_constraints_get_domain(
__isl_keep isl_schedule_constraints *sc);
__isl_give isl_set *isl_schedule_constraints_get_context(
__isl_keep isl_schedule_constraints *sc);
__isl_give isl_union_map *
isl_schedule_constraints_get_validity(
__isl_keep isl_schedule_constraints *sc);
__isl_give isl_union_map *
isl_schedule_constraints_get_coincidence(
__isl_keep isl_schedule_constraints *sc);
__isl_give isl_union_map *
isl_schedule_constraints_get_proximity(
191

__isl_keep isl_schedule_constraints *sc);
__isl_give isl_union_map *
isl_schedule_constraints_get_conditional_validity(
__isl_keep isl_schedule_constraints *sc);
__isl_give isl_union_map *
isl_schedule_constraints_get_conditional_validity_condition(
__isl_keep isl_schedule_constraints *sc);
An isl_schedule_constraints object can be read from input using the following functions.
#include 
__isl_give isl_schedule_constraints *
isl_schedule_constraints_read_from_str(isl_ctx *ctx,
const char *str);
__isl_give isl_schedule_constraints *
isl_schedule_constraints_read_from_file(isl_ctx *ctx,
FILE *input);
The contents of an isl_schedule_constraints object can be printed using the
following functions.
#include 
__isl_give isl_printer *
isl_printer_print_schedule_constraints(
__isl_take isl_printer *p,
__isl_keep isl_schedule_constraints *sc);
__isl_give char *isl_schedule_constraints_to_str(
__isl_keep isl_schedule_constraints *sc);
The following function computes a schedule directly from an iteration domain and
validity and proximity dependences and is implemented in terms of the functions described above. The use of isl_union_set_compute_schedule is discouraged.
#include 
__isl_give isl_schedule *isl_union_set_compute_schedule(
__isl_take isl_union_set *domain,
__isl_take isl_union_map *validity,
__isl_take isl_union_map *proximity);
The generated schedule represents a schedule tree. For more information on schedule trees, see §1.5.1.
Options
#include 
isl_stat isl_options_set_schedule_max_coefficient(

192

isl_ctx *ctx, int val);
int isl_options_get_schedule_max_coefficient(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_max_constant_term(
isl_ctx *ctx, int val);
int isl_options_get_schedule_max_constant_term(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_serialize_sccs(
isl_ctx *ctx, int val);
int isl_options_get_schedule_serialize_sccs(isl_ctx *ctx);
isl_stat isl_options_set_schedule_whole_component(
isl_ctx *ctx, int val);
int isl_options_get_schedule_whole_component(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_maximize_band_depth(
isl_ctx *ctx, int val);
int isl_options_get_schedule_maximize_band_depth(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_maximize_coincidence(
isl_ctx *ctx, int val);
int isl_options_get_schedule_maximize_coincidence(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_outer_coincidence(
isl_ctx *ctx, int val);
int isl_options_get_schedule_outer_coincidence(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_split_scaled(
isl_ctx *ctx, int val);
int isl_options_get_schedule_split_scaled(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_treat_coalescing(
isl_ctx *ctx, int val);
int isl_options_get_schedule_treat_coalescing(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_algorithm(
isl_ctx *ctx, int val);
int isl_options_get_schedule_algorithm(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_carry_self_first(
isl_ctx *ctx, int val);
int isl_options_get_schedule_carry_self_first(
isl_ctx *ctx);
isl_stat isl_options_set_schedule_separate_components(
isl_ctx *ctx, int val);
int isl_options_get_schedule_separate_components(
isl_ctx *ctx);
193

• schedule_max_coefficient
This option enforces that the coefficients for variable and parameter dimensions
in the calculated schedule are not larger than the specified value. This option
can significantly increase the speed of the scheduling calculation and may also
prevent fusing of unrelated dimensions. A value of -1 means that this option
does not introduce bounds on the variable or parameter coefficients. This option
has no effect on the Feautrier style scheduler.
• schedule_max_constant_term
This option enforces that the constant coefficients in the calculated schedule are
not larger than the maximal constant term. This option can significantly increase
the speed of the scheduling calculation and may also prevent fusing of unrelated
dimensions. A value of -1 means that this option does not introduce bounds on
the constant coefficients.
• schedule_serialize_sccs
If this option is set, then all strongly connected components in the dependence
graph are serialized as soon as they are detected. This means in particular that
instances of statements will only appear in the same band node if these statements belong to the same strongly connected component at the point where the
band node is constructed.
• schedule_whole_component
If this option is set, then entire (weakly) connected components in the dependence graph are scheduled together as a whole. Otherwise, each strongly connected component within such a weakly connected component is first scheduled
separately and then combined with other strongly connected components. This
option has no effect if schedule_serialize_sccs is set.
• schedule_maximize_band_depth
If this option is set, then the scheduler tries to maximize the width of the bands.
Wider bands give more possibilities for tiling. In particular, if the schedule_whole_component
option is set, then bands are split if this might result in wider bands. Otherwise, the effect of this option is to only allow strongly connected components
to be combined if this does not reduce the width of the bands. Note that if the
schedule_serialize_sccs options is set, then the schedule_maximize_band_depth
option therefore has no effect.
• schedule_maximize_coincidence
This option is only effective if the schedule_whole_component option is turned
off. If the schedule_maximize_coincidence option is set, then (clusters of)
strongly connected components are only combined with each other if this does
not reduce the number of coincident band members.
• schedule_outer_coincidence

194

If this option is set, then we try to construct schedules where the outermost
scheduling dimension in each band satisfies the coincidence constraints.
• schedule_algorithm
Selects the scheduling algorithm to be used. Available scheduling algorithms are
ISL_SCHEDULE_ALGORITHM_ISL and ISL_SCHEDULE_ALGORITHM_FEAUTRIER.
• schedule_split_scaled
If this option is set, then we try to construct schedules in which the constant
term is split off from the linear part if the linear parts of the scheduling rows for
all nodes in the graph have a common non-trivial divisor. The constant term is
then dropped and the linear part is reduced. This option is only effective when
the Feautrier style scheduler is being used, either as the main scheduler or as a
fallback for the Pluto-like scheduler.
• schedule_treat_coalescing
If this option is set, then the scheduler will try and avoid producing schedules that
perform loop coalescing. In particular, for the Pluto-like scheduler, this option
places bounds on the schedule coefficients based on the sizes of the instance
sets. For the Feautrier style scheduler, this option detects potentially coalescing
schedules and then tries to adjust the schedule to avoid the coalescing.
• schedule_carry_self_first
If this option is set, then the Feautrier style scheduler (when used as a fallback
for the Pluto-like scheduler) will first try to only carry self-dependences.
• schedule_separate_components
If this option is set then the function isl_schedule_get_map will treat set
nodes in the same way as sequence nodes.

1.5.4

AST Generation

This section describes the isl functionality for generating ASTs that visit all the elements in a domain in an order specified by a schedule tree or a schedule map. In
case the schedule given as a isl_union_map, an AST is generated that visits all the
elements in the domain of the isl_union_map according to the lexicographic order
of the corresponding image element(s). If the range of the isl_union_map consists
of elements in more than one space, then each of these spaces is handled separately
in an arbitrary order. It should be noted that the schedule tree or the image elements
in a schedule map only specify the order in which the corresponding domain elements
should be visited. No direct relation between the partial schedule values or the image
elements on the one hand and the loop iterators in the generated AST on the other hand
should be assumed.
Each AST is generated within a build. The initial build simply specifies the constraints on the parameters (if any) and can be created, inspected, copied and freed using
the following functions.
195

#include 
__isl_give isl_ast_build *isl_ast_build_alloc(
isl_ctx *ctx);
__isl_give isl_ast_build *isl_ast_build_from_context(
__isl_take isl_set *set);
__isl_give isl_ast_build *isl_ast_build_copy(
__isl_keep isl_ast_build *build);
__isl_null isl_ast_build *isl_ast_build_free(
__isl_take isl_ast_build *build);
The set argument is usually a parameter set with zero or more parameters. In fact,
when creating an AST using isl_ast_build_node_from_schedule, this set is required to be a parameter set. An isl_ast_build created using isl_ast_build_alloc
does not specify any parameter constraints. More isl_ast_build functions are described in §1.5.4 and §1.5.4. Finally, the AST itself can be constructed using one of
the following functions.
#include 
__isl_give isl_ast_node *isl_ast_build_node_from_schedule(
__isl_keep isl_ast_build *build,
__isl_take isl_schedule *schedule);
__isl_give isl_ast_node *
isl_ast_build_node_from_schedule_map(
__isl_keep isl_ast_build *build,
__isl_take isl_union_map *schedule);
Inspecting the AST
The basic properties of an AST node can be obtained as follows.
#include 
enum isl_ast_node_type isl_ast_node_get_type(
__isl_keep isl_ast_node *node);
The type of an AST node is one of isl_ast_node_for, isl_ast_node_if,
isl_ast_node_block, isl_ast_node_mark or isl_ast_node_user. An isl_ast_node_for
represents a for node. An isl_ast_node_if represents an if node. An isl_ast_node_block
represents a compound node. An isl_ast_node_mark introduces a mark in the AST.
An isl_ast_node_user represents an expression statement. An expression statement
typically corresponds to a domain element, i.e., one of the elements that is visited by
the AST.
Each type of node has its own additional properties.
#include 
__isl_give isl_ast_expr *isl_ast_node_for_get_iterator(
__isl_keep isl_ast_node *node);
__isl_give isl_ast_expr *isl_ast_node_for_get_init(
196

__isl_keep isl_ast_node *node);
__isl_give isl_ast_expr *isl_ast_node_for_get_cond(
__isl_keep isl_ast_node *node);
__isl_give isl_ast_expr *isl_ast_node_for_get_inc(
__isl_keep isl_ast_node *node);
__isl_give isl_ast_node *isl_ast_node_for_get_body(
__isl_keep isl_ast_node *node);
isl_bool isl_ast_node_for_is_degenerate(
__isl_keep isl_ast_node *node);
An isl_ast_for is considered degenerate if it is known to execute exactly once.
#include 
__isl_give isl_ast_expr *isl_ast_node_if_get_cond(
__isl_keep isl_ast_node *node);
__isl_give isl_ast_node *isl_ast_node_if_get_then(
__isl_keep isl_ast_node *node);
isl_bool isl_ast_node_if_has_else(
__isl_keep isl_ast_node *node);
__isl_give isl_ast_node *isl_ast_node_if_get_else(
__isl_keep isl_ast_node *node);
__isl_give isl_ast_node_list *
isl_ast_node_block_get_children(
__isl_keep isl_ast_node *node);
__isl_give isl_id *isl_ast_node_mark_get_id(
__isl_keep isl_ast_node *node);
__isl_give isl_ast_node *isl_ast_node_mark_get_node(
__isl_keep isl_ast_node *node);
isl_ast_node_mark_get_id returns the identifier of the mark. isl_ast_node_mark_get_node
returns the child node that is being marked.
#include 
__isl_give isl_ast_expr *isl_ast_node_user_get_expr(
__isl_keep isl_ast_node *node);
All descendants of a specific node in the AST (including the node itself) can be
visited in depth-first pre-order using the following function.
#include 
isl_stat isl_ast_node_foreach_descendant_top_down(
__isl_keep isl_ast_node *node,
isl_bool (*fn)(__isl_keep isl_ast_node *node,
void *user), void *user);

197

The callback function should return isl_bool_true if the children of the given
node should be visited and isl_bool_false if they should not. It should return
isl_bool_error in case of failure, in which case the entire traversal is aborted.
Each of the returned isl_ast_exprs can in turn be inspected using the following
functions.
#include 
enum isl_ast_expr_type isl_ast_expr_get_type(
__isl_keep isl_ast_expr *expr);
The type of an AST expression is one of isl_ast_expr_op, isl_ast_expr_id
or isl_ast_expr_int. An isl_ast_expr_op represents the result of an operation.
An isl_ast_expr_id represents an identifier. An isl_ast_expr_int represents an
integer value.
Each type of expression has its own additional properties.
#include 
enum isl_ast_op_type isl_ast_expr_get_op_type(
__isl_keep isl_ast_expr *expr);
int isl_ast_expr_get_op_n_arg(__isl_keep isl_ast_expr *expr);
__isl_give isl_ast_expr *isl_ast_expr_get_op_arg(
__isl_keep isl_ast_expr *expr, int pos);
isl_stat isl_ast_expr_foreach_ast_op_type(
__isl_keep isl_ast_expr *expr,
isl_stat (*fn)(enum isl_ast_op_type type,
void *user), void *user);
isl_stat isl_ast_node_foreach_ast_op_type(
__isl_keep isl_ast_node *node,
isl_stat (*fn)(enum isl_ast_op_type type,
void *user), void *user);
isl_ast_expr_get_op_type returns the type of the operation performed. isl_ast_expr_get_op_n_arg
returns the number of arguments. isl_ast_expr_get_op_arg returns the specified argument. isl_ast_expr_foreach_ast_op_type calls fn for each distinct
isl_ast_op_type that appears in expr. isl_ast_node_foreach_ast_op_type
does the same for each distinct isl_ast_op_type that appears in node. The operation type is one of the following.
isl_ast_op_and
Logical and of two arguments. Both arguments can be evaluated.
isl_ast_op_and_then
Logical and of two arguments. The second argument can only be evaluated if
the first evaluates to true.
isl_ast_op_or
Logical or of two arguments. Both arguments can be evaluated.
198

isl_ast_op_or_else
Logical or of two arguments. The second argument can only be evaluated if the
first evaluates to false.
isl_ast_op_max
Maximum of two or more arguments.
isl_ast_op_min
Minimum of two or more arguments.
isl_ast_op_minus
Change sign.
isl_ast_op_add
Sum of two arguments.
isl_ast_op_sub
Difference of two arguments.
isl_ast_op_mul
Product of two arguments.
isl_ast_op_div
Exact division. That is, the result is known to be an integer.
isl_ast_op_fdiv_q
Result of integer division, rounded towards negative infinity. The divisor is
known to be positive.
isl_ast_op_pdiv_q
Result of integer division, where dividend is known to be non-negative. The
divisor is known to be positive.
isl_ast_op_pdiv_r
Remainder of integer division, where dividend is known to be non-negative. The
divisor is known to be positive.
isl_ast_op_zdiv_r
Equal to zero iff the remainder on integer division is zero. The divisor is known
to be positive.
isl_ast_op_cond
Conditional operator defined on three arguments. If the first argument evaluates
to true, then the result is equal to the second argument. Otherwise, the result is
equal to the third argument. The second and third argument may only be evaluated if the first argument evaluates to true and false, respectively. Corresponds
to a ? b : c in C.
199

isl_ast_op_select
Conditional operator defined on three arguments. If the first argument evaluates
to true, then the result is equal to the second argument. Otherwise, the result is
equal to the third argument. The second and third argument may be evaluated
independently of the value of the first argument. Corresponds to a * b + (1 a) * c in C.
isl_ast_op_eq
Equality relation.
isl_ast_op_le
Less than or equal relation.
isl_ast_op_lt
Less than relation.
isl_ast_op_ge
Greater than or equal relation.
isl_ast_op_gt
Greater than relation.
isl_ast_op_call
A function call. The number of arguments of the isl_ast_expr is one more
than the number of arguments in the function call, the first argument representing
the function being called.
isl_ast_op_access
An array access. The number of arguments of the isl_ast_expr is one more
than the number of index expressions in the array access, the first argument representing the array being accessed.
isl_ast_op_member
A member access. This operation has two arguments, a structure and the name
of the member of the structure being accessed.
#include 
__isl_give isl_id *isl_ast_expr_get_id(
__isl_keep isl_ast_expr *expr);
Return the identifier represented by the AST expression.
#include 
__isl_give isl_val *isl_ast_expr_get_val(
__isl_keep isl_ast_expr *expr);
Return the integer represented by the AST expression.
200

Properties of ASTs
#include 
isl_bool isl_ast_expr_is_equal(
__isl_keep isl_ast_expr *expr1,
__isl_keep isl_ast_expr *expr2);
Check if two isl_ast_exprs are equal to each other.
Manipulating and printing the AST
AST nodes can be copied and freed using the following functions.
#include 
__isl_give isl_ast_node *isl_ast_node_copy(
__isl_keep isl_ast_node *node);
__isl_null isl_ast_node *isl_ast_node_free(
__isl_take isl_ast_node *node);
AST expressions can be copied and freed using the following functions.
#include 
__isl_give isl_ast_expr *isl_ast_expr_copy(
__isl_keep isl_ast_expr *expr);
__isl_null isl_ast_expr *isl_ast_expr_free(
__isl_take isl_ast_expr *expr);
New AST expressions can be created either directly or within the context of an
isl_ast_build.
#include 
__isl_give isl_ast_expr *isl_ast_expr_from_val(
__isl_take isl_val *v);
__isl_give isl_ast_expr *isl_ast_expr_from_id(
__isl_take isl_id *id);
__isl_give isl_ast_expr *isl_ast_expr_neg(
__isl_take isl_ast_expr *expr);
__isl_give isl_ast_expr *isl_ast_expr_address_of(
__isl_take isl_ast_expr *expr);
__isl_give isl_ast_expr *isl_ast_expr_add(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_sub(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_mul(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
201

__isl_give isl_ast_expr *isl_ast_expr_div(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_pdiv_q(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_pdiv_r(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_and(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2)
__isl_give isl_ast_expr *isl_ast_expr_and_then(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2)
__isl_give isl_ast_expr *isl_ast_expr_or(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2)
__isl_give isl_ast_expr *isl_ast_expr_or_else(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2)
__isl_give isl_ast_expr *isl_ast_expr_eq(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_le(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_lt(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_ge(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_gt(
__isl_take isl_ast_expr *expr1,
__isl_take isl_ast_expr *expr2);
__isl_give isl_ast_expr *isl_ast_expr_access(
__isl_take isl_ast_expr *array,
__isl_take isl_ast_expr_list *indices);
__isl_give isl_ast_expr *isl_ast_expr_call(
__isl_take isl_ast_expr *function,
__isl_take isl_ast_expr_list *arguments);
The function isl_ast_expr_address_of can be applied to an isl_ast_expr of
type isl_ast_op_access only. It is meant to represent the address of the isl_ast_expr_access.
The second argument of the functions isl_ast_expr_pdiv_q and isl_ast_expr_pdiv_r
202

should always evaluate to a positive number. The function isl_ast_expr_and_then
as well as isl_ast_expr_or_else are short-circuit versions of isl_ast_expr_and
and isl_ast_expr_or, respectively.
#include 
__isl_give isl_ast_expr *isl_ast_build_expr_from_set(
__isl_keep isl_ast_build *build,
__isl_take isl_set *set);
__isl_give isl_ast_expr *isl_ast_build_expr_from_pw_aff(
__isl_keep isl_ast_build *build,
__isl_take isl_pw_aff *pa);
__isl_give isl_ast_expr *
isl_ast_build_access_from_pw_multi_aff(
__isl_keep isl_ast_build *build,
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_ast_expr *
isl_ast_build_access_from_multi_pw_aff(
__isl_keep isl_ast_build *build,
__isl_take isl_multi_pw_aff *mpa);
__isl_give isl_ast_expr *
isl_ast_build_call_from_pw_multi_aff(
__isl_keep isl_ast_build *build,
__isl_take isl_pw_multi_aff *pma);
__isl_give isl_ast_expr *
isl_ast_build_call_from_multi_pw_aff(
__isl_keep isl_ast_build *build,
__isl_take isl_multi_pw_aff *mpa);
The set set and the domains of pa, mpa and pma should correspond to the schedule
space of build. The tuple id of mpa or pma is used as the array being accessed or
the function being called. If the accessed space is a nested relation, then it is taken to
represent an access of the member specified by the range of this nested relation of the
structure specified by the domain of the nested relation.
The following functions can be used to modify an isl_ast_expr.
#include 
__isl_give isl_ast_expr *isl_ast_expr_set_op_arg(
__isl_take isl_ast_expr *expr, int pos,
__isl_take isl_ast_expr *arg);
Replace the argument of expr at position pos by arg.
#include 
__isl_give isl_ast_expr *isl_ast_expr_substitute_ids(
__isl_take isl_ast_expr *expr,
__isl_take isl_id_to_ast_expr *id2expr);

203

The function isl_ast_expr_substitute_ids replaces the subexpressions of
expr of type isl_ast_expr_id by the corresponding expression in id2expr, if there
is any.
User specified data can be attached to an isl_ast_node and obtained from the
same isl_ast_node using the following functions.
#include 
__isl_give isl_ast_node *isl_ast_node_set_annotation(
__isl_take isl_ast_node *node,
__isl_take isl_id *annotation);
__isl_give isl_id *isl_ast_node_get_annotation(
__isl_keep isl_ast_node *node);
Basic printing can be performed using the following functions.
#include 
__isl_give isl_printer *isl_printer_print_ast_expr(
__isl_take isl_printer *p,
__isl_keep isl_ast_expr *expr);
__isl_give isl_printer *isl_printer_print_ast_node(
__isl_take isl_printer *p,
__isl_keep isl_ast_node *node);
__isl_give char *isl_ast_expr_to_str(
__isl_keep isl_ast_expr *expr);
__isl_give char *isl_ast_node_to_str(
__isl_keep isl_ast_node *node);
__isl_give char *isl_ast_expr_to_C_str(
__isl_keep isl_ast_expr *expr);
__isl_give char *isl_ast_node_to_C_str(
__isl_keep isl_ast_node *node);
The functions isl_ast_expr_to_C_str and isl_ast_node_to_C_str are convenience functions that return a string representation of the input in C format.
More advanced printing can be performed using the following functions.
#include 
__isl_give isl_printer *isl_ast_op_type_set_print_name(
__isl_take isl_printer *p,
enum isl_ast_op_type type,
__isl_keep const char *name);
isl_stat isl_options_set_ast_print_macro_once(
isl_ctx *ctx, int val);
int isl_options_get_ast_print_macro_once(isl_ctx *ctx);
__isl_give isl_printer *isl_ast_op_type_print_macro(
enum isl_ast_op_type type,
__isl_take isl_printer *p);
__isl_give isl_printer *isl_ast_expr_print_macros(
204

__isl_keep isl_ast_expr *expr,
__isl_take isl_printer *p);
__isl_give isl_printer *isl_ast_node_print_macros(
__isl_keep isl_ast_node *node,
__isl_take isl_printer *p);
__isl_give isl_printer *isl_ast_node_print(
__isl_keep isl_ast_node *node,
__isl_take isl_printer *p,
__isl_take isl_ast_print_options *options);
__isl_give isl_printer *isl_ast_node_for_print(
__isl_keep isl_ast_node *node,
__isl_take isl_printer *p,
__isl_take isl_ast_print_options *options);
__isl_give isl_printer *isl_ast_node_if_print(
__isl_keep isl_ast_node *node,
__isl_take isl_printer *p,
__isl_take isl_ast_print_options *options);
While printing an isl_ast_node in ISL_FORMAT_C, isl may print out an AST
that makes use of macros such as floord, min and max. The names of these macros
may be modified by a call to isl_ast_op_type_set_print_name. The user-specified
names are associated to the printer object. isl_ast_op_type_print_macro prints
out the macro corresponding to a specific isl_ast_op_type. If the print-macro-once
option is set, then a given macro definition is only printed once to any given printer
object. isl_ast_expr_print_macros scans the isl_ast_expr for subexpressions
where these macros would be used and prints out the required macro definitions. Essentially, isl_ast_expr_print_macros calls isl_ast_expr_foreach_ast_op_type
with isl_ast_op_type_print_macro as function argument. isl_ast_node_print_macros
does the same for expressions in its isl_ast_node argument. isl_ast_node_print,
isl_ast_node_for_print and isl_ast_node_if_print print an isl_ast_node
in ISL_FORMAT_C, but allow for some extra control through an isl_ast_print_options
object. This object can be created using the following functions.
#include 
__isl_give isl_ast_print_options *
isl_ast_print_options_alloc(isl_ctx *ctx);
__isl_give isl_ast_print_options *
isl_ast_print_options_copy(
__isl_keep isl_ast_print_options *options);
__isl_null isl_ast_print_options *
isl_ast_print_options_free(
__isl_take isl_ast_print_options *options);
__isl_give isl_ast_print_options *
isl_ast_print_options_set_print_user(
__isl_take isl_ast_print_options *options,
__isl_give isl_printer *(*print_user)(
205

__isl_take isl_printer *p,
__isl_take isl_ast_print_options *options,
__isl_keep isl_ast_node *node, void *user),
void *user);
__isl_give isl_ast_print_options *
isl_ast_print_options_set_print_for(
__isl_take isl_ast_print_options *options,
__isl_give isl_printer *(*print_for)(
__isl_take isl_printer *p,
__isl_take isl_ast_print_options *options,
__isl_keep isl_ast_node *node, void *user),
void *user);
The callback set by isl_ast_print_options_set_print_user is called whenever a node of type isl_ast_node_user needs to be printed. The callback set by
isl_ast_print_options_set_print_for is called whenever a node of type isl_ast_node_for
needs to be printed. Note that isl_ast_node_for_print will not call the callback set
by isl_ast_print_options_set_print_for on the node on which isl_ast_node_for_print
is called, but only on nested nodes of type isl_ast_node_for. It is therefore safe to
call isl_ast_node_for_print from within the callback set by isl_ast_print_options_set_print_for.
The following option determines the type to be used for iterators while printing the
AST.
isl_stat isl_options_set_ast_iterator_type(
isl_ctx *ctx, const char *val);
const char *isl_options_get_ast_iterator_type(
isl_ctx *ctx);
The AST printer only prints body nodes as blocks if these blocks cannot be safely
omitted. For example, a for node with one body node will not be surrounded with
braces in ISL_FORMAT_C. A block will always be printed by setting the following option.
isl_stat isl_options_set_ast_always_print_block(isl_ctx *ctx,
int val);
int isl_options_get_ast_always_print_block(isl_ctx *ctx);
Options
#include 
isl_stat isl_options_set_ast_build_atomic_upper_bound(
isl_ctx *ctx, int val);
int isl_options_get_ast_build_atomic_upper_bound(
isl_ctx *ctx);
isl_stat isl_options_set_ast_build_prefer_pdiv(isl_ctx *ctx,
int val);
int isl_options_get_ast_build_prefer_pdiv(isl_ctx *ctx);
206

isl_stat isl_options_set_ast_build_detect_min_max(
isl_ctx *ctx, int val);
int isl_options_get_ast_build_detect_min_max(
isl_ctx *ctx);
isl_stat isl_options_set_ast_build_exploit_nested_bounds(
isl_ctx *ctx, int val);
int isl_options_get_ast_build_exploit_nested_bounds(
isl_ctx *ctx);
isl_stat isl_options_set_ast_build_group_coscheduled(
isl_ctx *ctx, int val);
int isl_options_get_ast_build_group_coscheduled(
isl_ctx *ctx);
isl_stat isl_options_set_ast_build_separation_bounds(
isl_ctx *ctx, int val);
int isl_options_get_ast_build_separation_bounds(
isl_ctx *ctx);
isl_stat isl_options_set_ast_build_scale_strides(
isl_ctx *ctx, int val);
int isl_options_get_ast_build_scale_strides(
isl_ctx *ctx);
isl_stat isl_options_set_ast_build_allow_else(isl_ctx *ctx,
int val);
int isl_options_get_ast_build_allow_else(isl_ctx *ctx);
isl_stat isl_options_set_ast_build_allow_or(isl_ctx *ctx,
int val);
int isl_options_get_ast_build_allow_or(isl_ctx *ctx);
• ast_build_atomic_upper_bound
Generate loop upper bounds that consist of the current loop iterator, an operator
and an expression not involving the iterator. If this option is not set, then the
current loop iterator may appear several times in the upper bound. For example,
when this option is turned off, AST generation for the schedule
[n] -> { A[i] -> [i] : 0 <= i <= 100, n }
produces
for (int c0 = 0; c0 <= 100 && n >= c0; c0 += 1)
A(c0);
When the option is turned on, the following AST is generated
for (int c0 = 0; c0 <= min(100, n); c0 += 1)
A(c0);

207

• ast_build_prefer_pdiv
If this option is turned off, then the AST generation will produce ASTs that may
only contain isl_ast_op_fdiv_q operators, but no isl_ast_op_pdiv_q or
isl_ast_op_pdiv_r operators. If this option is turned on, then isl will try to
convert some of the isl_ast_op_fdiv_q operators to (expressions containing)
isl_ast_op_pdiv_q or isl_ast_op_pdiv_r operators.
• ast_build_detect_min_max
If this option is turned on, then isl will try and detect min or max-expressions
when building AST expressions from piecewise affine expressions.
• ast_build_exploit_nested_bounds
Simplify conditions based on bounds of nested for loops. In particular, remove
conditions that are implied by the fact that one or more nested loops have at least
one iteration, meaning that the upper bound is at least as large as the lower bound.
For example, when this option is turned off, AST generation for the schedule
[N,M] -> { A[i,j] -> [i,j] : 0 <= i <= N and
0 <= j <= M }
produces
if (M >= 0)
for (int c0 = 0; c0 <= N; c0 += 1)
for (int c1 = 0; c1 <= M; c1 += 1)
A(c0, c1);
When the option is turned on, the following AST is generated
for (int c0 = 0; c0 <= N; c0 += 1)
for (int c1 = 0; c1 <= M; c1 += 1)
A(c0, c1);
• ast_build_group_coscheduled
If two domain elements are assigned the same schedule point, then they may be
executed in any order and they may even appear in different loops. If this options
is set, then the AST generator will make sure that coscheduled domain elements
do not appear in separate parts of the AST. This is useful in case of nested AST
generation if the outer AST generation is given only part of a schedule and the
inner AST generation should handle the domains that are coscheduled by this
initial part of the schedule together. For example if an AST is generated for a
schedule
{ A[i] -> [0]; B[i] -> [0] }
208

then the isl_ast_build_set_create_leaf callback described below may
get called twice, once for each domain. Setting this option ensures that the callback is only called once on both domains together.
• ast_build_separation_bounds
This option specifies which bounds to use during separation. If this option is
set to ISL_AST_BUILD_SEPARATION_BOUNDS_IMPLICIT then all (possibly implicit) bounds on the current dimension will be used during separation. If this
option is set to ISL_AST_BUILD_SEPARATION_BOUNDS_EXPLICIT then only
those bounds that are explicitly available will be used during separation.
• ast_build_scale_strides
This option specifies whether the AST generator is allowed to scale down iterators of strided loops.
• ast_build_allow_else
This option specifies whether the AST generator is allowed to construct if statements with else branches.
• ast_build_allow_or
This option specifies whether the AST generator is allowed to construct if conditions with disjunctions.
AST Generation Options (Schedule Tree)

In case of AST construction from a schedule tree, the options that control how an AST
is created from the individual schedule dimensions are stored in the band nodes of the
tree (see §1.5.1).
In particular, a schedule dimension can be handled in four different ways, atomic,
separate, unroll or the default. This loop AST generation type can be set using isl_schedule_node_band_member_set
Alternatively, the first three can be selected by including a one-dimensional element
with as value the position of the schedule dimension within the band and as name one of
atomic, separate or unroll in the options set by isl_schedule_node_band_set_ast_build_options.
Only one of these three may be specified for any given schedule dimension within a
band node. If none of these is specified, then the default is used. The meaning of the
options is as follows.
atomic
When this option is specified, the AST generator will make sure that a given
domains space only appears in a single loop at the specified level.
For example, for the schedule tree
domain: "{ a[i] : 0 <= i < 10; b[i] : 0 <= i < 10 }"
child:
schedule: "[{ a[i] -> [i]; b[i] -> [i+1] }]"
options: "{ atomic[x] }"
209

the following AST will be generated
for (int c0 = 0; c0 <= 10; c0 += 1) {
if (c0 >= 1)
b(c0 - 1);
if (c0 <= 9)
a(c0);
}
On the other hand, for the schedule tree
domain: "{ a[i] : 0 <= i < 10; b[i] : 0 <= i < 10 }"
child:
schedule: "[{ a[i] -> [i]; b[i] -> [i+1] }]"
options: "{ separate[x] }"
the following AST will be generated
{
a(0);
for (int c0 = 1; c0 <= 9; c0 += 1) {
b(c0 - 1);
a(c0);
}
b(9);
}
If neither atomic nor separate is specified, then the AST generator may produce either of these two results or some intermediate form.
separate
When this option is specified, the AST generator will split the domain of the
specified schedule dimension into pieces with a fixed set of statements for which
instances need to be executed by the iterations in the schedule domain part. This
option tends to avoid the generation of guards inside the corresponding loops.
See also the atomic option.
unroll
When this option is specified, the AST generator will completely unroll the corresponding schedule dimension. It is the responsibility of the user to ensure that
such unrolling is possible. To obtain a partial unrolling, the user should apply
an additional strip-mining to the schedule and fully unroll the inner schedule
dimension.

210

The isolate option is a bit more involved. It allows the user to isolate a range of
schedule dimension values from smaller and greater values. Additionally, the user may
specify a different atomic/separate/unroll choice for the isolated part and the remaining
parts. The typical use case of the isolate option is to isolate full tiles from partial
tiles. The part that needs to be isolated may depend on outer schedule dimensions. The
option therefore needs to be able to reference those outer schedule dimensions. In particular, the space of the isolate option is that of a wrapped map with as domain the
flat product of all outer band nodes and as range the space of the current band node. The
atomic/separate/unroll choice for the isolated part is determined by an option that lives
in an unnamed wrapped space with as domain a zero-dimensional isolate space and
as range the regular atomic, separate or unroll space. This option may also be set
directly using isl_schedule_node_band_member_set_isolate_ast_loop_type.
The atomic/separate/unroll choice for the remaining part is determined by the regular
atomic, separate or unroll option. Since the isolate option references outer
schedule dimensions, its use in a band node causes any tree containing the node to be
considered anchored.
As an example, consider the isolation of full tiles from partial tiles in a tiling of a
triangular domain. The original schedule is as follows.
domain: "{ A[i,j] : 0 <= i,j and i + j <= 100 }"
child:
schedule: "[{ A[i,j] -> [floor(i/10)] }, \
{ A[i,j] -> [floor(j/10)] }, \
{ A[i,j] -> [i] }, { A[i,j] -> [j] }]"
The output is
for (int c0 = 0; c0 <= 10; c0 += 1)
for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
for (int c2 = 10 * c0;
c2 <= min(10 * c0 + 9, -10 * c1 + 100); c2 += 1)
for (int c3 = 10 * c1;
c3 <= min(10 * c1 + 9, -c2 + 100); c3 += 1)
A(c2, c3);
Isolating the full tiles, we have the following input
domain: "{ A[i,j] : 0 <= i,j and i + j <= 100 }"
child:
schedule: "[{ A[i,j] -> [floor(i/10)] }, \
{ A[i,j] -> [floor(j/10)] }, \
{ A[i,j] -> [i] }, { A[i,j] -> [j] }]"
options: "{ isolate[[] -> [a,b,c,d]] : 0 <= 10a,10b and \
10a+9+10b+9 <= 100 }"
and output

211

{
for (int c0 = 0; c0 <= 8; c0 += 1) {
for (int c1 = 0; c1 <= -c0 + 8; c1 += 1)
for (int c2 = 10 * c0;
c2 <= 10 * c0 + 9; c2 += 1)
for (int c3 = 10 * c1;
c3 <= 10 * c1 + 9; c3 += 1)
A(c2, c3);
for (int c1 = -c0 + 9; c1 <= -c0 + 10; c1 += 1)
for (int c2 = 10 * c0;
c2 <= min(10 * c0 + 9, -10 * c1 + 100); c2
for (int c3 = 10 * c1;
c3 <= min(10 * c1 + 9, -c2 + 100); c3 +=
A(c2, c3);
}
for (int c0 = 9; c0 <= 10; c0 += 1)
for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
for (int c2 = 10 * c0;
c2 <= min(10 * c0 + 9, -10 * c1 + 100); c2
for (int c3 = 10 * c1;
c3 <= min(10 * c1 + 9, -c2 + 100); c3 +=
A(c2, c3);

+= 1)
1)

+= 1)
1)

}
We may then additionally unroll the innermost loop of the isolated part
domain: "{ A[i,j] : 0 <= i,j and i + j <= 100 }"
child:
schedule: "[{ A[i,j] -> [floor(i/10)] }, \
{ A[i,j] -> [floor(j/10)] }, \
{ A[i,j] -> [i] }, { A[i,j] -> [j] }]"
options: "{ isolate[[] -> [a,b,c,d]] : 0 <= 10a,10b and \
10a+9+10b+9 <= 100; [isolate[] -> unroll[3]] }"
to obtain
{
for (int c0 = 0; c0 <= 8; c0 += 1) {
for (int c1 = 0; c1 <= -c0 + 8; c1 += 1)
for (int c2 = 10 * c0; c2 <= 10 * c0 + 9; c2 += 1) {
A(c2, 10 * c1);
A(c2, 10 * c1 + 1);
A(c2, 10 * c1 + 2);
A(c2, 10 * c1 + 3);
A(c2, 10 * c1 + 4);
A(c2, 10 * c1 + 5);
A(c2, 10 * c1 + 6);
212

A(c2, 10 * c1 + 7);
A(c2, 10 * c1 + 8);
A(c2, 10 * c1 + 9);
}
for (int c1 = -c0 + 9; c1 <= -c0 + 10; c1 += 1)
for (int c2 = 10 * c0;
c2 <= min(10 * c0 + 9, -10 * c1 + 100); c2 += 1)
for (int c3 = 10 * c1;
c3 <= min(10 * c1 + 9, -c2 + 100); c3 += 1)
A(c2, c3);
}
for (int c0 = 9; c0 <= 10; c0 += 1)
for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
for (int c2 = 10 * c0;
c2 <= min(10 * c0 + 9, -10 * c1 + 100); c2 += 1)
for (int c3 = 10 * c1;
c3 <= min(10 * c1 + 9, -c2 + 100); c3 += 1)
A(c2, c3);
}
AST Generation Options (Schedule Map)
In case of AST construction using isl_ast_build_node_from_schedule_map, the
options that control how an AST is created from the individual schedule dimensions
are stored in the isl_ast_build. They can be set using the following function.
#include 
__isl_give isl_ast_build *
isl_ast_build_set_options(
__isl_take isl_ast_build *build,
__isl_take isl_union_map *options);
The options are encoded in an isl_union_map. The domain of this union relation
refers to the schedule domain, i.e., the range of the schedule passed to isl_ast_build_node_from_schedule_map.
In the case of nested AST generation (see §1.5.4), the domain of options should refer
to the extra piece of the schedule. That is, it should be equal to the range of the wrapped
relation in the range of the schedule. The range of the options can consist of elements
in one or more spaces, the names of which determine the effect of the option. The
values of the range typically also refer to the schedule dimension to which the option
applies, with value 0 representing the outermost schedule dimension. In case of nested
AST generation (see §1.5.4), these values refer to the position of the schedule dimension within the innermost AST generation. The constraints on the domain elements of
the option should only refer to this dimension and earlier dimensions. We consider the
following spaces.
separation_class

213

This option has been deprecated. Use the isolate option on schedule trees
instead.
This space is a wrapped relation between two one dimensional spaces. The input space represents the schedule dimension to which the option applies and the
output space represents the separation class. While constructing a loop corresponding to the specified schedule dimension(s), the AST generator will try to
generate separate loops for domain elements that are assigned different classes.
If only some of the elements are assigned a class, then those elements that are
not assigned any class will be treated as belonging to a class that is separate from
the explicitly assigned classes. The typical use case for this option is to separate
full tiles from partial tiles. The other options, described below, are applied after
the separation into classes.
As an example, consider the separation into full and partial tiles of a tiling of a
triangular domain. Take, for example, the domain
{ A[i,j] : 0 <= i,j and i + j <= 100 }
and a tiling into tiles of 10 by 10. The input to the AST generator is then the
schedule
{ A[i,j] -> [([i/10]),[j/10],i,j] : 0 <= i,j and
i + j <= 100 }
Without any options, the following AST is generated
for (int c0 = 0; c0 <= 10; c0 += 1)
for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
for (int c2 = 10 * c0;
c2 <= min(-10 * c1 + 100, 10 * c0 + 9);
c2 += 1)
for (int c3 = 10 * c1;
c3 <= min(10 * c1 + 9, -c2 + 100);
c3 += 1)
A(c2, c3);
Separation into full and partial tiles can be obtained by assigning a class, say
0, to the full tiles. The full tiles are represented by those values of the first and
second schedule dimensions for which there are values of the third and fourth
dimensions to cover an entire tile. That is, we need to specify the following
option
{ [a,b,c,d] -> separation_class[[0]->[0]] :
exists b’: 0 <= 10a,10b’ and
10a+9+10b’+9 <= 100;
[a,b,c,d] -> separation_class[[1]->[0]] :
0 <= 10a,10b and 10a+9+10b+9 <= 100 }
214

which simplifies to
{ [a, b, c, d]
a >= 0
[a, b, c, d]
a >= 0

-> separation_class[[1] -> [0]] :
and b >= 0 and b <= 8 - a;
-> separation_class[[0] -> [0]] :
and a <= 8 }

With this option, the generated AST is as follows
{
for (int c0 = 0; c0 <= 8; c0 += 1) {
for (int c1 = 0; c1 <= -c0 + 8; c1 += 1)
for (int c2 = 10 * c0;
c2 <= 10 * c0 + 9; c2 += 1)
for (int c3 = 10 * c1;
c3 <= 10 * c1 + 9; c3 += 1)
A(c2, c3);
for (int c1 = -c0 + 9; c1 <= -c0 + 10; c1 += 1)
for (int c2 = 10 * c0;
c2 <= min(-10 * c1 + 100, 10 * c0 + 9);
c2 += 1)
for (int c3 = 10 * c1;
c3 <= min(-c2 + 100, 10 * c1 + 9);
c3 += 1)
A(c2, c3);
}
for (int c0 = 9; c0 <= 10; c0 += 1)
for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
for (int c2 = 10 * c0;
c2 <= min(-10 * c1 + 100, 10 * c0 + 9);
c2 += 1)
for (int c3 = 10 * c1;
c3 <= min(10 * c1 + 9, -c2 + 100);
c3 += 1)
A(c2, c3);
}
separate
This is a single-dimensional space representing the schedule dimension(s) to
which “separation” should be applied. Separation tries to split a loop into several pieces if this can avoid the generation of guards inside the loop. See also the
atomic option.
atomic
This is a single-dimensional space representing the schedule dimension(s) for
which the domains should be considered “atomic”. That is, the AST generator
215

will make sure that any given domain space will only appear in a single loop at
the specified level.
Consider the following schedule
{ a[i] -> [i] : 0 <= i < 10;
b[i] -> [i+1] : 0 <= i < 10 }
If the following option is specified
{ [i] -> separate[x] }
then the following AST will be generated
{
a(0);
for (int c0 = 1; c0 <= 9; c0 += 1) {
a(c0);
b(c0 - 1);
}
b(9);
}
If, on the other hand, the following option is specified
{ [i] -> atomic[x] }
then the following AST will be generated
for (int c0 = 0; c0 <= 10; c0 += 1) {
if (c0 <= 9)
a(c0);
if (c0 >= 1)
b(c0 - 1);
}
If neither atomic nor separate is specified, then the AST generator may produce either of these two results or some intermediate form.
unroll
This is a single-dimensional space representing the schedule dimension(s) that
should be completely unrolled. To obtain a partial unrolling, the user should
apply an additional strip-mining to the schedule and fully unroll the inner loop.

216

Fine-grained Control over AST Generation
Besides specifying the constraints on the parameters, an isl_ast_build object can be
used to control various aspects of the AST generation process. In case of AST construction using isl_ast_build_node_from_schedule_map, the most prominent way of
control is through “options”, as explained above.
Additional control is available through the following functions.
#include 
__isl_give isl_ast_build *
isl_ast_build_set_iterators(
__isl_take isl_ast_build *build,
__isl_take isl_id_list *iterators);
The function isl_ast_build_set_iterators allows the user to specify a list of
iterator isl_ids to be used as iterators. If the input schedule is injective, then the number of elements in this list should be as large as the dimension of the schedule space,
but no direct correspondence should be assumed between dimensions and elements. If
the input schedule is not injective, then an additional number of isl_ids equal to the
largest dimension of the input domains may be required. If the number of provided
isl_ids is insufficient, then additional names are automatically generated.
#include 
__isl_give isl_ast_build *
isl_ast_build_set_create_leaf(
__isl_take isl_ast_build *build,
__isl_give isl_ast_node *(*fn)(
__isl_take isl_ast_build *build,
void *user), void *user);
The isl_ast_build_set_create_leaf function allows for the specification of a
callback that should be called whenever the AST generator arrives at an element of the
schedule domain. The callback should return an AST node that should be inserted at
the corresponding position of the AST. The default action (when the callback is not set)
is to continue generating parts of the AST to scan all the domain elements associated to
the schedule domain element and to insert user nodes, “calling” the domain element, for
each of them. The build argument contains the current state of the isl_ast_build.
To ease nested AST generation (see §1.5.4), all control information that is specific to
the current AST generation such as the options and the callbacks has been removed
from this isl_ast_build. The callback would typically return the result of a nested
AST generation or a user defined node created using the following function.
#include 
__isl_give isl_ast_node *isl_ast_node_alloc_user(
__isl_take isl_ast_expr *expr);
#include 
__isl_give isl_ast_build *
217

isl_ast_build_set_at_each_domain(
__isl_take isl_ast_build *build,
__isl_give isl_ast_node *(*fn)(
__isl_take isl_ast_node *node,
__isl_keep isl_ast_build *build,
void *user), void *user);
__isl_give isl_ast_build *
isl_ast_build_set_before_each_for(
__isl_take isl_ast_build *build,
__isl_give isl_id *(*fn)(
__isl_keep isl_ast_build *build,
void *user), void *user);
__isl_give isl_ast_build *
isl_ast_build_set_after_each_for(
__isl_take isl_ast_build *build,
__isl_give isl_ast_node *(*fn)(
__isl_take isl_ast_node *node,
__isl_keep isl_ast_build *build,
void *user), void *user);
__isl_give isl_ast_build *
isl_ast_build_set_before_each_mark(
__isl_take isl_ast_build *build,
isl_stat (*fn)(__isl_keep isl_id *mark,
__isl_keep isl_ast_build *build,
void *user), void *user);
__isl_give isl_ast_build *
isl_ast_build_set_after_each_mark(
__isl_take isl_ast_build *build,
__isl_give isl_ast_node *(*fn)(
__isl_take isl_ast_node *node,
__isl_keep isl_ast_build *build,
void *user), void *user);
The callback set by isl_ast_build_set_at_each_domain will be called for
each domain AST node. The callbacks set by isl_ast_build_set_before_each_for
and isl_ast_build_set_after_each_for will be called for each for AST node.
The first will be called in depth-first pre-order, while the second will be called in depthfirst post-order. Since isl_ast_build_set_before_each_for is called before the
for node is actually constructed, it is only passed an isl_ast_build. The returned
isl_id will be added as an annotation (using isl_ast_node_set_annotation) to
the constructed for node. In particular, if the user has also specified an after_each_for
callback, then the annotation can be retrieved from the node passed to that callback using isl_ast_node_get_annotation. The callbacks set by isl_ast_build_set_before_each_mark
and isl_ast_build_set_after_each_mark will be called for each mark AST node
that is created, i.e., for each mark schedule node in the input schedule tree. The
first will be called in depth-first pre-order, while the second will be called in depth218

first post-order. Since the callback set by isl_ast_build_set_before_each_mark
is called before the mark AST node is actually constructed, it is passed the identifier of the mark node. All callbacks should NULL (or isl_stat_error) on failure.
The given isl_ast_build can be used to create new isl_ast_expr objects using
isl_ast_build_expr_from_pw_aff or isl_ast_build_call_from_pw_multi_aff.
Nested AST Generation
isl allows the user to create an AST within the context of another AST. These nested
ASTs are created using the same isl_ast_build_node_from_schedule_map function that is used to create the outer AST. The build argument should be an isl_ast_build
passed to a callback set by isl_ast_build_set_create_leaf. The space of the
range of the schedule argument should refer to this build. In particular, the space
should be a wrapped relation and the domain of this wrapped relation should be the
same as that of the range of the schedule returned by isl_ast_build_get_schedule
below. In practice, the new schedule is typically created by calling isl_union_map_range_product
on the old schedule and some extra piece of the schedule. The space of the schedule
domain is also available from the isl_ast_build.
#include 
__isl_give isl_union_map *isl_ast_build_get_schedule(
__isl_keep isl_ast_build *build);
__isl_give isl_space *isl_ast_build_get_schedule_space(
__isl_keep isl_ast_build *build);
__isl_give isl_ast_build *isl_ast_build_restrict(
__isl_take isl_ast_build *build,
__isl_take isl_set *set);
The isl_ast_build_get_schedule function returns a (partial) schedule for the
domains elements for which part of the AST still needs to be generated in the current
build. In particular, the domain elements are mapped to those iterations of the loops
enclosing the current point of the AST generation inside which the domain elements
are executed. No direct correspondence between the input schedule and this schedule
should be assumed. The space obtained from isl_ast_build_get_schedule_space
can be used to create a set for isl_ast_build_restrict to intersect with the current
build. In particular, the set passed to isl_ast_build_restrict can have additional
parameters. The ids of the set dimensions in the space returned by isl_ast_build_get_schedule_space
correspond to the iterators of the already generated loops. The user should not rely
on the ids of the output dimensions of the relations in the union relation returned by
isl_ast_build_get_schedule having any particular value.

1.6

Applications

Although isl is mainly meant to be used as a library, it also contains some basic
applications that use some of the functionality of isl. For applications that take one

219

or more polytopes or polyhedra as input, this input may be specified in either the isl
format or the PolyLib format.

1.6.1

isl_polyhedron_sample

isl_polyhedron_sample takes a polyhedron as input and prints an integer element
of the polyhedron, if there is any. The first column in the output is the denominator
and is always equal to 1. If the polyhedron contains no integer points, then a vector of
length zero is printed.

1.6.2

isl_pip

isl_pip takes the same input as the example program from the piplib distribution,
i.e., a set of constraints on the parameters, a line containing only -1 and finally a set of
constraints on a parametric polyhedron. The coefficients of the parameters appear in
the last columns (but before the final constant column). The output is the lexicographic
minimum of the parametric polyhedron. As isl currently does not have its own output
format, the output is just a dump of the internal state.

1.6.3

isl_polyhedron_minimize

isl_polyhedron_minimize computes the minimum of some linear or affine objective function over the integer points in a polyhedron. If an affine objective function is
given, then the constant should appear in the last column.

1.6.4

isl_polytope_scan

Given a polytope, isl_polytope_scan prints all integer points in the polytope.

1.6.5

isl_flow

Given an isl_union_access_info object as input, isl_flow prints out the corresponding dependences, as computed by isl_union_access_info_compute_flow.

1.6.6

isl_codegen

Given either a schedule tree or a sequence consisting of a schedule map, a context set
and an options relation, isl_codegen prints out an AST that scans the domain elements of the schedule in the order of their image(s) taking into account the constraints
in the context set.

1.6.7

isl_schedule

Given an isl_schedule_constraints object as input, isl_schedule prints out a
schedule that satisfies the given constraints.

220

Chapter 2

Implementation Details
2.1

Sets and Relations

Definition 2.1.1 (Polyhedral Set) A polyhedral set S is a finite union of basic sets
S
S = i S i , each of which can be represented using affine constraints
d

S i : Zn → 2Z : s 7→ S i (s) = { x ∈ Zd | ∃z ∈ Ze : Ax + Bs + Dz + c ≥ 0 },
with A ∈ Zm×d , B ∈ Zm×n , D ∈ Zm×e and c ∈ Zm .
d

Definition 2.1.2 (Parameter Domain of a Set) Let S ∈ Zn → 2Z be a set. The parameter domain of S is the set
pdom S B { s ∈ Zn | S (s) , ∅ }.
Definition 2.1.3 (Polyhedral Relation) A polyhedral relation R is a finite union of baS
d +d
sic relations R = i Ri of type Zn → 2Z 1 2 , each of which can be represented using
affine constraints
Ri = s 7→ Ri (s) = { x1 → x2 ∈ Zd1 × Zd2 | ∃z ∈ Ze : A1 x1 + A2 x2 + Bs + Dz + c ≥ 0 },
with Ai ∈ Zm×di , B ∈ Zm×n , D ∈ Zm×e and c ∈ Zm .
Definition 2.1.4 (Parameter Domain of a Relation) Let R ∈ Zn → 2Z
tion. The parameter domain of R is the set

d+d

be a rela-

pdom R B { s ∈ Zn | R(s) , ∅ }.
Definition 2.1.5 (Domain of a Relation) Let R ∈ Zn → 2Z
main of R is the polyhedral set

d+d

be a relation. The do-

dom R B s 7→ { x1 ∈ Zd1 | ∃x2 ∈ Zd2 : (x1 , x2 ) ∈ R(s) }.
Definition 2.1.6 (Range of a Relation) Let R ∈ Zn → 2Z
of R is the polyhedral set

d+d

be a relation. The range

ran R B s 7→ { x2 ∈ Zd2 | ∃x1 ∈ Zd1 : (x1 , x2 ) ∈ R(s) }.
221

d +d

Definition 2.1.7 (Composition of Relations) Let R ∈ Zn → 2Z 1 2 and S ∈ Zn →
d +d
2Z 2 3 be two relations, then the composition of R and S is defined as
S ◦ R B s 7→ { x1 → x3 ∈ Zd1 × Zd3 | ∃x2 ∈ Zd2 : x1 → x2 ∈ R(s) ∧ x2 → x3 ∈ S (s) }.
d+d

Definition 2.1.8 (Difference Set of a Relation) Let R ∈ Zn → 2Z be a relation. The
difference set (∆ R) of R is the set of differences between image elements and the corresponding domain elements,
∆ R B s 7→ { δ ∈ Zd | ∃x → y ∈ R : δ = y − x }

2.2

Simple Hull

It is sometimes useful to have a single basic set or basic relation that contains a given
set or relation. For rational sets, the obvious choice would be to compute the (rational)
convex hull. For integer sets, the obvious choice would be the integer hull. However,
isl currently does not support an integer hull operation and even if it did, it would be
fairly expensive to compute. The convex hull operation is supported, but it is also fairly
expensive to compute given only an implicit representation.
Usually, it is not required to compute the exact integer hull, and an overapproximation of this hull is sufficient. The “simple hull” of a set is such an overapproximation
and it is defined as the (inclusion-wise) smallest basic set that is described by constraints that are translates of the constraints in the input set. This means that the simple
hull is relatively cheap to compute and that the number of constraints in the simple hull
is no larger than the number of constraints in the input.
S
Definition 2.2.1 (Simple Hull of a Set) The simple hull of a set S = 1≤i≤v S i , with




_




d
e
n
Zd
Ai x + Bi s + Di z + ci ≥ 0 
x ∈ Z | ∃z ∈ Z :
S : Z → 2 : s 7→ S (s) = 




1≤i≤v

is the set
n

Zd

H:Z →2





^




d
e
A
x
+
B
s
+
D
z
+
c
+
K
≥
0
x
∈
Z
|
∃z
∈
Z
:
: s 7→ S (s) = 

i
i
i
i
i
,



1≤i≤v

with Ki the (component-wise) smallest non-negative integer vectors such that S ⊆ H.
The Ki can be obtained by solving a number of LP problems, one for each element
of each Ki . If any LP problem is unbounded, then the corresponding constraint is
dropped.

2.3
2.3.1

Parametric Integer Programming
Introduction

Parametric integer programming (P. Feautrier 1988) is used to solve many problems
within the context of the polyhedral model. Here, we are mainly interested in dependence analysis (P. Feautrier 1991) and in computing a unique representation for
222

existentially quantified variables. The latter operation has been used for counting elements in sets involving such variables (Boulet and Redon 1998; Verdoolaege, Beyls,
et al. 2005) and lies at the core of the internal representation of isl.
Parametric integer programming was first implemented in PipLib. An alternative method for parametric integer programming was later implemented in barvinok
Verdoolaege 2006. This method is not based on Feautrier’s algorithm, but on rational generating functions Barvinok and Woods 2003 and was inspired by the “digging”
technique of De Loera, Haws, et al. (2004) for solving non-parametric integer programming problems.
In the following sections, we briefly recall the dual simplex method combined with
Gomory cuts and describe some extensions and optimizations. The main algorithm
is applied to a matrix data structure known as a tableau. In case of parametric problems, there are two tableaus, one for the main problem and one for the constraints on
the parameters, known as the context tableau. The handling of the context tableau is
described in Section 2.3.7.

2.3.2

The Dual Simplex Method

Tableaus can be represented in several slightly different ways. In isl, the dual simplex
method uses the same representation as that used by its incremental LP solver based
on the primal simplex method. The implementation of this LP solver is based on that
of Simplify (Detlefs, G. Nelson, et al. 2005), which, in turn, was derived from the
work of C. G. Nelson (1980). In the original (C. G. Nelson 1980), the tableau was
implemented as a sparse matrix, but neither Simplify nor the current implementation
of isl does so.
Given some affine constraints on the variables, Ax + b ≥ 0, the tableau represents
the relationship between the variables x and non-negative variables

 y = Ax + b corresponding to the constraints. The initial tableau contains b A and expresses the
constraints y in the rows in terms of the variables x in the columns. The main operation defined on a tableau exchanges a column and a row variable and is called a
pivot. During this process, some coefficients may become rational. As in the PipLib
implementation, isl maintains a shared denominator per row. The sample value of a
tableau is one where each column variable is assigned zero and each row variable is
assigned the constant term of the row. This sample value represents a valid solution if
each constraint variable is assigned a non-negative value, i.e., if the constant terms of
rows corresponding to constraints are all non-negative.
The dual simplex method starts from an initial sample value that may be invalid,
but that is known to be (lexicographically) no greater than any solution, and gradually
increments this sample value through pivoting until a valid solution is obtained. In
P
particular, each pivot exchanges a row variable r = −n + i ai ci with negative sample
P
value −n with a column variable c j such that a j > 0. Since c j = (n + r − i, j ai ci )/a j ,
the new row variable will have a positive sample value n. If no such column can be
found, then the problem is infeasible. By always choosing the column that leads to
the (lexicographically) smallest increment in the variables x, the first solution found is
guaranteed to be the (lexicographically) minimal solution P. Feautrier 1988. In order
to be able to determine the smallest increment, the tableau is (implicitly) extended
223

with extra rows defining the original variables in terms of the column variables. If
we assume that all variables are non-negative, then we know that the zero vector is no
greater than the minimal solution and then the initial extended tableau looks as follows.
 1 c 
x  0 I 


r b A
Each column in this extended tableau is lexicographically positive and will remain so
because of the column choice explained above. It is then clear that the value of x will
increase in each step. Note that there is no need to store the extra rows explicitly. If a
given xi is a column variable, then the corresponding row is the unit vector ei . If, on
the other hand, it is a row variable, then the row already appears somewhere else in the
tableau.
In case of parametric problems, the sign of the constant term may depend on the
parameters. Each time the constant term of a constraint row changes, we therefore
need to check whether the new term can attain negative and/or positive values over
the current set of possible parameter values, i.e., the context. If all these terms can
only attain non-negative values, the current state of the tableau represents a solution.
If one of the terms can only attain non-positive values and is not identically zero, the
corresponding row can be pivoted. Otherwise, we pick one of the terms that can attain
both positive and negative values and split the context into a part where it only attains
non-negative values and a part where it only attains negative values.

2.3.3

Gomory Cuts

The solution found by the dual simplex method may have non-integral coordinates.
If so, some rational solutions (including the current sample value), can be cut off by
applying a (parametric) Gomory cut. Let r = b(p) + ha, ci be the row corresponding to
the first non-integral coordinate of x, with b(p) the constant term, an affine expression
in the parameters p, i.e., b(p) = hf, pi + g. Note that only row variables can attain
non-integral values as the sample value of the column variables is zero. Consider the
expression b(p) − db(p)e + h{a} , ci, with d·e the ceiling function and {·} the fractional
part. This expression is negative at the sample value since c = 0 and r = b(p) is
fractional, i.e., db(p)e > b(p). On the other hand, for each integral value of r and c ≥ 0,
the expression is non-negative because b(p) − db(p)e > −1. Imposing this expression
to be non-negative therefore does not invalidate any integral solutions, while it does
cut away the current fractional sample value. To be able to formulate this constraint,
a new variable q = b−b(p)c = − db(p)e is added to the context. This integral variable
is uniquely defined by the constraints 0 ≤ −d b(p) − d q ≤ d − 1, with d the common
denominator of f and g. In practice, the variable q0 = bh{− f } , pi + {−g}c is used instead
and the coefficients of the new constraint are adjusted accordingly. The sign of the
constant term of this new constraint need not be determined as it is non-positive by
construction. When several of these extra context variables are added, it is important
to avoid adding duplicates. Recent versions of PipLib also check for such duplicates.
224

2.3.4

Negative Unknowns and Maximization

There are two places in the above algorithm where the unknowns x are assumed to be
non-negative: the initial tableau starts from sample value x = 0 and c is assumed to be
non-negative during the construction of Gomory cuts. To deal with negative unknowns,
P. Feautrier (1991, Appendix A.2) proposed to use a “big parameter”, say M, that is
taken to be an arbitrarily large positive number. Instead of looking for the lexicographically minimal value of x, we search instead for the lexicographically minimal value of
x0 = M + x. The sample value x0 = 0 of the initial tableau then corresponds to x = −M,
which is clearly not greater than any potential solution. The sign of the constant term
of a row is determined lexicographically, with the coefficient of M considered first.
That is, if the coefficient of M is not zero, then its sign is the sign of the entire term.
Otherwise, the sign is determined by the remaining affine expression in the parameters.
If the original problem has a bounded optimum, then the final sample value will be of
the form M + v and the optimal value of the original problem is then v. Maximization
problems can be handled in a similar way by computing the minimum of M − x.
When the optimum is unbounded, the optimal value computed for the original problem will involve the big parameter. In the original implementation of PipLib, the big
parameter could even appear in some of the extra variables q created during the application of a Gomory cut. The final result could then contain implicit conditions on
the big parameter through conditions on such q variables. This problem was resolved
in later versions of PipLib by taking M to be divisible by any positive number. The
big parameter can then never appear in any q because {αM} = 0. It should be noted,
though, that an unbounded problem usually (but not always) indicates an incorrect formulation of the problem.
The original version of PipLib required the user to “manually” add a big parameter, perform the reformulation and interpret the result (P. Feautrier, Collard, et al. 2002).
Recent versions allow the user to simply specify that the unknowns may be negative or
that the maximum should be computed and then these transformations are performed
internally. Although there are some application, e.g., that of Paul Feautrier (1992),
where it is useful to have explicit control over the big parameter, negative unknowns
and maximization are by far the most common applications of the big parameter and
we believe that the user should not be bothered with such implementation issues. The
current version of isl therefore does not provide any interface for specifying big parameters. Instead, the user can specify whether a maximum needs to be computed
and no assumptions are made on the sign of the unknowns. Instead, the sign of the
unknowns is checked internally and a big parameter is automatically introduced when
needed. For compatibility with PipLib, the isl_pip tool does explicitly add nonnegativity constraints on the unknowns unless the Urs_unknowns option is specified.
Currently, there is also no way in isl of expressing a big parameter in the output. Even
though isl makes the same divisibility assumption on the big parameter as recent versions of PipLib, it will therefore eventually produce an error if the problem turns out
to be unbounded.

225

2.3.5

Preprocessing

In this section, we describe some transformations that are or can be applied in advance
to reduce the running time of the actual dual simplex method with Gomory cuts.
Feasibility Check and Detection of Equalities
Experience with the original PipLib has shown that Gomory cuts do not perform very
well on problems that are (non-obviously) empty, i.e., problems with rational solutions,
but no integer solutions. In isl, we therefore first perform a feasibility check on the
original problem considered as a non-parametric problem over the combined space of
unknowns and parameters. In fact, we do not simply check the feasibility, but we
also check for implicit equalities among the integer points by computing the integer
affine hull. The algorithm used is the same as that described in Section 2.3.7 below.
Computing the affine hull is fairly expensive, but it can bring huge benefits if any
equalities can be found or if the problem turns out to be empty.
Constraint Simplification
If the coefficients of the unknown and parameters in a constraint have a common factor,
then this factor should be removed, possibly rounding down the constant term. For
example, the constraint 2x − 5 ≥ 0 should be simplified to x − 3 ≥ 0. isl performs
such simplifications on all sets and relations. Recent versions of PipLib also perform
this simplification on the input.
Exploiting Equalities
If there are any (explicit) equalities in the input description, PipLib converts each into
a pair of inequalities. It is also possible to write r equalities as r + 1 inequalities (P.
Feautrier, Collard, et al. 2002), but it is even better to exploit the equalities to reduce
the dimensionality of the problem. Given an equality involving at least one unknown,
we pivot the row corresponding to the equality with the column corresponding to the
last unknown with non-zero coefficient. The new column variable can then be removed
completely because it is identically zero, thereby reducing the dimensionality of the
problem by one. The last unknown is chosen to ensure that the columns of the initial
tableau remain lexicographically positive. In particular, if the equality is of the form
P
b + i≤ j ai xi = 0 with a j , 0, then the (implicit) top rows of the initial tableau are
changed as follows
j




 0 I1

 0

I1





 −b/a j −ai /a j

j
j  0
1




0
I2
0
I2
Currently, isl also eliminates equalities involving only parameters in a similar way,
provided at least one of the coefficients is equal to one. The application of parameter
compression (see below) would obviate the need for removing parametric equalities.
226

Offline Symmetry Detection
Some problems, notably those of Bygde (2010), have a collection of constraints, say
bi (p) + ha, xi ≥ 0, that only differ in their (parametric) constant terms. These constant
terms will be non-negative on different parts of the context and this context may have
to be split for each of the constraints. In the worst case, the basic algorithm may have
to consider all possible orderings of the constant terms. Instead, isl introduces a new
parameter, say u, and replaces the collection of constraints by the single constraint
u + ha, xi ≥ 0 along with context constraints u ≤ bi (p). Any solution to the new
system is also a solution to the original system since ha, xi ≥ −u ≥ −bi (p). Conversely,
m = mini bi (p) satisfies the constraints on u and therefore extends a solution to the new
system. It can also be plugged into a new solution. See Section 2.3.6 for how this
substitution is currently performed in isl. The method described in this section can
only detect symmetries that are explicitly available in the input. See Section 2.3.9 for
the detection and exploitation of symmetries that appear during the course of the dual
simplex method.
Note that the replacement of the bi (p) by u may lose information if the parameters
that occur in bi (p) also occur in other constraints. The replacement is therefore currently only applied when all the parameters in all of the bi (p) only occur in a single
constraint, i.e., the one in which the parameter is removed. This is the case for the
examples from Bygde (2010) in Table 2.1. The version of isl that was used during the
experiments of Section 2.3.8 did not take into account this single-occurrence constraint.
Parameter Compression
It may in some cases be apparent from the equalities in the problem description that
there can only be a solution for a sublattice of the parameters. In such cases “parameter
compression” (Meister 2004; Meister and Verdoolaege 2008) can be used to replace
the parameters by alternative “dense” parameters. For example, if there is a constraint
2x = n, then the system will only have solutions for even values of n and n can be
replaced by 2n0 . Similarly, the parameters n and m in a system with the constraint
2n = 3m can be replaced by a single parameter n0 with n = 3n0 and m = 2n0 . It
is also possible to perform a similar compression on the unknowns, but it would be
more complicated as the compression would have to preserve the lexicographical order.
Moreover, due to our handling of equalities described above there should be no need for
such variable compression. Although parameter compression has been implemented in
isl, it is currently not yet used during parametric integer programming.

2.3.6

Postprocessing

The output of PipLib is a quast (quasi-affine selection tree). Each internal node in this
tree corresponds to a split of the context based on a parametric constant term in the main
tableau with indeterminate sign. Each of these nodes may introduce extra variables in
the context corresponding to integer divisions. Each leaf of the tree prescribes the
solution in that part of the context that satisfies all the conditions on the path leading
to the leaf. Such a quast is a very economical way of representing the solution, but it

227

would not be suitable as the (only) internal representation of sets and relations in isl.
Instead, isl represents the constraints of a set or relation in disjunctive normal form.
The result of a parametric integer programming problem is then also converted to this
internal representation. Unfortunately, the conversion to disjunctive normal form can
lead to an explosion of the size of the representation. In some cases, this overhead
would have to be paid anyway in subsequent operations, but in other cases, especially
for outside users that just want to solve parametric integer programming problems, we
would like to avoid this overhead in future. That is, we are planning on introducing
quasts or a related representation as one of several possible internal representations
and on allowing the output of isl_pip to optionally be printed as a quast.
Currently, isl also does not have an internal representation for expressions such
as mini bi (p) from the offline symmetry detection of Section 2.3.5. Assume that one of
these expressions has n bounds bi (p). If the expression does not appear in the affine
expression describing the solution, but only in the constraints, and if moreover, the
expression only appears with a positive coefficient, i.e., mini bi (p) ≥ f j (p), then each
of these constraints can simply be reduplicated n times, once for each of the bounds.
Otherwise, a conversion to disjunctive normal form leads to n cases, each described as
u = bi (p) with constraints bi (p) ≤ b j (p) for j > i and bi (p) < b j (p) for j < i. Note that
even though this conversion leads to a size increase by a factor of n, not detecting the
symmetry could lead to an increase by a factor of n! if all possible orderings end up
being considered.

2.3.7

Context Tableau

The main operation that a context tableau needs to provide is a test on the sign of an
affine expression over the elements of the context. This sign can be determined by
solving two integer linear feasibility problems, one with a constraint added to the context that enforces the expression to be non-negative and one where the expression is
negative. As already mentioned by P. Feautrier (1988), any integer linear feasibility
solver could be used, but the PipLib implementation uses a recursive call to the dual
simplex with Gomory cuts algorithm to determine the feasibility of a context. In isl,
two ways of handling the context have been implemented, one that performs the recursive call and one, used by default, that uses generalized basis reduction. We start with
some optimizations that are shared between the two implementations and then discuss
additional details of each of them.
Maintaining Witnesses
A common feature of both integer linear feasibility solvers is that they will not only
say whether a set is empty or not, but if the set is non-empty, they will also provide a
witness for this result, i.e., a point that belongs to the set. By maintaining a list of such
witnesses, we can avoid many feasibility tests during the determination of the signs
of affine expressions. In particular, if the expression evaluates to a positive number
on some of these points and to a negative number on some others, then no feasibility
test needs to be performed. If all the evaluations are non-negative, we only need to
check for the possibility of a negative value and similarly in case of all non-positive
228

evaluations. Finally, in the rare case that all points evaluate to zero or at the start, when
no points have been collected yet, one or two feasibility tests need to be performed
depending on the result of the first test.
When a new constraint is added to the context, the points that violate the constraint
are temporarily removed. They are reconsidered when we backtrack over the addition
of the constraint, as they will satisfy the negation of the constraint. It is only when
we backtrack over the addition of the points that they are finally removed completely.
When an extra integer division is added to the context, the new coordinates of the witnesses can easily be computed by evaluating the integer division. The idea of keeping
track of witnesses was first used in barvinok.
Choice of Constant Term on which to Split
Recall that if there are no rows with a non-positive constant term, but there are rows
with an indeterminate sign, then the context needs to be split along the constant term
of one of these rows. If there is more than one such row, then we need to choose
which row to split on first. PipLib uses a heuristic based on the (absolute) sizes of the
coefficients. In particular, it takes the largest coefficient of each row and then selects
the row where this largest coefficient is smaller than those of the other rows.
In isl, we take that row for which non-negativity of its constant term implies nonnegativity of as many of the constant terms of the other rows as possible. The intuition
behind this heuristic is that on the positive side, we will have fewer negative and indeterminate signs, while on the negative side, we need to perform a pivot, which may
affect any number of rows meaning that the effect on the signs is difficult to predict.
This heuristic is of course much more expensive to evaluate than the heuristic used by
PipLib. More extensive tests are needed to evaluate whether the heuristic is worthwhile.
Dual Simplex + Gomory Cuts
When a new constraint is added to the context, the first steps of the dual simplex method
applied to this new context will be the same or at least very similar to those taken on
the original context, i.e., before the constraint was added. In isl, we therefore apply
the dual simplex method incrementally on the context and backtrack to a previous state
when a constraint is removed again. An initial implementation that was never made
public would also keep the Gomory cuts, but the current implementation backtracks to
before the point where Gomory cuts are added before adding an extra constraint to the
context. Keeping the Gomory cuts has the advantage that the sample value is always
an integer point and that this point may also satisfy the new constraint. However, due
to the technique of maintaining witnesses explained above, we would not perform a
feasibility test in such cases and then the previously added cuts may be redundant,
possibly resulting in an accumulation of a large number of cuts.
If the parameters may be negative, then the same big parameter trick used in the
main tableau is applied to the context. This big parameter is of course unrelated to
the big parameter from the main tableau. Note that it is not a requirement for this
parameter to be “big”, but it does allow for some code reuse in isl. In PipLib, the

229

extra parameter is not “big”, but this may be because the big parameter of the main
tableau also appears in the context tableau.
Finally, it was reported by Galea (2009), who worked on a parametric integer programming implementation in PPL (Bagnara, Hill, et al. n.d.), that it is beneficial to add
cuts for all rational coordinates in the context tableau. Based on this report, the initial
isl implementation was adapted accordingly.
Generalized Basis Reduction
The default algorithm used in isl for feasibility checking is generalized basis reduction (Cook, Rutherford, et al. 1991). This algorithm is also used in the barvinok
implementation. The algorithm is fairly robust, but it has some overhead. We therefore
try to avoid calling the algorithm in easy cases. In particular, we incrementally keep
track of points for which the entire unit hypercube positioned at that point lies in the
context. This set is described by translates of the constraints of the context and if (rationally) non-empty, any rational point in the set can be rounded up to yield an integer
point in the context.
A restriction of the algorithm is that it only works on bounded sets. The affine hull
of the recession cone therefore needs to be projected out first. As soon as the algorithm
is invoked, we then also incrementally keep track of this recession cone. The reduced
basis found by one call of the algorithm is also reused as initial basis for the next call.
Some problems lead to the introduction of many integer divisions. Within a given
context, some of these integer divisions may be equal to each other, even if the expressions are not identical, or they may be equal to some affine combination of other
variables. To detect such cases, we compute the affine hull of the context each time
a new integer division is added. The algorithm used for computing this affine hull is
that of Karr (1976), while the points used in this algorithm are obtained by performing
integer feasibility checks on that part of the context outside the current approximation
of the affine hull. The list of witnesses is used to construct an initial approximation of
the hull, while any extra points found during the construction of the hull is added to
this list. Any equality found in this way that expresses an integer division as an integer
affine combination of other variables is propagated to the main tableau, where it is used
to eliminate that integer division.

2.3.8

Experiments

Table 2.1 compares the execution times of isl (with both types of context tableau)
on some more difficult instances to those of other tools, run on an Intel Xeon W3520
@ 2.66GHz. These instances are available in the testsets/pip directory of the isl
distribution. Easier problems such as the test cases distributed with PipLib can be
solved so quickly that we would only be measuring overhead such as input/output and
conversions and not the running time of the actual algorithm. We compare the following versions: piplib-1.4.0-5-g0132fd9, barvinok-0.32.1-73-gc5d7751,
isl-0.05.1-82-g3a37260 and PPL version 0.11.2.
The first test case is the following dependence analysis problem originating from
the Phideo project (Verhaegh 1995) that was communicated to us by Bart Kienhuis:
230

Phideo
e1
e3
e4
e5
e6
e7
e8
e9
vd
bouleti
difficult
cnt/sum
jcomplex

PipLib
TC
0.33s
0.14s
0.24s
0.12s
0.10s
0.03s
0.03s
OOM
0.04s
0.25s
OOM
TC
TC

barvinok
793m
3.5s
0.13s
9.1s
6.0s
6.8s
0.27s
0.18s
70m
0.10s
line
1.3s
max
max

isl cut
>999m
0.08s
0.10s
0.09s
0.06s
0.17s
0.04s
0.03s
2.6s
0.03s
0.06s
1.7s
2.2s
3.7s

isl gbr
2.7s
0.11s
0.10s
0.11s
0.14s
0.08s
0.04s
0.04s
0.94s
0.03s
0.06s
0.33s
2.2s
3.9s

PPL
372m
0.18s
0.17s
0.70s
0.17s
0.21s
0.03s
0.01s
22s
0.03s
0.15s
1.4s
OOM
OOM

Table 2.1: Comparison of Execution Times

lexmax { [j1,j2] -> [i1,i2,i3,i4,i5,i6,i7,i8,i9,i10] : 1 <= i1,j1
<= 8 and 1 <= i2,i3,i4,i5,i6,i7,i8,i9,i10 <= 2 and 1 <= j2
<= 128 and i1-1 = j1-1 and i2-1+2*i3-2+4*i4-4+8*i5-8+16*i6
-16+32*i7-32+64*i8-64+128*i9-128+256*i10-256=3*j2-3+66 };
This problem was the main inspiration for some of the optimizations in Section 2.3.7.
The second group of test cases are projections used during counting. The first nine
of these come from Seghir and Loechner (2006). The remaining two come from Verdoolaege, Beyls, et al. (2005) and were used to drive the first, Gomory cuts based, implementation in isl. The third and final group of test cases are borrowed from Bygde
(2010) and inspired the offline symmetry detection of Section 2.3.5. Without symmetry
detection, the running times are 11s and 5.9s. All running times of barvinok and isl
include a conversion to disjunctive normal form. Without this conversion, the final two
cases can be solved in 0.07s and 0.21s. The PipLib implementation has some fixed
limits and will sometimes report the problem to be too complex (TC), while on some
other problems it will run out of memory (OOM). The barvinok implementation does
not support problems with a non-trivial lineality space (line) nor maximization problems (max). The Gomory cuts based isl implementation was terminated after 1000
minutes on the first problem. The gbr version introduces some overhead on some of
the easier problems, but is overall the clear winner.

2.3.9

Online Symmetry Detection

Manual experiments on small instances of the problems of Bygde (2010) and an analysis of the results by the approximate MPA method developed by Bygde (2010) have
revealed that these problems contain many more symmetries than can be detected using the offline method of Section 2.3.5. In this section, we present an online detection
231

mechanism that has not been implemented yet, but that has shown promising results in
manual applications.
Let us first consider what happens when we do not perform offline symmetry detection. At some point, one of the bi (p) + ha, xi ≥ 0 constraints, say the jth constraint,
appears as a column variable, say c1 , while the other constraints are represented as rows
of the form bi (p) − b j (p) + c. The context is then split according to the relative order
of b j (p) and one of the remaining bi (p). The offline method avoids this split by replacing all bi (p) by a single newly introduced parameter that represents the minimum
of these bi (p). In the online method the split is similarly avoided by the introduction of a new parameter. In particular, a new parameter is introduced that represents
b j (p) − bi (p) + = max(b j (p) − bi (p), 0).
In general, let r = b(p) + ha, ci be a row of the tableau such that the sign of b(p) is
indeterminate and such that exactly one of the elements of a is a 1, while all remaining
P
elements are non-positive. That is, r = b(p) + c j − f with f = − i, j ai ci ≥ 0. We
introduce a new parameter t with context constraints t ≥ −b(p) and t ≥ 0 and replace
the column variable c j by c0 + t. The row r is now equal to b(p) + t + c0 − f . The
constant term of this row is always non-negative because any negative value of b(p) is
compensated by t ≥ −b(p) while and non-negative value remains non-negative because
t ≥ 0.
We need to show that this transformation does not eliminate any valid solutions
and that it does not introduce any spurious solutions. Given a valid solution for the
original problem, we need to find a non-negative value of c0 satisfying the constraints.
If b(p) ≥ 0, we can take t = 0 so that c0 = c j − t = c j ≥ 0. If b(p) < 0, we can take
t = −b(p). Since r = b(p) + c j − f ≥ 0 and f ≥ 0, we have c0 = c j + b(p) ≥ 0. Note that
these choices amount to plugging in t = |−b(p)|+ = max(−b(p), 0). Conversely, given a
solution to the new problem, we need to find a non-negative value of c j , but this is easy
since c j = c0 + t and both of these are non-negative.
Plugging in t = max(−b(p), 0) can be performed as in Section 2.3.6, but, as in the
case of offline symmetry detection, it may be better to provide a direct representation
for such expressions in the internal representation of sets and relations or at least in a
quast-like output format.

2.4

Coalescing

See Verdoolaege (2015) for details on integer set coalescing.

2.5
2.5.1

Transitive Closure
Introduction
d+d

Definition 2.5.1 (Power of a Relation) Let R ∈ Zn → 2Z
a positive number, then power k of relation R is defined as



if k = 1
R
k
R B

R ◦ Rk−1 if k ≥ 2.
232

be a relation and k ∈ Z≥1

(2.1)

d+d

Definition 2.5.2 (Transitive Closure of a Relation) Let R ∈ Zn → 2Z be a relation,
then the transitive closure R+ of R is the union of all positive powers of R,
[
R+ B
Rk .
k≥1

Alternatively, the transitive closure may be defined inductively as


R+ B R ∪ R ◦ R+ .

(2.2)

Since the transitive closure of a polyhedral relation may no longer be a polyhedral relation (Kelly, Pugh, et al. 1996), we can, in the general case, only compute an
approximation of the transitive closure. Whereas Kelly, Pugh, et al. (1996) compute
underapproximations, we, like Beletska, Barthou, et al. (2009), compute overapproximations. That is, given a relation R, we will compute a relation T such that R+ ⊆ T .
Of course, we want this approximation to be as close as possible to the actual transitive closure R+ and we want to detect the cases where the approximation is exact, i.e.,
where T = R+ .
For computing an approximation of the transitive closure of R, we follow the same
general strategy as Beletska, Barthou, et al. (2009) and first compute an approximation
of Rk for k ≥ 1 and then project out the parameter k from the resulting relation.
Example 2.5.3 As a trivial example, consider the relation R = { x → x + 1 }. The kth
power of this map for arbitrary k is
Rk = k 7→ { x → x + k | k ≥ 1 }.
The transitive closure is then
R+ = { x → y | ∃k ∈ Z≥1 : y = x + k }
= { x → y | y ≥ x + 1 }.

2.5.2

Computing an Approximation of Rk

There are some special cases where the computation of Rk is very easy. One such case
is that where R does not compose with itself, i.e., R ◦ R = ∅ or dom R ∩ ran R = ∅. In
this case, Rk is only non-empty for k = 1 where it is equal to R itself.
In general, it is impossible to construct a closed form of Rk as a polyhedral relation.
We will therefore need to make some approximations. As a first approximations, we
will consider each of the basic relations in R as simply adding one or more offsets to
a domain element to arrive at an image element and ignore the fact that some of these
offsets may only be applied to some of the domain elements. That is, we will only
consider the difference set ∆ R of the relation. In particular, we will first construct a
collection P of paths that move through a total of k offsets and then intersect domain
and range of this collection with those of R. That is,
K = P ∩ (dom R → ran R) ,

233

(2.3)

with
P = s 7→ { x → y | ∃ki ∈ Z≥0 , δi ∈ ki ∆i (s) : y = x +

X

δi ∧

i

X

ki = k > 0 }

(2.4)

i

and with ∆i the basic sets that compose the difference set ∆ R. Note that the number of
basic sets ∆i need not be the same as the number of basic relations in R. Also note that
since addition is commutative, it does not matter in which order we add the offsets and
so we are allowed to group them as we did in (2.4).
If all the ∆i s are singleton sets ∆i = { δi } with δi ∈ Zd , then (2.4) simplifies to
X
X
P = { x → y | ∃ki ∈ Z≥0 : y = x +
ki δi ∧
ki = k > 0 }
(2.5)
i

i

and then the approximation computed in (2.3) is essentially the same as that of Beletska, Barthou, et al. (2009). If some of the ∆i s are not singleton sets or if some of δi s
are parametric, then we need to resort to further approximations.
To ease both the exposition and the implementation, we will for the remainder of
this section work with extended offsets ∆0i = ∆i × { 1 }. That is, each offset is extended
with an extra coordinate that is set equal to one. The paths constructed by summing
such extended offsets have the length encoded as the difference of their final coordinates. The path P0 can then be decomposed into paths P0i , one for each ∆i ,


P0 = (P0m ∪ Id) ◦ · · · ◦ (P02 ∪ Id) ◦ (P01 ∪ Id) ∩ { x0 → y0 | yd+1 − xd+1 = k > 0 }, (2.6)
with
P0i = s 7→ { x0 → y0 | ∃k ∈ Z≥1 , δ ∈ k ∆0i (s) : y0 = x0 + δ }.
Note that each P0i contains paths of length at least one. We therefore need to take the
union with the identity relation when composing the P0i s to allow for paths that do not
contain any offsets from one or more ∆0i . The path that consists of only identity relations
is removed by imposing the constraint yd+1 − xd+1 > 0. Taking the union with the
identity relation means that that the relations we compose in (2.6) each consist of two
basic relations. If there are m disjuncts in the input relation, then a direct application of
the composition operation may therefore result in a relation with 2m disjuncts, which
is prohibitively expensive. It is therefore crucial to apply coalescing (Section 2.4) after
each composition.
Let us now consider how to compute an overapproximation of P0i . Those that correspond to singleton ∆i s are grouped together and handled as in (2.5). Note that this is
just an optimization. The procedure described below would produce results that are at
least as accurate. For simplicity, we first assume that no constraint in ∆0i involves any
existentially quantified variables. We will return to existentially quantified variables at
the end of this section. Without existentially quantified variables, we can classify the
constraints of ∆0i as follows
1. non-parametric constraints
A1 x + c1 ≥ 0

(2.7)

B2 s + c2 ≥ 0

(2.8)

2. purely parametric constraints

234

3. negative mixed constraints
A3 x + B3 s + c3 ≥ 0

(2.9)

such that for each row j and for all s,
∆0i (s) ∩ { δ0 | B3, j s + c3, j > 0 } = ∅
4. positive mixed constraints
A4 x + B4 s + c4 ≥ 0
such that for each row j, there is at least one s such that
∆0i (s) ∩ { δ0 | B4, j s + c4, j > 0 } , ∅
We will use the following approximation Qi for P0i :
Qi = s 7→ { x0 → y0 | ∃k ∈ Z≥1 , f ∈ Zd : y0 = x0 + (f, k) ∧
A1 f + kc1 ≥ 0 ∧ B2 s + c2 ≥ 0 ∧ A3 f + B3 s + c3 ≥ 0 }.

(2.10)

To prove that Qi is indeed an overapproximation of P0i , we need to show that for every
s ∈ Zn , for every k ∈ Z≥1 and for every f ∈ k ∆i (s) we have that (f, k) satisfies the
constraints in (2.10). If ∆i (s) is non-empty, then s must satisfy the constraints in (2.8).
Each element (f, k) ∈ k ∆0i (s) is a sum of k elements (f j , 1) in ∆0i (s). Each of these
elements satisfies the constraints in (2.7), i.e.,
h
i "f #
A1 c1 j ≥ 0.
1
The sum of these elements therefore satisfies the same set of inequalities, i.e., A1 f +
kc1 ≥ 0. Finally, the constraints in (2.9) are such that for any s in the parameter
domain of ∆, we have −r(s) B B3 s + c3 ≤ 0, i.e., A3 f j ≥ r(s) ≥ 0 and therefore also
A3 f ≥ r(s). Note that if there are no mixed constraints and if the rational relaxation
of ∆i (s), i.e., { x ∈ Qd | A1 x + c1 ≥ 0 }, has integer vertices, then the approximation
is exact, i.e., hQi = Pi0i . In this case, the vertices of ∆0i (s) generate the rational cone
{ x0 ∈ Qd+1 | A1 c1 x0 } and therefore ∆0i (s) is a Hilbert basis of this cone (Schrijver
1986, Theorem 16.4).
Note however that, as pointed out by De Smet (2010), if there are any mixed constraints, then the above procedure may not compute the most accurate affine approximation of k ∆i (s) with k ≥ 1. In particular, we only consider the negative mixed
constraints that happen to appear in the description of ∆i (s), while we should instead
consider all valid such constraints. It is also sufficient to consider those constraints
because any constraint that is valid for k ∆i (s) is also valid for 1 ∆i (s) = ∆i (s). Take
therefore any constraint ha, xi + hb, si + c ≥ 0 valid for ∆i (s). This constraint is also
valid for k ∆i (s) iff k ha, xi + hb, si + c ≥ 0. If hb, si + c can attain any positive value,
then ha, xi may be negative for some elements of ∆i (s). We then have k ha, xi < ha, xi
235

for k > 1 and so the constraint is not valid for k ∆i (s). We therefore need to impose
hb, si+c ≤ 0 for all values of s such that ∆i (s) is non-empty, i.e., b and c need to be such
that −hb, si − c ≥ 0 is a valid constraint of ∆i (s). That is, (b, c) are the opposites of the
coefficients of a valid constraint of ∆i (s). The approximation of k ∆i (s) can therefore be
obtained using three applications of Farkas’ lemma. The first obtains the coefficients
of constraints valid for ∆i (s). The second obtains the coefficients of constraints valid
for the projection of ∆i (s) onto the parameters. The opposite of the second set is then
computed and intersected with the first set. The result is the set of coefficients of constraints valid for k ∆i (s). A final application of Farkas’ lemma is needed to obtain the
approximation of k ∆i (s) itself.
Example 2.5.4 Consider the relation
n → { (x, y) → (1 + x, 1 − n + y) | n ≥ 2 }.
The difference set of this relation is
∆ = n → { (1, 1 − n) | n ≥ 2 }.
Using our approach, we would only consider the mixed constraint y−1+n ≥ 0, leading
to the following approximation of the transitive closure:
n → { (x, y) → (o0 , o1 ) | n ≥ 2 ∧ o1 ≤ 1 − n + y ∧ o0 ≥ 1 + x }.
If, instead, we apply Farkas’s lemma to ∆, i.e.,
D := [n] -> { [1, 1 - n] : n >= 2 };
CD := coefficients D;
CD;
we obtain
{ rat: coefficients[[c_cst, c_n] -> [i2, i3]] : i3 <= c_n and
i3 <= c_cst + 2c_n + i2 }
The pure-parametric constraints valid for ∆,
P := { [a,b] -> [] }(D);
CP := coefficients P;
CP;
are
{ rat: coefficients[[c_cst, c_n] -> []] : c_n >= 0 and 2c_n >= -c_cst }
Negating these coefficients and intersecting with CD,
NCP := { rat: coefficients[[a,b] -> []]
-> coefficients[[-a,-b] -> []] }(CP);
CK := wrap((unwrap CD) * (dom (unwrap NCP)));
CK;
236

we obtain
{ rat: [[c_cst, c_n] -> [i2, i3]] : i3 <= c_n and
i3 <= c_cst + 2c_n + i2 and c_n <= 0 and 2c_n <= -c_cst }
The approximation for k ∆,
K := solutions CK;
K;
is then
[n] -> { rat: [i0, i1] : i1 <= -i0 and i0 >= 1 and i1 <= 2 - n - i0 }
Finally, the computed approximation for R+ ,
T := unwrap({ [dx,dy] -> [[x,y] -> [x+dx,y+dy]] }(K));
R := [n] -> { [x,y] -> [x+1,y+1-n] : n >= 2 };
T := T * ((dom R) -> (ran R));
T;
is
[n] -> { [x, y] -> [o0, o1] : o1 <= x + y - o0 and
o0 >= 1 + x and o1 <= 2 - n + x + y - o0 and n >= 2 }
Existentially quantified variables can be handled by classifying them into variables
that are uniquely determined by the parameters, variables that are independent of the
parameters and others. The first set can be treated as parameters and the second as variables. Constraints involving the other existentially quantified variables are removed.
Example 2.5.5 Consider the relation
R = n → { x → y | ∃ α0 , α1 : 7α0 = −2 + n ∧ 5α1 = −1 − x + y ∧ y ≥ 6 + x }.
The difference set of this relation is
∆ = ∆ R = n → { x | ∃ α0 , α1 : 7α0 = −2 + n ∧ 5α1 = −1 + x ∧ x ≥ 6 }.
The existentially quantified variables can be defined in terms of the parameters and
variables as
%
$
%
$
−1 + x
−2 + n
and
α1 =
.
α0 =
7
5
α0 can therefore be treated as a parameter, while α1 can be treated as a variable. This
in turn means that 7α0 = −2+n can be treated as a purely parametric constraint, while
the other two constraints are non-parametric. The corresponding Q (2.10) is therefore
n → { (x, z) → (y, w) | ∃ α0 , α1 , k, f : k ≥ 1 ∧ y = x + f ∧ w = z + k ∧
7α0 = −2 + n ∧ 5α1 = −k + x ∧ x ≥ 6k }.
Projecting out the final coordinates encoding the length of the paths, results in the exact
transitive closure
R+ = n → { x → y | ∃ α0 , α1 : 7α1 = −2 + n ∧ 6α0 ≥ −x + y ∧ 5α0 ≤ −1 − x + y }.
237

The fact that we ignore some impure constraints clearly leads to a loss of accuracy.
In some cases, some of this loss can be recovered by not considering the parameters in
a special way. That is, instead of considering the set
∆ = ∆ R = s 7→ { δ ∈ Zd | ∃x → y ∈ R : δ = y − x }
we consider the set
∆0 = ∆ R0 = { δ ∈ Zn+d | ∃(s, x) → (s, y) ∈ R0 : δ = (s − s, y − x) }.
The first n coordinates of every element in ∆0 are zero. Projecting out these zero coordinates from ∆0 is equivalent to projecting out the parameters in ∆. The result is
obviously a superset of ∆, but all its constraints are of type (2.7) and they can therefore
all be used in the construction of Qi .
Example 2.5.6 Consider the relation
R = n → { (x, y) → (1 + x, 1 − n + y) | n ≥ 2 }.
We have
∆ R = n → { (1, 1 − n) | n ≥ 2 }
and so, by treating the parameters in a special way, we obtain the following approximation for R+ :
n → { (x, y) → (x0 , y0 ) | n ≥ 2 ∧ y0 ≤ 1 − n + y ∧ x0 ≥ 1 + x }.
If we consider instead
R0 = { (n, x, y) → (n, 1 + x, 1 − n + y) | n ≥ 2 }
then
∆ R0 = { (0, 1, y) | y ≤ −1 }
and we obtain the approximation
n → { (x, y) → (x0 , y0 ) | n ≥ 2 ∧ x0 ≥ 1 + x ∧ y0 ≤ x + y − x0 }.
If we consider both ∆ R and ∆ R0 , then we obtain
n → { (x, y) → (x0 , y0 ) | n ≥ 2 ∧ y0 ≤ 1 − n + y ∧ x0 ≥ 1 + x ∧ y0 ≤ x + y − x0 }.
Note, however, that this is not the most accurate affine approximation that can be obtained. That would be
n → { (x, y) → (x0 , y0 ) | y0 ≤ 2 − n + x + y − x0 ∧ n ≥ 2 ∧ x0 ≥ 1 + x }.

238

2.5.3

Checking Exactness

The approximation T for the transitive closure R+ can be obtained by projecting out
the parameter k from the approximation K (2.3) of the power Rk . Since K is an overapproximation of Rk , T will also be an overapproximation of R+ . To check whether
the results are exact, we need to consider two cases depending on whether R is cyclic,
where R is defined to be cyclic if R+ maps any element to itself, i.e., R+ ∩ Id , ∅. If R
is acyclic, then the inductive definition of (2.2) is equivalent to its completion, i.e.,


R+ = R ∪ R ◦ R+
is a defining property. Since T is known to be an overapproximation, we only need to
check whether
T ⊆ R ∪ (R ◦ T ) .
This is essentially Theorem 5 of Kelly, Pugh, et al. (1996). The only difference is
that they only consider lexicographically forward relations, a special case of acyclic
relations.
If, on the other hand, R is cyclic, then we have to resort to checking whether the
approximation K of the power is exact. Note that T may be exact even if K is not
exact, so the check is sound, but incomplete. To check exactness of the power, we
simply need to check (2.1). Since again K is known to be an overapproximation, we
only need to check whether
K 0 |yd+1 −xd+1 =1 ⊆ R0
K 0 |yd+1 −xd+1 ≥2 ⊆ R0 ◦ K 0 |yd+1 −xd+1 ≥1 ,
where R0 = { x0 → y0 | x → y ∈ R ∧ yd+1 − xd+1 = 1 }, i.e., R extended with path lengths
equal to 1.
All that remains is to explain how to check the cyclicity of R. Note that the exactness on the power is always sound, even in the acyclic case, so we only need to be
careful that we find all cyclic cases. Now, if R is cyclic, i.e., R+ ∩ Id , ∅, then, since T
is an overapproximation of R+ , also T ∩ Id , ∅. This in turn means that ∆ K 0 contains
a point whose first d coordinates are zero and whose final coordinate is positive. In the
implementation we currently perform this test on P0 instead of K 0 . Note that if R+ is
acyclic and T is not, then the approximation is clearly not exact and the approximation
of the power K will not be exact either.

2.5.4

Decomposing R into strongly connected components

If the input relation R is a union of several basic relations that can be partially ordered
then the accuracy of the approximation may be improved by computing an approximation of each strongly connected components separately. For example, if R = R1 ∪ R2
and R1 ◦ R2 = ∅, then we know that any path that passes through R2 cannot later pass
through R1 , i.e.,


R+ = R+1 ∪ R+2 ∪ R+2 ◦ R+1 .
(2.11)
We can therefore compute (approximations of) transitive closures of R1 and R2 separately. Note, however, that the condition R1 ◦ R2 = ∅ is actually too strong. If R1 ◦ R2
239

is a subset of R2 ◦ R1 then we can reorder the segments in any path that moves through
both R1 and R2 to first move through R1 and then through R2 .
This idea can be generalized to relations that are unions of more than two basic
relations by constructing the strongly connected components in the graph with as vertices the basic relations and an edge between two basic relations Ri and R j if Ri needs
to follow R j in some paths. That is, there is an edge from Ri to R j iff
Ri ◦ R j * R j ◦ Ri .

(2.12)

The components can be obtained from the graph by applying Tarjan’s algorithm (Tarjan
1972).
In practice, we compute the (extended) powers Ki0 of each component separately
and then compose them as in (2.6). Note, however, that in this case the order in
which we apply them is important and should correspond to a topological ordering of
the strongly connected components. Simply applying Tarjan’s algorithm will produce
topologically sorted strongly connected components. The graph on which Tarjan’s algorithm is applied is constructed on-the-fly. That is, whenever the algorithm checks
if there is an edge between two vertices, we evaluate (2.12). The exactness check is
performed on each component separately. If the approximation turns out to be inexact
for any of the components, then the entire result is marked inexact and the exactness
check is skipped on the components that still need to be handled.
It should be noted that (2.11) is only valid for exact transitive closures. If overapproximations are computed in the right hand side, then the result will still be an
overapproximation of the left hand side, but this result may not be transitively closed.
If we only separate components based on the condition Ri ◦ R j = ∅, then there is no
problem, as this condition will still hold on the computed approximations of the transitive closures. If, however, we have exploited (2.12) during the decomposition and
if the result turns out not to be exact, then we check whether the result is transitively
closed. If not, we recompute the transitive closure, skipping the decomposition. Note
that testing for transitive closedness on the result may be fairly expensive, so we may
want to make this check configurable.
Example 2.5.7 Consider the relation in example closure4 that comes with the Omega
calculator (Kelly, Maslov, et al. 1996a), R = R1 ∪ R2 , with
R1 = { (x, y) → (x, y + 1) | 1 ≤ x, y ≤ 10 }
R2 = { (x, y) → (x + 1, y) | 1 ≤ x ≤ 20 ∧ 5 ≤ y ≤ 15 }.
This relation is shown graphically in Figure 2.1. We have
R1 ◦ R2 = { (x, y) → (x + 1, y + 1) | 1 ≤ x ≤ 9 ∧ 5 ≤ y ≤ 10 }
R2 ◦ R1 = { (x, y) → (x + 1, y + 1) | 1 ≤ x ≤ 10 ∧ 4 ≤ y ≤ 10 }.
Clearly, R1 ◦ R2 ⊆ R2 ◦ R1 and so


(R1 ∪ R2 )+ = R+2 ◦ R+1 ∪ R+1 ∪ R+2 .

240

Figure 2.1: The relation from Example 2.5.7

Figure 2.2: The relation from Example 2.5.8

241

Example 2.5.8 Consider the relation on the right of Beletska, Barthou, et al. (2009,
Figure 2), reproduced in Figure 2.2. The relation can be described as R = R1 ∪ R2 ∪ R3 ,
with
R1 = n 7→ { (i, j) → (i + 3, j) | i ≤ 2 j − 4 ∧ i ≤ n − 3 ∧ j ≤ 2i − 1 ∧ j ≤ n }
R2 = n 7→ { (i, j) → (i, j + 3) | i ≤ 2 j − 1 ∧ i ≤ n ∧ j ≤ 2i − 4 ∧ j ≤ n − 3 }
R3 = n 7→ { (i, j) → (i + 1, j + 1) | i ≤ 2 j − 1 ∧ i ≤ n − 1 ∧ j ≤ 2i − 1 ∧ j ≤ n − 1 }.
The figure shows this relation for n = 7. Both R3 ◦ R1 ⊆ R1 ◦ R3 and R3 ◦ R2 ⊆ R2 ◦ R3 ,
which the reader can verify using the iscc calculator:
R1 := [n] -> { [i,j] -> [i+3,j] : i
j
R2 := [n] -> { [i,j] -> [i,j+3] : i
j
R3 := [n] -> { [i,j] -> [i+1,j+1] :

<= 2
<= 2
<= 2
<= 2
i <=
j <=

j
i
j
i
2
2

j
i

4
1
1
4
-

and i
and j
and i
and j
1 and
1 and

<= n
<= n
<= n
<= n
i <=
j <=

- 3
};
and
- 3
n n -

and

};
1 and
1 };

(R1 . R3) - (R3 . R1);
(R2 . R3) - (R3 . R2);
R3 can therefore be moved forward in any path. For the other two basic relations,
we have both R2 ◦ R1 * R1 ◦ R2 and R1 ◦ R2 * R2 ◦ R1 and so R1 and R2 form a
strongly connected component. By computing the power of R3 and R1 ∪ R2 separately
and composing the results, the power of R can be computed exactly using (2.5). As
explained by Beletska, Barthou, et al. (2009), applying the same formula to R directly,
without a decomposition, would result in an overapproximation of the power.

2.5.5

Partitioning the domains and ranges of R

The algorithm of Section 2.5.2 assumes that the input relation R can be treated as a
union of translations. This is a reasonable assumption if R maps elements of a given
abstract domain to the same domain. However, if R is a union of relations that map
between different domains, then this assumption no longer holds. In particular, when
an entire dependence graph is encoded in a single relation, as is done by, e.g., Barthou,
Cohen, et al. (2000, Section 6.1), then it does not make sense to look at differences
between iterations of different domains. Now, arguably, a modified Floyd-Warshall
algorithm should be applied to the dependence graph, as advocated by Kelly, Pugh,
et al. (1996), with the transitive closure operation only being applied to relations from
a given domain to itself. However, it is also possible to detect disjoint domains and
ranges and to apply Floyd-Warshall internally.
Let the input relation R be a union of m basic relations Ri . Let D2i be the domains of
Ri and D2i+1 the ranges of Ri . The first step is to group overlapping D j until a partition
is obtained. If the resulting partition consists of a single part, then we continue with the
algorithm of Section 2.5.2. Otherwise, we apply Floyd-Warshall on the graph with as
vertices the parts of the partition and as edges the Ri attached to the appropriate pairs of
vertices. In particular, let there be n parts Pk in the partition. We construct n2 relations
[
R pq B
Ri ,
i s.t. dom Ri ⊆P p ∧ran Ri ⊆Pq

242

Algorithm 1: The modified Floyd-Warshall algorithm of Kelly, Pugh, et al.
(1996)
Input: Relations R pq , 0 ≤ p, q < n
Output: Updated relations R pq such that each relation R pq contains all indirect
paths from p to q in the input graph
1
2
3
4
5
6

for r ∈ [0, n − 1] do
Rrr B R+rr
for p ∈ [0, n − 1] do
for q ∈ [0, n − 1] do
if p , r or q , r then

 

R pq B R pq ∪ Rrq ◦ R pr ∪ Rrq ◦ Rrr ◦ R pr

Figure 2.3: The relation (solid arrows) on the right of Figure 1 of Beletska, Barthou,
et al. (2009) and its transitive closure
apply Algorithm 1 and return the union of all resulting R pq as the transitive closure of R.
Each iteration of the r-loop in Algorithm 1 updates all relations R pq to include paths that
go from p to r, possibly stay there for a while, and then go from r to q. Note that paths
that “stay in r” include all paths that pass through earlier vertices since Rrr itself has
been updated accordingly in previous iterations of the outer loop. In principle, it would
be sufficient to use the R pr and Rrq computed in the previous iteration of the r-loop
in Line 6. However, from an implementation perspective, it is easier to allow either or
both of these to have been updated in the same iteration of the r-loop. This may result in
duplicate paths, but these can usually be removed by coalescing (Section 2.4) the result
of the union in Line 6, which should be done in any case. The transitive closure in
Line 2 is performed using a recursive call. This recursive call includes the partitioning
step, but the resulting partition will usually be a singleton. The result of the recursive
call will either be exact or an overapproximation. The final result of Floyd-Warshall is
therefore also exact or an overapproximation.
Example 2.5.9 Consider the relation on the right of Figure 1 of Beletska, Barthou,

243

et al. (2009), reproduced in Figure 2.3. This relation can be described as
{ (x, y) → (x2 , y2 ) | (3y = 2x ∧ x2 = x ∧ 3y2 = 3 + 2x ∧ x ≥ 0 ∧ x ≤ 3) ∨
(x2 = 1 + x ∧ y2 = y ∧ x ≥ 0 ∧ 3y ≥ 2 + 2x ∧ x ≤ 2 ∧ 3y ≤ 3 + 2x) }.
Note that the domain of the upward relation overlaps with the range of the rightward
relation and vice versa, but that the domain of neither relation overlaps with its own
range or the domain of the other relation. The domains and ranges can therefore be
partitioned into two parts, P0 and P1 , shown as the white and black dots in Figure 2.3,
respectively. Initially, we have
R00 = ∅
R01 = { (x, y) → (x + 1, y) | (x ≥ 0 ∧ 3y ≥ 2 + 2x ∧ x ≤ 2 ∧ 3y ≤ 3 + 2x) }
R10 = { (x, y) → (x2 , y2 ) | (3y = 2x ∧ x2 = x ∧ 3y2 = 3 + 2x ∧ x ≥ 0 ∧ x ≤ 3) }
R11 = ∅.
In the first iteration, R00 remains the same (∅+ = ∅). R01 and R10 are therefore also
unaffected, but R11 is updated to include R01 ◦ R10 , i.e., the dashed arrow in the figure.
This new R11 is obviously transitively closed, so it is not changed in the second iteration
and it does not have an effect on R01 and R10 . However, R00 is updated to include
R10 ◦ R01 , i.e., the dotted arrow in the figure. The transitive closure of the original
relation is then equal to R00 ∪ R01 ∪ R10 ∪ R11 .

2.5.6

Incremental Computation

In some cases it is possible and useful to compute the transitive closure of union of
basic relations incrementally. In particular, if R is a union of m basic maps,
[
R j,
R=
j

then we can pick some Ri and compute the transitive closure of R as

+
[

+
+
∗
∗

R = Ri ∪  Ri ◦ R j ◦ Ri  .

(2.13)

j,i

For this approach to be successful, it is crucial that each of the disjuncts in the argument
of the second transitive closure in (2.13) be representable as a single basic relation, i.e.,
without a union. If this condition holds, then by using (2.13), the number of disjuncts
in the argument of the transitive closure can be reduced by one. Now, R∗i = R+i ∪ Id, but
in some cases it is possible to relax the constraints of R+i to include part of the identity
relation, say on domain D. We will use the notation C(Ri , D) = R+i ∪ IdD to represent
this relaxed version of R+ . Kelly, Pugh, et al. (1996) use the notation R?i . C(Ri , D) can
be computed by allowing k to attain the value 0 in (2.10) and by using
P ∩ (D → D)
244

instead of (2.3). Typically, D will be a strict superset of both dom Ri and ran Ri . We
therefore need to check that domain and range of the transitive closure are part of
C(Ri , D), i.e., the part that results from the paths of positive length (k ≥ 1), are equal
to the domain and range of Ri . If not, then the incremental approach cannot be applied
for the given choice of Ri and D.
In order to be able to replace R∗ by C(Ri , D) in (2.13), D should be chosen to
include both dom R and ran R, i.e., such that IdD ◦R j ◦ IdD = R j for all j , i. Kelly,
Pugh, et al. (1996) say that they use D = dom Ri ∪ ran Ri , but presumably they mean
that they use D = dom R ∪ ran R. Now, this expression of D contains a union, so it not
directly usable. Kelly, Pugh, et al. (1996) do not explain how they avoid this union.
Apparently, in their implementation, they are using the convex hull of dom R ∪ ran R or
at least an approximation of this convex hull. We use the simple hull (Section 2.2) of
dom R ∪ ran R.
It is also possible to use a domain D that does not include dom R ∪ ran R, but then
we have to compose with C(Ri , D) more selectively. In particular, if we have
for each j , i either dom R j ⊆ D or dom R j ∩ ran Ri = ∅

(2.14)

for each j , i either ran R j ⊆ D or ran R j ∩ dom Ri = ∅

(2.15)

and, similarly,

then we can refine (2.13) to
 
 

 
 

 
 
 [
[
 
 

C ◦ R j  ∪ 
C ◦ R j ◦ C ∪ 
R+i ∪
 
 dom R ∩ran R =∅
dom R ⊆D
j
j
i
 
 

ran R j ⊆D

 
+
 

 

[
[
 

R j  .
R j ◦ C ∪ 


 dom R j ∩ran Ri =∅ 
dom R j ⊆D

ran R j ∩dom Ri =∅

ran R j ⊆D

If only property (2.14) holds, we can use
 


 

 [


R j ◦ C ∪ 
R+i ∪  R+i ∪ Id ◦ 
dom R j ⊆D

while if only property (2.15) holds, we can use

 
 [
 
[



+
Ri ∪ 
C ◦ R j  ∪ 
ran R j ⊆D

[

ran R j ∩dom Ri =∅

+ 
 
R j   ,

dom R j ∩ran Ri =∅

+







+
R j  ◦ Ri ∪ Id  .

ran R j ∩dom Ri =∅

It should be noted that if we want the result of the incremental approach to be
transitively closed, then we can only apply it if all of the transitive closure operations
involved are exact. If, say, the second transitive closure in (2.13) contains extra elements, then the result does not necessarily contain the composition of these extra
elements with powers of Ri .

245

2.5.7

An Omega-like implementation

While the main algorithm of Kelly, Pugh, et al. (1996) is designed to compute and
underapproximation of the transitive closure, the authors mention that they could also
compute overapproximations. In this section, we describe our implementation of an
algorithm that is based on their ideas. Note that the Omega library computes underapproximations (Kelly, Maslov, et al. 1996b, Section 6.4).
The main tool is Equation (2) of Kelly, Pugh, et al. (1996). The input relation R is
first overapproximated by a “d-form” relation
{ i → j | ∃α : L ≤ j − i ≤ U ∧ (∀p : j p − i p = M p α p ) },
where p ranges over the dimensions and L, U and M are constant integer vectors.
The elements of U may be ∞, meaning that there is no upper bound corresponding
to that element, and similarly for L. Such an overapproximation can be obtained by
computing strides, lower and upper bounds on the difference set ∆ R. The transitive
closure of such a “d-form” relation is
{ i → j | ∃α, k : k ≥ 1 ∧ k L ≤ j − i ≤ k U ∧ (∀p : j p − i p = M p α p ) }.

(2.16)

The domain and range of this transitive closure are then intersected with those of the
input relation. This is a special case of the algorithm in Section 2.5.2.
In their algorithm for computing lower bounds, the authors use the above algorithm
as a substep on the disjuncts in the relation. At the end, they say
If an upper bound is required, it can be calculated in a manner similar to
that of a single conjunct [sic] relation.
Presumably, the authors mean that a “d-form” approximation of the whole input relation should be used. However, the accuracy can be improved by also trying to apply
the incremental technique from the same paper, which is explained in more detail in
Section 2.5.6. In this case, C(Ri , D) can be obtained by allowing the value zero for k in
(2.16), i.e., by computing
{ i → j | ∃α, k : k ≥ 0 ∧ k L ≤ j − i ≤ k U ∧ (∀p : j p − i p = M p α p ) }.
In our implementation we take as D the simple hull (Section 2.2) of dom R ∪ ran R.
To determine whether it is safe to use C(Ri , D), we check the following conditions, as
proposed by Kelly, Pugh, et al. (1996): C(Ri , D) − R+i is not a union and for each j , i
the condition




C(Ri , D) − R+i ◦ R j ◦ C(Ri , D) − R+i = R j
holds.

246

Chapter 3

Further Reading
Verdoolaege (2016) describes the concepts behind isl in some detail, mainly focusing
on Presburger formulas, but also including some information on polyhedral compilation, especially on dependence analysis. Individual aspects of isl are described in the
following publications.
• Verdoolaege, Janssens, et al. (2009) introduce isl as a library for manipulating
sets of integers defined by linear inequalities and integer divisions that is used in
their equivalence checker.
• Verdoolaege (2010a) provides a more detailed description of isl at the time
and still stands as the official reference for isl. However, many features were
only added later on and one or more of the publications below may be more
appropriate as a reference to these features.
• Verdoolaege (2010b, Section 5.1) provides some details on the dataflow analysis
step, but also see Verdoolaege (2016, Chapter 6) and Verdoolaege and Cohen
(2016) for a more recent treatment.
• The concepts of structured and named spaces and the manipulation of sets containing elements in different spaces were introduced by Verdoolaege (2011).
• The transitive closure operation is described by Verdoolaege, Cohen, and Beletska (2011).
• The scheduler is briefly described by Verdoolaege, Juega, et al. (2013, Section
6.2) and Verdoolaege and Cohen (2016, Section 2.4).
• Schedule trees started out as “trees of bands” (Verdoolaege, Juega, et al. 2013,
Section 6.2), were formally introduced by Verdoolaege, Guelton, et al. (2014),
and were slightly refined by Grosser, Verdoolaege, et al. (2015).
• The coalescing operation is described by Verdoolaege (2015).
• The AST generator is described by Grosser, Verdoolaege, et al. (2015).

247

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Verdoolaege, Sven, Albert Cohen, and Anna Beletska (2011). “Transitive Closures of
Affine Integer Tuple Relations and Their Overapproximations”. In: Proceedings
of the 18th International Conference on Static Analysis. SAS’11. Venice, Italy:
Springer-Verlag, pp. 216–232. doi: 10.1007/978-3-642-23702-7_18.
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Verdoolaege, Sven, Serge Guelton, Tobias Grosser, and Albert Cohen (2014). “Schedule Trees”. In: Proceedings of the 4th International Workshop on Polyhedral Compilation Techniques. Vienna, Austria. doi: 10.13140/RG.2.1.4475.6001. [247]
Verdoolaege, Sven, Gerda Janssens, and Maurice Bruynooghe (2009). “Equivalence
checking of static affine programs using widening to handle recurrences”. In: Computer Aided Verification 21. Springer, pp. 599–613. doi: 10.1007/978- 3- 64202658-4_44.
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Verdoolaege, Sven, Juan Carlos Juega, Albert Cohen, José Ignacio Gómez, Christian
Tenllado, and Francky Catthoor (2013). “Polyhedral parallel code generation for
CUDA”. In: ACM Trans. Archit. Code Optim. 9.4, p. 54. doi: 10.1145/2400682.
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Verhaegh, Wim F. J. (1995). “Multidimensional Periodic Scheduling”. PhD thesis.
Technische Universiteit Eindhoven.
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