Picat Guide

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A User’s Guide to Picat
Version 2.4
Neng-Fa Zhou and Jonathan Fruhman
Copyright c
picat-lang.org, 2013-2018.
Last updated April 8, 2018
Preface
Despite the elegant concepts, new extensions (e.g., tabling and constraints), and successful appli-
cations (e.g., knowledge engineering, NLP, and search problems), Prolog has a bad reputation for
being old and difficult. Many ordinary programmers find the implicit non-directionality and non-
determinism of Prolog to be hard to follow, and the non-logical features, such as cuts and dynamic
predicates, are prone to misuses, leading to absurd codes. The lack of language constructs (e.g.,
loops) and libraries for programming everyday things is also considered a big weakness of Prolog.
The backward compatibility requirement has made it hopeless to remedy the language issues in
current Prolog systems, and there are urgent calls for a new language.
Several successors of Prolog have been designed, including Mercury, Erlang, Oz, and Curry.
The requirement of many kinds of declarations in Mercury has made the language difficult to
use; Erlang’s abandonment of non-determinism in favor of concurrency has made the language
unsuited for many applications despite its success in the telecom industry; Oz has never attained
the popularity that the designers sought, probably due to its unfamiliar syntax and implicit laziness;
Curry is considered too close to Haskell. All of these successors were designed in the 1990s, and
now the time is ripe for a new logic-based language.
Picat aims to be a simple, and yet powerful, logic-based programming language for a variety of
applications. Picat incorporates many declarative language features for better productivity of soft-
ware development, including explicit non-determinism, explicit unification, functions, constraints,
and tabling. Picat lacks Prolog’s non-logical features, such as the cut operator and dynamic predi-
cates, making Picat more reliable than Prolog. Picat also provides imperative language constructs
for programming everyday things. The system can be used for not only symbolic computations,
which is a traditional application domain of declarative languages, but also for scripting and mod-
eling tasks.
Picat is a general-purpose language that incorporates features from logic programming, func-
tional programming, and scripting languages. The letters in the name summarize Picat’s features:
•Pattern-matching: A predicate defines a relation, and can have zero, one, or multiple an-
swers. A function is a special kind of a predicate that always succeeds with one answer.
Picat is a rule-based language. Predicates and functions are defined with pattern-matching
rules.
•Intuitive: Picat provides assignment and loop statements for programming everyday things.
An assignable variable mimics multiple logic variables, each of which holds a value at a
different stage of computation. Assignments are useful for computing aggregates and are
used with the foreach loop for implementing list and array comprehensions.
•Constraints: Picat supports constraint programming. Given a set of variables, each of which
has a domain of possible values, and a set of constraints that limit the acceptable set of
assignments of values to variables, the goal is to find an assignment of values to the variables
that satisfies all of the constraints. Picat provides three solver modules: cp,sat, and mip.
These three modules follow the same interface, which allows for seamless switching from
one solver to another.
•Actors: Actors are event-driven calls. Picat provides action rules for describing event-
driven behaviors of actors. Events are posted through channels. An actor can be attached to
a channel in order to watch and to process its events.
•Tabling: Tabling can be used to store the results of certain calculations in memory, allow-
ing the program to do a quick table lookup instead of repeatedly calculating a value. As
i
computer memory grows, tabling is becoming increasingly important for offering dynamic
programming solutions for many problems. The planner module, which is implemented
by the use of tabling, has been shown to be a more efficient tool than ASP and PDDL for
solving many planning problems.
The support of explicit unification, explicit non-determinism, tabling, and constraints makes
Picat more suitable than functional and scripting languages for symbolic computations. Picat is
arguably more expressive than Prolog for scripting and modeling. With arrays, loops, and list and
array comprehensions, it is not rare to find problems for which Picat requires an order of magnitude
fewer lines of code to describe than Prolog. Picat is more scalable than Prolog. The use of pattern-
matching rather than unification facilitates indexing of rules. Picat is also more reliable than
Prolog. In addition to explicit non-determinism, explicit unification, and a simple static module
system, the lack of cuts, dynamic predicates, and operator overloading also improves the reliability
of the language. Picat is not as powerful as Prolog for metaprogramming and it’s impossible to
write a meta-interpreter for Picat in Picat itself. Nevertheless, this weakness can be remedied with
library modules for implementing domain-specific languages.
The Picat implementation is based on the B-Prolog engine. The current implementation is
already ready for many kinds of applications. It will also serve as a foundation for new additions,
including external language interfaces with C, Java, and Python, an external language interface
with MySql, threads, sockets, Web services, and language processing modules. Anybody is wel-
come to contribute. The C source code is available to registered developers and users. Please
contact picat@picat-lang.org.
Acknowledgements
The initial design of Picat was published in December 2012, and the first alpha version was
released in May 2013. The following people have contributed to the project by reviewing the
ideas, the design, the implementation, and/or the documentation: Roman Bart´
ak, Nikhil Barth-
wal, Mike Bionchik, Lei Chen, Veronica Dahl, Claudio Cesar de S´
a, Agostino Dovier, Sergii
Dymchenko, Julio Di Egidio, Christian Theil Have, H˚
akan Kjellerstrand, Annie Liu, Nuno Lopes,
Richard O’Keefe, Lorenz Schiffmann, Paul Tarau, and Jan Wielemaker. Special thanks to H˚
akan
Kjellerstrand, who has been programming in Picat and blogging about Picat since May 2013.
The system wouldn’t have matured so quickly without H˚
akan’s hundreds of programs. The
picat-lang.org web page was designed by Bo Yuan (Bobby) Zhou. The Picat project was
supported in part by the NSF under grant numbers CCF1018006 and CCF1618046.
The Picat implementation is based on the B-Prolog engine. It uses the following public domain
modules: token.c by Richard O’Keefe; getline.c by Chris Thewalt; bigint.c by Matt
McCutchen; Lingeling by Armin Biere; Espresso (by Berkeley). In addition, Picat also
provides interfaces to GLPK by Andrew Makhorin and Gurobi by Gurobi Optimization, Inc.
ii
Contents
1 Overview 1
1.1 DataTypes ..................................... 1
1.2 DefiningPredicates................................. 5
1.3 DefiningFunctions ................................. 6
1.4 AssignmentsandLoops............................... 7
1.5 Tabling ....................................... 10
1.6 Modules....................................... 11
1.7 Constraints ..................................... 12
1.8 Exceptions...................................... 13
1.9 Higher-OrderCalls ................................. 14
1.10ActionRules .................................... 15
1.11PrebuiltMaps.................................... 16
1.12ProgrammingExercises............................... 17
2 How to Use the Picat System 18
2.1 How to Use the Picat Interpreter . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.1 How to Enter and Quit the Picat Interpreter . . . . . . . . . . . . . . . . 18
2.1.2 How to Use the Command-line Editor . . . . . . . . . . . . . . . . . . . 19
2.1.3 How to Compile and Load Programs . . . . . . . . . . . . . . . . . . . . 19
2.1.4 HowtoRunPrograms ........................... 20
2.1.5 How to Run Programs Directly . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.6 Creating Standalone Executables . . . . . . . . . . . . . . . . . . . . . . 21
2.2 HowtoUsetheDebugger.............................. 22
3 Data Types, Operators, and Built-ins 24
3.1 Variables ...................................... 26
3.2 Atoms........................................ 26
3.3 Numbers....................................... 27
3.4 CompoundTerms.................................. 30
3.4.1 Lists..................................... 30
3.4.2 Strings ................................... 33
3.4.3 Structures.................................. 33
3.4.4 Arrays.................................... 34
3.4.5 Maps .................................... 35
3.4.6 Sets..................................... 36
3.4.7 Heaps.................................... 36
3.5 Equality Testing, Unification, and Term Comparison . . . . . . . . . . . . . . . 37
3.5.1 NumericalEquality............................. 38
3.5.2 OrderingofTerms ............................. 38
iii
3.6 Expressions..................................... 39
3.7 Higher-order Predicates and Functions . . . . . . . . . . . . . . . . . . . . . . . 39
3.8 Other Built-ins in the basic Module ....................... 41
4 Predicates and Functions 43
4.1 Predicates...................................... 43
4.2 Functions ...................................... 44
4.3 Patterns and Pattern-Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Goals ........................................ 45
4.5 PredicateFacts ................................... 47
4.6 TailRecursion.................................... 47
5 Assignments and Loops 49
5.1 Assignments..................................... 49
5.1.1 If-Else.................................... 49
5.2 TypesofLoops ................................... 50
5.2.1 ForeachLoops ............................... 50
5.2.2 Foreach Loops with Multiple Iterators . . . . . . . . . . . . . . . . . . . 51
5.2.3 WhileLoops ................................ 52
5.2.4 Do-whileLoops .............................. 53
5.3 List and Array Comprehensions . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.4 CompilationofLoops................................ 54
5.4.1 ListComprehensions............................ 56
6 Exceptions 58
6.1 Built-inExceptions ................................. 58
6.2 ThrowingExceptions................................ 59
6.3 Defining Exception Handlers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7 Tabling 60
7.1 TableDeclarations ................................. 60
7.2 TheTablingMechanism .............................. 62
8 The planner Module 65
8.1 Depth-BoundedSearch............................... 66
8.2 Depth-UnboundedSearch.............................. 68
8.3 Examples ...................................... 68
9 Modules 71
9.1 Module and Import Declarations . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2 Binding Calls to Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.3 Binding Higher-Order Calls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.4 LibraryModules .................................. 73
10 I/O 74
10.1OpeningaFile.................................... 74
10.2ReadingfromaFile................................. 75
10.2.1 EndofFile ................................. 77
10.3WritingtoaFile................................... 78
10.4 Flushing and Closing a File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
iv
10.5 Standard File Descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
11 Event-Driven Actors and Action Rules 81
11.1 Channels, Ports, and Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
11.2ActionRules .................................... 82
11.3LazyEvaluation................................... 84
11.4ConstraintPropagators ............................... 85
12 Constraints 86
12.1DomainVariables.................................. 87
12.2Tableconstraints .................................. 88
12.3ArithmeticConstraints ............................... 89
12.4BooleanConstraints................................. 90
12.5GlobalConstraints ................................. 91
12.6SolverInvocation .................................. 94
12.6.1 Common Solving Options . . . . . . . . . . . . . . . . . . . . . . . . . 95
12.6.2 Solving Options for cp ........................... 95
12.6.3 Solving Options for sat .......................... 96
12.6.4 Solving Options for mip .......................... 96
13 The os Module 98
13.1 The P ath Parameter ................................ 98
13.2Directories...................................... 98
13.2.1 The Current Working Directory . . . . . . . . . . . . . . . . . . . . . . 99
13.3 Modifying Files and Directories . . . . . . . . . . . . . . . . . . . . . . . . . . 99
13.3.1 Creation................................... 99
13.3.2 Deletion................................... 99
13.4 Obtaining Information about Files . . . . . . . . . . . . . . . . . . . . . . . . . 100
13.5EnvironmentVariables ............................... 101
A The math Module 102
A.1 Constants ...................................... 102
A.2 Functions ...................................... 102
A.2.1 Sign and Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.2.2 Rounding and Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.2.3 Exponents, Roots, and Logarithms . . . . . . . . . . . . . . . . . . . . . 103
A.2.4 Converting Between Degrees and Radians . . . . . . . . . . . . . . . . . 104
A.2.5 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.2.6 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.2.7 RandomNumbers ............................. 106
A.2.8 OtherBuilt-ins ............................... 107
B The sys Module 108
B.1 Compiling and Loading Programs . . . . . . . . . . . . . . . . . . . . . . . . . 108
B.2 TracingExecution.................................. 109
B.2.1 Debugging Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B.3 Information about the Picat System . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.3.1 Statistics .................................. 110
B.3.2 Time .................................... 112
B.3.3 Other System Information . . . . . . . . . . . . . . . . . . . . . . . . . 112
v
B.4 GarbageCollection ................................. 112
B.5 Quitting the Picat System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
C The util Module 114
C.1 UtilitiesonTerms.................................. 114
C.2 Utilities on Strings and Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
C.3 UtilitiesonMatrices ................................ 115
C.4 Utilities on Lists and Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
D The ordset Module 117
E The datetime Module 118
F Formats 119
F.1 FormattedPrinting ................................. 119
G External Language Interface with C 120
G.1 TermRepresentation ................................ 120
G.2 Fetching Arguments of Picat Calls . . . . . . . . . . . . . . . . . . . . . . . . . 120
G.3 TestingPicatTerms................................. 120
G.4 Converting Picat Terms into C . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
G.5 Manipulating and Writing Picat Terms . . . . . . . . . . . . . . . . . . . . . . . 122
G.6 BuildingPicatTerms ................................ 122
G.7 Registering C-defined Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . 122
H Appendix: Tokens 125
I Appendix: Grammar 129
J Appendix: Operators 134
K Appendix: The Library Modules 135
Index 139
vi
Chapter 1
Overview
Before we give an overview of the Picat language, let us briefly describe how to use the Picat
system. The Picat system provides an interactive programming environment for users to load,
debug, and execute programs. Users can start the Picat interpreter with the OS command picat.
OSP rompt picat
Once the interpreter is started, users can type a command line after the prompt Picat>. The
help command shows the usages of commands, and the halt command terminates the Picat
interpreter. Users can also use the picat command to run a program directly as follows:
OSP rompt picat F ile Arg1Arg2. . . Argn
where F ile (with or without the extension .pi) is the main file name of the program. The program
must define a predicate named main/0 or main/1. If the command line contains arguments
after the file name, then main/1 is executed. Otherwise, if the file name is not followed by any
arguments, then main/0 is executed. When main/1 executed, all of the arguments after the file
name are passed to the predicate as a list of strings.
1.1 Data Types
Picat is a dynamically-typed language, in which type checking occurs at runtime. A variable in
Picat is a value holder. A variable name is an identifier that begins with a capital letter or the
underscore. An attributed variable is a variable that has a map of attribute-value pairs attached to
it. A variable is free until it is bound to a value. A value in Picat can be primitive or compound.
A primitive value can be an integer, a real number, or an atom. A character can be represented
as a single-character atom. An atom name is an identifier that begins with a lower-case letter or a
single-quoted sequence of characters.
A compound value can be a list in the form [t1,. . .,tn]or a structure in the form $s(t1,. . .,tn)
where sstands for a structure name, nis called the arity of the structure, and each ti(1≤i≤n)
is a term which is a variable or a value. The preceding dollar symbol is used to distinguish a struc-
ture from a function call. Strings, arrays, maps, sets, and heaps are special compound values. A
string is a list of single-character atoms. An array takes the form {t1,. . .,tn}, which is a special
structure with the name ’{}’. A map is a hash-table represented as a structure that contains a set
of key-value pairs. A set is a special map where only keys are used. A heap is a complete binary
tree represented as an array. A heap can be a min-heap or a max-heap.
The function new struct(Name, IntOrList)returns a structure. The function new map(S)
returns a map that initially contains the pairs in list S, where each pair has the form Key =
V al. The function new set(S)returns a map set that initially contains the elements in list
1
S. A map-set is a map in which every key is mapped to the atom not a value. The function
new array(I1, I2, . . . , In)returns an n-dimensional array, where each Iiis an integer expres-
sion specifying the size of a dimension. An n-dimensional array is a one-dimensional array where
the arguments are (n-1)-dimensional arrays.
Example
Picat> V1 = X1, V2 = _ab, V3 = _ % variables
Picat> N1 = 12, N2 = 0xf3, N3 = 1.0e8 % numbers
Picat> A1 = x1, A2 = ’_AB’, A3 = ’’ % atoms
Picat> L = [a,b,c,d] % a list
Picat> write("hello"++"picat") % strings
[h,e,l,l,o,p,i,c,a,t]
Picat> print("hello"++"picat")
hellopicat
Picat> writef("%s","hello"++"picat") % formatted write
hellopicat
Picat> writef("%-5d %5.2f",2,2.0) % formatted write
2 2.00
Picat> S = $point(1.0,2.0) % a structure
Picat> S = new_struct(point,3) % create a structure
S = point(_3b0,_3b4,_3b8)
Picat> A = {a,b,c,d} % an array
Picat> A = new_array(3) % create an array
A = {_3b0,_3b4,_3b8}
Picat> M = new_map([one=1,two=2]) % create a map
M = (map)[two = 2,one = 1]
Picat> M = new_set([one,two,three]) % create a map set
M = (map)[two,one,three]
Picat> X = 1..2..10 % ranges
X = [1,3,5,7,9]
Picat> X = 1..5
X = [1,2,3,4,5]
Picat allows function calls in arguments. For this reason, it requires structures to be preceded
2
with a dollar symbol in order for them to be treated as data. Without the dollar symbol, the
command S=point(1.0,2.0) would call the function point(1.0,2.0) and bind Sto
its return value. In order to ensure safe interpretation of meta-terms in higher-order calls, Picat
forbids the creation of terms that contain structures with the name ’.’, index notations, array
comprehensions, list comprehensions, and loops.
For each type, Picat provides a set of built-in functions and predicates. The index notation
X[I], where Xreferences a compound value and Iis an integer expression, is a special function
that returns a single component of X. The index of the first element of a list or a structure is 1. In
order to facilitate type checking at compile time, Picat does not overload arithmetic operators for
other purposes, and requires an index expression to be an integer.
A list comprehension, which takes the following form, is a special functional notation for
creating lists:
[T:E1in D1,Cond1,. . .,Enin Dn,Condn]
where Tis an expression, each Eiis an iterating pattern, each Diis an expression that gives
a compound value, and the optional conditions Cond1,. . .,Condnare callable terms. This list
comprehension means that for every tuple of values E1∈D1,. . .,En∈Dn, if the conditions are
true, then the value of Tis added into the list.
An array comprehension takes the following form:
{T:E1in D1,Cond1,. . .,Enin Dn,Condn}
It is the same as:
to array([T:E1in D1,Cond1,. . .,Enin Dn,Condn])
The predicate put(Map, Key, V al)attaches the key-value pair Key=V al to the map Map,
where Key is a non-variable term, and V al is any term. The function get(Map, Key)returns
V al of the key-value pair Key=V al attached to Map. The predicate has key(Map, Key)
returns true iff Map contains a pair with the given key.
An attributed variable has a map attached to it. The predicate put attr(X, Key, V al)
attaches the key-value pair Key=V al to X. The function get attr(X, Key)returns V al of
the key-value pair Key=V al attached to X.
Example
Picat> integer(5)
yes
Picat> real(5)
no
Picat> var(X)
yes
Picat> X=5, var(X)
no
Picat> 5 != 2+2
yes
3
Picat> X = to_binary_string(5)
X = [’1’,’0’,’1’]
Picat> L = [a,b,c,d], X = L[2]
X=b
Picat> L = [(A,I) : A in [a,b], I in 1..2].
L = [(a,1),(a,2),(b,1),(b,2)]
Picat> put_attr(X,one,1), One = get_attr(X,one) % attributed var
One = 1
Picat> S = new_struct(point,3), Name = name(S), Len = length(S)
S = point(_3b0,_3b4,_3b8)
Name = point
Len = 3
Picat> S = new_array(2,3), S[1,1] = 11, D2 = length(S[2])
S = {{11,_93a0,_93a4},{_938c,_9390,_9394}}
D2 = 3
Picat> M = new_map(), put(M,one,1), One = get(M,one)
One = 1
Picat> M = new_set(), put(M,one), has_key(M,one).
Picat also allows OOP notations for calling predicates and functions. The notation A1.f(A2, . . . , Ak)
is the same as f(A1, A2, . . . , Ak), unless A1is an atom, in which case A1must be a module qual-
ifier for f. The notation A.f, where fis an atom, is the same as the call A.f().
Example
Picat> X = 5.to_binary_string()
X = [’1’,’0’,’1’]
Picat> X = 5.to_binary_string().length
X=3
Picat> X.put(one,1), One = X.get(one)
One = 1
Picat> X = math.pi
X=3.14159
Picat> S = new_struct(point,3), Name = S.name, Len = S.length
S = point(_3b0,_3b4,_3b8)
Name = point
Len = 3
Picat> S = new_array(2,3), S[1,1] = 11, D2 = S[2].length
4
S = {{11,_93a0,_93a4},{_938c,_9390,_9394}}
D2 = 3
Picat> M = new_map(), M.put(one,1), One = M.one.
One = 1
1.2 Defining Predicates
A predicate call either succeeds or fails, unless an exception occurs. A predicate call can return
multiple answers through backtracking. The built-in predicate true always succeeds, and the
built-in predicate fail (or false) always fails. A goal is made from predicate calls and state-
ments, including conjunction (A, B and A&& B), disjunction (A;Band A|| B), negation (not
A), if-then-else, foreach loops, and while loops.
A predicate is defined with pattern-matching rules. Picat has two types of rules: the non-
backtrackable rule Head, Cond => Body, and the backtrackable rule Head, Cond ?=> Body.
The Head takes the form p(t1, . . . , tn), where pis called the predicate name, and nis called the
arity. When n= 0, the parentheses can be omitted. The condition Cond, which is an optional
goal, specifies a condition under which the rule is applicable. For a call C, if Cmatches Head and
Cond succeeds, meaning that the condition evaluates to true, the rule is said to be applicable to C.
When applying a rule to call C, Picat rewrites Cinto Body. If the used rule is non-backtrackable,
then the rewriting is a commitment, and the program can never backtrack to C. If the used rule
is backtrackable, however, the program will backtrack to Conce Body fails, meaning that Body
will be rewritten back to C, and the next applicable rule will be tried on C.
Example
fib(0,F) => F=1.
fib(1,F) => F=1.
fib(N,F),N>1 => fib(N-1,F1),fib(N-2,F2),F=F1+F2.
fib(N,F) => throw $error(wrong_argument,fib,N).
A call matches the head fib(0,F) if the first argument is 0. The second argument can
be anything. For example, for the call fib(0,2), the first rule is applied, since fib(0, 2)
matches its head. However, when the body is executed, the call 2=1 fails.
The predicate fib/2 can also be defined using if-then-else as follows:
fib(N,F) =>
if (N=0; N=1) then
F=1
elseif N>1 then
fib(N-1,F1),fib(N-2,F2),F=F1+F2
else
throw $error(wrong_argument,fib,N)
end.
An if statement takes the form if Cond then Goal1else Goal2end.1The then part
can contain one or more elseif clauses. The else part can be omitted. In that case the else
part is assumed to be else true. The built-in throw Ethrows term Eas an exception.
1Picat also accepts Prolog-style if-then-else in the form (If->Then;Else) and mandates the presence of the
else-part.
5
Example
member(X,[Y|_]) ?=> X=Y.
member(X,[_|L]) => member(X,L).
The pattern [Y|_] matches any list. The backtrackable rule makes a call nondeterministic,
and the predicate can be used to retrieve elements from a list one at a time through backtracking.
Picat> member(X,[1,2,3])
X=1;
X=2;
X=3;
no
After Picat returns an answer, users can type a semicolon immediately after the answer to ask for
the next answer. If users only want one answer to be returned from a call, they can use once
Call to stop backtracking.
The version of member that checks if a term occurs in a list can be defined as follows:
membchk(X,[X|_]) => true.
membchk(X,[_|L]) => membchk(X,L).
The first rule is applicable to a call if the second argument is a list and the first argument of the call
is identical to the first element of the list.
Picat allows inclusion of predicate facts in the form p(t1,. . .,tn)in predicate definitions.
Facts are translated into pattern-matching rules before they are compiled. A predicate definition
that consists of facts can be preceded by an index declaration in the form index (M11, . . . , M1n)
. . . (Mm1, . . . , Mmn)where each Mij is either +(meaning indexed) or −(meaning not in-
dexed). For each index pattern (Mi1, . . . , Min), the compiler generates a version of the predicate
that indexes all of the +arguments.
Example
index (+,-) (-,+)
edge(a,b).
edge(a,c).
edge(b,c).
edge(c,b).
For a predicate of indexed facts, a matching version of the predicate is selected for a call. If no
matching version is available, Picat throws an exception. For example, for the call edge(X,Y),
if both Xand Yare free, then no version of the predicate matches this call and Picat throws an ex-
ception. If predicate facts are not preceded by any index declaration, then no argument is indexed.
1.3 Defining Functions
A function call always succeeds with a return value if no exception occurs. Functions are defined
with non-backtrackable rules in which the head is an equation F=X, where Fis the function
pattern in the form f(t1, . . . , tn)and Xholds the return value. When n= 0, the parentheses can
be omitted.
6
Example
fib(0)=F => F=1.
fib(1)=F => F=1.
fib(N)=F,N>1 => F=fib(N-1)+fib(N-2).
qsort([])=L => L=[].
qsort([H|T])=L => L = qsort([E : E in T, E=<H])++[H]++
qsort([E : E in T, E>H]).
A function call never fails and never succeeds more than once. For function calls such as fib(-1)
or fib(X), Picat raises an exception.
Picat allows inclusion of function facts in the form f(t1,. . .,tn)=Exp in function definitions.
Example
fib(0)=1.
fib(1)=1.
fib(N)=F,N>1 => F=fib(N-1)+fib(N-2).
qsort([])=[].
qsort([H|T]) =
qsort([E : E in T, E=<H])++[H]++qsort([E : E in T, E>H]).
Function facts are automatically indexed on all of the input arguments, and hence no index decla-
ration is necessary. Note that while a predicate call with no argument does not need parentheses, a
function call with no argument must be followed with parentheses, unless the function is module-
quantified, as in math.pi.
The fib function can also be defined as follows:
fib(N) = cond((N=0;N=1),1,fib(N-1)+fib(N-2)).
The conditional expression returns 1 if the condition (N=0;N=1) is true, and the value of
fib(N-1)+fib(N-2) if the condition is false.
1.4 Assignments and Loops
Picat allows assignments in rule bodies. An assignment takes the form LHS:=RHS, where LHS
is either a variable or an access of a compound value in the form X[...]. When LHS is an
access in the form X[I], the component of Xindexed Iis updated. This update is undone if
execution backtracks over this assignment.
Example
test => X=0, X:=X+1, X:=X+2, write(X).
In order to handle assignments, Picat creates new variables at compile time. In the above
example, at compile time, Picat creates a new variable, say X1, to hold the value of Xafter the
assignment X:=X+1. Picat replaces Xby X1 on the LHS of the assignment. It also replaces all of
the occurrences of Xto the right of the assignment by X1. When encountering X1:=X1+2, Picat
creates another new variable, say X2, to hold the value of X1 after the assignment, and replaces
the remaining occurrences of X1 by X2. When write(X2) is executed, the value held in X2,
which is 3, is printed. This means that the compiler rewrites the above example as follows:
7
test => X=0, X1=X+1, X2=X1+2, write(X2).
Picat supports foreach and while statements for programming repetitions. A foreach
statement takes the form
foreach (E1in D1,Cond1,. . .,Enin Dn,Condn)
Goal
end
where each iterator, Eiin Di, can be followed by an optional condition Condi. Within each
iterator, Eiis an iterating pattern, and Diis an expression that gives a compound value. The
foreach statement means that Goal is executed for every possible combination of values E1∈
D1,. . .,En∈Dnthat satisfies the conditions Cond1,. . .,Condn. A while statement takes the
form
while (Cond)
Goal
end
It repeatedly executes Goal as long as Cond succeeds. A variant of the while loop in the form of
do
Goal
while (Cond)
executes Goal one time before testing Cond.
A loop statement forms a name scope. Variables that occur only in a loop, but do not occur
before the loop in the outer scope, are local to each iteration of the loop. For example, in the
following rule:
p(A) =>
foreach (I in 1 .. A.length)
E = A[I],
writeln(E)
end.
the variables Iand Eare local, and each iteration of the loop has its own values for these variables.
Example
write_map(Map) =>
foreach (Key=Value in Map)
writef("%w=%w\n",Key,Value)
end.
sum_list(L)=Sum => % returns sum(L)
S=0,
foreach (X in L)
S:=S+X
end,
Sum=S.
read_list=List =>
8
L=[],
E=read_int(),
while (E != 0)
L := [E|L],
E := read_int()
end,
List=L.
The function read list reads a sequence of integers into a list, terminating when 0 is read. The
loop corresponds to the following sequence of recurrences:
L=[]
L1=[e1|L]
L2=[e2|L1]
. . .
Ln=[en|Ln−1]
List=Ln
Note that the list of integers is in reversed order. If users want a list in the same order as the input,
then the following loop can be used:
read_list=List =>
List=L,
E=read_int(),
while (E != 0)
L = [E|T],
L := T,
E := read_int()
end,
L=[].
This loop corresponds to the following sequence of recurrences:
L=[e1|L1]
L1=[e2|L2]
. . .
Ln−1=[en|Ln]
Ln=[]
Loop statements are compiled into tail-recursive predicates. For example, the second read list
function given above is compiled into:
read_list=List =>
List=L,
E=read_int(),
p(E,L,Lout),
Lout=[].
p(0,Lin,Lout) => Lout=Lin.
p(E,Lin,Lout) =>
Lin=[E|Lin1],
NE = read_int(),
p(NE,Lin1,Lout).
9
A list comprehension is first compiled into a foreach loop, and then the loop is compiled
into a call to a generated tail-recursive predicate. For example, the list comprehension
List = [(A,X) : A in [a,b], X in 1..2]
is compiled into the following loop:
List = L,
foreach(A in [a,b], X in 1..2)
L = [(A,X)|T],
L := T
end,
L = [].
1.5 Tabling
A predicate defines a relation where the set of facts is implicitly generated by the rules. The
process of generating the facts may never end and/or may contain a lot of redundancy. Tabling
can prevent infinite loops and redundancy by memorizing calls and their answers. In order to have
all calls and answers of a predicate or function tabled, users just need to add the keyword table
before the first rule.
Example
table
fib(0) = 1.
fib(1) = 1.
fib(N) = fib(N-1)+fib(N-2).
When not tabled, the function call fib(N) takes exponential time in N. When tabled, however, it
takes only linear time.
Users can also give table modes to instruct the system on what answers to table. Mode-directed
tabling is especially useful for dynamic programming problems. In mode-directed tabling, a plus-
sign (+) indicates input, a minus-sign (-) indicates output, max indicates that the corresponding
variable should be maximized, min indicates that the corresponding variable should be minimized,
and nt indicates that the corresponding argument is not tabled.2
Example
table(+,+,min)
edit([],[],D) => D=0.
edit([X|Xs],[X|Ys],D) =>
edit(Xs,Ys,D).
edit(Xs,[Y|Ys],D) ?=> % insert
edit(Xs,Ys,D1),
D=D1+1.
edit([X|Xs],Ys,D) => % delete
edit(Xs,Ys,D1),
D=D1+1.
2An nt argument can carry some information that is dependent on the input arguments but useful for the computa-
tion.
10
For a call edit(L1,L2,D), where L1 and L2 are given lists and Dis a variable, the rules can
generate all facts, each of which contains a different editing distance between the two lists. The
table mode table(+,+,min) tells the system to keep a fact with the minimal editing distance.
A tabled predicate can be preceded by both a table declaration and at most one index declara-
tion if it contains facts. The order of these declarations is not important.
1.6 Modules
A module is a source file with the extension .pi. A module begins with a module name declara-
tion and optional import declarations. A module declaration has the form:
module Name.
where Name must be the same as the main file name. A file that does not begin with a module
declaration is assumed to belong to the global module, and all of the predicates and functions that
are defined in such a file are visible to all modules as well as the top-level of the interpreter.
An import declaration takes the form:
import Name1,. . .,Namen.
where each Nameiis a module name. When a module is imported, all of its public predicates and
functions will be visible to the importing module. A public predicate or function in a module can
also be accessed by preceding it with a module qualifier, as in m.p(), but the module still must
be imported.
Atoms and structure names do not belong to any module, and are globally visible. In a module,
predicates and functions are assumed to be visible both inside and outside of the module, unless
their definitions are preceded by the keyword private.
Example
% in file my_sum.pi
module my_sum.
my_sum(L)=Sum =>
sum_aux(L,0,Sum).
private
sum_aux([],Sum0,Sum) => Sum=Sum0.
sum_aux([X|L],Sum0,Sum) => sum_aux(L,X+Sum0,Sum).
% in file test_my_sum.pi
module test_my_sum.
import my_sum.
go =>
writeln(my_sum([1,2,3,4])).
The predicate sum aux is private, and is never visible outside of the module. The following
shows a session that uses these modules.
11
Picat> load("test_my_sum")
Picat> go
10
The command load(File) loads a module file into the system. If the file has not been compiled,
then the load command compiles the file before loading it. If this module is dependent on other
modules, then the other modules are loaded automatically if they are not yet in the system.3When
a module is loaded, all of its public predicates and functions become visible to the interpreter.
The Picat module system is static, meaning that the binding of normal calls to their definitions
takes place at compile time. For higher-order calls, however, Picat may need to search for their
definitions at runtime. Several built-in modules are imported by default, including basic,io,
math, and sys. For a normal call that is not higher-order in a module, the Picat compiler searches
modules for a definition in the following order:
1. The implicitly imported built-in modules in the order from basic to sys.
2. The enclosing module of the call.
3. The explicitly imported modules in the order that they were imported.
4. The global module.
1.7 Constraints
Picat can be used as a modeling and solving language for constraint satisfaction and optimization
problems. A constraint program normally poses a problem in three steps: (1) generate variables;
(2) generate constraints over the variables; and (3) call solve to find a valuation for the variables
that satisfies the constraints, and possibly optimizes an objective function. Picat provides three
solver modules, including cp,sat, and mip.
Example
import cp.
go =>
Vars=[S,E,N,D,M,O,R,Y], % generate variables
Vars :: 0..9,
all_different(Vars), % generate constraints
S #!= 0,
M #!= 0,
1000*S+100*E+10*N+D+1000*M+100*O+10*R+E
#= 10000*M+1000*O+100*N+10*E+Y,
solve(Vars), % search
writeln(Vars).
In arithmetic constraints, expressions are treated as data, and it is unnecessary to enclose them
with dollar-signs.
The loops provided by Picat facilitate modeling of many constraint satisfaction and optimiza-
tion problems. The following program solves a Sudoku puzzle:
3Dependent modules must be in a path that is specified by the environment variable PICATPATH.
12
import cp.
sudoku =>
instance(N,A),
A :: 1..N,
foreach(Row in 1..N)
all_different(A[Row])
end,
foreach(Col in 1..N)
all_different([A[Row,Col] : Row in 1..N])
end,
M=floor(sqrt(N)),
foreach(Row in 1..M, Col in 1..M)
Square = [A[Row1,Col1] :
Row1 in (Row-1)*M+1..Row*M,
Col1 in (Col-1)*M+1..Col*M],
all_different(Square)
end,
solve(A),
foreach(I in 1..N) writeln(A[I]) end.
instance(N,A) =>
N=9,
A={{5,3,_,_,7,_,_,_,_},
{6,_,_,1,9,5,_,_,_},
{_,9,8,_,_,_,_,6,_},
{8,_,_,_,6,_,_,_,3},
{4,_,_,8,_,3,_,_,1},
{7,_,_,_,2,_,_,_,6},
{_,_,_,_,_,_,_,_,_},
{_,_,_,_,_,_,_,_,_},
{_,_,_,_,_,_,_,_,_},
{_,_,_,_,_,_,_,_,_}}.
Recall that variables that occur within a loop, and do not occur before the loop in the outer scope,
are local to each iteration of the loop. For example, in the third foreach statement of the
sudoku predicate, the variables Row,Col, and Square are local, and each iteration of the
loop has its own values for these variables.
1.8 Exceptions
An exception is an event that occurs during the execution of a program that requires a spe-
cial treatment. In Picat, an exception is just a term. Example exceptions thrown by the sys-
tem include divide by zero,file not found,number expected,interrupt, and
out of range. The exception interrupt(keyboard) is raised when ctrl-c is typed
during a program’s execution. The built-in predicate throw Exception throws Exception.
All exceptions, including those raised by built-ins and interruptions, can be caught by catchers.
A catcher is a call in the form: catch(Goal,Exception,Handler)which is equivalent to
Goal, except when an exception is raised during the execution of Goal that unifies Exception.
13
When such an exception is raised, all of the bindings that have been performed on variables in
Goal will be undone, and Handler will be executed to handle the exception.
The call call cleanup(Goal,Cleanup)is equivalent to call(Call), except that Cleanup
is called when Goal succeeds determinately (i.e., with no remaining choice point), when Goal
fails, or when Goal raises an exception.
1.9 Higher-Order Calls
A predicate or function is said to be higher-order if it takes calls as arguments. The built-ins call,
apply, and find all are higher-order. The predicate call(S,Arg1,. . .,Argn), where Sis
an atom or a structure, calls the predicate named by Swith the arguments that are specified in S
together with extra arguments Arg1,. . .,Argn. The function apply(S,Arg1,. . .,Argn)is sim-
ilar to call, except that apply returns a value. The function findall(T emplate,S)returns
a list of all possible solutions of call(S)in the form of T emplate. Other higher-order predi-
cates include call cleanup/2,catch/3,count all,freeze/2,not/1,maxof/2-3,
maxof inc/2-3,minof/2-3,minof inc/2-3,once/1,time/1,time out/3 and
time2/1. All of these higher-order predicates are defined in the basic module, except for
time/1,time2/1, and time out/3, which are defined in the sys module. Higher-order
calls cannot contain assignments or loops.
Example
Picat> S=$member(X), call(S,[1,2,3])
X=1;
X=2;
X=3;
no
Picat> L=findall(X,member(X,[1,2,3])).
L=[1,2,3]
Picat> Z=apply(’+’,1,2)
Z=3
Among the higher-order built-ins, findall is special in that it forms a name scope like a
loop. Local variables that occur in a findall call are not visible to subsequent calls in the body
or query.
The meta-call apply never returns a partially evaluated function. If the number of arguments
does not match the required number, then it throws an exception.
Example
map(_F,[]) = [].
map(F,[X|Xs])=[apply(F,X)|map(F,Xs)].
map2(_F,[],[]) = [].
map2(F,[X|Xs],[Y|Ys])=[apply(F,X,Y)|map2(F,Xs,Ys)].
fold(_F,Acc,[]) = Acc.
fold(F,Acc,[H|T])=fold(F, apply(F,H,Acc),T).
14
A call that is passed to apply is assumed to invoke a definition in a pre-imported built-in
module, the enclosing module in which apply occurs, an imported module of the enclosing
module, or the global module. Due to the overhead of runtime search, the use of higher-order calls
is discouraged. Whenever possible, recursion, loops, or list and array comprehensions should be
used instead.
1.10 Action Rules
Picat provides action rules for describing event-driven actors. An actor is a predicate call that can
be delayed, and can be activated later by events. Each time an actor is activated, an action can be
executed. A predicate for actors contains at least one action rule in the form:
Head, Cond, {Event}=> Body
where Head is an actor pattern, Cond is an optional condition, Event is a non-empty set of event
patterns separated by ’,’, and Body is an action. For an actor and an event, an action rule is
said to be applicable if the actor matches Head and Cond is true. A predicate for actors cannot
contain backtrackable rules.
An event channel is an attributed variable to which actors can be attached, and through which
events can be posted to actors. A channel has four ports: ins,bound,dom, and any. An event
pattern in Event specifies the port to which the actor is attached. The event pattern ins(X)
attaches the actor to the ins-port of channel X, and the actor will be activated when Xis in-
stantiated. The event pattern event(X,T)attaches the actor to the dom-port of channel X.
The built-in post event(X,T)posts an event term Tto the dom-port of channel X. After an
event is posted to a port of a channel, the actors attached to that port are activated. For an activated
actor, the system searches for an applicable rule and executes the rule body if it finds one. After
execution, the actor is suspended, waiting to be activated again by other events. Picat does not
provide a built-in for detaching actors from channels. An actor fails if no rule is applicable to it
when it is activated or the body of the applied rule fails. An actor becomes a normal call once a
normal non-backtrackable rule is applied to it.
Example
echo(X,Flag),var(Flag),{event(X,T)} => writeln(T).
echo(_X,_Flag) => writeln(done).
foo(Flag) => Flag=1.
When a call echo(X,Flag) is executed, where Flag is a variable, it is attached to the dom-port
of Xas an actor. The actor is then suspended, waiting for events posted to the dom-port. For this
actor definition, the command
echo(X,Flag), post_event(X,hello), post_event(X,picat).
prints out hello followed by picat. If the call foo(Flag) is inserted before the second call
to post event, then var(Flag) fails when the actor is activated the second time, causing the
second rule to be applied to the actor. Then, the output will be hello followed by done. Note
that events are not handled until a non-inline call is executed. Replacing foo(Flag) by Flag=1
will result in a different behavior because Flag=1 is an inline call.
15
1.11 Prebuilt Maps
Picat has three kinds of prebuilt maps: heap maps, global maps, and table maps. Prebuilt heap
maps are created on the heap immediately after the system is started. The built-in function
get heap map(ID)returns the heap map that is associated with ID, where ID must be a
ground term. If no heap map is associated with ID, then this function establishes an association
between ID and an unused heap map, and returns the map. A heap map is like a normal map.
Users use put to add key-value pairs into the map. Users use get to retrieve a value that is
associated with a key in the map. Changes to a heap map up to a choice point are undone when ex-
ecution backtracks to that choice point. The built-in function get heap map() returns the heap
map that is associated with a system-generated default identifier. There are an unlimited number
of prebuilt heap maps.
Global maps are created in the global area when the Picat system is started. The built-in
function get global map(ID)returns the global map that is associated with ID, where ID
must be a ground term. If no global map is associated with ID, then this function establishes an
association between ID and an unused global map, and returns the map. A big difference between
a global map and a heap map is that changes to the global map are not undone upon backtracking.
When a key-value pair is added into the global map, the variables in the value term are numbered
before they are copied to the global area. If the value term contains attributed variables, then the
attributes of the variables are not copied, and are therefore lost. When retrieving a value that is
associated with a key, the value term in the global area is copied back to the heap after all of
the numbered variables are unnumbered. The built-in function get global map() returns the
global map that is associated with a system-generated default identifier. The number of prebuilt
global maps is 97, and the system halts if a program requests more than 97 global maps.
Table maps are created in the table area when the Picat system is started. The built-in function
get table map(ID)returns the table map that is associated with ID, where ID must be a
ground term. If no table map is associated with ID, then this function establishes an association
between ID and an unused table map, and returns the map. Like the global map, changes to a
table map are not undone upon backtracking. Unlike the global map, however, keys and values
are hash-consed so that common ground sub-terms are not replicated in the table area. The built-
in function get table map() returns the table map that is associated with a system-generated
default identifier. The number of prebuilt table maps is 97, and the system halts if a program
requests more than 97 table maps.
The advantage of using prebuilt maps is that data can be accessed everywhere without being
passed as arguments, and the disadvantage is that it affects locality of data and thus the readabil-
ity of programs. In tabled programs, using prebuilt maps is discouraged because it may cause
unanticipated effects.
Example
go ?=>
get_heap_map(h1).put(one,1),
get_global_map(g1).put(one,1),
get_table_map(t1).put(one,1),
fail.
go =>
if (get_heap_map(h1).has_key(one)) then
writef("heap map h1 has key%n")
else
writef("heap map h1 has no key%n")
16
end,
if (get_global_map(g1).has_key(one)) then
writef("global map g1 has key%n")
else
writef("global map g1 has no key%n")
end,
if (get_table_map(t1).has_key(one)) then
writef("table map t1 has key%n")
else
writef("table map t1 has no key%n")
end.
For the call go, the output is:
heap map h1 has no key
global map g1 has key
table map t1 has key
The fail call in the first rule causes execution to backtrack to the second rule. After backtracking,
the pair added to the heap map by the first rule is lost, but the pair added to the global map and the
pair added to the table map remain.
1.12 Programming Exercises
Project Euler (projecteuler.net) is an excellent platform for practicing programming and
problem solving skills. You can find Picat solutions to some of the problems at picat-lang.
org/projects.html. Select five problems from the Project Euler problem set for which no
solutions have been posted, and write a program in Picat for each of them.
17
Chapter 2
How to Use the Picat System
2.1 How to Use the Picat Interpreter
The Picat system is written in both C and Picat. The Picat interpreter is provided as a single
standalone executable file, named picat.exe for Windows and picat for Unix. The Picat
interpreter provides an interactive programming environment for users to compile, load, debug,
and execute programs. In order to start the Picat interpreter, users first need to open an OS ter-
minal. In Windows, this can be done by selecting Start->Run and typing cmd or selecting
Start->Programs->Accessories->Command Prompt. In order to start the Picat in-
terpreter in any working directory, the environment variable path must be properly set to contain
the directory where the executable is located.
2.1.1 How to Enter and Quit the Picat Interpreter
The Picat interpreter is started with the OS command picat.
OSP rompt picat
where OSP rompt is the OS prompt. After the interpreter is started, it responds with the prompt
Picat>, and is ready to accept queries.
In general, the picat command takes the following form:
picat Options P icatMainF ileN ame A1A2. . . An
where P icatMainF ileN ame can have the extension .pi and can contain a path of directories,
and Options is a sequence of options of the following:
•-g InitGoal: This option makes Picat execute a specified initial query InitGoal rather
than the default main predicate.
•--help: Print out the help info.
•-log: The option -log makes the system print log information and warning messages.
•-path Directories:Directories is a semicolon-separated and double-quoted list of di-
rectories. This option sets the value of the environment variable PICATPATH before the
execution of the program. The Picat system will look for P icatM ainF ileName and the
related modules in these directories.
•-s Size: This option reserves Size words for the stack and the heap when the system is
started.
18
•--v:
•--version: This option makes Picat print the version number.
Once the interpreter is started, users can type a query after the prompt. For example,
Picat> X=1+1
X=2
Picat> printf("hello"++" picat")
hello picat
The halt predicate, or the exit predicate, terminates the Picat interpreter. An alternative
way to terminate the interpreter is to enter ctrl-d (control-d) when the cursor is located at the
beginning of an empty line.
2.1.2 How to Use the Command-line Editor
The Picat interpreter uses the getline program written by Chris Thewalt. The getline pro-
gram memorizes up to 100 of the most recent queries that the users have typed, and allows users
to recall past queries and edit the current query by using Emacs editing commands. The following
gives the editing commands:
ctrl-f Move the cursor one position forward.
ctrl-b Move the cursor one position backward.
ctrl-a Move the cursor to the beginning of the line.
ctrl-e Move the cursor to the end of the line.
ctrl-d Delete the character under the cursor.
ctrl-h Delete the character to the left of the cursor.
ctrl-k Delete the characters to the right of the cursor.
ctrl-u Delete the whole line.
ctrl-p Load the previous query in the buffer.
ctrl-n Load the next query in the buffer.
Note that the command ctrl-d terminates the interpreter if the line is empty and the cursor is
located in the beginning of the line.
2.1.3 How to Compile and Load Programs
A Picat program is stored in one or more text files with the extension name pi. A file name is a
string of characters. Picat treats both ’/’ and ’\’ as file name separators. Nevertheless, since
’\’ is used as the escape character in quoted strings, two consecutive backslashes must be used,
as in "c:\\work\\myfile.pi", if ’\’ is used as the separator.
For the sake of demonstration we assume the existence of a file named welcome.pi in the
current working directory that stores the following program:
main =>
print(" Welcome to PICAT’s world! \n ").
main(Args) =>
print(" Welcome to PICAT’s world! \n"),
foreach (Arg in Args)
printf("%s\n", Arg)
end.
19
•cl(F ileN ame): A program first needs to be compiled and loaded into the system be-
fore it can be executed. The built-in predicate cl(F ileN ame)compiles and loads the
source file named F ileN ame.pi. Note that if the full path of the file name is not given,
then the file is assumed to be in the current working directory. Also note that users do not
need to give the extension name. The system compiles and loads not only the source file
F ileN ame.pi, but also all of the module files that are either directly imported or indirectly
imported by the source file. The system searches for such dependent files in the directory
in which F ileN ame.pi resides or the directories that are stored in the environment vari-
able PICATPATH. For F ileName.pi, the cl command loads the generated byte-codes
without creating a byte-code file. For example,
Picat> cl("welcome")
Compiling:: welcome.pi
welcome.pi compiled in 4 milliseconds
•cl: The built-in predicate cl (with no argument) compiles and loads a program from the
console, ending when the end-of-file character (ctrl-z for Windows and ctrl-d for
Unix) is typed.
•compile(F ileN ame): The built-in predicate compile(F ileName)compiles the file
F ileN ame.pi and all of its dependent module files without loading the generated byte-
code files. The destination directory for the byte-code file is the same as the source file’s
directory. If the Picat interpreter does not have permission to write into the directory in
which a source file resides, then this built-in throws an exception. For example,
Picat> compile("welcome")
Compiling::welcome.pi
welcome.pi compiled in 4 milliseconds
•load(F ileN ame): The built-in predicate load(F ileN ame)loads the byte-code file
F ileN ame.qi and all of its dependent byte-code files. For F ileName and its dependent
file names, the system searches for a byte-code file in the directory in which F ileName.qi
resides or the directories that are stored in the environment variable PICATPATH. If the
byte-code file F ileN ame.qi does not exist but the source file F ileName.pi exists, then
this built-in compiles the source file and loads the byte codes without creating a qi file.
Picat> load("welcome")
loading...welcome.qi
2.1.4 How to Run Programs
After a program is loaded, users can query the program. For each query, the system executes
the program, and reports yes when the query succeeds and no when the query fails. When a
query that contains variables succeeds, the system also reports the bindings for the variables. For
example,
Picat> cl("welcome")
Compiling:: welcome.pi
welcome.pi compiled in 4 milliseconds
loading...
20
yes
Picat> main
Welcome to PICAT’s world!
yes
Users can ask the system to find the next solution by typing ’;’ after a solution if the query
has multiple solutions. For example,
Picat> member(X,[1,2,3])
X=1;
X=2;
X=3;
no
Users can force a program to terminate by typing ctrl-c, or by letting it execute the built-
in predicate abort. Note that when the system is engaged in certain tasks, such as garbage
collection, users may need to wait for a while in order to see the termination after they type
ctrl-c.
2.1.5 How to Run Programs Directly
Programs that define the main/0 predicate or the main/1 predicate can be run directly as a OS
command. For example,
$ picat welcome
Welcome to PICAT’s world!
$ picat welcome a b c
Welcome to PICAT’s world!
a
b
c
The ‘$’ sign is the prompt of the OS. It is assumed that the environment variable PATH has
been set to contain the directory of the executable picat (picat.exe for Windows), and the
environment variable PICATPATH has been set to contain the directory of the welcome.pi file
or the file is in the current working directory.
2.1.6 Creating Standalone Executables
It is possible to create a script that can be run as a standalone executable. For example, consider
the following script welcome.exe for Linux:
#!/bin/bash
picat welcome.pi
echo " Finished!"
Once the environment variables PATH and PICATPATH are set properly, and the script is set to
have the execution permission, it can be executed as follows:
21
$ welcome.exe
Welcome to PICAT’s world!
Finished!
2.2 How to Use the Debugger
The Picat system has three execution modes: non-trace mode,trace mode, and spy mode. In trace
mode, it is possible to trace the execution of a program, showing every call in every possible stage.
In order to trace the execution, the program must be recompiled while the system is in trace mode.
In spy mode, it is possible to trace the execution of individual functions and predicates that are spy
points. When the Picat interpreter is started, it runs in non-trace mode. The predicate debug or
trace changes the mode to trace. The predicate nodebug or notrace changes the mode to
non-trace.
In trace mode, the debugger displays execution traces of queries. An execution trace consists
of a sequence of call traces. Each call trace is a line that consists of a stage, the number of the call,
and the information about the call itself. For a function call, there are two possible stages: Call,
meaning the time at which the function is entered, and Exit, meaning the time at which the call
is completed with an answer. For a predicate call, there are two additional possible stages: Redo,
meaning a time at which execution backtracks to the call, and Fail, meaning the time at which
the call is completed with a failure. The information about a call includes the name of the call, and
the arguments. If the call is a function, then the call is followed by =and ?at the Call stage, and
followed by =V alue at the Exit stage, where V alue is the return value of the call.
Consider, for example, the following program:
p(X) ?=> X=a.
p(X) => X=b.
q(X) ?=> X=1.
q(X) => X=2.
Assume the program is stored in a file named myprog.pi. The following shows a trace for a
query:
Picat> debug
{Trace mode}
Picat> cl(myprog)
{Trace mode}
Picat> p(X),q(Y)
Call: (1) p(_328) ?
Exit: (1) p(a)
Call: (2) q(_378) ?
Exit: (2) q(1)
X=a
Y=1?;
Redo: (2) q(1) ?
Exit: (2) q(2)
X=a
Y=2?;
Redo: (1) p(a) ?
22
Exit: (1) p(b)
Call: (3) q(_378) ?
Exit: (3) q(1)
X=b
Y=1?;
Redo: (3) q(1) ?
Exit: (3) q(2)
X=b
Y=2?;
no
In trace mode, the debugger displays every call in every possible stage. Users can set spy points
so that the debugger only shows information about calls of the symbols that users are spying. Users
can use the predicate
spy $Name/N
to set the functor Name/Nas a spy point, where the arity Nis optional. If the functor is defined
in multiple loaded modules, then all these definitions will be treated as spy points. If no arity is
given, then any functor of Name is treated as a spy point, regardless of the arity.
After displaying a call trace, if the trace is for stage Call or stage Redo, then the debugger
waits for a command from the users. A command is either a single letter followed by a carriage-
return, or just a carriage-return. See Appendix B for the debugging commands.
23
Chapter 3
Data Types, Operators, and Built-ins
Picat is a dynamically-typed language, in which type checking occurs at runtime. A variable
gets a type once it is bound to a value. In Picat, variables and values are terms. A value can
be primitive or compound. A primitive value can be an integer, a real number, or an atom. A
compound value can be a list or a structure. Strings, arrays, and maps are special compound
values. This chapter describes the data types and the built-ins for each data type that are provided
by the basic module.
Many of the built-ins are given as operators. Table 3.1 shows all of the operators that are
provided by Picat. Unless the table specifies otherwise, the operators are left-associative. The
as-pattern operator (@) and the operators for composing goals, including not,once, conjunction
(,and &&), and disjunction (;and ||), will be described in Chapter 4 on Predicates and Func-
tions. The constraint operators (the ones that begin with #) will be described in Chapter 12 on
Constraints. In Picat, no new operators can be defined, and none of the existing operators can be
redefined.
The dot operator (.) is used in OOP notations for calling predicates and functions. It is
also used to qualify calls with a module name. The notation A1.f(A2, . . . , Ak)is the same as
f(A1, A2, . . . , Ak), unless A1is an atom, in which case A1must be a module qualifier for f. If
an atom needs to be passed as the first argument to a function or a predicate, then this notation
cannot be used. The notation A.Attr, where Attr does not have the form f(. . .), is the same
as the function call get(A, Attr). For example, the expression S.name returns the name, and
the expression S.arity returns the arity of Sif Sis a structure. Note that the dot operator is
left-associative. For example, the expression X.f().g() is the same as g(f(X)).
The following functions are provided for all terms:
•copy term(T erm1) = T erm2: This function copies T erm1into T erm2. If T erm1is
an attributed variable, then T erm2will not contain any of the attributes.
•hash code(T erm) = Code: This function returns the hash code of T erm. If T erm is
a variable, then the returned hash code is always 0.
•to codes(T erm) = Codes: This function returns a list of character codes of T erm.
•to fstring(F ormat,Args . . .): This function converts the arguments in the Args . . .
parameter into a string, according to the format string F ormat, and returns the string. The
number of arguments in Args . . . cannot exceed 10. Format characters are described in
Chapter 10.
•to string(T erm) = String: This function returns a string representation of T erm.
Other built-ins on terms are given in Sections 3.5 and 3.8.
24
Table 3.1: Operators in Picat
Precedence Operators
Highest .,@
** (right-associative)
unary +, unary -,˜
*,/,//,/<,/>,div,mod,rem
binary +, binary -
>>,<<
/\
ˆ
\/
..
++ (right-associative)
=,!=,:=,==,!==,=:=,<,=<,<=,>,>=,::,in,notin
#=,#!=,#<,#=<,#<=,#>,#>=,@<,@=<,@<=,@>,@>=
#˜
#/\
#ˆ
#\/
#=> (right-associative)
#<=>
not,once
,(right-associative), && (right-associative)
Lowest ;(right-associative), || (right-associative)
25
3.1 Variables
Variables in Picat, like variables in mathematics, are value holders. Unlike variables in imperative
languages, Picat variables are not symbolic addresses of memory locations. A variable is said to
be free if it does not hold any value. A variable is instantiated when it is bound to a value. Picat
variables are single-assignment, which means that after a variable is instantiated to a value, the
variable will have the same identity as the value. After execution backtracks over a point where a
binding took place, the value that was assigned to a variable will be dropped, and the variable will
be turned back into a free variable.
A variable name is an identifier that begins with a capital letter or the underscore. For example,
the following are valid variable names:
X1 _ _ab
The name _is used for anonymous variables. In a program, different occurrences of _are treated
as different variables. So the test _ == _ is always false.
The following two built-ins are provided to test whether a term is a free variable:
•var(T erm): This predicate is true if T erm is a free variable.
•nonvar(T erm): This predicate is true if T erm is not a free variable.
An attributed variable is a variable that has a map of attribute-value pairs attached to it. The
following built-ins are provided for attributed variables:
•attr var(T erm): This predicate is true if T erm is an attributed variable.
•dvar(T erm): This predicate is true if T erm is an attributed domain variable.
•bool dvar(T erm): This predicate is true if T erm is an attributed domain variable
whose lower bound is 0 and whose upper bound is 1.
•dvar or int(T erm): This predicate is true if T erm is an attributed domain variable or
an integer.
•get attr(X,Key) = V al: This function returns the V al of the key-value pair Key=V al
that is attached to X. It throws an error if Xhas no attribute named Key.
•get attr(X,Key,DefaultV al) = V al: This function returns V al of the key-
value pair Key=V al that is attached to X. It returns DefaultV al if Xdoes not have
the attribute named Key.
•put attr(X,Key,V al): This predicate attaches the key-value pair Key=V al to X,
where Key is a non-variable term, and V al is any term.
•put attr(X,Key): This predicate call is the same as put attr(X,Key,not a value).
3.2 Atoms
An atom is a symbolic constant. An atom name can either be quoted or unquoted. An unquoted
name is an identifier that begins with a lower-case letter, followed by an optional string of letters,
digits, and underscores. A quoted name is a single-quoted sequence of arbitrary characters. A
character can be represented as a single-character atom. For example, the following are valid atom
names:
26
x x_1 ’_’ ’\\’ ’a\’b\n’ ’_ab’ ’$%’
No atom name can last more than one line. An atom name cannot contain more than 1000 char-
acters. The backslash character ’\’ is used as the escape character. So, the name ’a\’b\n’
contains four characters: a,’,b, and \n.
The following built-ins are provided for atoms:
•atom(T erm): This predicate is true if T erm is an atom.
•atom chars(Atm) = String: This function returns string that contains the characters
of the atom Atm. It throws an error if Atm is not an atom.
•atom codes(Atm) = List: This function returns the list of codes of the characters of
the atom Atm. It throws an error if Atm is not an atom.
•atomic(T erm): This predicate is true if T erm is an atom or a number.
•char(T erm): This predicate is true if T erm is an atom and the atom is made of one
character.
•chr(Code) = Char: This function returns the UTF-8 character of the code point Code.
•digit(T erm): This predicate is true if T erm is an atom and the atom is made of one
digit.
•len(Atom) = Len: This function returns the number of characters in Atom. Note that
this function is overloaded in such a way that the argument can also be an array, a list, or a
structure.
•length(Atom) = Len: This function is the same as len(Atom).
•ord(Char) = Int: This function returns the code point of the UTF-8 character Char. It
throws an error if Char is not a single-character atom.
3.3 Numbers
A number can be an integer or a real number. An integer can be a decimal numeral, a binary
numeral, an octal numeral, or a hexadecimal numeral. In a numeral, digits can be separated by
underscores, but underscore separators are ignored by the tokenizer. For example, the following
are valid integers:
12 345 a decimal numeral
0b100 4 in binary notation
0o73 59 in octal notation
0xf7 247 in hexadecimal notation
A real number consists of an optional integer part, an optional decimal fraction preceded by
a decimal point, and an optional exponent. If an integer part exists, then it must be followed by
either a fraction or an exponent in order to distinguish the real number from an integer literal. For
example, the following are valid real numbers.
12.345 0.123 12-e10 0.12E10
27
Table 3.2: Arithmetic Operators
X** Ypower
+Xsame as X
-Xsign reversal
˜Xbitwise complement
X*Ymultiplication
X/Ydivision
X// Yinteger division, truncated
X/> Yinteger division (ceiling(X/Y))
X/< Yinteger division (floor(X/Y))
Xdiv Yinteger division, floored
Xmod Ymodulo, same as X- floor(Xdiv Y) * Y
Xrem Yremainder (X- (X// Y) * Y)
X+Yaddition
X-Ysubtraction
X>> Yright shift
X<< Yleft shift
X/\ Ybitwise and
XˆYbitwise xor
X\/ Ybitwise or
F rom .. Step .. T o A range (list) of numbers with a step
F rom .. T o A range (list) of numbers with step 1
X=:= Ypretty much (numerically) equal
28
Table 3.2 gives the meaning of each of the numeric operators in Picat, from the operator with
the highest precedence (**) to the one with the lowest precedence (..). Except for the power
operator **, which is right-associative, all of the arithmetic operators are left-associative.
In addition to the numeric operators, the basic module also provides the following built-ins
for numbers:
•between(F rom,T o,X)(nondet): If Xis bound to an integer, then this predicate
determines whether Xis between F rom and T o. Otherwise, if Xis unbound, then this
predicate nondeterministically selects Xfrom the integers that are between F rom and T o.
It is the same as member(X,F rom..T o).
•float(T erm): This predicate is true if T erm is a real number.
•int(T erm): This predicate is true if T erm is an integer.
•integer(T erm): The same as int(T erm).
•max(X,Y) = V al: This function returns the maximum of Xand Y, where Xand Y
are terms.
•maxint small() = Int: This function returns the maximum integer that is represented
in one word. All integers that are greater than this integer are represented as big integers.
•min(X,Y) = V al: This function returns the minimum of Xand Y, where Xand Y
are terms.
•minint small() = Int: This function returns the minimum integer that is represented
in one word. All integers that are smaller than this integer are represented as big integers.
•number(T erm): This predicate is true if T erm is a number.
•number chars(Num) = String: This function returns a list of characters of Num.
This function is the same as to fstring("%d",Num)if Num is an integer, and the
same as to fstring("%f",Num)if Num is a real number.
•number codes(Num) = List: This function returns a list of codes of the characters
of Num. It is the same as number chars(Num).to codes().
•real(T erm): This predicate is the same as float(T erm).
•to binary string(Int) = String: This function returns the binary representation of
the integer Int as a string.
•to float(NS) = Real: This function is the same as NS*1.0 if N S is a number, and
the same as parse term(NS)if NS is a string of digits.
•to hex string(Int) = String: This function returns the hexadecimal representation
of the integer Int as a string.
•to int(ANS) = Int: This function is the same as truncate(ANS)in the math
module if ANS is a number, the same as ord(ANS)-ord(’0’) if ANS is a digit
character, and the same as parse term(ANS)if AN S is a string.
•to integer(ANS) = Int: This function is the same as to int(AN S).
29
•to number(ANS) = Num: This function is the same as ANS if ANS is a num-
ber, the same as ord(AN S)-ord(’0’) if AN S is a digit character, and the same as
parse term(ANS)if ANS is a string.
•to oct string(Int) = String: This function returns the octal representation of the
integer Int as a string.
•to radix string(Int,Base) = String: This function returns the representation of
the integer Int of the numeral Base as a string, where Base must be greater than 1 and less
than 37. The call to oct string(Int)is the same as to radix string(Int,8).
•to real(NS) = Real: This function is the same as to float(N S).
The math module provides more numeric functions. See Appendix A.
3.4 Compound Terms
A compound term can be a list or a structure. Components of compound terms can be accessed
with subscripts. Let Xbe a variable that references a compound value, and let Ibe an integer
expression that represents a subscript. The index notation X[I]is a special function that returns
the Ith component of X, counting from the beginning. Subscripts begin at 1, meaning that X[1]
is the first component of X. An index notation can take multiple subscripts. For example, the
expression X[1,2] is the same as T[2], where Tis a temporary variable that references the
component that is returned by X[1]. The predicate compound(T erm)is true if T erm is a
compound term.
3.4.1 Lists
A list takes the form [t1,. . .,tn], where each ti(1≤i≤n) is a term. Let Lbe a list. The
expression L.length, which is the same as the functions get(L,length) and length(L),
returns the length of L. Note that a list is represented internally as a singly-linked list. Also note
that the length of a list is not stored in memory; instead, it is recomputed each time that the function
length is called.
The symbol ’|’ is not an operator, but a separator that separates the first element (so-called
car) from the rest of the list (so-called cdr). The cons notation [H|T]can occur in a pattern or
in an expression. When it occurs in a pattern, it matches any list in which Hmatches the car and
Tmatches the cdr. When it occurs in an expression, it builds a list from Hand T. The notation
[A1,A2,. . .,An|T]is a shorthand for [A1|[A2|. . .[An|T]. . .]. So [a,b,c] is the same
as [a|[b|[c|[]]]].
The basic module provides the following built-ins on lists, most of which are overloaded for
strings (3.4.2) and arrays (see 3.4.4).
•List1++ List2=List: This function returns the concatenated list of List1and List2.
•append(X,Y,Z)(nondet): This predicate is true if appending Yto Xcan create Z.
This predicate may backtrack if Xis not a complete list.1
•append(W,X,Y,Z)(nondet): This predicate is defined as:
append(W,X,Y,Z) => append(W,X,WX), append(WX,Y,Z).
1A list is complete if it is empty, or if its tail is complete. For example, [a,b,c] and [X,Y,Z] are complete, but
[a,b|T] is not complete if Tis a variable.
30
•avg(List) = V al: This function returns the average of all the elements in List. This
function throws an exception if List is not a list or any of the elements is not a number.
•delete(List,X) = ResList: This function deletes the first occurrence of Xfrom
List, returning the result in ResList. The built-in !=/2 is used to test if two terms are
different. No variables in List or Xwill be bound after this function call.
•delete all(List,X) = ResList: This function deletes all occurrences of Xfrom
List, returning the result in ResList. The built-in !=/2 is used to test if two terms are
different.
•first(List) = T erm: This function returns the first element of List.
•flatten(List) = ResList: This function flattens a list of nested lists into a list. For
example, flatten([[1],[2,[3]]]) returns [1,2,3].
•head(List) = T erm: This function returns the head of the list List. For example,
head([1,2,3]) returns 1.
•insert(List,Index,Elm) = ResList: This function inserts Elm into List at the
index Index, returning the result in ResList. After insertion, the original List is not
changed, and ResList is the same as
List.slice(1, Index-1)++[Elm|List.slice(Index,List.length)].
•insert all(List,Index,AList) = ResList: This function inserts all of the ele-
ments in AList into List at the index Index, returning the result in ResList. After inser-
tion, the original List is not changed, and ResList is the same as
List.slice(1, Index-1)++AList++List.slice(Index,List.length).
•insert ordered(List,T erm): This function inserts T erm into the ordered list List,
such that the resulting list remains sorted.
•insert ordered down(List,T erm): This function inserts T erm into the descen-
dantly ordered list List, such that the resulting list remains sorted down.
•last(List) = T erm: This function returns the last element of List.
•len(List) = Len: This function returns the number of elements in List. Note that this
function is overloaded in such a way that the argument can also be an atom, an array, or a
structure.
•length(List) = Len: This function is the same as len(List).
•list(T erm): This predicate is true if T erm is a list.
•max(List) = V al: This function returns the maximum value that is in List, where List
is a list of terms.
•membchk(T erm,List): This predicate is true if T erm is an element of List.
•member(T erm,List)(nondet): This predicate is true if T erm is an element of List.
When T erm is a variable, this predicate may backtrack, instantiating T erm to different
elements of List.
•min(List) = V al: This function returns the minimum value that is in List, where List
is a list or an array of terms.
31
•new list(N) = List: This function creates a new list that has Nfree variable argu-
ments.
•new list(N,InitV al) = List: This function creates a new list that has Narguments
all initialized to InitV al.
•nth(Index,List,Elem)(nondet): This predicate is true when Elem is the Index’th
element of List. Counting starts at 1. When Index is a variable, this predicate may back-
track, instantiating Index to a different integer between 1 and len(List).
•prod(List) = V al: This function returns the product of all of the values in List.
•remove dups(List) = ResList: This function removes all duplicate values from List,
retaining only the first occurrence of each value. The result is returned in ResList. Note that
an O(n2)algorithm is used in the implementation. If List is large, then sort remove dups(List)
may be faster than this function.
•reverse(List) = ResList: This function reverses the order of the elements in List,
returning the result in ResList.
•select(X,List,ResList)(nondet): This predicate nondeterministically selects an
element Xfrom List, and binds ResList to the list after Xis removed. On backtracking,
it selects the next element.
•sort(List) = SList: This function sorts the elements of List in ascending order, re-
turning the result in SList.
•sort(List,KeyIndex) = SList: This function sorts the elements of List by the key
index KeyIndex in ascending order, returning the result in SList. The elements of List
must be compound values and KeyIndex must be a positive integer that does not exceed
the length of any of the elements of List. This function is defined as follows:
sort(List,KeyIndex) = SList =>
List1 = [(E[KeyIndex],E) : E in List],
List2 = sort(List1),
SList = [E : (_,E) in List2].
•sort remove dups(List) = SList: This function is the same as the following, but is
faster.
sort(List).remove dups()
•sort remove dups(List,KeyIndex) = SList: This function is the same as the fol-
lowing, but is faster.
sort(List,KeyIndex).remove dups()
•sort down(List) = SList: This function sorts the elements of List in descending or-
der, returning the result in SList.
•sort down(List,KeyIndex) = SList: This function sorts the elements of List by
the key index KeyIndex in descending order, returning the result in SList.
32
•sort down remove dups(List) = SList: This function is the same as the following,
but is faster.
sort down(List).remove dups()
•sort down remove dups(List,KeyIndex) = SList: This function is the same as
the following, but is faster.
sort down(List,KeyIndex).remove dups()
•slice(List,F rom,T o) = SList: This function returns the sliced list of List from
index F rom through index T o.F rom must not be less than 1.
•slice(List,F rom) = SList: This function is the same as the following.
slice(List,F rom,List.length)
•sum(List) = V al: This function returns the sum of all of the values in List.
•tail(List) = T erm: This function returns the tail of the list List. For example, the call
tail([1,2,3]) returns [2,3].
•to array(List) = Array: This function converts the list List to an array. The ele-
ments of the array are in the same order as the elements of the list.
•zip(List1,List2,. . .,Listn) = List: This function makes a list of array tuples.
The jth tuple in the list takes the form {E1j, . . . , Enj }, where Eij is the jth element in
Listi. In the current implementation, ncan be 2, 3, or 4.
3.4.2 Strings
Astring is represented as a list of single-character atoms. For example, the string "hello" is the
same as the list [h,e,l,l,o]. In addition to the built-ins on lists, the following built-ins are
provided for strings:
•string(T erm): This predicate is true if T erm is a string.
•to lowercase(String) = LString: This function converts all uppercase alphabetic
characters into lowercase characters, returning the result in LString.
•to uppercase(String) = UString: This function converts all lowercase alphabetic
characters into uppercase characters, returning the result in UString.
3.4.3 Structures
A structure takes the form $s(t1,. . .,tn), where sis an atom, and nis called the arity of the
structure. The dollar symbol is used to distinguish a structure from a function call. The functor of
a structure comprises the name and the arity of the structure.
The following types of structures can never denote functions, meaning that they do not need
to be preceded by a $ symbol.
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Goals: (a,b),(a;b),not a,X=Y,X != 100,X>1
Constraints: X+Y #= 100,X #!= 1
Arrays: {2,3,4},{P1,P2,P3}
Picat disallows creation of the following types of structures:
Dot notations: math.pi,my module.f(a)
Index notations: X[1]+2 X[Y[I]]
Assignments: X:=Y+Z,X:=X+1
Ranges: 1..10,1..2..10
List comprehensions: [X : X in 1..5]
Array comprehensions: {X : X in 1..5}
If-then: if X>Y then Z=X else Z=Y end
Loops: foreach (X in L) writeln(X) end
The compiler will report a syntax error when it encounters any of these expressions within a term
constructor.
The following built-ins are provided for structures:
•arity(Struct) = Arity: This function returns the arity of Struct, which must be a
structure.
•len(Struct) = Arity: This function is the same as arity(Struct).
•name(Struct) = N ame: This function returns the name of Struct.
•new struct(Name,IntOrList) = Struct: This function creates a structure that
has the name Name. If IntOrList is an integer, N, then the structure has Nfree vari-
able arguments. Otherwise, if IntOrList is a list, then the structure contains the elements
in the list.
•struct(T erm): This predicate is true if T erm is a structure.
•to list(Struct) = List: This function returns a list of the components of the structure
Struct.
3.4.4 Arrays
An array takes the form {t1,. . .,tn}, which is a special structure with the name ’{}’and ar-
ity n. Note that, unlike a list, an array always has its length stored in memory, so the function
length(Array)always takes constant time. Also note that Picat supports constant-time access
of array elements, so the index notation A[I] takes constant time.
In addition to the built-ins for structures, the following built-ins are provided for arrays:
•array(T erm): This predicate is true if T erm is an array.
•new array(D1,. . .,Dn) = Arr: This function creates an n-dimensional array, where
each Diis an integer expression that specifies the size of a dimension. In the current imple-
mentation, ncannot exceed 10.
The following built-ins, which are originally provided for lists (see 3.4.1), are overloaded for
arrays:
•Array1++ Array2=Array
34
•avg(Array) = V al
•first(Array) = T erm
•last(Array) = T erm
•len(Array) = Len
•length(Array) = Len
•max(Array) = V al
•min(Array) = V al
•nth(Index,List,Elem)(nondet)
•reverse(Array) = ResArray
•slice(Array,F rom,T o) = SArray
•slice(Array,F rom) = SArray
•sum(Array) = V al
•sort(Array) = SArray
•sort(Array,KeyIndex) = SArray
•sort remove dups(Array) = SArray
•sort remove dups(Array,KeyIndex) = SArray
•sort down(Array) = SArray
•sort down(Array,KeyIndex) = SArray
•sort down remove dups(Array) = SArray
•sort down remove dups(Array,KeyIndex) = SArray
Note that many of the overloaded built-ins for arrays are not implemented efficiently, but are
provided for convenience. For example, sort(Array) is implemented as follows:
sort(Array) = Array.to_list().sort().to_array().
3.4.5 Maps
Amap is a hash-table that is represented as a structure that contains a set of key-value pairs. The
functor of the structure that is used for a map is not important. An implementation may ban access
to the name and the arity of the structure of a map. Maps must be created with the built-in function
new map, unless they are prebuilt (see Section 1.11). In addition to the built-ins for structures,
the following built-ins are provided for maps:
•clear(Map): This predicate clears the map Map. It throws an error if Map is not a map.
•get(Map,Key) = V al: This function returns V al of the key-value pair Key=V al in
Map. It throws an error if Map does not contain the key Key.
35
•get(Map,Key,DefaultV al) = V al: This function returns V al of the key-value
pair Key=V al in Map. It returns DefaultV al if Map does not contain Key.
•has key(Map,Key): This predicate is true if M ap contains a pair with Key.
•keys(X) = List: This function returns the list of keys of the pairs in Map.
•map(T erm): This predicate is true if T erm is a map.
•map to list(Map) = P airsList: This function returns a list of Key=V al pairs that
constitute Map.
•new map(IntOrP airsList) = M ap: This function creates a map with an initial capac-
ity or an initial list of pairs.
•new map(N,P airsList) = Map: This function creates a map with the initial capacity
N, the initial list of pairs P airsList, where each pair has the form Key=V al.
•put(Map,Key,V al): This predicate attaches the key-value pair Key=V al to Map,
where Key is a non-variable term, and V al is any term.
•put(Map,Key): This predicate is the same as put(M ap,Key, not a value).
•values(Map) = List: This function returns the list of values of the pairs in Map.
•size(Map) = Size: This function returns the number of pairs in Map.
Most of the built-ins are overloaded for attributed variables.
3.4.6 Sets
A set is a map where every key is associated with the atom not a value. All of the built-ins
for maps can be applied to sets. For example, the built-in predicate has key(Set,Elm)tests if
Elm is in Set. In addition to the built-ins on maps, the following built-ins are provided for sets:
•new set(IntOrKeysList) = Set: This function creates a set with an initial capacity
or an initial list of keys.
•new set(N,KeysList) = Set: This function creates a set with the initial capacity N
and the initial list of keys KeysList.
3.4.7 Heaps
A heap2is a complete binary tree represented as an array. A heap can be a min-heap or a max-
heap. In a min-heap, the value at the root of each subtree is the minimum among all the values in
the subtree. In a max-heap, the value at the root of each subtree is the maximum among all the
values in the subtree.
•heap is empty(Heap): This predicate is true if Heap is empty.
•heap pop(Heap) = Elm: This function removes the root element from the heap, and
returns the element. As the function updates the heap, it is not pure. The update will be
undone when execution backtracks over the call.
2Note that a heap, as a data structure, is different from the heap area, in which data, including heap maps, are stored.
36
•heap push(Heap,Elm): This predicate pushes Elm into Heap in a way that main-
tains the heap property. The update to Heap will be undone when execution backtracks over
the call.
•heap size(Heap) = Size: This function returns the size of Heap.
•heap to list(Heap) = List: This function returns a list of the elements in Heap.
•heap top(Heap) = Elm: This function returns the element at the root of the heap. If
Heap is a min-heap, then the element is guaranteed to be the minimum, and if Heap is a
max-heap, then the element is guaranteed to be the maximum.
•new max heap(IntOrList) = Heap: This function creates a max-heap. If IntOrList
is an integer,then it indicates the capacity. Otherwise, if IntOrList is a list, then the max-
heap contains the elements in the list in an order that maintains the heap property.
•new min heap(IntOrList) = Heap: This function creates a min-heap. If IntOrList
is an integer,then it indicates the capacity. Otherwise, if IntOrList is a list, then the min-
heap contains the elements in the list in an order that maintains the heap property.
Example
main =>
L = [1,3,2,4,5,3,6],
H = new_min_heap(L),
N = H.heap_size(),
S = [H.heap_pop() : _ in 1..N],
println(S).
3.5 Equality Testing, Unification, and Term Comparison
The equality test T1== T2is true if term T1and term T2are identical. Two variables are identi-
cal if they are aliases. Two primitive values are identical if they have the same type and the same
internal representation. Two lists are identical if the cars are identical and the cdrs are identical.
Two structures are identical if their functors are the same and their components are pairwise iden-
tical. The inequality test T1!== T2is the same as not T1== T2. Note that two terms can be
identical even if they are stored in different memory locations. Also note that it takes linear time
in the worst case to test whether two terms are identical, unlike in C-family languages, in which
the equality test operator == only compares addresses.
The unification T1=T2is true if term T1and term T2are already identical, or if they can be
made identical by instantiating the variables in the terms. The built-in T1!= T2is true if term T1
and term T2are not unifiable. The predicate bind vars(T erm,V al)binds all of the variables
in T erm to V al.
Example
Picat> X = 1
X=1
Picat> $f(a,b) = $f(a,b)
yes
Picat> [H|T] = [a,b,c]
37
H=a
T = [b,c]
Picat> $f(X,b) = $f(a,Y)
X=a
Y=b
Picat> bind_vars({X,Y,Z},a)
Picat> X = $f(X)
The last query illustrates the occurs-check problem. When binding Xto f(X), Picat does not
check if Xoccurs in f(X) for the sake of efficiency. This unification creates a cyclic term, which
can never be printed.
When a unification’s operands contain attributed variables, the implementation is more com-
plex. When a plain variable is unified with an attributed variable, the plain variable is bound to the
attributed variable. When two attributed variables, say Yand O, where Yis younger than O, are
unified, Yis bound to O, but Y’s attributes are not copied to O. Since garbage collection does not
preserve the seniority of terms, the result of the unification of two attributed variables is normally
unpredictable.
3.5.1 Numerical Equality
The numerical equality test T1=:= T2is true if term T1and term T2are pretty much the same
numerical value. This means that T1and T2must both be numbers. Whereas the test T1==
T2fails if one number is an integer and one number is a real number, the test T1=:= T2may
succeed. Consider the following examples.
Example
Picat> 1 == 1.0
no
Picat> 1 =:= 1.0
yes
In the first query, 1is an integer, while 1.0is a real number, so the equality test fails. However, the
second query, which is a numerical equality test, succeeds.
3.5.2 Ordering of Terms
Picat orders terms in the following way:
var <number <atom <structure and array <list and string
Variables are ordered by their addresses. Note that the ordering of variables may change after
garbage collection. Numbers are ordered by their numerical values. Atoms are ordered lexico-
graphically. Structures are first ordered lexicographically by their names; if their names are the
same, then they are ordered by their components. Arrays are ordered as structures with the special
name ’{}’. Lists and strings are ordered by their elements.
•T erm1@< T erm2: The term T erm1precedes the term T erm2in the standard order.
For example, a @< b succeeds.
38
•T erm1@=< T erm2: The term T erm1either precedes, or is identical to, the term T erm2
in the standard order. For example, a @=< b succeeds.
•T erm1@<= T erm2: This is the same as T erm1@=< T erm2.
•T erm1@> T erm2: The term T erm1follows the term T erm2in the standard order.
•T erm1@>= T erm2: The term T erm1either follows, or is identical to, the term T erm2
in the standard order.
3.6 Expressions
Expressions are made from variables, values, operators, and function calls. Expressions differ
from terms in the following ways:
•An expression can contain dot notations, such as math.pi.
•An expression can contain index notations, such as X[I].
•An expression can contain ranges, such as 1..2..100.
•An expression can contain list comprehensions, such as [X : X in 1..100].
•An expression can contain array comprehensions, such as {X : X in 1..100}.
A conditional expression, which takes the form cond(Cond,Exp1,Exp2), is a special kind
of function call that returns the value of Exp1if the condition Cond is true and the value of Exp2
if Cond is false.
Note that, except for conditional expressions in which the conditions are made of predicates,
no expressions can contain predicates. A predicate is true or false, but never returns any value.
3.7 Higher-order Predicates and Functions
A predicate or function is said to be higher-order if it takes calls as arguments. The basic module
has the following higher-order predicates and functions.
•apply(S,Arg1,. . .,Argn) = V al:Sis an atom or a structure. This function calls
the function that is named by Swith the arguments that are specified in S, together with
extra arguments Arg1,...,Argn. This function returns the value that Sreturns.
•call(S,Arg1,. . .,Argn):Sis an atom or a structure. This predicate calls the pred-
icate that is named by Swith the arguments that are specified in S, together with extra
arguments Arg1,...,Argn.
•call cleanup(Call,Cleanup): This predicate is the same as call(Call), except
that Cleanup is called when Call succeeds determinately (i.e., with no remaining choice
point), when Call fails, or when Call raises an exception.
•catch(Call,Exception,Handler): This predicate is the same as Call, except when
an exception that matches Exception is raised during the execution of Call. When such an
exception is raised, all of the bindings that have been performed on variables in Call will
be undone, and Handler will be executed to handle the exception.
39
•count all(Call) = Count: This function returns the number of all possible instances
of call(Call)that are true. For example, count all(member(X,[1,2,3])) re-
turns 3.
•findall(T emplate,Call) = Answers: This function returns a list of all possible
instances of call(Call)that are true in the form of T emplate. Note that T emplate is
assumed to be a term without function calls, and that Call is assumed to be a predicate call
whose arguments can contain function calls. Also note that, like a loop, findall forms
a name scope. For example, in findall(f(X),p(X,g(Y))),f(X) is a term even
though it is not preceded with $;g(Y) is a function call; the variables Xand Yare assumed
to be local to findall if they do not occur before in the outer scope.
•find all(T emplate,Call) = Answers: This function is the same as the above func-
tion.
•freeze(X,Call): This predicate delays the evaluation of Call until Xbecomes a non-
variable term.
•map(F uncOrList,ListOrF unc) = ResList: This function applies a given function
to every element of a given list and returns a list of the results. One of the arguments is a
function, and the other is a list. The order of the arguments is not important.
•map(F unc,List1,List2) = ResList: Let List1be [A1,. . .,An]and List2be
[B1,. . .,Bn]. This function applies the function F unc to every pair of elements (Ai, Bi)
by calling apply(F unc,Ai,Bi), and returns a list of the results.
•maxof(Call,Objective): This predicate finds a satisfiable instance of Call, such that
Objective has the maximum value. Here, Call is used as a generator, and Objective is an
expression to be maximized. For every satisfiable instance of Call,Objective must be a
ground expression. For maxof, search is restarted with a new bound each time that a better
answer is found.
•maxof(Call,Objective,ReportCall): This is the same as maxof(Call,Objective),
except that call(ReportCall)is executed each time that an answer is found.
•maxof inc(Call,Objective): This is the same as maxof(Call,Objective), except
that search continues rather than being restarted each time that a better solution is found.
•maxof inc(Call,Objective,ReportCall): This is the same as the previous predi-
cate, except that call(ReportCall)is executed each time that an answer is found.
•minof(Call,Objective): This predicate finds a satisfiable instance of Call, such that
Objective has the minimum value.
•minof(Call,Objective,ReportCall): This is the same as minof(Call,Objective),
except that call(ReportCall)is executed each time that an answer is found.
•minof inc(Call,Objective): This predicate is the same as minof(Call,Objective),
except that search continues rather than being restarted each time that a better solution is
found.
•minof inc(Call,Objective,ReportCall): This predicate is the same as the previ-
ous one, except that call(ReportCall)is executed each time that an answer is found.
40
•reduce(F unc,List) = Res: If List is a list that contains only one element, this func-
tion returns the element. If List contains at least two elements, then the first two elements
A1and A2are replaced with apply(F unc,A1,A2). This step is repeatedly applied to
the list until the list contains a single element, which is the final value to be returned. The
order of the arguments is not important, meaning that the first argument can be a list and the
second one can be a function.
•reduce(F unc,List,InitV al) = Res: This function is the same as
reduce(F unc,[InitV al|List]).
3.8 Other Built-ins in the basic Module
•acyclic term(T erm): This predicate is true if T erm is acyclic, meaning that T erm
does not contain itself.
•and to list(Conj) = List: This function converts Conj in the form (a1,. . .,an)
into a list in the form [a1,. . .,an].
•compare terms(T erm1,T erm2) = Res: This function compares T erm1and T erm2.
If T erm1< T erm2, then this function returns −1. If T erm1== T erm2, then this func-
tion returns 0. Otherwise, T erm1> T erm2, and this function returns 1.
•different terms(T erm1,T erm2): This constraint ensures that T erm1and T erm2
are different. This constraint is suspended when the arguments are not sufficiently instanti-
ated.
•get global map() = Map: This function returns the global map, which is shared by
all threads.
•get heap map() = M ap: This function returns the current thread’s heap map. Each
thread has its own heap map.
•get table map() = M ap: This function returns the current thread’s table map. Each
thread has its own table map. The table map is stored in the table area and both keys and
values are hash-consed (i.e., common sub-terms are shared).
•ground(T erm): This predicate is true if T erm is ground. A ground term does not contain
any variables.
•list to and(List) = Conj: This function converts List in the form [a1,. . .,an]
into a term in the form (a1,. . .,an).
•number vars(T erm,N0) = N1: This function numbers the variables in T erm by
using the integers starting from N0.N1is the next integer that is available after T erm
is numbered. Different variables receive different numberings, and the occurrences of the
same variable all receive the same numbering.
•parse radix string(String,Base) = Int: This function converts a radix String
of Base into a decimal integer Int, where Base must be greater than 1 and less than
37. For example, parse radix string("101",2) returns 5, which is the same as
parse term("0b101").
•parse term(String,T erm,V ars): This predicate uses the Picat parser to extract a
term T erm from String.V ars is a list of pairs, where each pair has the form Name=V ar.
41
•parse term(String) = T erm: This function converts String to a term.
•second(Compound) = T erm: This function returns the second argument of the com-
pound term Compound.
•subsumes(T erm1,T erm2): This predicate is true if T erm1subsumes T erm2.
•variant(T erm1,T erm2): This predicate is true if T erm2is a variant of T erm1.
•vars(T erm) = V ars: This function returns a list of variables that occur in T erm.
42
Chapter 4
Predicates and Functions
In Picat, predicates and functions are defined with pattern-matching rules. Picat has two types of
rules: the non-backtrackable rule
Head, Cond => Body.
and the backtrackable rule
Head, Cond ?=> Body.
Each rule is terminated by a dot (.) followed by a white space.
4.1 Predicates
Apredicate defines a relation, and can have zero, one, or multiple answers. Within a predicate, the
Head is a pattern in the form p(t1, . . . , tn), where pis called the predicate name, and nis called
the arity. When n= 0, the parentheses can be omitted. The condition Cond, which is an optional
goal, specifies a condition under which the rule is applicable. Cond cannot succeed more than
once. The compiler converts Cond to once Cond if would otherwise be possible for Cond to
succeed more than once.
For a call C, if Cmatches the pattern p(t1, . . . , tn)and Cond is true, then the rule is said to
be applicable to C. When applying a rule to call C, Picat rewrites Cinto Body. If the used rule
is non-backtrackable, then the rewriting is a commitment, and the program can never backtrack to
C. However, if the used rule is backtrackable, then the program will backtrack to Conce Body
fails, meaning that Body will be rewritten back to C, and the next applicable rule will be tried on
C.
A predicate is said to be deterministic if it is defined with non-backtrackable rules only, non-
deterministic if at least one of its rules is backtrackable, and globally deterministic if it is determin-
istic and all of the predicates in the bodies of the predicate’s rules are also globally deterministic.
A deterministic predicate that is not globally deterministic can still have more than one answer.
Example
append(Xs,Ys,Zs) ?=> Xs=[], Ys=Zs.
append(Xs,Ys,Zs) => Xs=[X|XsR], append(XsR,Ys,Zs).
min_max([H],Min,Max) => Min=H, Max=H.
min_max([H|T],Min,Max) =>
min_max(T,MinT,MaxT),
43
Min=min(MinT,H),
Max=max(MaxT,H).
The predicate append(Xs,Ys,Zs) is true if the concatenation of Xs and Ys is Zs. It defines
a relation among the three arguments, and does not assume directionality of any of the arguments.
For example, this predicate can be used to concatenate two lists, as in the call
append([a,b],[c,d],L)
this predicate can also be used to split a list nondeterministically into two sublists, as in the call
append(L1,L2,[a,b,c,d]); this predicate can even be called with three free variables, as
in the call append(L1,L2,L3).
The predicate min max(L,Min,Max) returns two answers through its arguments. It binds
Min to the minimum of list L, and binds Max to the maximum of list L. This predicate does not
backtrack. Note that a call fails if the first argument is not a list. Also note that this predicate
consumes linear space. A tail-recursive version of this predicate that consumes constant space will
be given below.
4.2 Functions
Afunction is a special kind of a predicate that always succeeds with one answer. Within a function,
the Head is an equation p(t1, . . . , tn)=X, where pis called the function name, and Xis an
expression that gives the return value. Functions are defined with non-backtrackable rules only.
For a call C, if Cmatches the pattern p(t1, . . . , tn)and Cond is true, then the rule is said to be
applicable to C. When applying a rule to call C, Picat rewrites the equation C=X0into (Body,
X0=X), where X0is a newly introduced variable that holds the return value of C.
Picat allows inclusion of function facts in the form p(t1,. . .,tn)=Exp in function definitions.
The function fact p(t1,. . .,tn)=Exp is shorthand for the rule:
p(t1,. . .,tn)=X=> X=Exp.
where Xis a new variable.
Although all functions can be defined as predicates, it is preferable to define them as functions
for two reasons. Firstly, functions often lead to more compact expressions than predicates, because
arguments of function calls can be other function calls. Secondly, functions are easier to debug
than predicates, because functions never fail and never return more than one answer.
Example
qequation(A,B,C) = (R1,R2),
D=B*B-4*A*C,
D >= 0
=>
NTwoC = -2*C,
R1 = NTwoC/(B+sqrt(D)),
R2 = NTwoC/(B-sqrt(D)).
rev([]) = [].
rev([X|Xs]) = rev(Xs)++[X].
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The function qequation(A,B,C) returns the pair of roots of A*X2+B*X+C=0. If the discrim-
inant B*B-4*A*Cis negative, then an exception will be thrown.
The function rev(L) returns the reversed list of L. Note that the function rev(L) takes
quadratic time and space in the length of L. A tail-recursive version that consumes linear time and
space will be given below.
4.3 Patterns and Pattern-Matching
The pattern p(t1, . . . , tn)in the head of a rule takes the same form as a structure. Function calls
are not allowed in patterns. Also, patterns cannot contain index notations, dot notations, ranges,
array comprehensions, or list comprehensions. Pattern matching is used to decide whether a rule
is applicable to a call. For a pattern Pand a term T, term Tmatches pattern Pif Pis identical
to T, or if Pcan be made identical to Tby instantiating P’s variables. Note that variables in the
term do not get instantiated after the pattern matching. If term Tis more general than pattern P,
then the pattern matching can never succeed.
Unlike calls in many committed-choice languages, calls in Picat are never suspended if they
are more general than the head patterns of the rules. A predicate call fails if it does not match the
head pattern of any of the rules in the predicate. A function call throws an exception if it does
not match the head pattern of any of the rules in the function. For example, for the function call
rev(L), where Lis a variable, Picat will throw the following exception:
unresolved function call(rev(L)).
A pattern can contain as-patterns in the form V@P attern, where Vis a new variable in the
rule, and P attern is a non-variable term. The as-pattern V@P attern is the same as P attern
in pattern matching, but after pattern matching succeeds, Vis made to reference the term that
matched P attern. As-patterns can avoid re-constructing existing terms.
Example
merge([],Ys) = Ys.
merge(Xs,[]) = Xs.
merge([X|Xs],Ys@[Y|_]) = [X|Zs], X<Y => Zs=merge(Xs,Ys).
merge(Xs,[Y|Ys]) = [Y|merge(Xs,Ys)].
In the third rule, the as-pattern Ys@[Y| ] binds two variables: Ys references the second argu-
ment, and Yreferences the car of the argument. The rule can be rewritten as follows without using
any as-pattern:
merge([X|Xs],[Y|Ys]) = [X|Zs], X<Y => Zs=merge(Xs,[Y|Ys]).
Nevertheless, this version is less efficient, because the cons [Y|Ys] needs to be re-constructed.
4.4 Goals
In a rule, both the condition and the body are goals. Queries that the users give to the interpreter
are also goals. A goal can take one of the following forms:
•true: This goal is always true.
45
•fail: This goal is always false. When fail occurs in a condition, the condition is false,
and the rule is never applicable. When fail occurs in a body, it causes execution to back-
track.
•false: This goal is the same as fail.
•p(t1, . . . , tn): This goal is a predicate call. The arguments t1, . . . , tnare evaluated in the
given order, and the resulting call is resolved using the rules in the predicate p/n. If the
call succeeds, then variables in the call may get instantiated. Many built-in predicates are
written in infix notation. For example, X=Y is the same as ’=’(X,Y).
•P,Q: This goal is a conjunction of goal Pand goal Q. It is resolved by first resolving
P, and then resolving Q. The goal is true if both Pand Qare true. Note that the order is
important: (P,Q) is in general not the same as (Q,P).
•P&& Q: This is the same as (P,Q).
•P;Q: This goal is a disjunction of goal Pand goal Q. It is resolved by first resolving P.
If Pis true, then the disjunction is true. If Pis false, then Qis resolved. The disjunction is
true if Qis true. The disjunction is false if both Pand Qare false. Note that a disjunction
can succeed more than once. Note also that the order is important: (P;Q) is generally not
the same as (Q;P).
•P|| Q: This is the same as (P;Q).
•not P: This goal is the negation of P. It is false if Pis true, and true if Pis false. Note
a negation goal can never succeed more than once. Also note that no variables can get
instantiated, no matter whether the goal is true or false.
•once P: This goal is the same as P, but can never succeed more than once.
•repeat: This predicate is defined as follows:
repeat ?=> true.
repeat => repeat.
The repeat predicate is often used to describe failure-driven loops. For example, the query
repeat,writeln(a),fail
repeatedly outputs ’a’ until ctrl-c is typed.
•if-then: An if-then statement takes the form
if Cond1then
Goal1
elseif Cond2then
Goal2
.
.
.
elseif Condnthen
Goaln
else
Goalelse
end
46
where the elseif and else clauses are optional. If the else clause is missing, then the
else goal is assumed to be true. For the if-then statement, Picat finds the first condition
Condithat is true. If such a condition is found, then the truth value of the if-then statement
is the same as Goali. If none of the conditions is true, then the truth value of the if-then
statement is the same as Goalelse. Note that no condition can succeed more than once.
•throw Exception: This predicate throws the term Exception. This predicate will be
detailed in Chapter 6 on Exceptions.
•Loops: Picat has three types of loop statements: foreach, while, and do-while. A loop
statement is true if and only if every iteration of the loop is true. The details of loops are
given in Chapter 5.
4.5 Predicate Facts
For an extensional relation that contains a large number of tuples, it is tedious to define such a
relation as a predicate with pattern-matching rules. It is worse if the relation has multiple keys.
In order to facilitate the definition of extensional relations, Picat allows the inclusion of predicate
facts in the form p(t1,. . .,tn)in predicate definitions. Facts and rules cannot co-exist in predicate
definitions and facts must be ground. A predicate definition that consists of facts must be preceded
by an index declaration in the form
index (M11, M12, . . . , M1n). . . (Mm1, Mm2, . . . , Mmn)
where each Mij is either +(meaning indexed) or −(meaning not indexed). Facts are translated
into pattern-matching rules before they are compiled.
Example
index (+,-) (-,+)
edge(a,b).
edge(a,c).
edge(b,c).
edge(c,b).
The predicate edge is translated into the following rules:
edge(a,Y) ?=> Y=b.
edge(a,Y) => Y=c.
edge(b,Y) => Y=c.
edge(c,Y) => Y=b.
edge(X,b) ?=> X=a.
edge(X,c) ?=> X=a.
edge(X,c) => X=b.
edge(X,b) => X=c.
4.6 Tail Recursion
A rule is said to be tail-recursive if the last call of the body is the same predicate as the head. The
last-call optimization enables last calls to reuse the stack frame of the head predicate if the frame
is not protected by any choice points. This optimization is especially effective for tail recursion,
47
because it converts recursion into iteration. Tail recursion runs faster and consumes less memory
than non-tail recursion.
The trick to convert a predicate (or a function) into tail recursion is to define a helper that
uses an accumulator parameter to accumulate the result. When the base case is reached, the
accumulator is returned. At each iteration, the accumulator is updated. Initially, the original
predicate (or function) calls the helper with an initial value for the accumulator parameter.
Example
min_max([H|T],Min,Max) =>
min_max_helper([H|T],H,Min,H,Max).
min_max_helper([],CMin,Min,CMax,Max) => Min=CMin, Max=CMax.
min_max_helper([H|T],CMin,Min,CMax,Max) =>
min_max_helper(T,min(CMin,H),Min,max(CMax,H),Max).
rev([]) = [].
rev([X|Xs]) = rev_helper(Xs,[X]).
rev_helper([],R) = R.
rev_helper([X|Xs],R) = rev_helper(Xs,[X|R]).
In the helper predicate min max helper(L,CMin,Min,CMax,Max),CMin and CMax are
accumulators: CMin is the current minimum value, and CMax is the current maximum value.
When Lis empty, the accumulators are returned by the unification calls Min=CMin and Max=CMax.
When Lis a cons [H|T], the accumulators are updated: CMin changes to min(CMin,H), and
CMax changes to max(CMax,H). The helper function rev helper(L,R) follows the same
idea: it uses an accumulator list to hold, in reverse order, the elements that have been scanned.
When Lis empty, the accumulator is returned. When Lis the cons [X|Xs], the accumulator R
changes to [X|R].
48
Chapter 5
Assignments and Loops
This chapter discusses variable assignments, loop constructs, and list and array comprehensions
in Picat. It describes the scope of an assigned variable, indicating where the variable is defined,
and where it is not defined. Finally, it shows how assignments, loops, and list comprehensions are
related, and how they are compiled.
5.1 Assignments
Picat variables are single-assignment, meaning that once a variable is bound to a value, the variable
cannot be bound again. In order to simulate imperative language variables, Picat provides the
assignment operator :=. An assignment takes the form LHS:=RHS, where LHS is either a
variable or an access of a compound value in the form X[...]. When LHS is an access in
the form X[I], the component of Xindexed Iis updated. This update is undone if execution
backtracks over this assignment.
Example
test => X = 0, X := X + 1, X := X + 2, write(X).
The compiler needs to give special consideration to the scope of a variable. The scope of a
variable refers to the parts of a program where a variable occurs.
Consider the test example. This example binds Xto 0. Then, the example tries to bind X
to X+1. However, Xis still in scope, meaning that Xis already bound to 0. Since Xcannot be
bound again, the compiler must perform extra operations in order to manage assignments that use
the := operator.
In order to handle assignments, Picat creates new variables at compile time. In the test
example, at compile time, Picat creates a new variable, say X1, to hold the value of Xafter the
assignment X:=X+1. Picat replaces Xby X1 on the LHS of the assignment. All occurrences
of Xafter the assignment are replaced by X1. When encountering X1 := X1 + 2, Picat creates
another new variable, say X2, to hold the value of X1 after the assignment, and replaces the
remaining occurrences of X1 by X2. When write(X2) is executed, the value held in X2, which
is 3, is printed. This means that the compiler rewrites the above example as follows:
test => X = 0, X1 = X + 1, X2 = X1 + 2, write(X2).
5.1.1 If-Else
This leads to the question: what does the compiler do if the code branches? Consider the following
code skeleton.
49
Example
if_ex(Z) =>
X=1,Y=2,
if Z > 0 then
X := X *Z
else
Y:=Y+Z
end,
println([X,Y]).
The if ex example performs exactly one assignment. At compilation time, the compiler
does not know whether or not Z>0evaluates to true. Therefore, the compiler does not know
whether to introduce a new variable for Xor for Y.
Therefore, when an if-else statement contains an assignment, the compiler rewrites the if-else
statement as a predicate. For example, the compiler rewrites the above example as follows:
if_ex(Z) =>
X=1,Y=2,
p(X, Xout, Y, Yout, Z),
println([Xout,Yout]).
p(Xin, Xout, Yin, Yout, Z), Z > 0 =>
Xout = X *Z,
Yout = Yin.
p(Xin, Xout, Yin, Yout) =>
Xout = Xin,
Yout = Y + Z.
One rule is generated for each branch of the if-else statement. For each variable Vthat occurs on
the LHS of an assignment statement that is inside of the if-else statement, predicate pis passed
two arguments, Vin and Vout. In the above example, Xand Yeach occur on the LHS of an
assignment statement. Therefore, predicate pis passed the parameters Xin,Xout,Yin, and
Yout.
5.2 Types of Loops
Picat has three types of loop statements for programming repetitions: foreach,while, and
do-while.
5.2.1 Foreach Loops
Aforeach loop has the form:
foreach (E1in D1,Cond1,. . .,Enin Dn,Condn)
Goal
end
Each Eiis an iterating pattern. Each Diis an expression that gives a compound value. Each
Condiis an optional condition on iterators E1through Ei.
Foreach loops can be used to iterate through compound values, as in the following examples.
50
Example
loop_ex1 =>
L = [17, 3, 41, 25, 8, 1, 6, 40],
foreach (E in L)
println(E)
end.
loop_ex2(Map) =>
foreach(Key=Value in Map)
writef("%w=%w\n", Key, Value)
end.
The loop ex1 example iterates through a list. The loop ex2 example iterates through a
map, where Key=Value is the iterating pattern.
The loop ex1 example can also be written, using a failure-driven loop, as follows.
Example
loop_ex1 =>
L = [17, 3, 41, 25, 8, 1, 6, 40],
( member(E, L),
println(E),
fail
;
true
).
Recall that the range Start..Step..End stands for a list of numbers. Ranges can be used as
compound values in iterators.
Example
loop_ex3 =>
foreach(E in 1 .. 2 .. 9)
println(E)
end.
Also recall that the function zip(List1,List2,. . .,Listn)returns a list of tuples. This
function can be used to simultaneously iterate over multiple lists.
Example:
loop_ex_parallel =>
foreach(Pair in zip(1..2, [a,b]))
println(Pair)
end.
5.2.2 Foreach Loops with Multiple Iterators
Each of the previous examples uses a single iterator. Foreach loops can also contain multiple
iterators.
51
Example:
loop_ex4 =>
L = [2, 3, 5, 10],
foreach(I in L, J in 1 .. 10, J mod I != 0)
printf("%d is not a multiple of %d%n", J, I)
end.
If a foreach loop has multiple iterators, then it is compiled into a series of nested foreach
loops in which each nested loop has a single iterator. In other words, a foreach loop with multiple
iterators executes its goal once for every possible combination of values in the iterators.
The foreach loop in loop ex4 is the same as the nested loop:
loop_ex5 =>
L = [2, 3, 5, 10],
foreach(I in L)
foreach(J in 1..10)
if J mod I != 0 then
printf("%d is not a multiple of %d%n", J, I)
end
end
end.
5.2.3 While Loops
Awhile loop has the form:
while (Cond)
Goal
end
As long as Cond succeeds, the loop will repeatedly execute Goal.
Example:
loop_ex6 =>
I = 1,
while (I <= 9)
println(I),
I:=I+2
end.
loop_ex7 =>
J = 6,
while (J <= 5)
println(J),
J:=J+1
end.
loop_ex8 =>
E = read_int(),
while (E mod 2 == 0; E mod 5 == 0)
52
println(E),
E := read_int()
end.
loop_ex9 =>
E = read_int(),
while (E mod 2 == 0, E mod 5 == 0)
println(E),
E := read_int()
end.
The while loop in loop ex6 prints all of the odd numbers between 1and 9. It is similar to
the foreach loop
foreach(I in 1 .. 2 .. 9)
println(I)
end.
The while loop in loop ex7 never executes its goal. Jbegins at 6, so the condition J <= 5
is never true, meaning that the body of the loop does not execute.
The while loop in loop ex8 demonstrates a compound condition. The loop executes as long
as the value that is read into Eis either a multiple of 2or a multiple of 5.
The while loop in loop ex9 also demonstrates a compound condition. Unlike in loop ex8,
in which either condition must be true, in loop ex9, both conditions must be true. The loop
executes as long as the value that is read into Eis both a multiple of 2and a multiple of 5.
5.2.4 Do-while Loops
Ado-while loop has the form:
do
Goal
while (Cond)
A do-while loop is similar to a while loop, except that a do-while loop executes Goal one time
before testing Cond. The following example demonstrates the similarities and differences between
do-while loops and while loops.
Example
loop_ex10 =>
J = 6,
do
println(J),
J:=J+1
while (J <= 5).
Unlike loop ex7,loop ex10 executes its body once. Although Jbegins at 6, the do-while
loop prints J, and increments Jbefore evaluating the condition J <= 5.
53
5.3 List and Array Comprehensions
Alist comprehension is a special functional notation for creating lists. List comprehensions have
a similar format to foreach loops.
[T:E1in D1,Cond1,. . .,Enin Dn,Condn]
Tis an expression. Each Eiis an iterating pattern. Each Diis an expression that gives a compound
value. Each Condiis an optional condition on iterators E1through Ei.
An array comprehension takes the following form:
{T:E1in D1,Cond1,. . .,Enin Dn,Condn}
It is the same as:
to array([T:E1in D1,Cond1,. . .,Enin Dn,Condn])
Example
picat> L = [(A, I) : A in [a, b], I in 1 .. 2].
L = [(a , 1),(a , 2),(b , 1),(b , 2)]
picat> L = {(A, I) : A in [a, b], I in 1 .. 2}.
L = {(a , 1),(a , 2),(b , 1),(b , 2)}
5.4 Compilation of Loops
Variables that occur in a loop, but do not occur before the loop in the outer scope, are local to each
iteration of the loop. For example, in the rule
p(A) =>
foreach (I in 1 .. A.length)
E = A[I],
println(E)
end.
the variables Iand Eare local, and each iteration of the loop has its own values for these variables.
Consider the example:
Example
while_test(N) =>
I = 1,
while (I <= N)
I:=I+1,
println(I)
end.
In this example, the while loop contains an assignment statement. As mentioned above, at compi-
lation time, Picat creates new variables in order to handle assignments. One new variable is created
for each assignment. However, when this example is compiled, the compiler does not know the
number of times that the body of the while loop can be executed. This means that the compiler
does not know how many times the assignment I:=I+1will occur, and the compiler is
54
unable to create new variables for this assignment. In order to solve this problem, the compiler
compiles while loops into tail-recursive predicates.
In the while test example, the while loop is compiled into:
while_test(N) =>
I = 1,
p(I, N).
p(I, N), I <= N =>
I1=I+1,
println(I1),
p(I1, N).
p(_, _) => true.
Note that the first rule of the predicate p(I, N) has the same condition as the while loop.
The second rule, which has no condition, terminates the while loop, because the second rule is only
executed if I>N. The call p(I1, N) is the tail-recursive call, with I1 storing the modified
value.
Suppose that a while loop modifies a variable that is then used outside of the while loop. For
each modified variable Vthat is used after the while loop, predicate pis passed two arguments, Vin
and Vout. Then, a predicate that has the body true is not sufficient to terminate the compiled
while loop. Instead, a predicate fact must be used, as in the next example.
The next example demonstrates a loop that has multiple accumulators, and that modifies values
which are then used outside of the loop.
Example
min_max([H|T], Min, Max) =>
LMin = H,
LMax = H,
foreach (E in T)
LMin := min(LMin, E),
LMax := max(LMax, E)
end,
Min = LMin,
Max = LMax.
This loop finds the minimum and maximum values of a list. The loop is compiled to:
min_max([H|T], Min, Max) =>
LMin = H,
LMax = H,
p(T, LMin, LMin1, LMax, LMax1),
Min = LMin1,
Max = LMax1.
p([], MinIn, MinOut, MaxIn, MaxOut) =>
MinOut = MinIn,
MaxOut= MaxIn.
p([E|T], MinIn, MinOut, MaxIn, MaxOut) =>
Min1 = min(MinIn, E),
55
Max1 = max(MaxIn, E),
p(T, Min1, MinOut, Max1, MaxOut).
Notice that there are multiple accumulators: MinIn and MaxIn. Since the min max predicate
returns two values, the accumulators each have an “in” variable (MinIn and Maxin) and an
“out” variable (MinOut and MaxOut). If the first parameter of predicate pis an empty list, then
MinOut is set to the value of MinIn, and MaxOut is set to the value of MaxIn.
Foreach and do-while loops are compiled in a similar manner to while loops.
Nested Loops
As mentioned above, variables that only occur within a loop are local to each iteration of the loop.
In nested loops, variables that are local to the outer loop are global to the inner loop. In other
words, if a variable occurs in the outer loop, then the variable is also visible in the inner loop.
However, variables that are local to the inner loop are not visiable to the outer loop.
For example, consider the nested loops:
nested =>
foreach (I in 1 .. 10)
printf("Numbers between %d and %d ", I, I *I),
foreach (J in I .. I *I)
printf("%d ", J)
end,
nl
end.
Variable Iis local to the outer foreach loop, and is global to the inner foreach loop. Therefore,
iterator Jis able to iterate from Ito I*Iin the inner foreach loop. Iterator Jis local to the
inner loop, and does not occur in the outer loop.
Since a foreach loop with Niterators is converted into Nnested foreach loops, the order of the
iterators matters.
5.4.1 List Comprehensions
List comprehensions are compiled into foreach loops.
Example
comp_ex =>
L = [(A, X) : A in [a, b], X in 1 .. 2].
This list comprehension is compiled to:
comp_ex =>
List = L,
foreach (A in [a, b], X in 1 .. 2)
L = [(A, X) | T],
L := T
end,
L = [].
56
Example
make_list1 =>
L = [Y : X in 1..5],
write(L).
make_list2 =>
Y = Y,
L = [Y : X in 1..5],
write(L).
Suppose that a user would like to create a list [Y, Y, Y, Y, Y]. The make list1 predicate
incorrectly attempts to make this list; instead, it outputs a list of 5 different variables since Yis
local. In order to make all five variables the same, make list2 makes variable Yglobal, by
adding the line Y=Yto globalize Y.
57
Chapter 6
Exceptions
An exception is an event that occurs during the execution of a program. An exception requires a
special treatment. In Picat, an exception is just a term. A built-in exception is a structure, where
the name denotes the type of the exception, and the arguments provide other information about the
exception, such as the source, which is the goal or function that raised the exception.
6.1 Built-in Exceptions
Picat throws many types of exceptions. The following are some of the built-in exceptions:1
•zero divisor(Source):Source divides a number by zero.
•domain error(V al,Source):Source receives a value V al that is unexpected in the
domain.
•existence error(Entity,Source):Source tries to use Entity, such as a file, a func-
tion, or a solver, that does not exist.
•interrupt(Source): The execution is interrupted by a signal. For an interrupt caused
by ctrl-c,Source is keyboard.
•io error(ENo,EMsg,Source): An I/O error with the number ENo and message
EMsg occurs in Source.
•load error(F Name,Source): An error occurs while loading the byte-code file named
F Name. This error is caused by the malformatted byte-code file.
•out of memory(Area): The system runs out of memory while expanding Area, which
can be: stack heap,trail,program,table, or findall.
•out of bound(EIndex,Source):Source tries to access an element of a compound
value using the index EIndex, which is out of bound. An index is out of bound if it is less
than or equal to zero, or if it is greater than the length of the compound value.
•syntax error(String,Source):String cannot be parsed into a value that is expected
by Source. For example, read int() throws this exception if it reads in a string "a"
rather than an integer.
•unresolved function call(F Call): No rule is applicable to the function call F Call.
1In the current implementation, exceptions thrown by B-Prolog are not compliant with the documentation.
58
•Type expected(EArg,Source): The argument EArg in Source is not an expected
type or value, where T ype can be var,nonvar,dvar,atom,integer,real,number,
list,map, etc.
6.2 Throwing Exceptions
The built-in predicate throw Exception throws Exception. After an exception is thrown, the
system searches for a handler for the exception. If none is found, then the system displays the
exception and aborts the execution of the current query. It also prints the backtrace of the stack
if it is in debug mode. For example, for the function call open("abc.txt"), the following
message will be displayed if there is no file that is named "abc.txt".
*** error(existence_error(source_sink,abc.txt),open)
6.3 Defining Exception Handlers
All exceptions, including those raised by built-ins and interruptions, can be caught by catchers. A
catcher is a call in the form:
catch(Goal,Exception,RecoverGoal)
which is equivalent to Goal, except when an exception is raised during the execution of Goal
that unifies Exception. When such an exception is raised, all of the bindings that have been
performed on variables in Goal will be undone, and RecoverGoal will be executed to handle
the exception. Note that Exception is unified with a renamed copy of the exception before
RecoverGoal is executed. Also note that only exceptions that are raised by a descendant call of
Goal can be caught.
The call call cleanup(Call,Cleanup) is equivalent to call(Call), except that
Cleanup is called when Call succeeds determinately (i.e., with no remaining choice point),
when Call fails, or when Call raises an exception.
59
Chapter 7
Tabling
The Picat system is a term-rewriting system. For a predicate call, Picat selects a matching rule
and rewrites the call into the body of the rule. For a function call C, Picat rewrites the equation
C=Xwhere Xis a variable that holds the return value of C. Due to the existence of recursion in
programs, the term-rewriting process may never terminate. Consider, for example, the following
program:
reach(X,Y) ?=> edge(X,Y).
reach(X,Y) => reach(X,Z),edge(Z,Y).
where the predicate edge defines a relation, and the predicate reach defines the transitive closure
of the relation. For a query such as reach(a,X), the program never terminates due to the
existence of left-recursion in the second rule. Even if the rule is converted to right-recursion, the
query may still not terminate if the graph that is represented by the relation contains cycles.
Another issue with recursion is redundancy. Consider the following problem: Starting in the
top left corner of a N×Ngrid, one can either go rightward or downward. How many routes are
there through the grid to the bottom right corner? The following gives a program in Picat for the
problem:
route(N,N,_Col) = 1.
route(N,_Row,N) = 1.
route(N,Row,Col) = route(N,Row+1,Col)+route(N,Row,Col+1).
The function call route(20,1,1) returns the number of routes through a 20×20 grid. The
function call route(N,1,1) takes exponential time in N, because the same function calls are
repeatedly spawned during the execution, and are repeatedly resolved each time that they are
spawned.
7.1 Table Declarations
Tabling is a memoization technique that can prevent infinite loops and redundancy. The idea
of tabling is to memorize the answers to subgoals and use the answers to resolve their variant
descendants. In Picat, in order to have all of the calls and answers of a predicate or function
tabled, users just need to add the keyword table before the first rule.
Example
table
reach(X,Y) ?=> edge(X,Y).
60
reach(X,Y) => reach(X,Z),edge(Z,Y).
table
route(N,N,_Col) = 1.
route(N,_Row,N) = 1.
route(N,Row,Col) = route(N,Row+1,Col)+route(N,Row,Col+1).
With tabling, all queries to the reach predicate are guaranteed to terminate, and the function call
route(N,1,1) takes only N2time.
For some problems, such as planning problems, it is infeasible to table all answers, because
there may be an infinite number of answers. For some other problems, such as those that require
the computation of aggregates, it is a waste to table non-contributing answers. Picat allows users
to provide table modes to instruct the system about which answers to table. For a tabled predicate,
users can give a table mode declaration in the form (M1, M2, . . . , Mn), where each Miis one
of the following: a plus-sign (+) indicates input, a minus-sign (-) indicates output, max indicates
that the corresponding variable should be maximized, and min indicates that the corresponding
variable should be minimized. The last mode Mncan be nt, which indicates that the argument is
not tabled. Two types of data can be passed to a tabled predicate as an nt argument: (1) global
data that are the same to all the calls of the predicate, and (2) data that are functionally dependent
on the input arguments. Input arguments are assumed to be ground. Output arguments, including
min and max arguments, are assumed to be variables. An argument with the mode min or max
is called an objective argument. Only one argument can be an objective to be optimized. As an
objective argument can be a compound value, this limit is not essential, and users can still specify
multiple objective variables to be optimized. When a table mode declaration is provided, Picat
tables only one optimal answer for the same input arguments.
Example
table(+,+,-,min)
sp(X,Y,Path,W) ?=>
Path = [(X,Y)],
edge(X,Y,W).
sp(X,Y,Path,W) =>
Path = [(X,Z)|Path1],
edge(X,Z,Wxz),
sp(Z,Y,Path1,W1),
W = Wxz+W1.
The predicate edge(X,Y,W) specifies a weighted directed graph, where Wis the weight of the
edge between node Xand node Y. The predicate sp(X,Y,Path,W) states that Path is a path
from Xto Ywith the minimum weight W. Note that whenever the predicate sp/4 is called, the
first two arguments must always be instantiated. For each pair, the system stores only one path
with the minimum weight.
The following program finds a shortest path among those with the minimum weight for each
pair of nodes:
table (+,+,-,min).
sp(X,Y,Path,WL) ?=>
Path = [(X,Y)],
WL = (Wxy,1),
61
edge(X,Y,Wxy).
sp(X,Y,Path,WL) =>
Path = [(X,Z)|Path1],
edge(X,Z,Wxz),
sp(Z,Y,Path1,WL1),
WL1 = (Wzy,Len1),
WL = (Wxz+Wzy,Len1+1).
For each pair of nodes, the pair of variables (W,Len) is minimized, where Wis the weight, and
Len is the length of a path. The built-in function compare terms(T1,T2)is used to compare
answers. Note that the order is important. If the term would be (Len,W), then the program would
find a shortest path, breaking a tie by selecting one with the minimum weight.
The tabling system is useful for offering dynamic programming solutions for planning prob-
lems. The following shows a tabled program for general planning problems:
table (+,-,min)
plan(S,Plan,Len),final(S) => Plan=[],Len=0.
plan(S,Plan,Len) =>
action(Action,S,S1),
plan(S1,Plan1,Len1),
Plan = [Action|Plan1],
Len = Len1+1.
The predicate action(Action,S,S1) selects an action and performs the action on state Sto
generate another state, S1.
Example
The program shown in Figure 7.1 solves the Farmer’s problem: The farmer wants to get his goat,
wolf, and cabbage to the other side of the river. His boat isn’t very big, and it can only carry him
and either his goat, his wolf, or his cabbage. If he leaves the goat alone with the cabbage, then
the goat will gobble up the cabbage. If he leaves the wolf alone with the goat, then the wolf will
gobble up the goat. When the farmer is present, the goat and cabbage are safe from being gobbled
up by their predators.
7.2 The Tabling Mechanism
The Picat tabling system employs the so-called linear tabling mechanism, which computes fix-
points by iteratively evaluating looping subgoals. The system uses a data area, called the table
area, to store tabled subgoals and their answers. The tabling area can be initialized with the fol-
lowing built-in predicate:
•initialize table: This predicate initializes the table area.
This predicate clears up the table area. It’s users’ responsibility to ensure that no data in the table
area are referenced by any part of the application.
Linear tabling relies on the following three primitive operations to access and update the table
area.
Subgoal lookup and registration: This operation is used when a tabled subgoal is encountered
during execution. It looks up the subgoal table to see if there is a variant of the subgoal.
62
go =>
S0=[s,s,s,s],
plan(S0,Plan,_),
writeln(Plan.reverse()).
table (+,-,min)
plan([n,n,n,n],Plan,Len) => Plan=[], Len=0.
plan(S,Plan,Len) =>
Plan=[Action|Plan1],
action(S,S1,Action),
plan(S1,Plan1,Len1),
Len=Len1+1.
action([F,F,G,C],S1,Action) ?=>
Action=farmer_wolf,
opposite(F,F1),
S1=[F1,F1,G,C],
not unsafe(S1).
action([F,W,F,C],S1,Action) ?=>
Action=farmer_goat,
opposite(F,F1),
S1=[F1,W,F1,C],
not unsafe(S1).
action([F,W,G,F],S1,Action) ?=>
Action=farmer_cabbage,
opposite(F,F1),
S1=[F1,W,G,F1],
not unsafe(S1).
action([F,W,G,C],S1,Action) ?=>
Action=farmer_alone,
opposite(F,F1),
S1=[F1,W,G,C],
not unsafe(S1).
index (+,-) (-,+)
opposite(n,s).
opposite(s,n).
unsafe([F,W,G,_C]),W==G,F!==W => true.
unsafe([F,_W,G,C]),G==C,F!==G => true.
Figure 7.1: A program for the Farmer’s problem.
63
If not, it inserts the subgoal (termed a pioneer or generator) into the subgoal table. It also
allocates an answer table for the subgoal and its variants. Initially, the answer table is empty.
If the lookup finds that there already is a variant of the subgoal in the table, then the record
that is stored in the table is used for the subgoal (called a consumer). Generators and con-
sumers are handled differently. In linear tabling, a generator is resolved using rules, and a
consumer is resolved using answers; a generator is iterated until the fixed point is reached,
and a consumer fails after it exhausts all of the existing answers.
Answer lookup and registration: This operation is executed when a rule succeeds in generating
an answer for a tabled subgoal. If a variant of the answer already exists in the table, then
it does nothing; otherwise, it inserts the answer into the answer table for the subgoal, or
it tables the answer according to the mode declaration. Picat uses the lazy consumption
strategy (also called the local strategy). After an answer is processed, the system backtracks
to produce the next answer.
Answer return: When a consumer is encountered, an answer is returned immediately, if an an-
swer exists. On backtracking, the next answer is returned. A generator starts consuming
its answers after it has exhausted all of its rules. Under the lazy consumption strategy, a
top-most looping generator does not return any answer until it is complete.
64
Chapter 8
The planner Module
The planner module provides several predicates for solving planning problems. Given an initial
state, a final state, and a set of possible actions, a planning problem is to find a plan that transforms
the initial state to the final state. In order to use the planner module to solve a planning problem,
users have to provide the condition for the final states and the state transition diagram through the
following global predicates:
•final(S): This predicate succeeds if Sis a final state.
•final(S,P lan,Cost): A final state can be reached from Sby the action sequence in
P lan with Cost. If this predicate is not given, then the system assumes the following
definition:
final(S,Plan,Cost) => Plan=[], Cost=0, final(S).
•action(S,NextS,Action,ActionCost): This predicate encodes the state transition
diagram of the planning problem. The state Scan be transformed into NextS by performing
Action. The cost of Action is ActionCost. If the plan’s length is the only interest, then
ActionCost should be 1.
A state is normally a ground term. As all states are tabled during search, it is of paramount
importance to find a good representation for states such that terms among states can be shared as
much as possible.
In addition to the two required predicates given above, users can optionally provide the fol-
lowing global procedures to assist Picat in searching for plans:
•heuristic(S): This function returns an estimation of the resource amount needed to
transform Sinto a goal state. This estimation is said to be admissible if it never exceeds
the real cost. All heuristic estimations must be admissible in order for the planner to find
optimal plans. This function is used by the planner to check the heuristic estimation before
each state expansion.
•sequence(P,Action): This predicate binds Action to a viable action form based on the
current partial plan P. Note that the actions in list Pare in reversed order, with the most
recent action occurring first in the list, and the first action occurring last in the list. The
planner calls sequence/2 to find an action for expanding the current state before calling
action/4. For example,
sequence([move(R,X,Y)|_], Action) ?=> Action = $move(R,Y,_).
65
sequence([move(R,_,_)|_], Action) ?=> Action = $jump(R).
sequence([move(R,_,_)|_], Action) => Action = $wait(R).
sequence(_, _) => true.
These sequence rules ban robots from moving in an interleaving fashion; a robot must con-
tinue to move until it takes the action jump or wait before another robot can start moving.
The last rule sequence( , ) => true is necessary; it permits any action to be taken
if the partial plan is empty, or if the most recent action in the partial plan is not move.
8.1 Depth-Bounded Search
Depth-bounded search amounts to exploring the search space, taking into account the current
available resource amount. A new state is only explored if the available resource amount is non-
negative. When depth-bounded search is used, the function current resource() can be
used to retrieve the current resource amount. If the heuristic estimate of the cost to travel from the
current state to the final state is greater than the available resource amount, then the current state
fails.
•plan(S,Limit,P lan,Cost): This predicate, if it succeeds, binds P lan to a plan that
can transform state Sto a final state that satisfies the condition given by final/1 or
final/3.Cost is the cost of P lan, which cannot exceed Limit, which is a given non-
negative integer.
•plan(S,Limit,P lan): If the second argument is an integer, then this predicate is the
same as the plan/4 predicate, except that the plan’s cost is not returned.
•plan(S,P lan,P lanCost): If the second argument is a variable, then this predicate is
the same as the plan/4 predicate, except that the limit is assumed to be 268435455.
•plan(S,P lan): This predicate is the same as the plan/4 predicate, except that the limit
is assumed to be 268435455, and that the plan’s cost is not returned.
•best plan(S,Limit,P lan,Cost): This predicate finds an optimal plan by using the
following algorithm: It first calls plan/4 to find a plan of 0 cost. If no plan is found, then
it increases the cost limit to 1 or double the cost limit. Once a plan is found, the algorithm
uses binary search to find an optimal plan.
•best plan(S,Limit,P lan): If the second argument is an integer, then this predicate is
the same as the best plan/4 predicate, except that the plan’s cost is not returned.
•best plan(S,P lan,P lanCost): If the second argument is a variable, then this pred-
icate is the same as the best plan/4 predicate, except that the limit is assumed to be
268435455.
•best plan(S,P lan): This predicate is the same as the best plan/4 predicate, except
that the limit is assumed to be 268435455, and that the plan’s cost is not returned.
•best plan nondet(S,Limit,P lan,Cost): This predicate is the same as
best plan(S,Limit,P lan,Cost), except that it allows multiple best plans to be re-
turned through backtracking.
•best plan nondet(S,Limit,P lan): If the second argument is an integer, then this
predicate is the same as the best plan nondet/4 predicate, except that the plan’s cost
is not returned.
66
•best plan nondet(S,P lan,P lanCost): If the second argument is a variable, then
this predicate is the same as the best plan nondet/4 predicate, except that the limit is
assumed to be 268435455.
•best plan nondet(S,P lan): This predicate is the same as the best plan nondet/4
predicate, except that the limit is assumed to be 268435455, and that the plan’s cost is not
returned.
•best plan bb(S,Limit,P lan,Cost): This predicate, if it succeeds, binds P lan to an
optimal plan that can transform state Sto a final state. Cost is the cost of P lan, which can-
not exceed Limit, which is a given non-negative integer. The branch-and-bound algorithm
is used to find an optimal plan.
•best plan bb(S,Limit,P lan): If the second argument is an integer, then this pred-
icate is the same as the best plan bb/4 predicate, except that the plan’s cost is not
returned.
•best plan bb(S,P lan,P lanCost): If the second argument is a variable, then this
predicate is the same as the best plan bb/4 predicate, except that the limit is assumed
to be 268435455.
•best plan bb(S,P lan): This predicate is the same as the best plan bb/4 predi-
cate, except that the limit is assumed to be 268435455, and that the plan’s cost is not re-
turned.
•best plan bin(S,Limit,P lan,Cost): This predicate, if it succeeds, binds P lan to
an optimal plan that can transform state Sto a final state. Cost is the cost of P lan, which
cannot exceed Limit, which is a given non-negative integer. The branch-and-bound algo-
rithm is used with binary search to find an optimal plan.
•best plan bin(S,Limit,P lan): If the second argument is an integer, then this pred-
icate is the same as the best plan bin/4 predicate, except that the plan’s cost is not
returned.
•best plan bin(S,P lan,P lanCost): If the second argument is a variable, then this
predicate is the same as the best plan bin/4 predicate, except that the limit is assumed
to be 268435455.
•best plan bin(S,P lan): This predicate is the same as the best plan bin/4 pred-
icate, except that the limit is assumed to be 268435455, and that the plan’s cost is not re-
turned.
•current resource()=Amount: This function returns the current available resource
amount of the current node. If the current execution path was not initiated by one of the
calls that performs resource-bounded search, then 268435455 is returned. This function can
be used to check the heuristics. If the heuristic estimate of the cost to travel from the current
state to a final state is greater than the available resource amount, then the current state can
be failed.
•current plan()=P lan: This function returns the current plan that has transformed the
initial state to the current state. If the current execution path was not initiated by one of the
calls that performs resource-bounded search, then [] is returned.
67
•current resource plan cost(Amount,P lan,Cost): This predicate retrieves the
attributes of the current node in the search tree, including the resource amount, the path to
the node, and its cost.
•is tabled state(S): This predicate succeeds if the state Shas been explored before
and has been tabled.
8.2 Depth-Unbounded Search
In contrast to depth-bounded search, depth-unbounded search does not take into account the avail-
able resource amount. A new state can be explored even if no resource is available for the explo-
ration. The advantage of depth-unbounded search is that failed states are never re-explored.
•plan unbounded(S,Limit,P lan,Cost): This predicate, if it succeeds, binds P lan
to a plan that can transform state Sto a final state. Cost is the cost of P lan, which cannot
exceed Limit, which is a given non-negative integer.
•plan unbounded(S,Limit,P lan)If the second argument is an integer, then this pred-
icate is the same as the plan unbounded/4 predicate, except that the plan’s cost is not
returned.
•plan unbounded(S,P lan,P lanCost): If the second argument is a variable, then this
predicate is the same as the plan unbounded/4 predicate, except that the limit is as-
sumed to be 268435455.
•plan unbounded(S,P lan): This predicate is the same as the above predicate, except
that the limit is assumed to be 268435455.
•best plan unbounded(S,Limit,P lan,Cost): This predicate, if it succeeds, binds
P lan to an optimal plan that can transform state Sto a final state. Cost is the cost of P lan,
which cannot exceed Limit, which is a given non-negative integer.
•best plan unbounded(S,Limit,P lan)If the second argument is an integer, then
this predicate is the same as the best plan unbounded/4 predicate, except that the
plan’s cost is not returned.
•best plan unbounded(S,P lan,P lanCost): If the second argument is a variable,
then this predicate is the same as the best plan unbounded/4 predicate, except that
the limit is assumed to be 268435455.
•best plan unbounded(S,P lan): This predicate is the same as the above predicate,
except that the limit is assumed to be 268435455.
8.3 Examples
The program shown in Figure 8.1 solves the Farmer’s problem by using the planner module.
Figure 8.2 gives a program for the 15-puzzle problem. A state is represented as a list of
sixteen locations, each of which takes the form (Ri,Ci), where Riis a row number and Ciis
a column number. The first element in the list gives the position of the empty square, and the
remaining elements in the list give the positions of the numbered tiles from 1 to 15. The function
heuristic(Tiles) returns the Manhattan distance between the current state and the final
state.
68
import planner.
go =>
S0=[s,s,s,s],
best_plan(S0,Plan),
writeln(Plan).
final([n,n,n,n]) => true.
action([F,F,G,C],S1,Action,ActionCost) ?=>
Action=farmer_wolf,
ActionCost = 1,
opposite(F,F1),
S1=[F1,F1,G,C],
not unsafe(S1).
action([F,W,F,C],S1,Action,ActionCost) ?=>
Action=farmer_goat,
ActionCost = 1,
opposite(F,F1),
S1=[F1,W,F1,C],
not unsafe(S1).
action([F,W,G,F],S1,Action,ActionCost) ?=>
Action=farmer_cabbage,
ActionCost = 1,
opposite(F,F1),
S1=[F1,W,G,F1],
not unsafe(S1).
action([F,W,G,C],S1,Action,ActionCost) =>
Action=farmer_alone,
ActionCost = 1,
opposite(F,F1),
S1=[F1,W,G,C],
not unsafe(S1).
index (+,-) (-,+)
opposite(n,s).
opposite(s,n).
unsafe([F,W,G,_C]),W==G,F!==W => true.
unsafe([F,_W,G,C]),G==C,F!==G => true.
Figure 8.1: A program for the Farmer’s problem using planner.
69
import planner.
main =>
InitS = [(1,2),(2,2),(4,4),(1,3),
(1,1),(3,2),(1,4),(2,4),
(4,2),(3,1),(3,3),(2,3),
(2,1),(4,1),(4,3),(3,4)],
best_plan(InitS,Plan),
foreach (Action in Plan)
println(Action)
end.
final(State) => State=[(1,1),(1,2),(1,3),(1,4),
(2,1),(2,2),(2,3),(2,4),
(3,1),(3,2),(3,3),(3,4),
(4,1),(4,2),(4,3),(4,4)].
action([P0@(R0,C0)|Tiles],NextS,Action,Cost) =>
Cost = 1,
(R1 = R0-1, R1 >= 1, C1 = C0, Action = up;
R1 = R0+1, R1 =< 4, C1 = C0, Action = down;
R1 = R0, C1 = C0-1, C1 >= 1, Action = left;
R1 = R0, C1 = C0+1, C1 =< 4, Action = right),
P1 = (R1,C1),
slide(P0,P1,Tiles,NTiles),
NextS = [P1|NTiles].
% slide the tile at P1 to the empty square at P0
slide(P0,P1,[P1|Tiles],NTiles) =>
NTiles = [P0|Tiles].
slide(P0,P1,[Tile|Tiles],NTiles) =>
NTiles=[Tile|NTilesR],
slide(P0,P1,Tiles,NTilesR).
heuristic([_|Tiles]) = Dist =>
final([_|FTiles]),
Dist = sum([abs(R-FR)+abs(C-FC) :
{(R,C),(FR,FC)} in zip(Tiles,FTiles)]).
Figure 8.2: A program for the 15-puzzle
70
Chapter 9
Modules
A module is a bundle of predicate and function definitions that are stored in one file. A module
forms a name space. Two definitions can have the same name if they reside in different modules.
Because modules avoid name clashes, they are very useful for managing source files of large
programs.
9.1 Module and Import Declarations
In Picat, source files must have the extension name ".pi". A module is a source file that begins
with a module name declaration in the form:
module Name.
where Name must be the same as the main file name. A file that does not begin with a module
declaration is assumed to belong to the default global module. The following names are reserved
for system modules and should not be used to name user’s modules: bp,basic,cp,glb,io,
math,mip,planner,ordset,sat,sys,os, and util.
In order to use symbols that are defined in another module, users must explicitly import them
with an import declaration in the form:
import Name1,. . .,Namen.
where each imported Nameiis a module name. For each imported module, the compiler first
searches for it in the search path that is specified by the environment variable PICATPATH. If no
module is found, the compiler gives an error message. Several modules are imported by default,
including basic,io,math, and sys.
The import relation is not transitive. Suppose that there are three modules: A,B, and C. If A
imports Band Bimports C, then Astill needs to import Cin order to reference C’s symbols.
The built-in command cl("xxx") compiles the file xxx.pi and loads the generated code
into the interpreter. The built-in command load("xxx") loads the bytecode file xxx.qi. It
compiles the source file xxx.pi only when necessary. The load command also imports the
public symbols defined in the module to the interpreter. This allows users to use these symbols on
the command line without explicitly importing the symbols. If the file xxx.pi imports modules,
those module files will be compiled and loaded when necessary.
9.2 Binding Calls to Definitions
The Picat system has a global symbol table for atoms, a global symbol table for structure names,
and a global symbol table for modules. For each module, Picat maintains a symbol table for the
71
public predicate and function symbols defined in the module. Private symbols that are defined in a
module are compiled away, and are never stored in the symbol table. While predicate and function
symbols can be local to a module, atoms and structures are always global.
The Picat module system is static, meaning that the binding of normal (or non-higher-order)
calls to their definitions takes place at compile time. For each call, the compiler first searches
the default modules for a definition that has the same name as the call. If no definition is found,
then the compiler searches for a definition in the enclosing module. If no definition is found, the
compiler searches the imported modules in the order that they were imported. If no definition is
found in any of these modules, then the compiler will issue an warning1, assuming the symbol is
defined in the global module.
It is possible for two imported modules to contain different definitions that have the same name.
When multiple names match a call, the order of the imported items determines which definition is
used. Picat allows users to use qualified names to explicitly select a definition. A module-qualified
call is a call preceded by a module name and ’.’ without intervening whitespace.
Example
% qsort.pi
module qsort.
sort([]) = [].
sort([H|T]) =
sort([E : E in T, E=<H])++[H]++sort([E : E in T, E>H]).
% isort.pi
module isort.
sort([]) = [].
sort([H|T]) = insert(H,sort(T)).
private
insert(X,[]) = [X].
insert(X,Ys@[Y|_]) = Zs, X=<Y => Zs=[X|Ys].
insert(X,[Y|Ys]) = [Y|insert(X,Ys)].
The module qsort.pi defines a function named sort using quick sort, and the module isort
defines a function of the same name using insertion sort. In the following session, both modules
are used.
picat> load("qsort")
picat> load("isort")
picat> L=sort([2,1,3])
L = [1,2,3]
picat> L=qsort.sort([2,1,3])
L = [1,2,3]
picat> L=isort.sort([2,1,3])
L = [1,2,3]
1A warning is issued instead of an error. This allows users to test incomplete programs with missing definitions.
72
As sort is also defined in the basic module, which is preloaded, that function is used for the
command L=sort([2,1,3]).
Module names are just atoms. Consequently, it is possible to bind a variable to a module name.
Nevertheless, in a module-qualified call M.C, the module name can never be a variable. Recall
that the dot notation is also used to access attributes and to call predicates and functions. The
notation M.C is treated as a call or an attribute if Mis not an atom.
Suppose that users want to define a function named generic sort(M,L) that sorts list L
using the sort function defined in module M. Users cannot just call M.sort(L), since Mis a
variable. Users can, however, select a function based on the value held in Mby using function facts
as follows:
generic_sort(qsort,L) = qsort.sort(L).
generic_sort(isort,L) = isort.sort(L).
9.3 Binding Higher-Order Calls
Because Picat forbids variable module qualifiers and terms in dot notations, it is impossible to
create module-qualified higher-order terms. For a higher-order call, if the compiler knows the
name of the higher-order term, as in findall(X,member(X, L)), then it searches for a def-
inition for the name, just like it does for a normal call. However, if the name is unknown, as in
apply(F,X,Y), then the compiler generates code to search for a definition. For a higher-order
call to call/N or apply/N, the runtime system searches modules in the following order:
1. The implicitly imported built-in modules basic,io,math, and sys.
2. The enclosing module of the higher-order call.
3. The explicitly imported modules in the enclosing module in the order that they were im-
ported.
4. The global module.
As private symbols are compiled away at compile time, higher-order terms can never reference
private symbols. Due to the overhead of runtime search, the use of higher-order calls is discour-
aged.
9.4 Library Modules
Picat comes with a library of standard modules, described in separate chapters. The function
sys.loaded modules() returns a list of modules that are currently in the system.
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Chapter 10
I/O
Picat has an io module for reading input from files and writing output to files. The io module is
imported by default.
The io module contains functions and predicates that read from a file, write to a file, reposition
the read/write pointer within a file, redirect input and output, and create temporary files and pipes.
The io module uses file descriptors to read input from files, and to write output to files. A
file descriptor is a structure that encodes file descriptor data, including an index in a file descriptor
table that stores information about opened files. The following example reads data from one file,
and writes the data into another file.
Example
rw =>
Reader = open("input_file.txt"),
Writer = open("output_file.txt", write),
L = read_line(Reader),
while (L != end_of_file)
println(Writer, L),
flush(Writer),
L := read_line(Reader)
end,
close(Reader),
close(Writer).
10.1 Opening a File
There are two functions for opening a file. Both of them are used in the previous example.
•open(Name) = F D: The Name parameter is a filename that is represented as a string.
This function opens the file with a default read mode.
•open(Name,Mode) = F D: The Mode parameter is one of the four atoms: read,
write, or append. The read atom is used for reading from a file; if the file does not
exist, or the program tries to write to the file, then the program will throw an error. The
write atom is used for reading from a file and writing to a file; if the file already exists,
then the file will be overwritten. The append atom is similar to the write atom; however, if
the file already exists, then data will be appended at the end of the pre-existing file.
74
10.2 Reading from a File
The io module has at least one function for reading data into each primitive data type. It also has
functions for reading tokens, strings, and bytes. Recall that strings are stored as lists of single-
character atoms.
The read functions in the io module take a file descriptor as the first parameter. This file
descriptor is the same descriptor that the open function returns. The parameter can be omitted if
it is the standard input file stdin.
•read int(F D) = Int: This function reads a single integer from the file that is repre-
sented by F D. It throws an input mismatch exception if F D is at the end of the file or
the next token at F D is not an integer.
•read int() = Int: This function is the same as read int(stdin).
•read real(F D) = Real: This function reads a single real number from the file that is
represented by F D. It throws an input mismatch exception if F D is at the end of the
file or the next token at F D is not a number.
•read real() = Real: This function is the same as read real(stdin).
•read char(F D) = V al: This function reads a single UTF-8 character from the file that
is represented by F D. It returns end of file if F D is at the end of the file.
•read char() = V al: This function is the same as read char(stdin).
•read char(F D,N) = String: This function reads up to NUTF-8 characters from
the file that is represented by F D. It returns a string that contains the characters that were
read.
•read char code(F D) = V al: This function reads a single UTF-8 character from the
file that is represented by F D and returns its code point. It returns -1 if F D is at the end of
the file.
•read char code() = V al: This function is the same as read char code(stdin).
•read char code(F D,N) = List: This function reads up to NUTF-8 characters
from the file that is represented by F D. It returns a list of code points of the characters that
were read.
•read picat token(F D,T okenT ype,T okenV alue): This predicate reads a single
Picat token from the file that is represented by F D.T okenT ype is the type and T okenV alue
is the value of the token. T okenT ype is one of the following: atom,end of file,
end of rule,integer,punctuation,real,string,underscore, and var.
•read picat token(T okenT ype,T okenV alue): This predicate reads a token from
stdin.
•read picat token(F D) = T okenV alue: This function reads a single Picat token
from the file that is represented by F D and returns the token value.
•read picat token() = T okenV alue: This function is the same as the above, except
that it reads from stdin.
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•read term(F D) = T erm: This function reads a single Picat term from the file that is
represented by F D. The term must be followed by a dot ‘.’ and at least one whitespace
character. This function consumes the dot symbol. The whitespace character is not stored
in the returned string.
•read term() = T erm: This function is the same as read term(stdin).
•read line(F D) = String: This function reads a string from the file that is represented
by F D, stopping when either a newline (‘\r\n’ on Windows, and ‘\n’ on Unix) is read, or
the end of file atom is returned. The newline is not stored in the returned string.
•read line() = String: This function is the same as read line(stdin).
•readln(F D) = String: This function does the same thing as read line.
•readln() = String: This function is the same as readln(stdin).
•read byte(F D) = V al: This function reads a single byte from the file that is repre-
sented by F D.
•read byte() = V al: This function is the same as read byte(stdin).
•read byte(F D,N) = List: This function reads up to Nbytes from the file that is
represented by F D. It returns the list of bytes that were read.
•read file bytes(F ile) = List: This function reads an entire byte file into a list.
•read file bytes() = List: This function reads an entire byte file from the console
into a list.
•read file chars(F ile) = String: This function reads an entire character file into a
string.
•read file chars() = String: This function reads an entire character file from the
console into a string.
•read file codes(F ile) = List: This function reads UTF-8 codes of an entire char-
acter file into a list.
•read file codes() = List: This function reads UTF-8 codes of an entire character
file from the console into a list.
•read file lines(F ile) = Lines: This function reads an entire character file into a
list of line strings.
•read file lines() = Lines: This function reads an entire character file from the
console into a list of line strings.
•read file terms(F ile) = Lines: This function reads an entire text file into a list of
terms. In the file, each term must be terminated by ’.’ followed by at least one white space.
•read file terms() = Lines: This function reads an entire text file from the console
into a list of terms.
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There are cases when the read char(F D,N), and read byte(F D,N)functions
will read fewer than Nvalues. One case occurs when the end of the file is encountered. Another
case occurs when reading from a pipe. If a pipe is empty, then the read functions wait until
data is written to the pipe. As soon as the pipe has data, the read functions read the data. If a
pipe has fewer than Nvalues when a read occurs, then these three functions will return a string
that contains all of the values that are currently in the pipe, without waiting for more values. In
order to determine the actual number of elements that were read, after the functions return, use
length(List)to check the length of the list that was returned.
The io module also has functions that peek at the next value in the file without changing the
current file location. This means that the next read or peek function will return the same value,
unless the read/write pointer is repositioned or the file is modified.
•peek char(F D) = V al
•peek byte(F D) = V al
10.2.1 End of File
The end of a file is detected through the end of file atom. If the input function returns a single
value, and the read/write pointer is at the end of the file, then the end of file atom is returned.
If the input function returns a list, then the end-of-file behavior is more complex. If no other values
have been read into the list, then the end of file atom is returned. However, if other values
have already been read into the list, then reaching the end of the file causes the function to return
the list, and the end of file atom will not be returned until the next input function is called.
Instead of checking for end of file, the at end of stream predicate can be used to
monitor a file descriptor for the end of a file.
•at end of stream(F D): The at end of stream predicate is demonstrated in the
following example.
Example
rw =>
Reader = open("file1.txt"),
Writer = open("file2.txt", write),
while (not at_end_of_stream(Reader))
L := read_line(Reader),
println(Writer, L),
flush(Writer)
end,
close(Reader),
close(Writer).
The advantage of using the at end of stream predicate instead of using the end of file
atom is that at end of stream immediately indicates that the end of the file was reached, even
if the last read function read values into a list. In the first example in this chapter, which used
the end of file atom, an extra read line function was needed before the end of the file was
detected. In the above example, which used at end of stream,read line was only called
if there was data remaining to be read.
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10.3 Writing to a File
The write and print predicates take a file descriptor as the first parameter. The file descriptor
is the same descriptor that the open function returns. If the file descriptor is stdout, then the
parameter can be omitted.
•write(F D,T erm): This predicate writes T erm to a file. Single-character lists are
treated as strings. Strings are double-quoted, and atoms are single-quoted when necessary.
This predicate does not print a newline, meaning that the next write will begin on the same
line.
•write(T erm): This predicate is the same as write(stdout,T erm).
•write byte(F D,Bytes): This predicate writes a single byte or a list of bytes to a file.
•write byte(Bytes): This predicate is the same as write byte(stdout,Bytes).
•write char(F D,Chars): This predicate writes a single character or a list of charac-
ters to a file. The characters are not quoted. When writing a single-character atom Char,
write char(F D,Char)is the same as print(F D,Char), but write char is
faster than print.
•write char(Chars): This predicate is the same as write char(stdout,Chars).
•write char code(F D,Codes): This predicate writes a single character or a list of
characters of the given code or list of codes to a file.
•write char code(Codes): This predicate is the same as the above, except that it writes
to stdout.
•writeln(F D,T erm): This predicate writes T erm and a newline, meaning that the
next write will begin on the next line.
•writeln(T erm): This predicate is the same as writeln(stdout,T erm).
•writef(F D,F ormat,Args . . .): This predicate is used for formatted writing, where
the F ormat parameter contains format characters that indicate how to print each of the
arguments in the Args parameter. The number of arguments in Args . . . cannot exceed 10.
Note that these predicates write both primitive values and compound values.
The writef predicate includes a parameter that specifies the string that is to be formatted.
The F ormat parameter is a string that contains format characters. Format characters take the form
%[flags][width][.precision]specifier. Only the percent sign and the specifier are
mandatory. Flags can be used for justification and padding. The width is the minimum number of
characters that are to be printed. The precision is the number of characters that are to be printed
after the number’s radix point. Note that the width includes all characters, including the radix
point and the characters that follow it. The specifier indicates the type of data that is to be written.
A specifier can be one of the C format specifiers %%,%c,1%d,%e,%E,%f,%g,%G,%i,%o,%s,
%u,%x, and %X. In addition, Picat uses the specifier %n for newlines, and uses %w for terms. For
details, see Appendix F.
1The specifier %c can only be used to print ASCII characters. Use the specifier %w to print UTF-8 characters.
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Example
formatted_print =>
FD = open("birthday.txt",write),
Format1 = "Hello, %s. Happy birthday! ",
Format2 = "You are %d years old today. ",
Format3 = "That is %.2f%% older than you were last year",
writef(FD, Format1, "Bob"),
writef(FD, Format2, 7),
writef(FD, Format3, 7.0 / 6.0),
close(FD).
This writes “Hello, Bob. Happy birthday! You are 7 years old today.
That is 1.17% older than you were last year”.
The io module also has the three print predicates.
•print(F D,T erm): This predicate prints T erm to a file. Unlike the write predicates,
the print predicates do not place quotes around strings and atoms.
•print(T erm): This predicate is the same as print(stdout,T erm).
•println(F D,T erm)This predicate prints T erm and a newline.
•println(T erm)This predicate is the same as println(stdout,T erm).
•printf(F D,F ormat,Args . . .): This predicate is the same as writef, except that
printf uses print to display the arguments in the Args parameter, while writef uses
write to display the arguments in the Args parameter.
The following example demonstrates the differences between the write and print predi-
cates.
Example
picat> write("abc")
[a,b,c]
picat> write([a,b,c])
[a,b,c]
picat> write(’a@b’)
’a@b’
picat> writef("%w %s%n",[a,b,c],"abc")
[a,b,c] abc
picat> print("abc")
abc
picat> print([a,b,c])
abc
picat> print(’a@b’)
a@b
picat> printf("%w %s%n",[a,b,c],"abc")
abc abc
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10.4 Flushing and Closing a File
The io module has one predicate to flush a file stream, and one predicate to close a file stream.
•flush(F D): This predicate causes all buffered data to be written without delay.
•close(F D): This predicate causes the file to be closed, releasing the file’s resources, and
removing the file from the file descriptor table. Any further attempts to write to the file
descriptor without calling open will cause an error to be thrown.
10.5 Standard File Descriptors
The atoms stdin,stdout, and stderr represent the file descriptors for standard input, stan-
dard output, and standard error. These atoms allow the program to use the input and output func-
tions of the io module to read from and to write to the three standard streams.
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Chapter 11
Event-Driven Actors and Action Rules
Many applications require event-driven computing. For example, an interactive GUI system needs
to react to UI events such as mouse clicks on UI components; a Web service provider needs to
respond to service requests; a constraint propagator for a constraint needs to react to updates to the
domains of the variables in the constraint. Picat provides action rules for describing event-driven
actors. An actor is a predicate call that can be delayed and can be activated later by events. Actors
communicate with each other through event channels.
11.1 Channels, Ports, and Events
An event channel is an attributed variable to which actors can be attached, and through which
events can be posted to actors. A channel has four ports, named ins,bound,dom, and any,
respectively. Many built-ins in Picat post events. When an attributed variable is instantiated,
an event is posted to the ins-port of the variable. When the lower bound or upper bound of a
variable’s domain changes, an event is posted to the bound-port of the variable. When an inner
element E, which is neither the lower or upper bound, is excluded from the domain of a variable,
Eis posted to the dom-port of the variable. When an arbitrary element E, which can be the lower
or upper bound or an inner element, is excluded from the domain of a variable, Eis posted to the
any-port of the variable. The division of a channel into ports facilitates speedy handling of events.
For better performance, the system posts an event to a port only when there are actors attached to
the port. For example, if no actor is attached to a domain variable to handle exclusions of domain
elements, then these events will never be posted.
The built-in post event(X,T)posts the event term Tto the dom-port of the channel
variable X.
The following built-ins are used to post events to one of a channel’s four ports:
•post event ins(X): posts an event to the ins-port of the channel X.
•post event bound(X): posts an event to the bound-port of the channel X.
•post event dom(X,T): posts the term Tto the dom-port of the channel X.
•post event any(X,T): posts the event Tto the any-port of the channel of X.
The call post event(X,T)is the same as post event dom(X,T). This means that the
dom-port of a finite domain variable has two uses: posting exclusions of inner elements from the
domain, and posting general term events.
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11.2 Action Rules
Picat provides action rules for describing the behaviors of actors. An action rule takes the follow-
ing form:
Head, Cond, {Event}=> Body
where Head is an actor pattern, Cond is an optional condition, Event is a non-empty set of event
patterns separated by ’,’, and Body is an action. For an actor that is activated by an event, an
action rule is said to be applicable if the actor matches Head and Cond is true. A predicate for
actors is defined with action rules and non-backtrackable rules. It cannot contain backtrackable
rules.
Unlike rules for a normal predicate or function, in which the conditions can contain any pred-
icates, the conditions of the rules in a predicate for actors must be conjunctions of inline test
predicates, such as type-checking built-ins (e.g., integer(X)and var(X)) and comparison
built-ins (e.g., equality test X== Y, disequality test X!== Y, and arithmetic comparison X
> Y ). This restriction ensures that no variables in an actor can be changed while the condition is
executed.
For an actor that is activated by an event, the system searches the definition sequentially from
the top for an applicable rule. If no applicable rule is found, then the actor fails. If an applicable
rule is found, the system executes the body of the rule. If the body fails, then the actor also
fails. The body cannot succeed more than once. The system enforces this by converting Body
into ‘once Body’if Body contains calls to nondeterministic predicates. If the applied rule is
an action rule, then the actor is suspended after the body is executed, meaning that the actor is
waiting to be activated again. If the applied rule is a normal non-backtrackable rule, then the actor
vanishes after the body is executed. For each activation, only the first applicable rule is applied.
For a call and an action rule ‘Head, Cond, {Event}=> Body’, the call is registered as an
actor if the call matches Head and Cond evaluates to true. The event pattern Event implicitly
specifies the ports to which the actor is attached, and the events that the actor watches. The
following event patterns are allowed in Event:
•event(X,T): This is the general event pattern. The actor is attached to the dom-ports of
the variables in X. The actor will be activated by events posted to the dom-ports. Tmust
be a variable that does not occur before event(X,T)in the rule.
•ins(X): The actor is attached to the ins-ports of the variables in X. The actor will be
activated when a variable in Xis instantiated.
•bound(X): The actor is attached to the bound-ports of the variables in X. The actor will
be activated when the lower bound or upper bound of the domain of a variable in Xchanges.
•dom(X): The actor is attached to the dom-ports of the variables in X. The actor will be
activated when an inner value is excluded from the domain of a variable in X. The actor is
not interested in what value is actually excluded.
•dom(X,E): This is the same as dom(X), except the actor is interested in the value E
that is excluded. Emust be a variable that does not occur before dom(X,E)in the rule.
•dom any(X): The actor is attached to the any-ports of the variables in X. The actor will
be activated when an arbitrary value, including the lower bound value and the upper bound
value, is excluded from the domain of a variable in X. The actor is not interested in what
value is actually excluded.
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•dom any(X,E): This is the same as dom any(X), except the actor is interested in
the value Ethat is actually excluded. Emust be a variable that does not occur before
dom any(X,E)in the rule.
In an action rule, multiple event patterns can be specified. After a call is registered as an actor
on the channels, it will be suspended, waiting for events, unless the atom generated occurs in
Event, in which case the actor will be suspended after Body is executed.
Each thread has an event queue. After events are posted, they are added into the queue. Events
are not handled until execution enters or exits a non-inline predicate or function. In other words,
only non-inline predicates and functions can be interrupted, and inline predicates, such as X=
Y, and inline functions, such as X+Y, are never interrupted by events.
Example
Consider the following action rule:
p(X), {event(X,T)} => writeln(T).
The following gives a query and its output:
Picat> p(X), X.post_event(ping), X.post_event(pong)
ping
pong
The call p(X) is an actor. After X.post event(ping), the actor is activated and the body of
the action rule is executed, giving the output ping. After X.post event(pong), the actor is
activated again, outputting pong.
There is no primitive for killing actors or explicitly detaching actors from channels. As de-
scribed above, an actor never disappears as long as action rules are applied to it. An actor vanishes
only when a normal rule is applied to it. Consider the following example.
p(X,Flag),
var(Flag),
{event(X,T)}
=>
writeln(T),
Flag=1.
p(_,_) => true.
An actor defined here can only handle one event posting. After it handles an event, it binds the
variable Flag. When a second event is posted, the action rule is no longer applicable, causing the
second rule to be selected.
One question arises here: what happens if a second event is never posted to X? In this case, the
actor will stay forever. If users want to immediately kill the actor after it is activated once, then
users have to define it as follows:
p(X,Flag),
var(Flag),
{event(X,O),ins(Flag)},
=>
write(O),
Flag=1.
p(_,_) => true.
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In this way, the actor will be activated again after Flag is bound to 1, and will be killed after the
second rule is applied to it.
11.3 Lazy Evaluation
The built-in predicate freeze(X,Goal) is equivalent to ‘once Goal’, but its evaluation is
delayed until Xis bound to a non-variable term. The predicate is defined as follows:
freeze(X,Goal),var(X),{ins(X)} => true.
freeze(X,Goal) => call(Goal).
For the call freeze(X,Goal), if Xis a variable, then Xis registered as an actor on the ins-port
of X, and Xis then suspended. Whenever Xis bound, the event ins is posted to the ins-port of
X, which activates the actor freeze(X,Goal). The condition var(X) is checked. If true, the
actor is suspended again; otherwise, the second rule is executed, causing the actor to vanish after
it is rewritten into once Goal.
The built-in predicate different terms(T1,T2)is a disequality constraint on terms T1
and T2. The constraint fails if the two terms are identical; it succeeds whenever the two terms are
found to be different; it is delayed if no decision can be made because the terms are not sufficiently
instantiated. The predicate is defined as follows:
import cp.
different_terms(X,Y) =>
different_terms(X,Y,1).
different_terms(X,Y,B),var(X),{ins(X),ins(Y)} => true.
different_terms(X,Y,B),var(Y),{ins(X)} => true.
different_terms([X|Xs],[Y|Ys],B) =>
different_terms(X,Y,B1),
different_terms(Xs,Ys,B2),
B #= (B1 #\/ B2).
different_terms(X,Y,B),struct(X),struct(Y) =>
writeln(X), writeln(Y),
if (X.name !== Y.name; X.length !== Y.length) then
B=1
else
Bs = new_list(X.length),
foreach(I in 1 .. X.length)
different_terms(X[I],Y[I],Bs[I])
end,
max(Bs) #= B
end.
different_terms(X,Y,B),X==Y => B=0.
different_terms(X,Y,B) => B=1.
The call different terms(X,Y,B) is delayed if either Xor Yis a variable. The delayed call
watches ins(X) and ins(Y) events. Once both Xand Ybecome non-variable, the action rule
becomes inapplicable, and one of the subsequent rules will be applied. If Xand Yare lists, then
they are different if the heads are different (B1), or if the tails are different (B2). This relationship
is represented as the Boolean constraint B #= (B1 #\/ B2). If Xand Yare both structures,
84
then they are different if the functor is different, or if any pair of arguments of the structures is
different.
11.4 Constraint Propagators
A constraint propagator is an actor that reacts to updates of the domains of the variables in a
constraint.The following predicate defines a propagator for maintaining arc consistency on Xfor
the constraint X+Y #= C:
import cp.
x_in_c_y_ac(X,Y,C),var(X),var(Y),
{dom(Y,Ey)}
=>
fd_set_false(X,C-Ey).
x_in_c_y_ac(X,Y,C) => true.
Whenever an inner element Ey is excluded from the domain of Y, this propagator is triggered to
exclude C-Ey, which is the support of Ey, from the domain of X. For the constraint X+Y #= C,
users need to generate two propagators, namely, x in c y ac(X,Y,C) and x in c y ac(Y,X,C),
to maintain the arc consistency. Note that in addition to these two propagators, users also need to
generate propagators for maintaining interval consistency, because dom(Y,Ey) only captures
exclusions of inner elements, and does not capture bounds. The following propagator maintains
interval consistency for the constraint:
import cp.
x_add_y_eq_c_ic(X,Y,C),var(X),var(Y),
{generated,ins(X),ins(Y),bound(X),bound(Y)}
=>
X :: C-Y.max .. C-Y.min,
Y :: C-X.max .. C-X.min.
x_add_y_eq_c_ic(X,Y,C),var(X) =>
X = C-Y.
x_add_y_eq_c_ic(X,Y,C) =>
Y = C-X.
When both Xand Yare variables, the propagator x add y eq c ic(X,Y,C) is activated when-
ever Xand Yare instantiated, or whenever the bounds of their domains are updated. The body
maintains the interval consistency of the constraint X+Y #= C. The body is also executed when
the propagator is generated. When either Xor Ybecomes non-variable, the propagator becomes a
normal call, and vanishes after the variable Xor Yis solved.
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Chapter 12
Constraints
Picat provides three solver modules, including cp,sat, and mip, for modeling and solving con-
straint satisfaction and optimization problems (CSPs). All three of these modules implement the
same set of constraints on integer-domain variables. The mip module also supports real-domain
variables. In order to use a solver, users must first import the module. In order to make the sym-
bols defined in a module available to the top level of the interpreter, users can use the built-in
load to load the module or any module that imports the module. As the three modules have the
same interface, this chapter describes the three modules together. Figure 12.1 shows the constraint
operators that are provided by Picat. Unless it is explicitly specified otherwise, the built-ins that
are described in this chapter appear in all three modules. In the built-ins that are presented in this
chapter, an integer-domain variable can also be an integer, unless it is explicitly specified to only
be a variable.
Table 12.1: Constraint operators in Picat
Precedence Operators
Highest ::,notin,#=,#!=,#<,#=<,#<=,#>,#>=
#˜
#/\
#ˆ
#\/
#=>
Lowest #<=>
A constraint program normally poses a problem in three steps: (1) generate variables; (2)
generate constraints over the variables; and (3) call solve to find a valuation for the variables
that satisfies the constraints and possibly optimizes an objective function.
Example
This program in Figure 12.1 imports the cp module in order to solve the N-queens problem. The
same program runs with the SAT solver if sat is imported, or runs with the LP/MIP solver if mip
is imported. The predicate Qs :: 1..N declares the domains of the variables. The operator
#!= is used for inequality constraints. In arithmetic constraints, expressions are treated as terms,
and it is unnecessary to enclose them with dollar-signs. The predicate solve(Qs) calls the solver
in order to solve the array of variables Qs. For cp,solve([ff],Qs), which always selects a
86
import cp.
queens(N) =>
Qs=new_array(N),
Qs :: 1..N,
foreach (I in 1..N-1, J in I+1..N)
Qs[I] #!= Qs[J],
abs(Qs[I]-Qs[J]) #!= J-I
end,
solve(Qs),
writeln(Qs).
Figure 12.1: A program for N-queens.
variable that has the smallest domain (the so-called first-fail principle), can be more efficient than
solve(Qs).
12.1 Domain Variables
A domain variable is an attributed variable that has a domain attribute. The Boolean domain
is treated as a special integer domain where 1 denotes true and 0 denotes false. Domain
variables are declared with the built-in predicate V ars :: Exp.
•V ars :: Exp: This predicate restricts the domain or domains of V ars to Exp.V ars
can be either a single variable, a list of variables, or an array of variables. For integer-domain
variables, Exp must result in a list of integer values. For real-domain variables for the mip
module, Exp must be an interval in the form L..U, where Land Uare real values.
Domain variables, when being created, are usually represented internally by using intervals.
An interval turns to a bit vector when a hole occurs in the interval. The following built-in predicate
can be used to reset the range or access the current range.
•fd vector min max(Min,Max): When the arguments are integers, this predicate
specifies the range of bit vectors; when the arguments are variables, this predicate binds
them to the current bounds of the range. The default range is -3200..3200.
The following built-ins are provided for domain variables.
•V ars notin Exp: This predicate excludes values Exp from the domain or domains of
V ars, where V ars and Exp are the same as in V ars :: Exp. This constraint cannot
be applied to real-domain variables.
•fd degree(F DV ar) = Degree: This function returns the number of propagators that
are attached to F DV ar. This built-in is only provided by cp.
•fd disjoint(F DV ar1,F DV ar2): This predicate is true if F DV ar1’s domain and
F DV ar2’s domain are disjoint.
•fd dom(F DV ar) = List: This function returns the domain of F DV ar as a list, where
F DV ar is an integer-domain variable. If F DV ar is an integer, then the returned list con-
tains the integer itself.
87
•fd false(F DV ar,Elm): This predicate is true if the integer Elm is not an element
in the domain of F DV ar.
•fd max(F DV ar) = Max: This function returns the upper bound of the domain of
F DV ar, where F DV ar is an integer-domain variable.
•fd min(F DV ar) = Min: This function returns the lower bound of the domain of
F DV ar, where F DV ar is an integer-domain variable.
•fd min max(F DV ar,Min,Max): This predicate binds Min to the lower bound
of the domain of F DV ar, and binds Max to the upper bound of the domain of F DV ar,
where F DV ar is an integer-domain variable.
•fd next(F DV ar,Elm) = NextElm: This function returns the next element of Elm
in F DV ar’s domain. It throws an exception if Elm has no next element in F DV ar’s do-
main.
•fd prev(F DV ar,Elm) = P revElm: This function returns the previous element
of Elm in F DV ar’s domain. It throws an exception if Elm has no previous element in
F DV ar’s domain.
•fd set false(F DV ar,Elm): This predicate excludes the element Elm from the
domain of F DV ar. If this operation results in a hole in the domain, then the domain
changes from an interval representation into a bit-vector representation, no matter how big
the domain is. This built-in is only provided by cp.
•fd size(F DV ar) = Size: This function returns the size of the domain of F DV ar,
where F DV ar is an integer-domain variable.
•fd true(F DV ar,Elm): This predicate is true if the integer Elm is an element in the
domain of F DV ar.
•new dvar() = F DV ar: This function creates a new domain variable with the default
domain, which has the bounds -72057594037927935..72057594037927935 on
64-bit computers and -268435455..268435455 on 32-bit computers.
12.2 Table constraints
Atable constraint, or an extensional constraint, over a tuple of variables specifies a set of tuples
that are allowed (called positive) or disallowed (called negative) for the variables. A positive
constraint takes the form table in(DV ars,R), where DV ars is either a tuple of variables
{X1, . . . , Xn}or a list of tuples of variables, and Ris a list of tuples of integers in which each tuple
takes the form {a1, . . . , an}. A negative constraint takes the form table notin(DV ars,R).
Example
The following example solves a toy crossword puzzle. One variable is used for each cell in the
grid, so each slot corresponds to a tuple of variables. Each word is represented as a tuple of
integers, and each slot takes on a set of words of the same length as the slot. Recall that the
function ord(Char)returns the code of Char, and that the function chr(Code)returns the
character of Code.
88
import cp.
crossword(Vars) =>
Vars=[X1,X2,X3,X4,X5,X6,X7],
Words2=[{ord(’I’),ord(’N’)},
{ord(’I’),ord(’F’)},
{ord(’A’),ord(’S’)},
{ord(’G’),ord(’O’)},
{ord(’T’),ord(’O’)}],
Words3=[{ord(’F’),ord(’U’),ord(’N’)},
{ord(’T’),ord(’A’),ord(’D’)},
{ord(’N’),ord(’A’),ord(’G’)},
{ord(’S’),ord(’A’),ord(’G’)}],
table_in([{X1,X2},{X1,X3},{X5,X7},{X6,X7}], Words2),
table_in([{X3,X4,X5},{X2,X4,X6}],Words3),
solve(Vars),
writeln([chr(Code) : Code in Vars]).
12.3 Arithmetic Constraints
An arithmetic constraint takes the form
Exp1Rel Exp2
where Exp1and Exp2are arithmetic expressions, and Rel is one of the constraint operators:
#=,#!=,#<,#=<,#<=,#>, or #>=. The operators #=< and #<= are the same, meaning less
than or equal to. An arithmetic expression is made from integers, variables, arithmetic functions,
and constraints. The following arithmetic functions are allowed: +(addition), -(subtraction),
*(multiplication), /(truncated integer division), // (truncated integer division), count,div
(floored integer division), mod,** (power), abs,min,max, and sum. Except for index notations,
array comprehensions and list comprehensions, which are interpreted as function calls as in normal
expressions, expressions in arithmetic constraints are treated as terms, and it is unnecessary to
enclose them with dollar-signs. In addition to the numeric operators, the following functions are
allowed in constraints:
•cond(BoolConstr,T henExp,ElseExp): This expression is the same as
BoolConstr*T henExp+(1-BoolConstr)*ElseExp.
•count(V,DV ars): The number of times Voccurs in DV ars, where DV ars is a list of
domain variables.
•min(DV ars): The minimum of DV ars, where DV ars is a list of domain variables.
•max(DV ars): The maximum of DV ars, where DV ars is a list of domain variables.
•min(Exp1,Exp2): The minimum of Exp1and Exp2.
•max(Exp1,Exp2): The maximum of Exp1and Exp2.
•sum(DV ars): The sum of DV ars, where DV ars is a list of domain variables.
•prod(DV ars): The product of DV ars, where DV ars is a list of domain variables.
When a constraint occurs in an arithmetic expression, it is evaluated to 1 if it is satisfied and 0 if it
is not satisfied.
89
Example
import mip.
go =>
M={{0,3,2,3,0,0,0,0},
{0,0,0,0,0,0,5,0},
{0,1,0,0,0,1,0,0},
{0,0,2,0,2,0,0,0},
{0,0,0,0,0,0,0,5},
{0,4,0,0,2,0,0,1},
{0,0,0,0,0,2,0,3},
{0,0,0,0,0,0,0,0}},
maxflow(M,1,8).
maxflow(M,Source,Sink) =>
N=M.length,
X=new_array(N,N),
foreach(I in 1..N, J in 1..N)
X[I,J] :: 0..M[I,J]
end,
foreach(I in 1..N, I!=Source, I!=Sink)
sum([X[J,I] : J in 1..N]) #= sum([X[I,J] : J in 1..N])
end,
Total #= sum([X[Source,I] : I in 1..N]),
Total #= sum([X[I,Sink] : I in 1..N]),
solve([$max(Total)],X),
writeln(Total),
writeln(X).
This program uses MIP to solve the maximum integer flow problem. Given the capacity matrix
Mof a directed graph, the start vertex Source, and the destination vertex Sink, the predicate
maxflow(M,Source,Sink) finds a maximum flow from Source to Sink over the graph.
When two vertices are not connected by an arc, the capacity is given as 0. The first foreach loop
specifies the domains of the variables. For each variable X[I,J], the domain is restricted to
integers between 0 and the capacity, M[I,J]. If the capacity is 0, then the variable is immediately
instantiated to 0. The next foreach loop posts the conservation constraints. For each vertex I, if it
is neither the source nor the sink, then its total incoming flow amount
sum([X[J,I] : J in 1..N])
is equal to the total outgoing flow amount
sum([X[I,J] : J in 1..N]).
The total flow amount is the total outgoing amount from the source, which is the same as the total
incoming amount to the sink.
12.4 Boolean Constraints
A Boolean constraint takes one of the following forms:
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#˜ BoolExp
BoolExp #/\ BoolExp
BoolExp #ˆ BoolExp
BoolExp #\/ BoolExp
BoolExp #=> BoolExp
BoolExp #<=> BoolExp
BoolExp is either a Boolean constant (0 or 1), a Boolean variable (an integer-domain variable
with the domain [0,1]), an arithmetic constraint, a domain constraint (in the form of V ar ::
Domain or V ar notin Domain), or a Boolean constraint. As shown in Table 12.1, the oper-
ator #˜ has the highest precedence, and the operator #<=> has the lowest precedence. Note that
the Boolean constraint operators have lower precedence than the arithmetic constraint operators.
So the constraint
X #!= 3 #/\ X#!= 5 #<=> B
is interpreted as
((X #!= 3) #/\ (X#!= 5)) #<=> B.
The Boolean constraint operators are defined as follows.
•#˜ BoolExp: This constraint is 1 iff BoolExp is equal to 0.
•BoolExp1#/\ BoolExp2: This constraint is 1 iff both BoolExp1and BoolExp2are
1.
•BoolExp1#ˆ BoolExp2: This constraint is 1 iff exactly one of BoolExp1and BoolExp2
is 1.
•BoolExp1#\/ BoolExp2: This constraint is 1 iff BoolExp1or BoolExp2is 1.
•BoolExp1#=> BoolExp2: This constraint is 1 iff BoolExp1implies BoolExp2.
•BoolExp1#<=> BoolExp2: This constraint is 1 iff BoolExp1and BoolExp2are equiv-
alent.
12.5 Global Constraints
A global constraint is a constraint over multiple variables. A global constraint can normally be
translated into a set of smaller constraints, such as arithmetic and Boolean constraints. If the cp
module is used, then global constraints are not translated into smaller constraints; rather, they are
compiled into special propagators that maintain a certain level of consistency for the constraints.
In Picat, constraint propagators are encoded as action rules. If the sat module is used, then global
constraints are translated into smaller constraints before being translated further into conjunctive
normal form. If the mip module is used, then global constraints are decomposed into equality and
disequality constraints.
Picat provides the following global constraints.
•all different(F DV ars): This constraint ensures that each pair of variables in the
list or array F DV ars is different. This constraint is compiled into a set of inequality con-
straints. For each pair of variables V1and V2in F DV ars,all different(F DV ars)
generates the constraint V1#!= V2.
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•all distinct(F DV ars): This constraint is the same as all different, but for the
cp module it maintains a higher level of consistency. For some problems, this constraint is
faster and requires fewer backtracks than all different, and, for some other problems,
this constraint is slower due to the overhead of consistency checking.
•all different except 0(F DV ars): This constraint is true if all non-zero values in
F DV ars are different.
•assignment(F DV ars1,F DV ars2): This constraint ensures that F DV ars2is a
dual assignment of F DV ars1, i.e., if the ith element of F DV ars1is j, then the jth ele-
ment of F DV ars2is i. The constraint can be defined as:
assignment(Xs,Ys) =>
N = Xs.length,
(var(Ys) -> Ys = new_list(N); true),
Xs :: 1..N,
Ys :: 1..N,
foreach(I in 1..N, J in 1..N)
X[I] #= J #<=> Y[J] #= I
end.
•at least(N,L,V): This constraint succeeds if there are at least Nelements in Lthat
are equal to V, where Nand Vmust be integer-domain variables, and Lmust be a list of
integer-domain variables.
•at most(N,L,V): This constraint succeeds if there are at most Nelements in Lthat
are equal to V, where Nand Vmust be integer-domain variables, and Lmust be a list of
integer-domain variables.
•circuit(F DV ars): Let F DV ars be a list of variables [X1, X2, . . . , XN], where each
Xihas the domain 1..N. A valuation X1=v1,X2=v2,. . .,Xn=vnsatisfies the
constraint if 1->v1, 2->v2, ..., n->vnforms a Hamiltonian cycle. This constraint
ensures that each variable has a different value, and that the graph that is formed by the
assignment does not contain any sub-cycles. For example, for the constraint
circuit([X1,X2,X3,X4])
[3,4,2,1] is a solution, but [2,1,4,3] is not, because the graph 1->2, 2->1,
3->4, 4->3 contains two sub-cycles.
•count(V,F DV ars,Rel,N): In this constraint, Vand Nare integer-domain vari-
ables, F DV ars is a list of integer-domain variables, and Rel is an arithmetic constraint
operator (#=,#!=,#>,#>=,#<,#=<, or #<=). Let Count be the number of elements in
F DV ars that are equal to V. The constraint is true iff Count Rel N is true. This constraint
can be defined as follows:
count(V,L,Rel,N) =>
sum([V #= E : E in L]) #= Count,
call(Rel,Count,N).
•count(V,F DV ars,N): This constraint is the same as count(V,F DV ars,#=,
N).
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•cumulative(Starts,Durations,Resources,Limit): This constraint is useful
for describing and solving scheduling problems. The arguments Starts,Durations, and
Resources are lists of integer-domain variables of the same length, and Limit is an integer-
domain variable. Let Starts be [S1,S2,. . .,Sn],Durations be [D1,D2,. . .,
Dn], and Resources be [R1,R2,. . .,Rn]. For each job i,Sirepresents the start
time, Direpresents the duration, and Rirepresents the units of resources needed. Limit
is the limit on the units of resources available at any time. This constraint ensures that the
limit cannot be exceeded at any time.
•decreasing(L): The sequence (an array or a list) Lis in (non-strictly) decreasing order.
•decreasing strict(L): The sequence (an array or a list) Lis in strictly decreasing
order.
•diffn(RectangleList): This constraint ensures that no two rectangles in RectangleList
overlap with each other. A rectangle in an n-dimensional space is represented by a list of
2×nelements [X1,X2,. . .,Xn,S1,S2,. . .,Sn], where Xiis the starting
coordinate of the edge in the ith dimension, and Siis the size of the edge.
•disjunctive tasks(T asks):T asks is a list of terms. Each term has the form
disj tasks(S1,D1,S2,D2), where S1and S2are two integer-domain variables, and
D1and D2are two positive integers. This constraint is equivalent to posting the disjunc-
tive constraint S1+D1#=< S2#\/ S2+D2#=< S1for each term in T asks; however the
constraint may be more efficient, because it converts the disjunctive tasks into global con-
straints.
•element(I,List,V): This constraint is true if the Ith element of List is V, where I
and Vare integer-domain variables, and List is a list of integer-domain variables.
•exactly(N,L,V): This constraint succeeds if there are exactly Nelements in Lthat
are equal to V, where Nand Vmust be integer-domain variables, and Lmust be a list of
integer-domain variables.
•global cardinality(List,P airs): Let List be a list of integer-domain variables
[X1,. . .,Xd], and P airs be a list of pairs [K1-V1,. . .,Kn-Vn], where each key
Kiis a unique integer, and each Viis an integer-domain variable. The constraint is true if
every element of List is equal to some key, and, for each pair Ki-Vi, exactly Vielements
of List are equal to Ki. This constraint can be defined as follows:
global_cardinality(List,Pairs) =>
foreach($Key-V in Pairs)
sum([B : E in List, B#<=>(E#=Key)]) #= V
end.
•increasing(L): The sequence (an array or a list) Lis in (non-strictly) increasing order.
•increasing strict(L): The sequence (an array or a list) Lis in strictly increasing
order.
•lex le(L1,L2): The sequence (an array or a list) L1is lexicographically less than or
equal to L2.
•lex lt(L1,L2): The sequence (an array or a list) L1is lexicographically less than L2.
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•matrix element(Matrix,I,J,V): This constraint is true if the entry at <I,J> in
Matrix is V, where I,J, and Vare integer-domain variables, and Matrix is an two-
dimensional array of integer-domain variables.
•neqs(NeqList):NeqList is a list of inequality constraints of the form X#!= Y, where
Xand Yare integer-domain variables. This constraint is equivalent to the conjunction of
the inequality constraints in NeqList, but it extracts all distinct constraints from the
inequality constraints.
•nvalue(N,List): The number of distinct values in List is N, where List is a list of
integer-domain variables.
•regular(L, Q, S, M, Q0, F ): Given a finite automaton (DFA or NFA) of Qstates num-
bered 1, 2, . . .,Qwith input 1..S, transition matrix M, initial state Q0(1 ≤Q0≤Q), and
a list of accepting states F, this constraint is true if the list Lis accepted by the automaton.
The transition matrix Mrepresents a mapping from 1..Q ×1..S to 0..Q, where 0denotes
the error state. For a DFA, every entry in Mis an integer, and for an NFA, entries can be a
list of integers.
•scalar product(A,X,P roduct): The scalar product of Aand Xis P roduct, where
Aand Xare lists or arrays of integer-domain variables, and P roduct is an integer-domain
variable. Aand Xmust have the same length.
•scalar product(A,X,Rel,P roduct): The scalar product of Aand Xhas the rela-
tion Rel with P roduct, where Rel is one of the following operators: #=,#!=,#>=,#>,
#=< (#<=), and #<.
•serialized(Starts,Durations): This constraint describes a set of non-overlapping
tasks, where Starts and Durations are lists of integer-domain variables, and the lists have
the same length. Let Os be a list of 1s that has the same length as Starts. This constraint is
equivalent to cumulative(Starts,Durations,Os,1).
•subcircuit(F DV ars): This constraint is the same as circuit(F DV ars), except
that not all of the vertices are required to be in the circuit. If the ith element of F DV ars is
i, then the vertex iis not part of the circuit.
12.6 Solver Invocation
•solve(Options,V ars): This predicate calls the imported solver to label the variables
V ars with values, where Options is a list of options for the solver. The options will be de-
tailed below. This predicate can backtrack in order to find multiple solutions. The cp mod-
ule allows incremental labeling of variables, and some variables that occur in constraints
but are not passed to solve may remain uninstantiated after a call to solve. The user
is responsible for having all the variables that need to be instantiated passed to solve. In
constrast, the sat and mip modules do not support incremental labeling of variables.
•solve(V ars): This predicate is the same as solve([], V ars).
•indomain(V ar): This predicate is only accepted by cp. It is the same as solve([],
[V ar]).
•solve all(Options,V ars) = Solutions: This function returns all the solutions that
satisfy the constraints.
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•solve all(V ars) = Solutions: This function is the same as solve all([],V ars).
•solve suspended(Options): After solve(V ars)has successfully labeled V ars,
some constraints may remain suspended and not completely checked because not all of the
decision variables are included in V ars. The solve suspended(Options)predicate
labels all remaining variables in the suspended constraints. This predicate is only provided
by the cp module.
•solve suspended: This predicate is the same as solve suspended([]).
•indomain down(V ar): This predicate is the same as solve([down], [V ar]). It
is only accepted by cp.
12.6.1 Common Solving Options
The following options are accepted by all three of the solvers.1
•$min(V ar): Minimize the variable V ar.
•$max(Exp): Maximize the variable V ar.
•$report(Call): Execute Call each time a better answer is found while searching for an
optimal answer. This option cannot be used if the mip module is used.
12.6.2 Solving Options for cp
The cp module also accepts the following options:
•backward: The list of variables is reversed first.
•constr: Variables are first ordered by the number of attached constraints.
•degree: Variables are first ordered by degree, i.e., the number of connected variables.
•down: Values are assigned to variables from the largest to the smallest.
•ff: The first-fail principle is used: the leftmost variable with the smallest domain is selected.
•ffc: The same as with the two options: ff and constr.
•ffd: The same as with the two options: ff and degree.
•forward: Choose variables in the given order, from left to right.
•inout: The variables are reordered in an inside-out fashion. For example, the variable list
[X1,X2,X3,X4,X5] is rearranged into the list [X3,X2,X4,X1,X5].
•label(CallName): This option informs the CP solver that once a variable Vis selected,
the user-defined call CallN ame(V)is used to label V, where CallName must be defined
in the same module, an imported module, or the global module.
•leftmost: The same as forward.
•max: First, select a variable whose domain has the largest upper bound, breaking ties by
selecting a variable with the smallest domain.
1Note that the preceding dollar sign indicates that it is followed by a term, not a function call.
95
•min: First, select a variable whose domain has the smallest lower bound, breaking ties by
selecting a variable with the smallest domain.
•rand: Both variables and values are randomly selected when labeling.
•rand var: Variables are randomly selected when labeling.
•rand val: Values are randomly selected when labeling.
•reverse split: Bisect the variable’s domain, excluding the lower half first.
•split: Bisect the variable’s domain, excluding the upper half first.
•updown: Values are assigned to variables from the values that are nearest to the middle of
the domain.
12.6.3 Solving Options for sat
•dump: Dump the CNF code to stdout.
•dump(F ile): Dump the CNF code to F ile.
•seq: Use sequential search to find an optimal answer.
•split: Use binary search to find an optimal answer (default).
•$nvars(NV ars): The number of variables in the CNF code is NV ars.
•$ncls(NCls): The number of clauses in the CNF code is NCls.
•$threads(N): Use Nthreads to solve the generated CNF code. In the current imple-
mentation, the parallel version of lingeling, plingeling, will be used if this option is
given.
•threads: The same as $threads(8).
12.6.4 Solving Options for mip
•dump: Dump the constraints in CPLEX format to stdout.
•dump(F ile): Dump the CPLEX format to F ile.
•glpk: Instruct Picat to use the GLPK MIP solver. Picat uses the following command to call
the GLPK solver:
glpsol --lp -o SolF ile T mpF ile
where SolF ile is a solution file, and T mpF ile is a file that stores the CPLEX-format con-
straints. Picat throws existence error if the command glpsol is not available.
•gurobi: Instruct Picat to use the Gurobi MIP solver. Picat uses the following command to
call the Gurobi solver:
gurobi cl ResultFile=SolF ile T mpF ile
96
where SolF ile is a file for the solution, and T mpF ile is a file that stores the CPLEX-
format constraints. Picat throws existence error if the command gurobi cl is not
available.
•tmp(F ile): Dump the CPLEX format to F ile rather than the default file ’ tmp.lp’, before
calling the mip solver. The name F ile must be a string or an atom that has the extension
name ’.lp’. When this file name is specified, the mip solver will save the solution into a file
name that has the same main name as F ile but the extension name ’.sol’.
97
Chapter 13
The os Module
Picat has an os module for manipulating files and directories. In order to use any of the functions
or predicates, users must import the module.
13.1 The P ath Parameter
Many of the functions and predicates in this module have a P ath parameter. This parameter is
a string or an atom, representing the path of a file or directory. This path can be an absolute
path, from the system’s root directory, or a relative path, from the current file location. Different
systems use different separator characters to separate directories in different levels of the directory
hierarchy. For example, Windows uses ‘\’ and Unix uses ‘/’. The following function outputs a
single character, representing the character that the current system uses as a file separator.
•separator() = V al
13.2 Directories
The os module includes functions for reading and modifying directories. The following example
shows how to list all of the files in a directory tree, using a depth-first directory traversal.
Example
import os.
traverse(Dir), directory(Dir) =>
List = listdir(Dir),
printf("Inside %s%n",Dir),
foreach(File in List)
printf(" %s%n",File)
end,
foreach (File in List, File != ".", File != "..")
FullName = Dir ++ [separator()] ++ File,
traverse(FullName)
end.
traverse(_Dir) => true.
The following function can be used to read the contents of a directory:
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•listdir(P ath) = List: This function returns a list of all of the files and directories
that are contained inside the directory specified by P ath. If P ath is not a directory, then
an error is thrown. The returned list contains strings, each of which is the name of a file or
directory.
The above example also uses the directory predicate, which will be discussed in Section 13.4.
13.2.1 The Current Working Directory
The os module includes two functions that obtain the program’s current working directory:
•cwd() = P ath
•pwd() = P ath
The os module also includes two predicates to change the program’s current working direc-
tory:
•cd(P ath)
•chdir(P ath)
If the cd and chdir predicates cannot move to the directory specified by P ath, the functions
throw an error. This can occur if P ath does not exist, if P ath is not a directory, or if the program
does not have permission to access P ath.
13.3 Modifying Files and Directories
13.3.1 Creation
The os module contains a number of predicates for creating new files and directories:
•mkdir(P ath): This predicate creates a new directory at location P ath. The directory will
be created with a default permission list of [rwu, rwg, ro]. If the program does not
have permission to write to the parent directory of P ath, this predicate will throw an error.
An error will also occur if the parent directory does not exist.
•rename(Old,New): This renames a file or a directory from Old to N ew. This predi-
cate will throw an error if Old does not exist. An error will also occur if the program does
not have permission to write to Old or N ew.
•cp(F romP ath,T oP ath): This copies a file from F romP ath to T oP ath. This predi-
cate will throw an error if F romP ath does not exist or F romP ath is a directory. An error
will also occur if the program does not have permission to read from F romP ath, or if it
does not have permission to write to T oP ath.
13.3.2 Deletion
The os module contains a number of predicates for deleting files and directories.
•rm(P ath): This deletes a file. An error will be thrown if the file does not exist, if the
program does not have permission to delete the file, or if P ath refers to a directory, a hard
link, a symbolic link, or a special file type.
•rmdir(P ath): This deletes a directory. An error will be thrown if the directory does
not exist, the program does not have permission to delete the directory, the directory is not
empty, or if P ath does not refer to a directory.
99
13.4 Obtaining Information about Files
The os module contains a number of functions that retrieve file status information, and predicates
that test the type of a file. These predicates will all throw an error if the program does not have
permission to read from P ath.
•readable(P ath): Is the program allowed to read from the file?
•writable(P ath): Is the program allowed to write to the file?
•executable(P ath): Is the program allowed to execute the file?
•size(P ath) = Int: If P ath is not a symbolic link, then this function returns the number
of bytes contained in the file to which P ath refers. If P ath is a symbolic link, then this
function returns the path size of the symbolic link. Because the function size/1 is defined
in the basic module for returning the size of a map, this function requires an explicit
module qualifier os.size(P ath).
•file base name(P ath) = List: This function returns a string containing the base
name of P ath. For example, the base name of “a/b/c.txt” is “c.txt”.
•file directory name(P ath) = List: This function returns a string containing the
path of the directory that contains P ath. For example, the directory name of “a/b/c.txt”
is “a/b/”.
•exists(P ath): Is P ath an existing file or directory?
•file(P ath): Does P ath refer to a regular file? This predicate is true if P ath is neither a
directory nor a special file, such as a socket or a pipe.
•file exists(P ath): This tests whether P ath exists, and, if it exists, whether P ath
refers to a regular file.
•directory(P ath): Does P ath refer to a directory?
The following example shows how to use a few of the predicates.
Example
import os.
test_file(Path) =>
if (not exists(Path)) then
printf("%s does not exist %n",Path)
elseif (directory(Path)) then
println("Directory")
elseif (file(Path)) then
println("File")
else
println("Unknown")
end.
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13.5 Environment Variables
•env exists(Name): This predicate succeeds if Name is an environment variable in
the system.
•getenv(Name) = String: This function returns the value of the environment variable
Name as a string. This function will throw an error if the environment variable Name does
not exist.
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Appendix A
The math Module
Picat provides a math module, which has common mathematical constants and functions. The
math module is imported by default.
A.1 Constants
The math module provides two constants.
•e = 2.71828182845904523536
•pi = 3.14159265358979323846
A.2 Functions
The math module contains mathematical functions that serve a number of different purposes.
Note that the arguments must all be numbers. If the arguments are not numbers, then Picat will
throw an error.
A.2.1 Sign and Absolute Value
The following functions deal with the positivity and negativity of numbers.
•sign(X) = V al: This function determines whether Xis positive or negative. If Xis
positive, then this function returns 1. If Xis negative, then this function returns −1. If Xis
0, then this function returns 0.
•abs(X) = V al: This function returns the absolute value of X. If X≥0, then this
function returns X. Otherwise, this function returns −X.
Example
Picat> Val1 = sign(3), Val2 = sign(-3), Val3 = sign(0)
Val1 = 1
Val2 = -1
Val3 = 0
Picat> Val = abs(-3)
Val = 3
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A.2.2 Rounding and Truncation
The math module includes the following functions for converting a real number into the integers
that are closest to the number.
•ceiling(X) = V al: This function returns the closest integer that is greater than or
equal to X.
•floor(X) = V al: This function returns the closest integer that is less than or equal to
X.
•round(X) = V al: This function returns the integer that is closest to X.
•truncate(X) = V al: This function removes the fractional part from a real number.
•modf(X)=(IntV al,F ractV al): This function splits a real number into its integer
part and its fractional part.
Example
Picat> Val1 = ceiling(-3.2), Val2 = ceiling(3)
Val1 = -3
Val2 = 3
Picat> Val1 = floor(-3.2), Val2 = floor(3)
Val1 = -4
Val2 = 3
Picat> Val1 = round(-3.2), Val2 = round(-3.5), Val3 = round(3.5)
Val1 = -3
Val2 = -4
Val3 = 4
Picat> Val1 = truncate(-3.2), Val2 = truncate(3)
Val1 = -3
Val2 = 3
Picat> IF = modf(3.2)
IF = (3.0 , 0.2)
A.2.3 Exponents, Roots, and Logarithms
The following functions provide exponentiation, root, and logarithmic functions. Note that, in the
logarithmic functions, if X≤0, then an error is thrown.
•pow(X,Y) = V al: This function returns XY. It does the same thing as X∗ ∗Y.
•pow mod(X,Y,Z) = V al: This function returns XYmod Z. All of the arguments
must be integers, and Ymust not be negative.
•exp(X) = V al: This function returns eX.
•sqrt(X) = V al: This function returns the square root of X. Note that the math module
does not support imaginary numbers. Therefore, if X < 0, this function throws an error.
•log(X) = V al: This function returns loge(X).
•log10(X) = V al: This function returns log10(X).
103
•log2(X) = V al: This function returns log2(X).
•log(B,X) = V al: This function returns logB(X).
Example
Picat> P1 = pow(2, 5), P2 = exp(2)
P1 = 32
P2 = 7.38906
Picat> S = sqrt(1)
S = 1.0
Picat> E = log(7), T = log10(7), T2 = log2(7), B = log(7, 7)
E = 1.94591
T = 0.845098
T2 = 2.80735
B = 1.0
A.2.4 Converting Between Degrees and Radians
The math module has two functions to convert between degrees and radians.
•to radians(Degree) = Radian: This function converts from degrees to radians.
•to degrees(Radian) = Degree: This function converts from radians to degrees.
Example
Picat> R = to_radians(180)
R = 3.14159
Picat> D = to_degrees(pi)
D = 180.0
A.2.5 Trigonometric Functions
The math module provides the following trigonometric functions.
•sin(X) = V al: This function returns the sine of X, where Xis given in radians.
•cos(X) = V al: This function returns the cosine of X, where Xis given in radians.
•tan(X) = V al: This function returns the tangent of X, where Xis given in radians. If
the tangent is undefined, such as at pi / 2, then this function throws an error.
•sec(X) = V al: This function returns the secant of X, where Xis given in radians. If
cos(X)is 0, then this function throws an error.
•csc(X) = V al: This function returns the cosecant of X, where Xis given in radians. If
sin(X)is 0, then this function throws an error.
•cot(X) = V al: This function returns the cotangent of X, where Xis given in radians.
If tan(X)is 0, then this function throws an error.
•asin(X) = V al: This function returns the arc sine of X, in radians. The returned value
is in the range [-pi / 2,pi / 2]. Xmust be in the range [−1,1]; otherwise, this
function throws an error.
104
•acos(X) = V al: This function returns the arc cosine of X, in radians. The returned
value is in the range [0,pi]. Xmust be in the range [−1,1]; otherwise, this function
throws an error.
•atan(X) = V al: This function returns the arc tangent of X, in radians. The returned
value is in the range [-pi / 2,pi / 2].
•atan2(X,Y) = V al: This function returns the arc tangent of Y/X, in radians. X
and Yare coordinates. The returned value is in the range [-pi,pi].
•asec(X) = V al: This function returns the arc secant of X, in radians. The returned
value is in the range [0,pi]. Xmust be in the range (−∞,−1] or [1,∞); otherwise, this
function throws an error.
•acsc(X) = V al: This function returns the arc cosecant of X, in radians. The returned
value is in the range [-pi / 2,pi / 2]. Xmust be in the range (−∞,−1] or [1,∞);
otherwise, this function throws an error.
•acot(X) = V al: This function returns the arc cotangent of X, in radians. The returned
value is in the range [-pi / 2,pi / 2].
Example
Picat> S = sin(pi), C = cos(pi), T = tan(pi)
S = 0.0
C = -1.0
T = 0.0
Picat> S = asin(0), C = acos(0), T = atan(0), T2 = atan2(-10, 10)
S = 0.0
C = 1.5708
T = 0.0
T2 = -0.785398
Picat> S = sec(pi / 4), C = csc(pi / 4), T = cot(pi / 4)
S = 1.41421
C = 1.41421
T = 1.0
Picat> S = asec(2), C = acsc(2), T = acot(0)
S = 1.0472
C = 0.5236
T = 1.5708
A.2.6 Hyperbolic Functions
The math module provides the following hyperbolic functions.
•sinh(X) = V al: This function returns the hyperbolic sine of X.
•cosh(X) = V al: This function returns the hyperbolic cosine of X.
•tanh(X) = V al: This function returns the hyperbolic tangent of X.
•sech(X) = V al: This function returns the hyperbolic secant of X.
105
•csch(X) = V al: This function returns the hyperbolic cosecant of X. If Xis 0, then this
function throws an error.
•coth(X) = V al: This function returns the hyperbolic cotangent of X. If Xis 0, then
this function throws an error.
•asinh(X) = V al: This function returns the arc hyperbolic sine of X.
•acosh(X) = V al: This function returns the arc hyperbolic cosine of X. If X < 1, then
this function throws an error.
•atanh(X) = V al: This function returns the arc hyperbolic tangent of X.Xmust be in
the range (−1,1); otherwise, this function throws an error.
•asech(X) = V al: This function returns the arc hyperbolic secant of X.Xmust be in
the range (0,1]; otherwise, this function throws an error.
•acsch(X) = V al: This function returns the arc hyperbolic cosecant of X. If Xis 0,
then this function throws an error.
•acoth(X) = V al: This function returns the arc hyperbolic cotangent of X.Xmust be
in the range (−∞,−1) or (1,∞); otherwise, this function throws an error.
Example
Picat> S = sinh(pi), C = cosh(pi), T = tanh(pi)
S = 11.54874
C = 11.59195
T = 0.99627
Picat> S = sech(pi / 4), C = csch(pi / 4), T = coth(pi / 4)
S = 0.75494
C = 1.15118
T = 1.52487
Picat> S = asinh(0), C = acosh(1), T = atanh(0)
S = 0.0
C = 0.0
T = 0.0
A.2.7 Random Numbers
The following functions provide access to a random number generator.
•random() = V al: This function returns a random integer.
•random2() = V al: This function returns a random integer, using an environment-dependent
seed.
•rand max() = V al: This function returns the maximum random integer.
•random(Seed) = V al: This function returns a random integer. At the same time, it
changes the seed of the random number generator.
•random(Low,High) = V al: This function returns a random integer in the range
Low..High.
106
•frand() = V al: This function returns a random real number between 0.0 and 1.0, inclu-
sive.
•frand(Low,High) = V al: This function returns a random real number between Low
and High, inclusive.
A.2.8 Other Built-ins
•even(N): This predicate is true if Nis an even integer.
•gcd(A,B): This function returns the greatest common divisor of integer Aand integer B.
•odd(N): This predicate is true if Nis an odd integer.
•prime(N): This predicate is true if Nis a prime number.
•primes(N) = List: This function returns a list of prime numbers that are less than or
equal to N.
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Appendix B
The sys Module
The sys module, which is imported by default, contains built-ins that are relevant to the Picat
system. The built-ins in the sys module perform operations that include compiling programs,
tracing execution, and displaying statistics and information about the Picat system.
B.1 Compiling and Loading Programs
The sys module includes a number of built-ins for compiling programs and loading them into
memory.
•compile(F ileN ame): This predicate compiles the file F ileN ame.pi and all of its de-
pendent files without loading the generated byte-code files. The destination directory for the
byte-code file is the same as the source file’s directory. If the Picat interpreter does not have
permission to write into the directory in which a source file resides, then this built-in throws
an exception. If F ileN ame.pi imports modules, then these module files are also com-
piled. The system searches for these module files in the directory in which F ileN ame.pi
resides or the directories that are stored in the environment variable PICATPATH.
•compile bp(F ileN ame): This predicate translates the Picat file F ileN ame.pi into a
B-Prolog file F ileN ame.pl. If the file is dependent on other Picat files, then those files
are compiled using compile/1. The destination directory for the B-Prolog file is the same
as the source file’s directory. If the Picat interpreter does not have permission to write into
the directory in which a source file resides, then this built-in throws an exception.
•load(F ileN ame): This predicate loads the byte-code file F ileN ame.qi and all of
its dependent byte-code files into the system for execution. For F ileN ame, the system
searches for a byte-code file in the directory specified by F ileN ame or the directories
that are stored in the environment variable PICATPATH. For the dependent file names, the
system searches for a byte-code file in the directory in which F ileName.qi resides or the
directories that are stored in the environment variable PICATPATH. If the byte-code file
F ileN ame.qi does not exist, but the source file F ileN ame.pi exists, then this built-in
compiles the source file and loads the byte codes without creating a qi file. Note that, for
the dependent files, if the byte-code file does not exist, but the source file exists, then the
source file will be compiled.
•cl(F ileN ame): This predicate compiles and loads the source file named F ileN ame.pi.
Note that the extension .pi does not need to be given. The system also compiles and loads
all of the module files that are either directly imported or indirectly imported by the source
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file. The system searches for such dependent files in the directory in which F ileName.pi
resides or the directories that are stored in the environment variable PICATPATH.
•cl: This predicate compiles and loads a program from the console, ending when the end-
of-file character (ctrl-z for Windows and ctrl-d for Unix) is typed.
•cl facts(F acts): This predicate compiles and loads facts into the system. The argument
F acts is a list of ground facts.
•cl facts(F acts,IndexInfo): This predicate compiles and loads facts into the sys-
tem. The argument F acts is a list of ground facts. The argument IndexInf o is a list of
indexing information in the form p(M1,M2, ..., Mn). Each Mican either be +,
which indicates that the argument is input, or -, which indicates that the argument is output.
•cl facts table(F acts): This predicate is the same as cl facts/1, except that the
facts are all tabled.
•cl facts table(F acts,IndexInfo): This predicate is the same as cl facts/2,
except that the facts are all tabled.
B.2 Tracing Execution
The Picat system has three execution modes: non-trace mode,trace mode, and spy mode. In
trace mode, it is possible to trace the execution of a program, showing every call in every possible
stage. In order to trace the execution, the program must be recompiled while the system is in trace
mode. In spy mode, it is possible to trace the execution of individual functions and predicates. The
following predicates are used to switch between non-trace mode and trace mode.
•trace: This predicate switches the execution mode to trace mode.
•notrace: This predicate switches the execution mode to non-trace mode.
•debug: This predicate switches the execution mode to trace mode.
•nodebug: This predicate switches the execution mode to non-trace mode.
•spy(P oint): This predicate places a spy point on P oint, which is a function or a predicate,
optionally followed by an arity. The creation of a spy point switches the Picat system to spy
mode.
•nospy: This predicate removes all spy points, and switches the execution mode to non-trace
mode.
•abort: This predicate terminates the current program. This can be used in all three execu-
tion modes.
B.2.1 Debugging Commands
In trace mode, the system displays a message when a function or a predicate is entered (Call),
exited (Exit), re-entered (Redo) or has failed (Fail). After a function or a predicate is entered
or re-entered, the system waits for a command from the user. A command is a single letter followed
by a carriage-return, or may simply be a carriage-return. The following commands are available:
•+: create a spy point.
109
•-: remove a spy point.
•<: reset the print depth to 10.
•<i: reset the print depth to i.
•a: abort, quit debugging, moving control to the top level.
•<cr> : A carriage return causes the system to show the next call trace.
•c: creep, show the next call trace.
•h: help, display the debugging commands.
•?: help, display the debugging commands.
•l: leap, be silent until a spy point is encountered.
•n: nodebug, prevent the system from displaying debugging messages for the remainder of
the program.
•r: repeat, continue to creep or leap without intervention.
•s: skip, be silent until the call is completed (Exit or Fail).
•t: backtrace, show the backtrace leading to the current call.
•t i : backtrace, show the backtrace from the call numbered ito the current call.
•u: undo what has been done to the current call and redo it.
•u i : undo what has been done to the call numbered iand redo it.
B.3 Information about the Picat System
The sys module contains a number of built-ins that display information about the Picat system.
This information includes statistics about the system, including the memory that is used, and the
amount of time that it takes to perform a goal.
B.3.1 Statistics
The following built-ins display statistics about the memory that Picat system uses.
•statistics: This predicate displays the number of bytes that are allocated to each data
area, and the number of bytes that are already in use.
•statistics(Key,V alue): The statistics concerning Key are V alue. This predicate
gives multiple solutions upon backtracking. Keys include runtime,program,heap,
control,trail,table,gc,backtracks, and gc time. The values for most of
the keys are lists of two elements. For the key runtime, the first element denotes the
amount of time in milliseconds that has elapsed since Picat started, and the second element
denotes the amount of time that has elapsed since the previous call to statistics/2
was executed. For the key gc, the number indicates the number of times that the garbage
collector has been invoked. For the key backtracks, the number indicates the number
of backtracks that have been done during the labeling of finite domain variables since Picat
was started. For all other keys, the first element denotes the size of memory in use, and the
110
second element denotes the size of memory that is still available in the corresponding data
area.
•statistics all() = List: This function returns a list of lists that are in the form
[Key, Value]. The list contains all of the keys that statistics/2 can display, to-
gether with their corresponding values.
Example
Picat> statistics
Stack+Heap: 8,000,000 bytes
Stack in use: 1,156 bytes
Heap in use: 28,592 bytes
Program: 8,000,000 bytes
In use: 1,448,436 bytes
Symbols: 5,300
Trail: 4,000,000 bytes
In use: 936 bytes
Memory manager:
GC: Call(0), Time(0 ms)
Expansions: Stack+Heap(0), Program(0), Trail(0), Table(0)
Picat> statistics(Key, Value)
Key = runtime
Value = [359947,66060]?;
Key = program
Value = [1451656,6548344]?;
Key = heap
Value = [34112,7964524]?;
Key = control
Value = [1360,7964524]?;
Key = trail
Value = [1496,3998504]?;
Key = table
Value = [0,4000000]?;
Key = table_blocks
Value = 1?;
key = gc
Value = 0?;
111
Key = backtracks
V=0?;
Key = gc_time
Value = 0
Picat> L = statistics_all()
L = [[runtime, [359947,66060]], [program, [1451656,6548344]],
[heap, [34112,7964524]], [control, [1360,7964524]],
[trail, [1496,3998504]], [table, [0,4000000]],
[table_blocks,1],[gc, 0], [backtracks, 0], [gc_time, 0]]
B.3.2 Time
The following predicates display the amount of CPU time that it takes to perform a goal.
•time(Goal): This predicate calls Goal, and reports the number of seconds of CPU time
that were consumed by the execution.
•time2(Goal): This predicate calls Goal, and reports the number of seconds of CPU
time that were consumed by the execution, and the number of backtracks that have been
performed in labeling finite-domain variables during the execution of Goal.
•time out(Goal,Limit,Result): This predicate is logically equivalent to once Goal,
except that is imposes a time limit, in milliseconds, on the evaluation. If Goal is not finished
when Limit expires, then the evaluation will be aborted, and Result will be unified with
the atom time out. If Goal succeeds within the time limit, then Result will be unified
with the atom success. Note that time-out may be delayed or never occur because of the
execution of an external C function.
B.3.3 Other System Information
•help: This predicate displays the usages of some of the commands that the system accepts.
•loaded modules() = List: This function returns a list of the modules that are cur-
rently loaded in the Picat system. This list includes library modules and user-defined mod-
ules. By default, this function returns [basic,sys,io,math].
•nolog: This predicate turns off the logging flag.
•picat path() = P ath: This function returns the directories that are stored in the en-
vironment variable PICATPATH. If the environment variable PICATPATH does not exist,
then this function throws an error.
•command(String) = Int: This function sends the command String to the OS and re-
turns the status that is returned from the OS.
B.4 Garbage Collection
Picat incorporates an incremental garbage collector for the control stack and the heap. The garbage
collector is active by default. The sys module includes the following predicates for garbage
collection.
112
•garbage collect: This predicate starts the garbage collector.
•garbage collect(Size): This predicate calls the garbage collector. If there are less
than Size words on the control stack and heap after garbage collection, then it invokes the
memory manager to expand the stack and heap so that there are Size words on the control
stack and heap.
B.5 Quitting the Picat System
The following predicates can be used to terminate the Picat interpreter.
•exit
•halt
113
Appendix C
The util Module
The util module provides general useful utility functions and predicates. This module is ex-
pected to be expanded in the future. This module must be imported before use.
C.1 Utilities on Terms
•replace(T erm,Old,New) = NewT erm: This function returns a copy of T erm,
replacing all of the occurrences of Old in T erm by New.
•replace at(T erm,Index,New) = NewT erm: This function returns a copy of T erm,
replacing the argument at Index by New.T erm must be a compound term.
•find first of(T erm,P attern) = Index: This function returns the first index at
which the argument unifies with P attern. If there is no argument that unifies with P attern,
then this function returns -1. T erm must be either a list or a structure.
•find last of(T erm,P attern) = Index: This function returns the last index at which
the argument unifies with P attern. If there is no argument that unifies with P attern, then
this function returns -1. T erm must be either a list or a structure.
C.2 Utilities on Strings and Lists
•chunks of(List,K) = ListOfLists: This function splits List into chunks, each of
which has length K, and returns the list of the chunks. The last chunk may have less than
Kelements if the length of List is not a multiple of K.
•drop(List,K) = List: This function returns the suffix of List after the first Kelements
are dropped, or [] if Kis greater than the length of List.
•find(String,SubString,F rom,T o): This predicate searches for an occurrence of
SubString in String, and binds F rom to the starting index and T o to the ending index.
On backtracking, this predicate searches for the next occurrence of SubString.
•find ignore case(String,SubString,F rom,T o): This predicate is the same as
find(String,SubString,F rom,T o), except that it is case insensitive.
•join(T okens) = String: This function is the same as join(T okens," " ).
•join(T okens,Separator) = String: This function concatenates the tokens T okens
into a string, adding Separator, which is a string or an atom, between each two tokens.
114
•lstrip(List) = List: This function is the same as lstrip(List," \t\n\r").
•lstrip(List,Elms) = List: This function returns a copy of List with leading ele-
ments in Elms removed.
•rstrip(List) = List: This function is the same as rstrip(List," \t\n\r").
•rstrip(List,Elms) = List: This function returns a copy of List with trailing ele-
ments in Elms removed.
•split(String) = T okens: This function is the same as split(String," \t\n\r"),
which uses white spaces as split characters.
•split(String,Separators) = T okens: This function splits String into a list of to-
kens, using characters in the string Separators as split characters. Recall that a string is a
list of characters. A token is a string, so the return value is a list of lists of characters.
•strip(List) = List: This function is the same as strip(List," \t\n\r").
•strip(List,Elms) = List: This function returns a copy of List with leading and trail-
ing elements in Elms removed.
•take(List,K) = List: This function returns the prefix of List of length K, or List
itself if Kis greater than the length of List.
C.3 Utilities on Matrices
An array matrix is a two-dimensional array. The first dimension gives the number of the rows and
the second dimension gives the number of the columns. A list matrix represents a matrix as a list
of lists.
•matrix multi(A,B): This function returns the product of matrices Aand B. Both A
and Bmust be array matrices.
•transpose(A) = B: This function returns the transpose of the matrix A.Acan be an
array matrix or a list matrix. If Ais an array matrix, then the returned transpose is also an
array. If Ais a list matrix, then the returned transpose is also a list.
•rows(A) = List: This function returns the rows of the matrix Aas a list.
•columns(A) = List: This function returns the columns of the matrix Aas a list.
•diagonal1(A) = List: This function returns primary diagonal of the matrix Aas a list.
•diagonal2(A) = List: This function returns secondary diagonal of the matrix Aas a
list.
•array matrix to list matrix(A) = List: This function converts the array ma-
trix Ato a list matrix.
•array matrix to list(A) = List: This function converts the array matrix Ato a
flat list, row by row.
115
C.4 Utilities on Lists and Sets
•permutation(List,P): This predicate generates a permutation Pof List. This predi-
cate is non-deterministic. On backtracking, it generates the next permutation.
•permutations(List) = P s: This function returns all the permutations of List.
•nextto(E1,E2,List): This predicate is true if E1follows E2in List. This predicate
is non-deterministic.
116
Appendix D
The ordset Module
An ordered set is represented as a sorted list that does not contain duplicates. The ordset module
provides useful utility functions and predicates on ordered sets. This module must be imported
before use.
•delete(OSet,Elm) = OSet1: This function returns of a copy of OSet that does not
contain the element Elm.
•disjoint(OSet1,OSet2): This predicate is true when OSet1and OSet2have no ele-
ment in common.
•insert(OSet,Elm) = OSet1: This function returns a copy of OSet with the element
Elm inserted.
•intersection(OSet1,OSet2)=OSet3: This function returns an ordered set that con-
tains elements which are in both OSet1and OSet2.
•new ordset(List) = OSet: This function returns an ordered set that contains the ele-
ments of List.
•ordset(T erm): This predicate is true if T erm is an ordered set.
•subset(OSet1,OSet2): This predicate is true if OSet1is a subset of OSet2.
•subtract(OSet1,OSet2)=OSet3: This function returns an ordered set that contains all
of the elements of OSet1which are not in OSet2.
•union(OSet1,OSet2)=OSet3: This function returns an ordered set that contains all of
the elements which are present in either OSet1or OSet2.
117
Appendix E
The datetime Module
Picat’s datetime module provides built-ins for retrieving the date and time. This module must
be imported before use.
•current datetime() = DateT ime: This function returns the current date and time
as a structure in the form
date time(Y ear,Month,Day,Hour,Minute,Second)
where the arguments are all integers, and have the following meanings and ranges.
Argument Meaning Range
Y ear years since 1900 an integer
Month months since January 0-11
Day day of the month 1-31
Hour hours since midnight 0-23
Minute minutes after the hour 0-59
Second seconds after the minute 0-60
In the Month argument, 0represents January, and 11 represents December. In the Hour
argument, 0represents 12 AM, and 23 represents 11 PM. In the Second argument, the value
60 represents a leap second.
•current date() = Date: This function returns the current date as a structure in the
form date(Y ear,Month,Day), where the arguments have the meanings and ranges
that are defined above.
•current day() = W Day: This function returns the number of days since Sunday, in
the range 0 to 6.
•current time() = T ime: This function returns the current time as a structure in the
form time(Hour,Minute,Second), where the arguments have the meanings and
ranges that are defined above.
118
Appendix F
Formats
F.1 Formatted Printing
The following table shows the specifiers that can be used in formats for the writef,printf,
and to fstring.
Specifier Output
%% Percent Sign
%c Character
%d Signed Decimal Integer
%e Scientific Notation, with Lowercase e
%E Scientific Notation, with Uppercase E
%f Decimal Real Number
%g Shorter of %e and %f
%G Shorter of %E and %f
%i Signed Decimal Integer
%n Platform-independent Newline
%o Unsigned Octal Integer
%s String
%u Unsigned Decimal Integer
%w Term
%x Unsigned Lowercase Hexadecimal Integer
%X Unsigned Uppercase Hexadecimal Integer
119
Appendix G
External Language Interface with C
Picat has an interface with C, through which Picat programs can call deterministic predicates that
are written as functions in C. C programs that use this interface must include the file "picat.h"
in the directory Picat/Emulator. In order to make C-defined predicates available to Picat,
users have to re-compile Picat’s C source code together with the newly-added C functions.
G.1 Term Representation
Picat’s C interface provides functions for accessing, manipulating, and building Picat terms. In
order to understand these functions, users need to know how terms are represented in Picat’s
virtual machine.
A term is represented by a word that contains a value and a tag. A word has 32 bits or 64 bits,
depending on the underlying CPU and OS. The tag in a word distinguishes the type of the term.
The value of a term is an address, except when the term is an integer (in which case, the value
represents the integer itself). The location to which the address points is dependent on the type
of the term. In a reference, the address points to the referenced term. An unbound variable is
represented by a self-referencing pointer. In an atom, the address points to the record for the atom
symbol in the symbol table. In a structure, f(t1, . . . , tn), the address points to a block of n+ 1
consecutive words, where the first word points to the record for the functor, f/n, in the symbol
table, and the remaining nwords store the components of the structure. Arrays, floating-point
numbers, and big integers are represented as special structures. Picat lists are singly-linked lists.
In a list, [H|T], the address points to a block of two consecutive words, where the first word stores
the car, H, and the second word stores the cdr, T.
G.2 Fetching Arguments of Picat Calls
A C function that defines a Picat predicate should not take any argument. The following function
is used in order to fetch arguments in the current Picat call.
•TERM picat get call arg(int i, int arity): Fetch the ith argument, where
arity is the arity of the predicate, and imust be an integer between 1 and arity. The
validity of the arguments is not checked, and an invalid argument may cause fatal errors.
G.3 Testing Picat Terms
The following functions are provided for testing Picat terms. They return PICAT TRUE when
they succeed and PICAT FALSE when they fail.
120
•int picat is var(TERM t): Term tis a variable.
•int picat is attr var(TERM t): Term tis an attributed variable.
•int picat is dvar(TERM t): Term tis an attributed domain variable.
•int picat is bool dvar(TERM t): Term tis an attributed Boolean variable.
•int picat is integer(TERM t): Term tis an integer.
•int picat is float(TERM t): Term tis a floating-point number.
•int picat is atom(TERM t): Term tis an atom.
•int picat is nil(TERM t): Term tis nil, i.e., the empty list [].
•int picat is list(TERM t): Term tis a list.
•int picat is string(TERM t): Term tis a string.
•int picat is structure(TERM t): Term tis a structure (but not a list).
•int picat is array(TERM t): Term tis an array.
•int picat is compound(TERM t): True if either picat is list(t)
or picat is structure(t) is true.
•int picat is identical(TERM t1, TERM t2):t1 and t2 are identical. This
function is equivalent to the Picat call t1==t2.
•int picat is unifiable(TERM t1, TERM t2):t1 and t2 are unifiable. This
is equivalent to the Picat call not(not(t1=t2)).
G.4 Converting Picat Terms into C
The following functions convert Picat terms to C. If a Picat term does not have the expected type,
then the global C variable exception, which is of type Term, is assigned a term. A C program
that uses these functions must check exception in order to see whether data are converted
correctly. The converted data are only correct when exception is (TERM)NULL.
•long picat get integer(TERM t): Convert the Picat integer tinto C. The term
tmust be an integer; otherwise exception is set to integer expected and 0 is
returned. Note that precision may be lost if tis a big integer.
•double picat get float(TERM t): Convert the Picat float tinto C. The term t
must be a floating-point number; otherwise exception is set to number expected,
and 0.0 is returned.
•(char *) picat get atom name(TERM t): Return a pointer to the string that is
the name of atom t. The term tmust be an atom; otherwise, exception is set to
atom expected, and NULL is returned.
•(char *) picat get struct name(TERM t): Return a pointer to the string that
is the name of structure t. The term tmust be a structure; otherwise, exception is set to
structure expected, and NULL is returned.
121
•int picat get struct arity(TERM t): Return the arity of term t. The term t
must be a structure; otherwise, exception is set to structure expected and 0 is
returned.
G.5 Manipulating and Writing Picat Terms
•int picat unify(TERM t1,TERM t2): Unify two Picat terms t1 and t2. The
result is PICAT TRUE if the unification succeeds, and PICAT FALSE if the unification
fails.
•TERM picat get arg(int i,TERM t): Return the ith argument of term t. The
term tmust be compound, and imust be an integer that is between 1 and the arity of
t; otherwise, exception is set to compound expected, and the Picat integer 0is
returned.
•TERM picat get car(TERM t): Return the car of the list t. The term tmust be a
non-empty list; otherwise exception is set to list expected, and the Picat integer 0
is returned.
•TERM picat get cdr(TERM t): Return the cdr of the list t. The term tmust be a
non-empty list; otherwise exception is set to list expected, and the Picat integer 0
is returned.
•void picat write term(TERM t): Send term tto the standard output stream.
G.6 Building Picat Terms
•TERM picat build var(): Return a free Picat variable.
•TERM picat build integer(long i): Return a Picat integer whose value is i.
•TERM picat build float(double f): Return a Picat float whose value is f.
•TERM picat build atom(char *name): Return a Picat atom whose name is name.
•TERM picat build nil(): Return an empty Picat list.
•TERM picat build list(): Return a Picat list whose car and cdr are free variables.
•TERM picat build structure(char *name, int arity): Return a Picat struc-
ture whose functor is name, and whose arity is arity. The structure’s arguments are all
free variables.
•TERM picat build array(int n): Return a Picat array whose size is n. The array’s
arguments are all free variables.
G.7 Registering C-defined Predicates
The following function registers a predicate that is defined by a C function.
insert_cpred(char *name, int arity, int (*func)())
122
The first argument is the predicate name, the second argument is the arity, and the third argument is
the name of the function that defines the predicate. The function that defines the predicate cannot
take any argument. As described above, picat get call arg(i,arity) is used to fetch
arguments from the Picat call.
For example, the following registers a predicate whose name is "p", and whose arity is 2.
extern int p();
insert_cpred("p", 2, p)
The C function’s name does not need to be the same as the predicate name.
Predicates that are defined in C should be registered after the Picat engine is initialized, and
before any call is executed. One good place for registering predicates is the Cboot() function
in the file cpreds.c, which registers all of the C-defined built-ins of Picat. After registration,
the predicate can be called. All C-defined predicates must be explicitly called with the module
qualifier bp, as in bp.p(a,X).
Example
Consider the Picat predicate:
p(a,X) => X = $f(a).
p(b,X) => X = [1].
p(c,X) => X = 1.2.
where the first argument is given and the second is unknown. The following steps show how to
define this predicate in C, and how to make it callable from Picat.
Step 1 . Write a C function to implement the predicate. The following shows a sample:
#include "picat.h"
p(){
TERM a1, a2, a, b, c, f1, l1, f12;
char *name_ptr;
/*prepare Picat terms */
a1 = picat_get_call_arg(1, 2); /*first argument */
a2 = picat_get_call_arg(2, 2); /*second argument */
a = picat_build_atom("a");
b = picat_build_atom("b");
c = picat_build_atom("c");
f1 = picat_build_structure("f", 1); /*f(a) */
picat_unify(picat_get_arg(1, f1), a);
l1 = picat_build_list(); /*[1] */
picat_unify(picat_get_car(l1), picat_build_integer(1));
picat_unify(picat_get_cdr(l1), picat_build_nil());
f12 = picat_build_float(1.2); /*1.2 */
/*code for the rules */
if (!picat_is_atom(a1))
return PICAT_FALSE;
123
name_ptr = picat_get_name(a1);
switch (*name_ptr){
case ’a’:
return (picat_unify(a1, a) ?
picat_unify(a2, f1) : PICAT_FALSE);
case ’b’:
return (picat_unify(a1, b) ?
picat_unify(a2, l1) : PICAT_FALSE);
case ’c’:
return (picat_unify(a1, c) ?
picat_unify(a2, f12) : PICAT_FALSE);
default: return PICAT_FALSE;
}
}
Step 2 . Insert the following two lines into Cboot() in cpreds.c:
extern int p();
insert_cpred("p", 2, p);
Step 3 . Modify the make file, if necessary, and recompile the system. Now, p/2 is in the group
of built-ins in Picat.
Step 4 . Use bp.p(...) to call the predicate.
picat> bp.p(a,X)
X = f(a)
124
Appendix H
Appendix: Tokens
/*Picat lexical rules
[...] means optional
{...} means 0, 1, or more occurrences
"..." means as-is
/*... */ comment
Tokens to be returned:
Token-type lexeme
=====================
ATOM a string of chars of the atom name
VARIABLE a string of chars of the variable name
INTEGER an integer literal
FLOAT a float literal
STRING a string of chars
OPERATOR a string of chars in the operator
SEPARATOR one of "(" ")" "{" "}" "[" "]"
*/
line_terminator ->
the LF character, also known as "newline"
the CR character, also known as "return"
the CR character followed by the LF character
input_char ->
unicode_input_char but not CR or LF
comment ->
traditional_comment
end_of_line_comment
traditional_comment ->
"/*" comment_tail
comment_tail ->
"*" comment_tail_star
not_star comment_tail
comment_tail_star ->
"/"
"*" comment_tail_star
not_star_not_slash comment_tail
not_star ->
input_char but not "*"
line_terminator
not_star_not_slash ->
input_char but not "*" or "/"
line_terminator
end_of_line_comment ->
125
"%" {input_char} line_terminator
white_space ->
the SP character, also known as "space"
the HT character, also known as "horizontal tab"
the FF character, also known as "form feed"
line_terminator
token ->
atom_token
variable_token
integer_literal
real_literal
string_literal
operator_token
separator_token
atom_token ->
small_letter {alphanumeric_char}
single_quoted_token
variable_token ->
anonymous_variable
named_variable
anonymous variable ->
"_"
named_variable ->
"_" alphanumeric {alphanumeric}
capital_letter {alphanumeric}
alphanumeric ->
alpha_char
decimal_digit
alpha_char ->
underscore_char
letter
letter ->
small_letter
capital_letter
single_quoted_token ->
"’" {string_char} "’"
string_literal ->
"\"" {string_char} "\""
string_char ->
input_char
escape_sequence
integer_literal ->
decimal_numeral
hex_numeral
octal_numeral
binary_numeral
decimal_numeral ->
decimal_digit [decimal_digits_and_underscores]
decimal_digits_and_underscores ->
decimal_digit_or_underscore
decimal_digits_and_underscores decimal_digit_or_underscore
decimal_digit_or_underscore ->
decimal_digit
"_"
126
hex_numeral ->
"0x" hex_digits
"0X" hex_digits
hex_digits ->
hex_digit [hex_digits_and_underscores]
hex_digits_and_underscores ->
hex_digit_or_underscore
hex_digits_and_underscores hex_digit_or_underscore
hex_digit_or_underscore ->
hex_digit
"_"
octal_numeral ->
"0O" octal_digits
"0o" octal_digits
octal_digits ->
octal_digit [octal_digits_and_underscores]
octal_digits_and_underscores ->
octal_digit_or_underscore
octal_digits_and_underscores octal_digit_or_underscore
octal_digit_or_underscore ->
octal_digit
"_"
binary_numeral ->
"0b" binary_digits
"0B" binary_digits
binary_digits:
binary_digit [binary_digits_and_underscores]
binary_digits_and_underscores ->
binary_digit_or_underscore
binary_digits_and_underscores binary_digit_or_underscore
binary_digit_or_underscore:
binary_digit
"_"
real_literal ->
decimal_numeral "." decimal_numeral [exponent_part]
exponent_part ->
exponent_indicator signed_integer
exponent_indicator ->
"e"
"E"
signed_integer ->
[sign] decimal_numeral
sign ->
"+"
"-"
separator ->
one of "(" ")" "{" "}" "[" "]"
operator ->
one of
"=" "!=" ">" ">=" "<" "<=" "=<" ".." "!"
"," ";" ":" "::" "." ". " (dot-whitespace)
127
"=>" "?=>" "==" "!==" ":=" "|" "$" "@"
"/\" "\/" "˜" "ˆ" "<<" ">>"
"+" "-" "*" "**" "/" "/>" "/<" "ˆ"
"#=" "#!=" "#>" "#>=" "#<" "#<=" "#=<"
"#/\" "#\/" "#˜" "#ˆ" "#=>" "#<=>"
"@>" "@>=" "@<" "@<=" "@=<"
small_letter ->
one of "a" "b" ... "z"
capital_letter ->
one of "A" "B" ... "Z"
decimal_digit ->
one of "0" "1" "2" "3" "4" "5" "6" "7" "8" "9"
hd ->
hex_digit
hex_digit ->
one of
"0" "1" "2" "3" "4" "5" "6" "7" "8" "9"
"a" "b" "c" "d" "e" "f" "A" "B" "C" "D" "E" "F"
octal_digit ->
one of "0" "1" "2" "3" "4" "5" "6" "7"
binary_digit ->
one of "0" "1"
escape_sequence ->
"\b" /*\u0008: backspace BS */
"\t" /*\u0009: horizontal tab HT */
"\n" /*\u000a: linefeed LF */
"\f" /*\u000c: form feed FF */
"\r" /*\u000d: carriage return CR */
"\"" /*\u0022: double quote " */
"\’" /*\u0027: single quote ’ */
"\\" /*\u005c: backslash \ */
unicode_escape
unicode_escape ->
"\u" hd hd hd hd
"\U" hd hd hd hd hd hd hd hd
128
Appendix I
Appendix: Grammar
/*Picat syntax rules
[...] means optional
{...} means 0, 1, or more occurrences
(a | b) means choice
"..." means a token
%... one-line comment
input tokens:
atom
variable
integer
float
operator
separator
eor is "." followed by a white space or eof
*/
program ->
[module_declaration]
{import_declaration}
program_body
program_body ->
{predicate_definition | function_definition | actor_definition}
module_declaration ->
"module" atom eor
import_declaration ->
import import_item {"," import_item} eor
import_item ->
atom
predicate_definition ->
{predicate_directive} predicate_rule_or_fact {predicate_rule_or_fact}
function_definition ->
{function_directive} function_rule_or_fact {function_rule_or_fact}
actor_definition ->
["private"] action_rule {(action_rule
| nonbacktrackable_predicate_rule)}
function_directive ->
"private"
"table"
predicate_directive ->
"private"
"table" ["(" table_mode {"," table_mode} ")" ]
"index" index_declaration {"," index_declaration}
129
index_declaration ->
"(" index_mode {"," index_mode} ")"
index_mode ->
"+"
"-"
table_mode ->
"+"
"-"
"min"
"max"
"nt"
predicate_rule_or_fact ->
predicate_rule
predicate_fact
function_rule_or_fact ->
function_rule
function_fact
predicate_rule ->
head ["," condition] ("=>" | "?=>") body eor
nonbacktrackable_predicate_rule ->
head ["," condition] "=>" body eor
predicate_fact ->
head eor
head ->
atom ["(" [term {"," term}] ")"]
function_rule ->
head "=" expression ["," condition] "=>" body eor
function_fact ->
head "=" argument eor
action_rule ->
head ["," condition] "," "{" event_pattern "}" => body eor
event_pattern ->
term {’,’ term}
condition -> goal
body -> goal
goal ->
disjunctive_goal
argument ->
negative_goal
disjunctive_goal ->
disjunctive_goal ";" conjunctive_goal
conjunctive_goal
conjunctive_goal ->
conjunctive_goal "," negative_goal
negative_goal
negative_goal ->
"not" negative_goal
equiv_constr
equiv_constr ->
equiv_constr "#<=>" impl_constr
130
impl_constr
impl_constr ->
impl_constr "#=>" or_constr
or_constr
or_constr ->
or_constr "#\/" xor_constr
xor_constr
xor_constr ->
xor_constr "#ˆ" and_constr
and_constr
and_constr ->
and_constr "#/\" not_constr
not_constr
not_constr ->
"#˜" not_constr
enclosed_goal
enclosed_goal ->
"if" goal "then" goal {"elseif" goal "then" goal} "else" goal "end"
"foreach" "(" iterator {"," (iterator | condition)} ")" goal "end"
"while" "(" goal ")" ["do"] goal "end"
"do" goal "while" "(" goal ")"
expression {bin_rel_op expression}
bin_rel_op ->
"="
"!="
":="
"=="
"!=="
">"
">="
"<"
"=<"
"<="
"::"
"in"
"notin"
"#="
"#!="
"#>"
"#>="
"#<"
"#=<"
"#<="
"@>"
"@>="
"@<"
"@=<"
"@<="
expression ->
concat_expression
concat_expression ->
range_expression ["++" concat_expression]
range_expression ->
or_expression [".." or_expression [".." or_expression]]
or_expression ->
xor_expression
or_expression "\/" xor_expression
xor_expression ->
131
and_expression
xor_expression "ˆ" and_expression % bit-wise xor
and_expression ->
shift_expression
and_expression "/\" shift_expression
shift_expression ->
additive_expression
shift_expr ( "<<" | ">>" ) additive_expression
additive_expression ->
multiplicative_expression
additive_expression "+" multiplicative_expression
additive_expression "-" multiplicative_expression
multiplicative_expression ->
unary_expression
multiplicative_expression "*" unary_expression
multiplicative_expression "/" unary_expression
multiplicative_expression "//" unary_expression
multiplicative_expression "/>" unary_expression
multiplicative_expression "/<" unary_expression
multiplicative_expression "div" unary_expression
multiplicative_expression "mod" unary_expression
multiplicative_expression "rem" unary_expression
unary_expression ->
power_expression
"+" unary_expression
"-" unary_expression
"˜" unary_expression % bit-wise complement
power_expression ->
primary_expression ["**" unary_expression]
primary_expression ->
"(" goal ")"
variable "[" argument ["," argument] "]" % subscript notation
variable "@" term ["@"] % as-pattern
variable
integer
float
atom_or_call
list_expression
array_expression
function_call
term_constructor
primary_expression "." atom_or_call % dot-notation
atom_or_call ->
atom ["(" [argument {"," argument}] ")"]
list_expression ->
"[]"
"[" argument list_expression_suffix "]"
list_expression_suffix ->
":" iterator {"," (iterator | condition)} % list comprehension
{"," argument} ["|" argument]
array_expression ->
"{}"
"{" argument array_expression_suffix "}"
array_expression_suffix ->
":" iterator {"," (iterator | condition)} % array comprehension
{"," argument}
132
function_call ->
[primary_expression "."] atom "(" [argument {"," argument}] ")"
variable_list ->
"[" [variable {"," variable}] "]"
term_constructor ->
"$" goal ["$"]
/*a term has the same form as a goal except that it cannot contain loops
or if-then-else. Note that subscript notations, range expressions, dot
notations, and list comprehensions are still treated as functions in
term constructors */
133
Appendix J
Appendix: Operators
Precedence Operators
Highest .,@
** (right-associative)
unary +, unary -,˜
*,/,//,/<,/>,div,mod,rem
binary +, binary -
>>,<<
/\
ˆ
\/
..
++ (right-associative)
=,!=,:=,==,!==,=:=,<,=<,<=,>,>=,::,in,notin
#=,#!=,#<,#=<,#<=,#>,#>=,@<,@=<,@<=,@>,@>=
#˜
#/\
#ˆ
#\/
#=> (right-associative)
#<=>
not,once
,(right-associative), && (right-associative)
Lowest ;(right-associative), || (right-associative)
134
Appendix K
Appendix: The Library Modules
Module basic (imported by default)
•X!= Y
•X!== Y
•X:= Y
•X<Y
•X<= Y
•X=Y
•X=< Y
•X=:= Y
•X== Y
•X>Y
•X>= Y
•X@< Y
•X@<= Y
•X@=< Y
•X@> Y
•X@>= Y
•T erm1++ T erm2=List
•[X:Iin D,... ] = List
•L.. U=List
•L.. Step .. U=List
•-X=Y
•+X=Y
•X+Y=Z
•X-Y=Z
•X*Y=Z
•X/Y=Z
•X// Y=Z
•Xdiv Y=Z
•X/< Y=Z
•X/> Y=Z
•X** Y=Z
•Xmod Y=Z
•Xrem Y=Z
•˜X=Y
•X\/ Y=Z
•X/\ Y=Z
•XˆY=Z
•X<< Y=Z
•X>> Y=Z
•V ar[Index1,...,Indexn]
•Goal1,Goal2
•Goal1&& Goal2
•Goal1;Goal2
•Goal1|| Goal2
•acyclic term(T erm)
•and to list(Conj) = List
•append(X,Y,Z)(nondet)
•append(X,Y,Z,T)(nondet)
•apply(S,Arg1,...,Argn) = V al
•arity(Struct) = Arity
•array(T erm)
•atom(T erm)
•atom chars(Atm) = String
•atom codes(Atm) = List
•atomic(T erm)
•attr var(T erm)
•avg(ListOrArray) = V al
•between(F rom,T o,X)(nondet)
•bind vars(T erm,V al)
•call(S,Arg1,...,Argn)
•call cleanup(S,Cleanup)
•catch(S,Exception,Handler)
•char(T erm)
•chr(Code) = Char
•clear(Map)
•compare terms(T erm1,T erm2) = Res
•compound(T erm)
•copy term(T erm1) = T erm2
•count all(Call) = Int
•delete(List,X) = ResList
•delete all(List,X) = ResList
•different terms(T erm1,T erm2)
•digit(Char)
•dvar(T erm)
•dvar or int(T erm)
•fail
•false
•find all(T emplate,Call) = List
•findall(T emplate,Call) = List
•first(Compound) = T erm
•flatten(List1) = List2
•float(T erm)
•fold(F,ACC,List) = Res
•freeze(X,Goal)
•get(Map,Key) = V al
•get(Map,Key,Def aultV al)=V al
•get attr(AttrV ar,Key) = V al
•get attr(AttrV ar,Key,Def aultV al)=V al
•get global map(ID) = M ap
•get global map() = M ap
135
•get heap map(ID) = M ap
•get heap map() = M ap
•get table map(ID) = M ap
•get table map() = M ap
•ground(T erm)
•handle exception(T erm,T erm)
•has key(Map,Key)
•hash code(T erm) = Int
•head(List) = T erm
•heap is empty(Heap)
•heap pop(Heap) = Elm
•heap push(Heap,Elm)
•heap size(Heap) = Size
•heap to list(Heap) = List
•heap top(Heap) = Elm
•insert(List,Index,Elm) = ResList
•insert all(List,Index,AList) = ResList
•insert ordered(List,T erm) = R
•insert ordered down(List,T erm) = R
•int(T erm)
•integer(T erm)
•is(Exp,Exp)
•keys(Map) = List
•last(Compound) = T erm
•len(T erm) = Len
•length(T erm) = Len
•list(T erm)
•list to and(List) = Conj
•lowercase(Char)
•map(F unc,List1,List2) = ResList
•map(F uncOrList,ListOrF unc) = ResList
•map(T erm)
•map to list(Map) = List
•max(ListOrArray) = V al
•max(X,Y) = V al
•maxint small() = Int
•maxof(Call,Objective)
•maxof(Call,Objective,ReportCall)
•maxof inc(Call,Objective)
•maxof inc(Call,Objective,ReportCall)
•membchk(T erm,List)
•member(T erm,List)(nondet)
•min(ListOrArray) = V al
•min(X,Y) = V al
•minint small() = Int
•minof(Call,Objective)
•minof(Call,Objective,ReportCall)
•minof inc(Call,Objective)
•minof inc(Call,Objective,ReportCall)
•name(Struct) = N ame
•new array(D1,...,Dn) = Arr
•new list(N) = List
•new list(N,InitV al) = List
•new map(Int,P airsList) = M ap
•new map(IntOrP airsList) = Map
•new max heap(IntOrList) = Heap
•new min heap(IntOrList) = Heap
•new set(Int,ElmsList) = M ap
•new set(IntOrElmsList) = M ap
•new struct(Name,IntOrList) = Struct
•nonvar(T erm)
•not Call
•nth(I,ListOrArray,V al)(nondet)
•number(T erm)
•number chars(Num) = String
•number codes(Num) = List
•number vars(T erm,N0) = N1
•once Call
•ord(Char) = Int
•parse radix string(String,Base) = Int
•parse term(String) = T erm
•parse term(String,T erm,V ars)
•post event(X,Event)
•post event any(X,Event)
•post event bound(X)
•post event dom(X,Event)
•post event ins(X)
•prod(ListOrArray) = V al
•put(Map,Key)
•put(Map,Key,V al)
•put attr(V ar,Key)
•put attr(V ar,Key,V al)
•real(T erm)
•reduce(F unc,List) = Res
•reduce(F unc,List,InitV al) = Res
•remove dups(ListOrArray) = ResList
•repeat (nondet)
•reverse(ListOrArray) = Res
•second(Compound) = T erm
•select(X,List,ResList)(nondet)
•size(Map) = Size
•slice(ListOrArray,F rom)
•slice(ListOrArray,F rom,T o)
•sort(ListOrArray) = Sorted
•sort(ListOrArray,KeyIndex) = Sorted
•sort down(ListOrArray) = Sorted
•sort down(ListOrArray,KeyIndex) = Sorted
•sort down remove dups(ListOrArray) = Sorted
•sort down remove dups(ListOrArray,KeyIndex) =
Sorted
•sort remove dups(ListOrArray) = Sorted
•sort remove dups(ListOrArray,KeyIndex) = Sorted
•sorted(ListOrArray)
•sorted down(ListOrArray)
•string(T erm)
•struct(T erm)
•subsumes(T erm1,T erm2)
•sum(ListOrArray) = V al
•tail(List) = T erm
•throw E
•to array(List) = Array
•to atom(String) = Atom
•to binary string(Int) = String
•to codes(T erm) = List
•to fstring(F ormat,Args . . .) = String
•to hex string(Int) = String
•to int(NumOrCharOrStr) = Int
136
•to integer(NumOrCharOrStr) = Int
•to list(Struct) = List
•to lowercase(String) = LString
•to number(NumOrCharOrStr) = N umber
•to oct string(Int) = String
•to radix string(Int,Base) = String
•to real(NumOrStr) = Real
•to string(T erm) = String
•to uppercase(String) = U String
•true
•uppercase(Char)
•values(Map) = List
•var(T erm)
•variant(T erm1,T erm2)
•vars(T erm) = V ars
•zip(List1,List2) = List
•zip(List1,List2,List3) = List
•zip(List1,List2,List3,List4) = List
Module math (imported by default)
•abs(X) = V al
•acos(X) = V al
•acosh(X) = V al
•acot(X) = V al
•acoth(X) = V al
•acsc(X) = V al
•acsch(X) = V al
•asec(X) = V al
•asech(X) = V al
•asin(X) = V al
•asinh(X) = V al
•atan(X) = V al
•atan2(X,Y) = V al
•atanh(X) = V al
•ceiling(X) = V al
•cos(X) = V al
•cosh(X) = V al
•cot(X) = V al
•coth(X) = V al
•csc(X) = V al
•csch(X) = V al
•e() = 2.71828182845904523536
•even(Int)
•exp(X) = V al
•floor(X) = V al
•frand() = V al
•frand(Low,High) = V al
•gcd(A,B) = V al
•log(X) = V al
•log(B,X) = V al
•log10(X) = V al
•log2(X) = V al
•modf(X)=(IntV al,F ractV al)
•odd(Int)
•pi() = 3.14159265358979323846
•pow(X,Y) = V al
•pow mod(X,Y,Z) = V al
•prime(Int)
•primes(Int) = List
•rand max() = V al
•random = V al
•random(Low,High) = V al
•random(Seed) = V al
•random2() = Int
•round(X) = V al
•sec(X) = V al
•sech(X) = V al
•sign(X) = V al
•sin(X) = V al
•sinh(X) = V al
•sqrt(X) = V al
•tan(X) = V al
•tanh(X) = V al
•to degrees(Radian) = Degree
•to radians(Degree) = Radian
•truncate(X) = V al
Module io (imported by default)
•at end of stream(F D)
•close(F D)
•flush(F D)
•flush()
•nl(F D)
•nl()
•open(Name) = F D
•open(Name,Mode) = F D
•peek byte(F D) = V al
•peek char(F D) = V al
•print(F D,T erm)
•print(T erm)
•printf(F D,F ormat,Args . . .)
•println(F D,T erm)
•println(T erm)
•read atom(F D) = Atom
•read atom() = Atom
•read byte(F D) = V al
•read byte(F D,N) = List
•read byte() = V al
•read char(F D) = V al
•read char(F D,N) = String
•read char() = V al
•read char code(F D) = V al
•read char code(F D,N) = List
•read char code() = V al
•read file bytes(F ile) = List
•read file bytes() = List
•read file chars(F ile) = String
•read file chars() = String
•read file codes(F ile) = List
•read file codes() = List
•read file lines(F ile) = List
•read file lines() = List
•read file terms(F ile) = List
•read file terms() = List
•read int(F D) = Int
•read int() = Int
•read line(F D) = String
137
•read line() = String
•read number(F D) = N umber
•read number() = N umber
•read picat token(F D) = T okenV alue
•read picat token(F D,T okenT ype,T okenV alue)
•read picat token(T okenT ype,T okenV alue)
•read picat token() = T okenV alue
•read real(F D) = Real
•read real() = Real
•read term(F D) = T erm
•read term() = T erm
•readln(F D) = String
•readln() = String
•write(F D,T erm)
•write(T erm)
•write byte(Bytes)
•write byte(F D,Bytes)
•write char(Chars)
•write char(F D,Chars)
•write char code(Codes)
•write char code(F D,Codes)
•writef(F D,F ormat,Args . . .)
•writeln(F D,T erm)
•writeln(T erm)
Module ordset
•delete(OSet,Elm) = OSet1
•disjoint(OSet1,OSet2)
•insert(OSet,Elm) = OSet1
•intersection(OSet1,OSet2)=OSet3
•new ordset(List)
•ordset(T erm)
•subset(OSet1,OSet2)
•subtract(OSet1,OSet2)=OSet3
•union(OSet1,OSet2)=OSet3
Module os
•cd(P ath)
•chdir(P ath)
•cp(F romP ath,T oP ath)
•cwd() = P ath
•directory(P ath)
•dir
•env exists(Name)
•executable(P ath)
•exists(P ath)
•file(P ath)
•file base name(P ath) = String
•file directory name(P ath) = String
•file exists(P ath)
•getenv(EnvString) = String
•listdir(P ath) = List
•ls
•mkdir(P ath)
•pwd() = P ath
•readable(P ath)
•rename(Old,N ew)
•rm(P ath)
•rmdir(P ath)
•separator() = V al
•size(P ath) = Int
•writable(P ath)
Modules cp,sat, and mip
•#˜X
•X#!= Y
•X#/\ Y
•X#< Y
•X#<= Y
•X#<=> Y
•X#= Y
•X#=< Y
•X#=> Y
•X#> Y
•X#>= Y
•X#\/ Y
•X#ˆ Y
•V ars :: Exp
•V ars notin Exp
•all different(F DV ars)
•all different except 0(F DV ars)
•all distinct(F DV ars)
•assignment(F DV ars1,F DV ars2)
•at least(N,L,V):
•at most(N,L,V):
•circuit(F DV ars)
•count(V,F DV ars,N)
•count(V,F DV ars,Rel,N)
•cumulative(Ss,Ds,Rs,Limit)
•decreasing(L)
•decreasing strict(L)
•diffn(RectangleList)
•disjunctive tasks(T asks)(cp only)
•element(I,List,V)
•exactly(N,L,V):
•fd degree(F DV ar) = Degree (cp only)
•fd disjoint(DV ar1,DV ar2)
•fd dom(F DV ar) = List
•fd false(F DV ar,Elm)
•fd max(F DV ar) = M ax
•fd min(F DV ar) = M in
•fd min max(F DV ar,M in,M ax)
•fd next(F DV ar,Elm) = N extElm
•fd prev(F DV ar,Elm) = P revElm
•fd set false(F DV ar,Elm)(cp only)
•fd size(F DV ar) = Size
•fd true(F DV ar,Elm)
•fd vector min max(Min,Max)
•global cardinality(List,P airs)
•increasing(L)
•increasing strict(L)
•indomain(V ar)(nondet) (cp only)
•indomain down(V ar)(nondet) (cp only)
•lex le(L1,L2)
•lex lt(L1,L2)
•matrix element(Matrix,I,J,V)
•neqs(NeqList)(cp only)
•new dvar() = F DV ar
•new fd var() = F DV ar
•regular(X, Q, S, D, Q0, F )
•scalar product(A,X,P roduct)
•scalar product(A,X,Rel,P roduct)
•serialized(Starts,Durations)
•solve(Options,V ars)(nondet)
•solve(V ars)(nondet)
•solve all(Options,V ars) = List
•solve all(V ars) = List
•solve suspended (cp only)
•solve suspended(Options)(cp only)
•subcircuit(F DV ars)
•table in(DV ars,R)
•table notin(DV ars,R)
138
Module planner
•best plan(S,Limit,P lan)
•best plan(S,Limit,P lan,Cost)
•best plan(S,P lan)
•best plan(S,P lan,P lanCost)
•best plan bb(S,Limit,P lan)
•best plan bb(S,Limit,P lan,Cost)
•best plan bb(S,P lan)
•best plan bb(S,P lan,P lanCost)
•best plan bin(S,Limit,P lan)
•best plan bin(S,Limit,P lan,Cost)
•best plan bin(S,P lan)
•best plan bin(S,P lan,P lanCost)
•best plan nondet(S,Limit,P lan)(nondet)
•best plan nondet(S,Limit,P lan,Cost)(nondet)
•best plan nondet(S,P lan)(nondet)
•best plan nondet(S,P lan,P lanCost)(nondet)
•best plan unbounded(S,Limit,P lan)
•best plan unbounded(S,Limit,P lan,Cost)
•best plan unbounded(S,P lan)
•best plan unbounded(S,P lan,P lanCost)
•current plan()=P lan
•current resource()=Amount
•current resource plan cost(Amount,P lan,Cost)
•is tabled state(S)
•plan(S,Limit,P lan)
•plan(S,Limit,P lan,Cost)
•plan(S,P lan)
•plan(S,P lan,P lanCost)
•plan unbounded(S,Limit,P lan)
•plan unbounded(S,Limit,P lan,Cost)
•plan unbounded(S,P lan)
•plan unbounded(S,P lan,P lanCost)
Module datetime
•current datetime() = DateT ime
•current day() = W Day
•current date() = Date
•current time() = T ime
Module sys (imported by default)
•abort
•cl
•cl(F ile)
•cl facts(F acts)
•cl facts(F acts,IndexInf o)
•cl facts table(F acts)
•cl facts table(F acts,IndexInf o)
•command(String)
•compile(F ile)
•debug
•exit
•garbage collect(Size)
•garbage collect
•halt
•initialize table
•load(F ile)
•loaded modules()
•nodebug
•nospy
•notrace
•spy F unctor
•statistics(Name,V alue)(nondet)
•statistics
•time(Goal)
•time2(Goal)
•time out(Goal,Limit,Res)
•trace
Module util
•array matrix to list(Matrix) = List
•array matrix to list matrix(AMatrix) = LMatrix
•chunks of(List,K) = ListOf Lists
•find(String,SubString,F rom,T o)(nondet)
•find first of(T erm,P attern) = Index
•find ignore case(String,SubString,F rom,T o)(nondet)
•find last of(T erm,P attern) = Index
•join(W ords) = String
•join(W ords,Separator) = String
•list matrix to array matrix(LMatrix) = AMatrix
•lstrip(List) = List
•lstrip(List,Elms) = List
•matrix multi(MatrixA,MatrixB) = M atrixC
•permutation(List,P erm)(nondet)
•permutations(List) = Lists
•power set(List) = Lists
•replace(T erm,Old,New) = NewT erm
•replace at(T erm,Index,New) = NewT erm
•rstrip(List) = List
•rstrip(List,Elms) = List
•split(List) = W ords
•split(List,Separators) = W ords
•strip(List) = List
•strip(List,Elms) = List
•take(List,K) = List
•transpose(Matrix) = T ransposed
139
Index
abort/0, 21, 109, 139
abs/1, 102, 137
acos/1, 105, 137
acosh/1, 106, 137
acot/1, 105, 137
acoth/1, 106, 137
acsc/1, 105, 137
acsch/1, 106, 137
action/4, 65
acyclic term/1, 41, 135
all different/1, 91, 92, 138
all different except 0/1, 92, 138
all distinct/1, 92, 94, 138
and to list/1, 41, 135
any-port, 15, 81, 82
append/3, 30, 44, 135
append/4, 30, 135
apply, 14, 39, 73, 135
arity/1, 24, 34, 135
array/1, 34, 135
array matrix to list/1, 115, 139
array matrix to list matrix/1, 115, 139
asec/1, 137
asech/1, 106, 137
asin/1, 104, 105, 137
asinh/1, 106, 137
assignment/2, 92, 138
at end of stream/1, 77, 137
atan/1, 105, 137
atan2/2, 105, 137
atanh/1, 106, 137
atleast/3, 92, 138
atmost/3, 92, 138
atom/1, 27, 135
atom chars/1, 27, 135
atom codes/1, 27, 135
atomic/1, 27, 135
attr var/1, 26, 135
avg/1, 31, 35, 135
best plan/2, 66, 139
best plan/3, 66, 139
best plan/4, 66, 139
best plan bb/2, 67, 139
best plan bb/3, 67, 139
best plan bb/4, 67, 139
best plan bin/2, 67, 139
best plan bin/3, 67, 139
best plan bin/4, 67, 139
best plan nondet/2, 67, 139
best plan nondet/3, 67, 139
best plan nondet/4, 66, 139
best plan unbounded/2, 68, 139
best plan unbounded/3, 68, 139
best plan unbounded/4, 68, 139
between/3, 29, 135
bind vars/2, 37, 135
bool dvar/1, 26
bound-port, 15, 81, 82
call cleanup/2, 14, 39, 59, 135
call, 14, 39, 135
catch/3, 14, 39, 59, 135
cd/1, 99, 138
ceiling/1, 103, 137
char/1, 27, 135
chdir/1, 99, 138
chr/1, 27, 88, 135
chunks of/2, 114, 139
circuit/1, 92, 94, 138
cl/0, 20, 109, 139
cl/1, 20, 71, 108, 139
cl facts/1, 109, 139
cl facts/2, 109, 139
cl facts table/1, 109, 139
cl facts table/2, 109
clear/1, 35, 135
close/1, 80, 137
command/1, 139
compare terms/2, 41, 135
compile/1, 20, 108, 139
compile bp/1, 108
compound/1, 30, 135
copy term/1, 24, 135
140
cos/1, 104, 137
cosh/1, 105, 137
cot/1, 104, 137
coth/1, 106, 137
count/3, 92, 138
count/4, 92, 138
count all/1, 135
count all/2, 14, 40
cp/2, 99, 138
csc/1, 104, 137
csch/1, 106, 137
cumulative/4, 93, 94, 138
current date/0, 118, 139
current datetime/0, 118, 139
current day/0, 118, 139
current plan/0, 67, 139
current resource/0, 67, 139
current resource plan cost/3, 68, 139
current time/0, 118, 139
cwd/0, 99, 138
debug/0, 22, 109, 139
decreasing/1, 93, 138
decreasing strict/1, 93, 138
delete/2, 31, 117, 135, 138
delete all/2, 31, 135
diagonal1/1, 115
diagonal2/1, 115
different terms/2, 41, 84, 135
diffn/1, 93, 138
digit/1, 27, 135
dir/0, 138
directory/1, 99, 100, 138
disjoint/2, 117, 138
disjunctive tasks/1, 93, 138
dom-port, 15, 81, 82
drop/2, 114
dvar/1, 26, 135
dvar or int/1, 26, 135
element/3, 93, 138
end of file, 77
env exists/1, 101, 138
even/1, 107, 137
exactly/3, 93, 138
executable/1, 100, 138
exists/1, 100, 138
exit/0, 19, 113, 139
exp/1, 103, 137
e, 102, 137
fail, 5, 17, 46, 135
false, 5, 46, 135
fd degree/1, 87, 138
fd disjoint/2, 87, 138
fd dom/1, 87, 138
fd false/2, 88, 138
fd max/1, 88, 138
fd min/1, 88, 138
fd min max/3, 88, 138
fd next/2, 88, 138
fd prev/2, 88, 138
fd set false/2, 88, 138
fd size/1, 88, 138
fd true/2, 88, 138
fd vector min max/2, 87, 138
file/1, 100, 138
file base name/1, 100, 138
file directory name/1, 100, 138
file exists/1, 100, 138
final/1, 65
final/3, 65
find/4, 114, 139
find all/2, 40
find all, 14, 135
find first of/2, 114, 139
find ignore case/4, 114, 139
find last of/2, 114, 139
findall/2, 14, 40
findall, 14, 73, 135
first/1, 31, 35, 135
flatten/1, 31, 135
float/1, 29, 135
floor/1, 103, 137
flush/0, 137
flush/1, 80, 137
fold/3, 135
frand/0, 107, 137
frand/2, 107, 137
freeze/2, 14, 40, 84, 135
garbage collect/0, 113, 139
garbage collect/1, 113, 139
gcd/2, 107, 137
get/2, 3, 16, 24, 30, 35, 135
get/3, 36, 135
get attr/2, 3, 26, 135
get attr/3, 26, 135
get global map/0, 16, 41, 135
get global map/1, 16, 135
get heap map/0, 16, 41, 136
get heap map/1, 16, 136
141
get table map/0, 16, 41, 136
get table map/1, 16, 136
getenv/1, 101, 138
global cardinality/2, 93, 138
glpk, 96
ground/1, 41, 136
gurobi, 96
halt/0, 1, 19, 113, 139
handle exception/2, 136
has key/2, 3, 36, 136
hash code/1, 24, 136
head/1, 31, 136
heap is empty/1, 36, 136
heap pop/1, 36, 136
heap push/2, 37, 136
heap size/1, 37, 136
heap to list/1, 37, 136
heap top/1, 37, 136
help/0, 1
heuristic/1, 65
import, 11, 71
increasing/1, 93, 138
increasing strict/1, 93, 138
index, 6, 47
indomain/1, 94, 138
indomain down/1, 95, 138
initialize table/0, 62, 139
insert/2, 117, 138
insert/3, 31, 136
insert all/3, 31, 136
insert ordered/2, 31
insert ordered/3, 136
insert ordered down/2, 31
insert ordered down/3, 136
ins-port, 15, 81, 82, 84
int/1, 29, 136
integer/1, 29, 82, 136
intersection/2, 117, 138
is/2, 136
is tabled state/1, 68, 139
join/1, 114, 139
join/2, 114, 139
keys/1, 36, 136
last/1, 31, 35, 136
len/1, 27, 31, 34, 35
length/1, 27, 30, 31, 35, 77, 136
lex le/2, 93, 138
lex lt/2, 93, 138
list/1, 31, 136
list matrix to array matrix/1, 139
list to and/1, 41, 136
listdir/1, 99, 138
load/1, 12, 20, 71, 108, 139
loaded modules/0, 73, 139
log/1, 103, 137
log/2, 104, 137
log10/1, 103, 137
log2/1, 104, 137
lowercase/1, 136
ls/0, 138
lstrip/1, 115, 139
lstrip/2, 115, 139
map/1, 36, 136
map/2, 40, 136
map/3, 40, 136
map to list/1, 36, 136
matrix element/4, 94, 138
matrix multi/2, 115, 139
max/1, 31, 35, 136
max/2, 29, 48, 136
maxint small/0, 29, 136
maxof/2, 14, 40, 136
maxof/3, 14, 40, 136
maxof inc/2, 14, 40, 136
maxof inc/3, 14, 40, 136
membchk/2, 31, 136
member/2, 6, 29, 31, 73, 136
min/1, 31, 35, 136
min/2, 29, 48, 136
minint small/0, 29, 136
minof/2, 14, 40, 136
minof/3, 14, 40, 136
minof inc/2, 14, 40, 136
minof inc/3, 14, 40, 136
mkdir/1, 99, 138
modf/1, 103, 137
module, 11, 71
name/1, 24, 34, 136
neqs/1, 94, 138
new array, 2, 34, 136
new dvar/0, 88, 138
new fd var/0, 138
new list/1, 32, 136
new list/2, 32, 136
new map/1, 1, 35, 36, 136
new map/2, 36, 136
new max heap/1, 37, 136
new min heap/1, 37, 136
142
new ordset/1, 117, 138
new set/1, 1, 36, 136
new set/2, 36, 136
new struct/2, 1, 34, 136
nextto/3, 116
nl/0, 137
nl/1, 137
nodebug/0, 22, 109, 139
nonvar/1, 26, 136
nospy/0, 109
nospy, 139
not/1, 14, 136
notrace/0, 22, 109
notrace, 139
not, 24, 46
nth/3, 32, 35, 136
number/1, 29, 136
number chars/1, 29, 136
number codes/1, 29, 136
number vars/2, 41, 136
nvalue/2, 94
odd/1, 107, 137
once/1, 6, 14, 136
once, 24, 43, 46, 82, 84, 112
open/1, 59, 74, 75, 78, 80, 137
open/2, 74, 75, 78, 80, 137
ord/1, 27, 88, 136
ordset/1, 117, 138
parse radix string/2, 41, 136
parse term/1, 42, 136
parse term/3, 41, 136
peek byte/1, 77, 137
peek char/1, 77, 137
perm/1, 139
permutation/2, 116, 139
permutations/1, 116
picat, 1, 18
pi, 7, 39, 102, 104, 105, 137
plan/2, 66, 139
plan/3, 66, 139
plan/4, 66, 139
plan unbounded/2, 68, 139
plan unbounded/3, 68, 139
plan unbounded/4, 68, 139
post event/2, 15, 81, 136
post event any/2, 81, 136
post event bound/1, 81, 136
post event dom/2, 81, 136
post event ins/1, 81, 136
pow/2, 103, 137
pow mod/3, 103, 137
power set/1, 139
prime/1, 107, 137
primes/1, 107, 137
print/1, 79, 137
print/2, 79, 137
printf, 79, 119, 137
println/1, 79, 137
println/2, 79, 137
prod/1, 32, 136
put/2, 26, 36, 136
put/3, 3, 16, 36, 136
put attr/2, 136
put attr/3, 3, 26, 136
pwd/0, 99, 138
rand max/0, 106, 137
random/0, 106, 137
random/1, 106, 137
random/2, 106, 137
random2/0, 106, 137
read atom/0, 137
read atom/1, 137
read byte/0, 76, 137
read byte/1, 76, 137
read byte/2, 76, 77, 137
read char/0, 75, 137
read char/1, 75, 137
read char/2, 75, 77, 137
read char code/0, 75, 137
read char code/1, 75, 137
read char code/2, 75, 137
read file bytes/0, 76, 137
read file bytes/1, 76, 137
read file chars/0, 76, 137
read file chars/1, 76, 137
read file codes/0, 76, 137
read file codes/1, 76, 137
read file lines/0, 76, 137
read file lines/1, 76, 137
read file terms/0, 76, 137
read file terms/1, 76, 137
read int/0, 58, 75, 137
read int/1, 75, 137
read line/0, 76, 138
read line/1, 76, 77, 137
read number/0, 138
read number/1, 138
read picat token/0, 75, 138
143
read picat token/1, 75, 138
read picat token/2, 75, 138
read picat token/3, 75, 138
read real/0, 75, 138
read real/1, 75, 138
read term/0, 76, 138
read term/1, 76, 138
readable/1, 100, 138
readln/0, 76, 138
readln/1, 76, 138
real/1, 29, 136
reduce/2, 41, 136
reduce/3, 41, 136
regular/6, 94, 138
remove dups/1, 32, 136
rename/2, 99, 138
repeat/0, 46, 136
replace/3, 114, 139
replace at/3, 114, 139
reverse/1, 32, 35, 136
rm/1, 99, 138
rmdir/1, 99, 138
round/1, 103, 137
rows/1, 115
rstrip/1, 115, 139
rstrip/2, 115, 139
scalar product/3, 94, 138
scalar product/4, 94, 138
sec/1, 104, 137
sech/1, 105, 137
second/1, 42, 136
select/3, 32, 136
separator/0, 98, 138
sequence/2, 65
serialized/2, 94, 138
sign/1, 102, 137
sin/1, 104, 137
sinh/1, 105, 137
size/1, 36, 100, 136, 138
slice/2, 33, 35, 136
slice/3, 33, 35, 136
solve/1, 12, 86, 87, 94, 138
solve/2, 12, 86, 94, 95, 138
solve all/1, 95, 138
solve all/2, 94, 138
solve suspended/0, 95, 138
solve suspended/1, 95, 138
sort/1, 32, 35, 136
sort/2, 32, 35, 136
sort down/1, 32, 35, 136
sort down/2, 136
sort down remove dups/1, 33, 35, 136
sort down remove dups/2, 136
sort remove dups/1, 32, 35, 136
sort remove dups/2, 32, 35, 136
sorted/1, 136
sorted down/1, 136
split/1, 115, 139
split/2, 115, 139
spy/1, 23, 109, 139
sqrt/1, 103, 137
statistics/0, 110, 139
statistics/2, 110, 111, 139
stderr, 80
stdin, 80
stdout, 80
string/1, 33, 136
strip/1, 115, 139
strip/2, 115, 139
struct/1, 34, 136
subcircuit/1, 94, 138
subset/2, 117, 138
subsumes/2, 42, 136
subtract/2, 117, 138
sum/1, 33, 136
table in/2, 88, 138
table notin/2, 88, 138
table, 10, 11, 60
tail/1, 33, 136
take/2, 115, 139
tan/1, 104, 137
tanh/1, 105, 137
throw, 5, 13, 47, 59, 136
time/1, 14, 112, 139
time2/1, 112, 139
time out/3, 14, 112, 139
to array/1, 33, 136
to atom/1, 136
to binary string/1, 29, 136
to codes/1, 24, 29, 136
to degrees/1, 104, 137
to float/1, 29
to fstring/2, 29
to fstring, 24, 119, 136
to hex string/1, 29, 136
to int/1, 29, 136
to integer/1, 29, 137
to list/1, 34, 137
144
to lowercase/1, 33, 137
to number/1, 30, 137
to oct string/1, 30, 137
to radians/1, 104, 137
to radix string/2, 30, 137
to real/1, 30, 137
to string/1, 24, 137
to uppercase/1, 33, 137
trace/0, 22, 109
trace, 139
transpose/1, 115, 139
true, 5, 45, 50, 55, 137
truncate/1, 29, 103, 137
union/2, 117, 138
uppercase/1, 137
values/1, 36, 137
var/1, 15, 26, 82, 84, 137
variant/2, 42, 137
vars/1, 42, 137
writable/1, 100, 138
write/1, 7, 49, 78, 79, 138
write/2, 78, 79, 138
write byte/1, 78, 138
write byte/2, 78, 138
write char/1, 78, 138
write char/2, 78, 138
write char code/1, 78, 138
write char code/2, 78, 138
writef, 78, 79, 119, 138
writeln/1, 78, 138
writeln/2, 78, 138
write, 79
zip/2, 137
zip/3, 137
zip/4, 137
zip, 33, 51
=/2, 37
=:=/2, 38
==/2, 37
accumulator, 48, 55, 56
action rule, 15, 81–84, 91
anonymous variable, 26
arity, 1, 5, 24, 33–35, 43
array, 1, 2, 24, 33, 34
array comprehension, 3, 54
as-pattern, 45
assignment, 7, 49, 50, 54, 55
atom, 1, 11, 14, 24, 26, 27, 33
attributed variable, 1, 3, 15, 24, 26, 36, 38
backtrackable rule, 5, 6, 15, 43
call trace, 22, 23
car, 30, 37, 45
cdr, 30, 37
command/1, 112
complete list, 30
compound value, 1, 3, 7, 8, 24, 30, 49–51, 54
cons, 30, 45, 48
constraint, 12, 86
debugging, 1
do-while loop, 8, 50, 53, 56
environment variable, 20
exception, 5–7, 13, 14, 58, 59
execution trace, 22
extensional constraint, 88
failure-driven loop, 46, 51
file descriptor, 74, 75, 77, 78, 80
file descriptor table, 74, 80
file name, 19, 20, 71
first-fail principle, 87, 95
foreach loop, 5, 8, 10, 13, 50–54, 56
free variable, 1, 6, 26, 32, 34
function, 1–4, 6, 7, 10–12, 14, 39, 43–45, 48
function fact, 7, 44, 73
functor, 33, 35, 37
garbage collector, 110, 112, 113
global map, 16
global module, 11, 71
goal, 5, 43, 45–47
ground, 41
handler, 59
hard link, 99
heap, 1, 36
heap map, 16
help/0, 112
higher-order call, 3, 12, 14, 39, 73
if statement, 5, 46, 50
imperative, 49
index declaration, 6, 7, 11, 47
instantiated variable, 15, 26, 31, 37, 41
integer, 1, 3, 24, 27, 29
interrupt, 13, 58
iterator, 3, 8, 50–52, 54, 56
145
last-call optimization, 47
linear tabling, 62, 64
list, 1, 3, 6, 9, 11, 14, 24, 27, 29–31, 33, 34, 36,
37, 42
list comprehension, 3, 10, 49, 54, 56
local variable, 8, 13, 54, 56
map, 1, 3, 16, 24, 26, 35, 36, 41
map-set, 2
max-heap, 1, 36
min-heap, 1, 36
mode-directed tabling, 10, 11, 61
module file, 12, 20
nested loop, 52, 56
nolog/0, 112
non-backtrackable rule, 5, 6, 15, 43, 44
non-trace mode, 22, 109
number, 1, 24, 27, 29
occurs-check problem, 38
parallel, 96
picat path/0, 112
prebuilt map, 16
predicate, 3–7, 9–12, 14, 15, 39, 43–48
predicate fact, 6, 47
primitive value, 1, 24, 37
scope, 8, 13, 49, 54
set, 1, 36
single-assignment, 26, 49
spy mode, 22, 109
spy point, 23
standalone, 21
statistics all/0, 111
string, 1, 24, 26, 27, 29, 30, 33
structure, 1–3, 11, 14, 24, 30, 33–35, 37, 58
symbol table, 71, 72
symbolic link, 99, 100
table constraint, 88
table map, 16
tabling, 10, 11, 16, 60–62, 64
tail recursion, 9, 10, 44, 45, 47, 48, 55
term, 1, 3, 5, 6, 13, 15
threads, 96
trace mode, 22, 23, 109
while loop, 5, 8, 50, 52–56
146

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