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cplint Manual SWI-Prolog Version Fabrizio Riguzzi fabrizio.riguzzi@unife.it January 20, 2018 1 Introduction cplint is a suite of programs for reasoning with LPADs/CP-logic programs [23], [24], [21], [22]. It contains modules for both inference and learning. cplint is available in two versions, one for Yap Prolog and one for SWI-Prolog. They differ slightly in the features offered. This manual is about the SWI-Prolog version. You can find the manual for the Yap version at http://ds.ing.unife.it/ ~friguzzi/software/cplint/manual.html. 2 Installation cplint is distributed as a pack of SWI-Prolog. To install it, use ?- pack_install(cplint). Moreover, in order to make sure you have a foreign library that matches your architecture, run ?- pack_rebuild(cplint). 3 Syntax cplint permits the definition of discrete probability distributions and continuous probaility densities. 3.1 Discrete Probability Distributions LPAD and CP-logic programs consist of a set of annotated disjunctive clauses. Disjunction in the head is represented with a semicolon and atoms in the head are separated from probabilities by a colon. For the rest, the usual syntax of Prolog is used. A general CP-logic clause has the form 1 h1:p1 ; ... ; hn:pn :- Body. where Body is a conjunction of goals as in Prolog. No parentheses are necessary. The pi are numeric expressions. It is up to the user to ensure that the numeric expressions are legal, i.e. that they sum up to less than one. If the clause has an empty body, it can be represented like this h1:p1 ; ... ; hn:pn. If the clause has a single head with probability 1, the annotation can be omitted and the clause takes the form of a normal prolog clause, i.e. h1 :- Body. stands for h1:1 :- Body. The coin example of [24] is represented as (file coin.pl) heads(Coin):1/2 ; tails(Coin):1/2 :toss(Coin),\+biased(Coin). heads(Coin):0.6 ; tails(Coin):0.4 :toss(Coin),biased(Coin). fair(Coin):0.9 ; biased(Coin):0.1. toss(coin). The first clause states that if we toss a coin that is not biased it has equal probability of landing heads and tails. The second states that if the coin is biased it has a slightly higher probability of landing heads. The third states that the coin is fair with probability 0.9 and biased with probability 0.1 and the last clause states that we toss a coin with certainty. Moreover, the bodies of rules may contain built-in predicates, predicates from the libraries lists, apply and clpr/nf_r plus the predicate average/2 that, given a list of numbers, computes its arithmetic mean. The body of rules may also contain the predicate prob/2 that computes the probability of an atom, thus allowing nested probability computations. For example (meta.pl) a:0.2:prob(b,P), P>0.2. is a valid rule. Moreover, the probabilistic annotations can be variables, as in (flexprob.pl)) 2 red(Prob):Prob. draw_red(R, G):Prob is R/(R + G), red(Prob). Variables in probabilistic annotations must be ground when resolution reaches the end of the body, otherwise an exception is raised. Alternative ways of specifying probability distribution include A:discrete(Var,D):-Body. or A:finite(Var,D):-Body. where A is an atom containg variable Var and D is a list of couples Value:Prob assigning probability Prob to Value. Moreover, you can use A:uniform(Var,D):-Body. where A is an atom containg variable Var and D is a list of values each taking the same probability (1 over the length of D). 3.1.1 ProbLog Syntax You can also use ProbLog [6] syntax, so a general clause takes the form p1::h1 ; ... ; pn::hn :- Body where the pi are numeric expressions. 3.1.2 PRISM Syntax You can also use PRISM [19] syntax, so a program is composed of a set of regular Prolog rules whose body may contain calls to the msw/2 predicate (multi-ary switch). A call msw(term,value) means that a random variable associated to term assumes value value. The admissible values for a discrete random variable are specified using facts for the values/2 predicate of the form values(T,L). where T is a term (possibly containing variables) and L is a list of values. The distribution over values is specified using directives for set_sw/2 of the form :- set_sw(T,LP). where T is a term (possibly containing variables) and LP is a list of probability values. Remember that in PRISM each call to msw/2 refers to a different random variable, i.e., no memoing is performed, differently from the case of LPAD/CP-Logic/ProbLog. For example, the coin example above in PRISM syntax becomes (coinmsw.pl) 3 values(throw(_),[heads,tails]). :- set_sw(throw(fair),[0.5,0.5]). :- set_sw(throw(biased),[0.6,0.4]). values(fairness,[fair,biased]). :- set_sw(fairness,[0.9,0.1]). res(Coin,R):- toss(Coin),fairness(Coin,Fairness),msw(throw(Fairness),R). fairness(_Coin,Fairness) :- msw(fairness,Fairness). toss(coin). 3.2 Continuous Probability Densities cplint handles continuous random variables as well with its sampling inference module. To specify a probability density on an argument Var of an atom A you can used rules of the form A:Density:- Body where Density is a special atom identifying a probability density on variable Var and Body (optional) is a regular clause body. Allowed Density atoms are • uniform(Var,L,U): Var is uniformly distributed in [L, U ] • gaussian(Var,Mean,Variance): Var follows a Gaussian distribution with mean Mean and variance Variance. The distribution can be multivariate if Mean is a list and Variance a list of lists representing the mean vector and the covariance matrix. In this case the values of Var are lists of real values with the same length as that of Mean • dirichlet(Var,Par): Var is a list of real numbers following a Dirichlet distribution with α parameters specified by the list Par • gamma(Var,Shape,Scale) Var follows a gamma distribution with shape parameter Shape and scale parameter Scale. • beta(Var,Alpha,Beta) Var follows a beta distribution with parameters Alpha and Beta. • poisson(Var,Lambda) Var follows a Poisson distribution with parameter Lambda. • binomial(Var,N,P) Var follows a binomial distribution with parameters N and P. • geometric(Var,P) Var follows a geometric distribution with parameter P. For example g(X): gaussian(X,0, 1). states that argument X of g(X) follows a Gaussian distribution with mean 0 and variance 1, while 4 g(X): gaussian(X,[0,0], [[1,0],[0,1]]). states that argument X of g(X) follows a Gaussian multivariate distribution with mean vector [0, 0] and covariance matrix 1 0 0 1 . For example, gaussian mixture.pl defines a mixture of two Gaussians: heads:0.6;tails:0.4. g(X): gaussian(X,0, 1). h(X): gaussian(X,5, 2). mix(X) :- heads, g(X). mix(X) :- tails, h(X). The argument X of mix(X) follows a distribution that is a mixture of two Gaussian, one with mean 0 and variance 1 with probability 0.6 and one with mean 5 and variance 2 with probability 0.4. The parameters of the distribution atoms can be taken from the probabilistic atom, the example gauss mean est.pl value(I,X) :mean(M), value(I,M,X). mean(M): gaussian(M,1.0, 5.0). value(_,M,X): gaussian(X,M, 2.0). states that for an index I the continuous variable X is sampled from a Gaussian whose variance is 2 and whose mean is sampled from a Guassian with mean 1 and variance 5. Any operation is allowed on continuous random variables. The example below (kalman filter.pl) encodes a Kalman filter: kf(N,O, T) :init(S), kf_part(0, N, S,O,T). kf_part(I, N, S,[V|RO], T) :I < N, NextI is I+1, trans(S,I,NextS), emit(NextS,I,V), kf_part(NextI, N, NextS,RO, T). kf_part(N, N, S, [],S). trans(S,I,NextS) :{NextS =:= E + S}, trans_err(I,E). 5 emit(NextS,I,V) :{NextS =:= V+X}, obs_err(I,X). init(S):gaussian(S,0,1). trans_err(_,E):gaussian(E,0,2). obs_err(_,E):gaussian(E,0,1). Continuous random variables are involved in arithmetic expressions (in trans/3 and emit/3). It is often convenient, as in this case, to use CLP(R) constraints (by including the directive :- use_module(library(clpr)).) as in this way the expressions can be used in multiple directions and the same clauses can be used both to sample and to evaluate the weight the sample on the basis of evidence, otherwise different clauses have to be written. In case random variables are not sufficiently instantiated to exploit expressions for inferring the values of other variables, inference will return an error. 3.2.1 Distributional Clauses Syntax You can also use the syntax of Distributional Clauses (DC) [13]. Continuous random variables are represented in this case by term whose distribution can be specified with density atoms as in T~Density’ := Body. Here := replaces the implication symbol, T is a term and Density’ is one of the density atoms above witthout the Var argument, because T itself represents a random variables. In the body of clauses you can use the infix operator ~= to equate a term representing a random variable with a logical variable or a constant as in T ~= X. Internally cplint transforms the terms representing random variables into atoms with an extra argument for holding the variable. DC can be used to represent also discrete distributions using T~uniform(L) := Body. T~finite(D) := Body. where L is a list of values and D is a list of couples P:V with P a probability and V a value. If Body is empty, as in regular Prolog, the implication symbol := can be omitted. The Indian GPA problem from http://www.robots.ox.ac.uk/~fwood/anglican/ examples/viewer/?worksheet=indian-gpain distributional clauses syntax (https: //github.com/davidenitti/DC/blob/master/examples/indian-gpa.pl) takes the form (indian gpadc.pl): is_density_A:0.95;is_discrete_A:0.05. % the probability distribution of GPA scores for American students is % continuous with probability 0.95 and discrete with probability 0.05 agpa(A): beta(A,8,2) :- is_density_A. % the GPA of American students follows a beta distribution if the 6 % distribution is continuous american_gpa(G) : finite(G,[4.0:0.85,0.0:0.15]) :- is_discrete_A. % the GPA of American students is 4.0 with probability 0.85 and 0.0 % with % probability 0.15 if the % distribution is discrete american_gpa(A):- agpa(A0), A is A0*4.0. % the GPA of American students is obtained by rescaling the value of % agpa % to the (0.0,4.0) interval is_density_I : 0.99; is_discrete_I:0.01. % the probability distribution of GPA scores for Indian students is % continuous with probability 0.99 and discrete with probability % 0.01 igpa(I): beta(I,5,5) :- is_density_I. % the GPA of Indian students follows a beta distribution if the % distribution is continuous indian_gpa(I): finite(I,[0.0:0.1,10.0:0.9]):- is_discrete_I. % the GPA of Indian students is 10.0 with probability 0.9 and 0.0 % with % probability 0.1 if the % distribution is discrete indian_gpa(I) :- igpa(I0), I is I0*10.0. % the GPA of Indian students is obtained by rescaling the value % of igpa % to the (0.0,4.0) interval nation(N) : finite(N,[a:0.25,i:0.75]). % the nation is America with probability 0.25 and India with % probability 0.75 student_gpa(G):- nation(a),american_gpa(G). % the GPA of the student is given by american_gpa if the nation is % America student_gpa(G) :- nation(i),indian_gpa(G). % the GPA of the student is given by indian_gpa if the nation %is India See 4 Semantics The semantics of LPADs for the case of programs without functions symbols can be given as follows. An LPAD defines a probability distribution over normal logic programs called worlds. A world is obtained from an LPAD by first grounding it, by selecting a single head atom for each ground clause and by including in the world the 7 clause with the selected head atom and the body. The probability of a world is the product of the probabilities associated to the heads selected. The probability of a ground atom (the query) is given by the sum of the probabilities of the worlds where the query is true. If the LPAD contains function symbols, the definition is more complex, see [15, 20, 17]. For the semantics of programs with continuous random variables, see [10] that defines the probability space for N continuous random variables by considering the Borel σ-algebra over RN and defines a Lebesgue measure on this set as the probability measure. The probability space is lifted to cover the entire program using the least model semantics of constraint logic programs. Alternatively, [13] defines the semantics of distributional clauses by resorting to a stochastic T p operator. cplint allows more freedom than distributional clauses in the use of continuous random variables in expressions, for example kalman filter.pl would not be allowed by distributional clauses. 5 Inference cplint answers queries using the module pita or mcintyre. The first performs the program transformation technique of [18]. Differently from that work, techniques alternative to tabling and answer subsumption are used. The latter performs approximate inference by sampling using a different program transformation technique and is described in [16]. Only mcintyre is able to handle continuous random variables. For answering queries, you have to prepare a Prolog file where you first load the inference module (for example pita), initialize it with a directive (for example :- pita) and then enclose the LPAD clauses in :-begin_lpad. or :-begin_plp. and :-end_lpad. or :-end_plp. For example, the coin program above can be stored in coin.pl for performing inference with pita as follows :- use_module(library(pita)). :- pita. :- begin_lpad. heads(Coin):1/2 ; tails(Coin):1/2:toss(Coin),\+biased(Coin). heads(Coin):0.6 ; tails(Coin):0.4:toss(Coin),biased(Coin). fair(Coin):0.9 ; biased(Coin):0.1. toss(coin). :- end_lpad. The same program for mcintyre is :- use_module(library(mcintyre)). 8 :- mc. :- begin_lpad. heads(Coin):1/2 ; tails(Coin):1/2:toss(Coin),\+biased(Coin). heads(Coin):0.6 ; tails(Coin):0.4:toss(Coin),biased(Coin). fair(Coin):0.9 ; biased(Coin):0.1. toss(coin). :- end_lpad. You can have also (non-probabilistic) clauses outside :-begin/end_lpad. These are considered as database clauses. In pita subgoals in the body of probabilistic clauses can query them by enclosing the query in db/1. For example (testdb.pl) :- use_module(library(pita)). :- pita. :- begin_lpad. sampled_male(X):0.5:db(male(X)). :- end_lpad. male(john). male(david). You can also use findall/3 on subgoals defined by database clauses (persons.pl) :- use_module(library(pita)). :- pita. :- begin_lpad. male:M/P; female:F/P:findall(Male,male(Male),LM), findall(Female,female(Female),LF), length(LM,M), length(LF,F), P is F+M. :- end_lpad. male(john). male(david). female(anna). female(elen). female(cathy). Aggregate predicates on probabilistic subgoals are not implemented due to their high computational cost (if the aggregation is over n atoms, the values of the aggregation are potentially 2n ). The Yap version of cplint includes reasoning algorithms that 9 allows aggregate predicates on probabilistic subgoals, see http://ds.ing.unife.it/ ~friguzzi/software/cplint/manual.html. In mcintyre you can query database clauses in the body of probabilistic clauses without any special syntax. You can also use findall/3. To run a query, you can simply load the Prolog file, for example coin.pl, as ?- [coin]. Note that supplying coin.pl as an argument to the swipl command currently returns errors due to bad interaction between pita and the top-level. The program is loaded correctly anyway but it is recommended to load it from the top-level to avoid these errors. 5.1 Unconditional Queries The unconditional probability of an atom can be asked using pita with the predicate prob(:Query:atom,-Probability:float). as in ?- prob(heads(coin),P). If the query is non-ground, prob/2 returns in backtracking the succesful instantiations together with their probability. When using mcintyre, the predicate for querying is mc_prob(:Query:atom,-Probability:float,+Options:list). as in Options is a list of options, the following are recognised by mc_prob/3: • bar(-BarChar:dict) BarChart is a dict for rendering with c3 as a bar chart with a bar for the number of successes and a bar for the number of failures. ?- mc_prob(heads(coin),P,[]). With mcintyre, you can also take a given number of sample with mc_sample(:Query:atom,+Samples:int,-Probability:float,Options:list). Options is a list of options, the following are recognised by mc_sample/4: • successes(-Successes:int) Number of succeses • failures(-Failures:int) Number of failueres as in (coinmc.pl) ?- mc_sample(heads(coin),1000,P,[successes(S),failures(F)]). 10 that samples heads(coin) 1000 times and returns in S the number of successes, in F the number of failures and in P the estimated probability (S/1000). Differently from exact inference, in approximate inference the query can be a conjunction of atoms. If you are just interested in the probability, you can use mc_sample(:Query:atom,+Samples:int,-Probability:float,Options:list) as in (coinmc.pl) ?- mc_sample(heads(coin),1000,Prob,[]). that samples heads(coin) 1000 times and returns the estimated probability that a sample is true (i.e., that a sample succeeds). Moreover, you can sample arguments of queries with mc_sample_arg(:Query:atom,+Samples:int,?Arg:var,-Values:list,+Options:list). The predicate samples Query a number of Samples times. Arg should be a variable in Query. The predicate returns in Values a list of couples L-N where L is the list of values of Arg for which Query succeeds in a world sampled at random and N is the number of samples returning that list of values. If L is the empty list, it means that for that sample the query failed. If L is a list with a single element, it means that for that sample the query is determinate. If, in all couples L-N, L is a list with a single element, it means that the clauses in the program are mutually exclusive, i.e., that in every sample, only one clause for each subgoal has the body true. This is one of the assumptions taken for programs of the PRISM system [20]. For example pcfglr.pl and plcg.pl satisfy this constraint while markov chain.pl and var obj.pl don’t. Options is a list of options, the following are recognised by mc_sample_arg/5: • successes(-Successes:int) Number of succeses • failures(-Failures:int) Number of failueres • bar(-BarChar:dict) BarChart is a dict for rendering with c3 as a bar chart with a bar for the number of successes and a bar for the number of failures. An example of use of the above predicate is ?- mc_sample_arg(reach(s0,0,S),50,S,Values,[]). of markov chain.pl that takes 50 samples of L in findall(S,(reach(s0,0,S),L). You can sample arguments of queries also with mc_sample_arg_raw(:Query:atom,+Samples:int,?Arg:var,-Values:list). that samples Query a number of Samples times The predicate returns in Values a list of values of Arg returned as the first answer by Query in a world sampled at random. The value is failure if the query fails. The predicate 11 mc_sample_arg_first(:Query:atom,+Samples:int,?Arg:var,-Values:list,+Options:list). samples Query a number of Samples times and returns in Values a list of couples V-N where V is the value of Arg returned as the first answer by Query in a world sampled at random and N is the number of samples returning that value. V is failure if the query fails. mc_sample_arg_first/5 differs from mc_sample_arg/5 because the first just computes the first answer of the query for each sampled world. Options is a list of options, the following are recognised by mc_sample_arg_first/5: • bar(-BarChar:dict) BarChart is a dict for rendering with c3 as a bar chart with a bar for the number of successes and a bar for the number of failures. The predicate mc_sample_arg_one(:Query:atom,+Samples:int,?Arg:var,-Values:list,+Options:list) samples Query a number of Samples times and returns in Values a list of couples V-N where V is a value sampled with uniform probability from those returned by Query in a world sampled at random and N is the number of samples returning that value. V is failure if the query fails. Options is a list of options, the following are recognised by mc_sample_arg_one/5: • bar(-BarChar:dict) BarChart is a dict for rendering with c3 as a bar chart with a bar for the number of successes and a bar for the number of failures. Finally, you can compute expectations with mc_expectation(:Query:atom,+N:int,?Arg:var,-Exp:float). that computes the expected value of Arg in Query by sampling. It takes N samples of Query and sums up the value of Arg for each sample. The overall sum is divided by N to give Exp. An example of use of the above predicate is ?- mc_expectation(eventually(elect,T),1000,T,E). of pctl slep.pl that returns in E the expected value of T by taking 1000 samples. 5.1.1 Drawing BDDs With pita, you can obtain the BDD for a query with the predicates bdd_dot_file(:Query:atom,+FileName:string,-Var:list) bdd_dot_string(:Query:atom,-DotString:string,-Var:list) The first write the BDD to a file, the latter returns it as a string. The BDD is represented in the dot format of graphviz. Solid edges indicate 1-children, dashed edges indicate 0-children and dotted edges indicate 0-children with negation applied to the sub BDD. Each level of the BDD is associated to a variable of the form XI J indicated on the left: I indicates the multivalued variable index and J the index of 12 the Boolean variable of rule I. The hexadecimal number in each node is part of its address in memory and is not significant. The table Var contains the associations between the rule groundings and the multivalued variables: the first column contains contains the multivalued variable index, the second column contains the rule index, corresponding to its position in the program, and the last column contains the list of constants grounding the rule, each replacing a variable in the order of appearance in the rule. The BDD can be drawn in cplint on SWISH by using the graphviz renderer. 5.2 Conditional Queries on Discrete Variables The conditional probability of an atom query given another atom evidence can be asked using pita with the predicate prob(:Query:atom,:Evidence:atom,-Probability:float). as in ?- prob(heads(coin),biased(coin),P). If the query/evidence are non-ground, prob/3 returns in backtracking ground instantiations together with their probability. The query and the evidence can be conjunctions of literals (positive or negative). When using mcintyre, you can ask conditional queries with rejection sampling or with Metropolis-Hastings Markov Chain Monte Carlo. In rejection sampling [25], you first query the evidence and, if the query is successful, query the goal in the same sample, otherwise the sample is discarded. In Metropolis-Hastings MCMC, mcintyre follows the algorithm proposed in [12] (the non adaptive version). A Markov chain is built by building an initial sample and by generating successor samples. The initial sample is built by randomly sampling choices so that the evidence is true. This is done with a backtracking meta-interpreter that starts with the goal and randomizes the order in which clauses are selected during the search so that the initial sample is unbiased. Each time the meta-interpreter encounters a probabilistic choice, it first checks whether a value has already been sampled, if not, it takes a sample and records it. If a failure is obtained, the meta-interpreter backtracks to other clauses but without deleting samples. Then the goal is queries using regular MCINTYRE. A successor sample is obtained by deleting a fixed number (parameter Lag) of sampled probabilistic choices. Then the evidence is queried using regular MCINTYRE starting with the undeleted choices. If the query succeeds, the goal is queried using 0 regular MCINTYRE. The sample is accepted with a probability of min{1, N N1 } where N0 is the number of choices sampled in the previous sample and N1 is the number of choices sampled in the current sample. In [12] the lag is always 1 but the proof in [12] that the above acceptance probability yields a valid Metropolis-Hastings algorithm holds also when forgetting more than one sampled choice, so the lag is user defined in cplint. 13 Then the number of successes of the query is increased by 1 if the query succeeded in the last accepted sample. The final probability is given by the number of successes over the total number of samples. You can take a given number of sample with rejection sampling using mc_rejection_sample(:Query:atom,:Evidence:atom,+Samples:int,-Probability:float,+Options:list) Options is a list of options, the following are recognised by mc_sample_arg/5: • successes(-Successes:int) Number of succeses • failures(-Failures:int) Number of failueres as in (coinmc.pl) ?- mc_rejection_sample(heads(coin),biased(coin),1000,P,[successes(S),failures(F)]). that takes 1000 samples where biased(coin) is true and returns in S the number of samples where heads(coin) is true, in F the number of samples where heads(coin) is false and in P the estimated probability (S/1000). The query and the evidence can be conjunctions of literals. You can take a given number of sample with Metropolis-Hastings MCMC using mc_mh_sample(:Query:atom,:Evidence:atom,+Samples:int, -Probability:float,+Options:list). where Lag (that is set with the options, default value 1) is the number of sampled choices to forget before taking a new sample. Options is a list of options, the following are recognised by mc_mh_sample/5: • mix(+Mix:int) The first Mix samples are discarded (mixing time), default value 0 • lag(+Lag:int) lag between each sample, Lag sampled choices are forgotten, default value 1 • successes(-Successes:int) Number of succeses • failures(-Failures:int) Number of failueres • bar(-BarChar:dict) BarChart is a dict for rendering with c3 as a bar chart with a bar for the number of successes and a bar for the number of failures. With Mix specified it takes Mix+Samples samples and discards the first Mix. For example (arithm.pl) ?- mc_mh_sample(eval(2,4),eval(1,3),10000,P,[successes(T), failures(F)]). takes 10000 accepted samples and returns in T the number of samples where eval(2,4) is true, in F the number of samples where eval(2,4) is false and in P the estimated probability (T/10000). Moreover, you can sample arguments of queries with rejection sampling and MetropolisHastings MCMC using 14 mc_rejection_sample_arg(:Query:atom,:Evidence:atom, +Samples:int,?Arg:var,-Values:list,+Options:list). Options is a list of options, the following are recognised by mc_rejection_sample_arg/6: • successes(-Successes:int) Number of succeses • failures(-Failures:int) Number of failueres • bar(-BarChar:dict) BarChart is a dict for rendering with c3 as a bar chart with a bar for the number of successes and a bar for the number of failures. mc_mh_sample_arg(:Query:atom,:Evidence:atom, +Samples:int,?Arg:var,-Values:list,+Options:list). Options is a list of options, the following are recognised by mc_mh_sample_arg/6: • mix(+Mix:int) The first Mix samples are discarded (mixing time), default value 0 • lag(+Lag:int) lag between each sample, Lag sampled choices are forgotten, default value 1 • successes(-Successes:int) Number of succeses • failures(-Failures:int) Number of failueres • bar(-BarChar:dict) BarChart is a dict for rendering with c3 as a bar chart with a bar for the number of successes and a bar for the number of failures. that return the distribution of values for Arg in Query in Samples of Query given that Evidence is true. Mix indicates the number of mixing samples. The predicate returns in Values a list of couples L-N where L is the list of values of Arg for which Query succeeds in a world sampled at random where Evidence is true and N is the number of samples returning that list of values. An example of use of the above predicates is ?- mc_mh_sample_arg(eval(2,Y),eval(1,3),1000,Y,V,[]). of arithm.pl. Finally, you can compute expectations with mc_expectation(:Query:atom,+N:int,?Arg:var,-Exp:float). that computes the expected value of Arg in Query by sampling. It takes N samples of Query and sums up the value of Arg for each sample. The overall sum is divided by N to give Exp. An example of use of the above predicate is ?- mc_expectation(eventually(elect,T),1000,T,E). 15 of pctl slep.pl that returns in E the expected value of T by taking 1000 samples. To compute conditional expectations, use mc_mh_expectation(:Query:atom,:Evidence:atom,+N:int, ?Arg:var,-Exp:float,+Options:list). Options is a list of options, the following are recognised by mc_mh_expectation/6: • mix(+Mix:int) The first Mix samples are discarded (mixing time), default value 0 • lag(+Lag:int) lag between each sample, Lag sampled choices are forgotten, default value 1 mc_rejection_expectation(:Query:atom,:Evidence:atom,+N:int, ?Arg:var,-Exp:float). as in ?- mc_mh_expectation(eval(2,Y),eval(1,3),1000,Y,E,[]). of arithm.pl that computes the expectation of argument Y of eval(2,Y) given that eval(1,3) is true by taking 1000 samples using Metropolis-Hastings MCMC. 5.3 Conditional Queries on Continuous Variables When you have continuous random variables, you may be interested in sampling arguments of goals representing continuous random variables. In this way you can build a probability density of the sampled argument. When you do not have evidence or you have evidence on atoms not depending on continuous random variables, you can use the above predicates for sampling arguments. For example ?- mc_sample_arg(value(0,X),1000,X,L,[]). from (gauss mean est.pl)) samples 1000 values for X in value(0,X) and returns them in L. When you have evidence on ground atoms that have continuous values as arguments, you cannot use rejection sampling or Metropolis-Hastings, as the probability of the evidence is 0. For example, the probability of sampling a specific value from a Gaussian is 0. Continuous variables have probability densities instead of distributions as discrete variables. In this case, you can use likelihood weighting or particle filtering [9, 11, 13] to obtain samples of continuous arguments of a goal. For each sample to be taken, likelihood weighting uses a meta-interpreter to find a sample where the goal is true, randomizing the choice of clauses when more than one resolves with the goal in order to obtain an unbiased sample. This meta-interpreter is similar to the one used to generate the first sample in Metropolis-Hastings. Then a different meta-interpreter is used to evaluate the weight of the sample. This meta-interpreter starts with the evidence as the query and a weight of 1. Each time the 16 meta-interpreter encounters a probabilistic choice over a continuous variable, it first checks whether a value has already been sampled. If so, it computes the probability density of the sampled value and multiplies the weight by it. If the value has not been sampled, it takes a sample and records it, leaving the weight unchanged. In this way, each sample in the result has a weight that is 1 for the prior distribution and that may be different from the posterior distribution, reflecting the influence of evidence. In particle filtering, the evidence is a list of atoms. Each sample is weighted by the likelihood of an element of the evidence and constitutes a particle. After weighting, particles are resampled and the next element of the evidence is considered. The predicate mc_lw_sample(:Query:atom,:Evidence:atom,+Samples:int,-Prob:float) samples Query a number of Samples times given that Evidence (a conjunction of atoms is allowed here). is true. The predicate returns in Prob the probability that the query is true. It performs likelihood weighting: each sample is weighted by the likelihood of evidence in the sample. For example ?- mc_lw_sample(nation(a),student_gpa(4.0),1000,PPost). from indian gpa.pl samples 1000 the query nation(a) given that student_gpa(4.0) has been observed. The predicate mc_lw_sample_arg(:Query:atom,:Evidence:atom,+N:int,?Arg:var,-ValList) returns in ValList a list of couples V-W where V is a value of Arg for which Query succeeds and W is the weight computed by likelihood weighting according to Evidence (a conjunction of atoms is allowed here). For example ?- mc_lw_sample_arg(value(0,X),(value(1,9),value(2,8)),100,X,L). from gauss mean est.pl samples 100 values for X in value(0,X) given that value(1,9) and value(2,8) have been observed. You can compute conditional expectations using likelihood weighting with mc_lw_expectation(:Query:atom,Evidence:atom,+N:int,?Arg:var,-Exp:float). that computes the expected value of Arg in Query given that Evidence is true. It takes N samples of Arg in Query, weighting each according to the evidence, and returns their weighted average. The predicate mc_particle_sample_arg(:Query:atom,+Evidence:list, +Samples:int,?Arg:var,-Values:list) samples argument Arg of Query using particle filtering given that Evidence is true. Evidence is a list of goals and Query can be either a single goal or a list of goals. When Query is a single goal, the predicate returns in Values a list of couples V-W where V is 17 a value of Arg for which Query succeeds in a particle in the last set of particles and W is the weight of the particle. For each element of Evidence, the particles are obtained by sampling Query in each current particle and weighting the particle by the likelihood of the evidence element. When Query is a list of goals, Arg is a list of variables, one for each query of Query and Arg and Query must have the same length of Evidence. Values is then list of the same length of Evidence and each of its elements is a list of couples V-W where V is a value of the corresponding element of Arg for which the corresponding element of Query succeeds in a particle and W is the weight of the particle. For each element of Evidence, the particles are obtained by sampling the corresponding element of Query in each current particle and weighting the particle by the likelihood of the evidence element. For example ?-[O1,O2,O3,O4]=[-0.133, -1.183, -3.212, -4.586], mc_particle_sample_arg([kf_fin(1,T1),kf_fin(2,T2),kf_fin(3,T3),kf_fin(4,T4)], [kf_o(1,O1),kf_o(2,O2),kf_o(3,O3),kf_o(4,O4)],100,[T1,T2,T3,T4],[F1,F2,F3,F4]). from kalman filter.pl performs particle filtering for a Kalman filter with four observations. For each observation, the value of the state at the same time point is sampled. The list of samples is returned in [F1,F2,F3,F4], with each element being the sample for a time point. The predicate mc_particle_sample(:Query:atom,:Evidence:list, +Samples:int,-Prob:float) samples Query a number of Samples times given that Evidence is true using particle filtering. Evidence is a list of goals. The predicate returns in Prob the probability that the query is true. You can compute conditional expectations using particle filtering with mc_particle_expectation(:Query:atom,Evidence:atom,+N:int,?Arg:var,-Exp:float). that computes the expected value of Arg in Query given that Evidence is true. It uses N particles. 5.4 Causal Inference pita and mcintyre support causal reasoning, i.e., computing the effect of actions using the do-calculus [14]. Actions in this setting are represented as literals of action predicates, that must be declared as such with the directive :- action predicate1/arity1,...,predicaten/arityn. 18 When performing causal reasoning, action literals must be enclosed in the do/1 functor and included in the evidence conjunction. More than one action can be included (each with in a separate do/1 term) and actions and observations can be freely mixed. All conditional inference goals can be used except those for particle filtering. For example ?- prob(recovery,do(drug),P). from simpson.swinb computes the probability of recovery of a patient given that the action of administering a drug has been performed. 5.5 Graphing the Results In cplint on SWISH you can draw graphs for visualizing the results either with C3.js or with R. Similar predicates are avaiiable for the two methods. There are two types of graphs: those that represent individual probability values with a bar chart and those that visualize the results of sampling arguments. 5.5.1 Using C3.js You can draw the probability of a query being true and being false as a bar chart with prob_bar(:Query:atom,-Probability:dict) as in ?- prob_bar(heads(coin),P). if you include :- use_rendering(c3). before :- pita. P will be instantiated with a dict for rendering with c3. It will be shown as a bar chart with a bar for the probability of heads(coin) true and a bar for the probability of heads(coin) false. You can obtain the BDD for a query with the predicate bdd_dot_string(:Query:atom,-DotString:string,-Var:list) if you include :- use_rendering(graphviz). before :- pita. Solid edges indicate 1-children, dashed edges indicate 0-children and dotted edges indicate 0-children with negation applied to the sub BDD. Each level of the BDD is associated to a variable of the form XI J indicated on the left: I indicates the multivalued variable index and J the index of the Boolean variable of rule I. The hexadecimal number in each node is part of its address in memory and is not significant. The table =Var= contains the associations between the rule groundings and the multivalued variables. For example (coin.pl) ?- bdd_dot_string(heads(coin),BDD,Var). 19 returns the BDD for the query heads(coin) and the list of associations between rule groundings and multivalued variables. When using mcintyre, you can use mc_prob(:Query:atom,-Probability:dict,+Options:list). as in ?- mc_prob(heads(coin),P,[bar(Chart)]). to obtain a chart representation of the probability. You can obtain a bar chart of the samples with ?- mc_sample(heads(coin),1000,P,[bar(Chart)]). that returns in Chart a diagram with one bar for the number of successes and one bar for the number of failures. For visualizing the results of sampling arguments you can use mc_sample(:Query:atom,+Samples:int,?Arg:var,+Probability:float,+Options:list). mc_sample_arg_first(:Query:atom,+Samples:int,?Arg:var,-Values:list,+Options:list). mc_rejection_sample_arg(:Query:atom,:Evidence:atom,+Samples:int, ?Arg:var,-Values:list,+Options:list). mc_mh_sample_arg(:Query:atom,:Evidence:atom,+Samples:int, ?Arg:var,-Values:list,+Options:list). with option bar(Chart) that return in Chart a bar chart with a bar for each possible sampled value whose size is the number of samples returning that value. An example is ?- mc_sample_arg(reach(s0,0,S),50,S,ValList,[bar(Chart)]). of markov chain.pl. Drawing a graph is particularly interesting when sampling values for continuous arguments of goals. In this case, you can use the samples to draw the probability density function of the argument. The predicate histogram(+List:list,+NBins:int,-Chart:dict,+Options:list) draws a histogram of the samples in List dividing the domain in NBins bins. List must be a list of couples of the form [V]-W or V-W where V is a sampled value and W is its weight. This is the format of the list of samples returned by argument sampling predicates. The predicate density(+List:list,+NBins:int,-Chart:dict,+Options:list) draws a line chart of the density the samples in List dividing the domain in NBins bins. List must be as for histogram/4. Options is a list of options, the following are recognised by density/4: 20 • min(+Min:float) the minimum value of domain, default value the minimum in List • max(+Max:float) the maximum value of domain, default value the maximum in List In this way you can specify the limits of the X axis. The predicate densities(+PriorList:list,+PostList:list,+NBins:int,-Chart:dict) draws a line chart of the density of two sets of samples, usually prior and post observations. The samples in PriorList and PostList can be either couples [V]-W or V-W where V is a value and W its weight. The lines are drawn dividing the domain in NBins bins. For example ?- mc_sample_arg(value(0,X),1000,X,L0,[]), histogram(L0,40,Chart,[]). from gauss mean est.pl takes 1000 samples of argument X of value(0,X) and draws the density of the samples using an histogram. Instead ?- mc_sample_arg(value(0,Y),1000,Y,L0,[]), mc_lw_sample_arg(value(0,X), (value(1,9),value(2,8)),1000,X,L), densities(L0,L,NBins,Chart). from gauss mean est.pl takes 1000 amples of argument X of value(0,X) before and after observing (value(1,9),value(2,8) and draws the prior and posterior densities of the samples using a line chart. 5.5.2 Using R You have to load library cplint r (a SWI-Prolog pack) with :- use_module(library(cplint_r)). Then you can use predicates prob_bar_r/1, prob_bar_r/2 mc_prob_bar_r/3 mc_sample_bar_r/2 mc_sample_arg_bar_r/3 mc_sample_arg_first_bar_r/3 mc_rejection_sample_arg_bar_r/4 mc_mh_sample_arg_bar_r/5 mc_mh_sample_arg_bar_r/6 21 as direct analogues of those for C3.js. You also have histogram_r(+List:list,+NBins:int) is det that works as histogram/3 but does not return the graph as an argument as the graph is printed with a different mechanism. density_r(+List:list) is det is like density/3 but does not require the number of bins in input, they are determined by R. densities_r(+PriorList:list,+PostList:list) is det is like densities/4 but does not require the number of bins in input. See https://frnmst.github.io/cplint_r/cplint_r.html for a more detailed manual. See gauss mean est R.pl for an example of use of these predicates. ml 5.6 Parameters The inference modules have a number of parameters in order to control their behavior. They can be set with the directive :- set_pita(, ). or :- set_mc( , ). after initialization (:-pita. or :-mc.) but outside :-begin/end_lpad. The current value can be read with ?- setting_pita( ,Value). or ?- setting_mc( ,Value). from the top-level. The available parameters common to both pita and mcintyre are: • epsilon_parsing: if (1 - the sum of the probabilities of all the head atoms) is larger than epsilon_parsing, then pita adds the null event to the head. Default value 0.00001. • single_var: determines how non ground clauses are treated: if true, a single random variable is assigned to the whole non ground clause, if false, a different random variable is assigned to every grounding of the clause. Default value false. 22 Moreover, pita has the parameters • depth_bound: if true, the depth of the derivation of the goal is limited to the value of the depth parameter. Default value false. • depth: maximum depth of derivations when depth_bound is set to true. Default value 5. If depth_bound is set to true, derivations are depth-bounded so you can query also programs containing infinite loops, for example programs where queries have an infinite number of explanations. However the probability that is returned is guaranteed only to be a lower bound, see for example markov chaindb.pl mcintyre has the parameters • min_error: minimal width of the binomial proportion confidence interval for the probability of the query. When the confidence interval for the probability of the query is below min_error, the computation stops. Default value 0.01. • k: the number of samples to take before checking whether the the binomial proportion confidence interval is below min_error. Default value 1000. max_samples: the maximum number of samples to take. This is used when the probability of the query is very close to 0 or 1. In fact mcintyre also checks for the validity of the the binomial proportion confidence interval: if less than 5 failures or successes are sampled, even if the width of the confidence interval is less than min_error, the computation continues. This would lead to non-termination in cases where the probability is 0 or 1. max_samples ensures termination. Default value 10e4. The example markov chain.pl shows that mcintyre can perform inference in presence of an infinite number of explanations for the goal. Differently from pita, no depth bound is necessary, as the probability of selecting the infinite computation branch is 0. However, also mcintyre may not terminate if loops not involving probabilistic predicates are present. If you want to set the seed of the random number generator, you can use SWI-Prolog predicates setrand/1 and getrand/1, see SWI-Prolog manual. 5.7 Tabling You can also use tabling in inference to speed up the computation and/or avoid loops, see the SWI-Prolog manual. To do so you have to use the tabling library module and declare some of the predicates as tabled. The tabling declarations go after the :-pita. or :- mc. directives. For example, to compute the probability of paths in undirected graphs you can use the program (path tabling.swinb) :- use_module(library(pita)). :- use_module(library(tabling)). 23 :- pita. :- table path/2. :- begin_lpad. path(X,X). path(X,Y):path(X,Z),edge(Z,Y). edge(X,Y):-arc(X,Y). edge(X,Y):-arc(Y,X). arc(a,b):0.2. arc(b,e):0.5. arc(a,c):0.3. arc(c,d):0.4. arc(d,e):0.4. arc(a,e):0.1. :- end_lpad. Then you can compute the probability that a and e are connected with prob(path(a,e),Prob). This programs has loops so if you run the above query without tabling pita would loop forever. You can use tabling with both pita and mcintyre. 6 Learning The following learning algorithms are available: • EMBLEM (EM over Bdds for probabilistic Logic programs Efficient Mining): an implementation of EM for learning parameters that computes expectations directly on BDDs [3], [1], [2] • SLIPCOVER (Structure LearnIng of Probabilistic logic programs by searChing OVER the clause space): an algorithm for learning the structure of programs by searching the clause space and the theory space separately [4] • LEMUR (LEarning with a Monte carlo Upgrade of tRee search): an algorithm for learning the structure of programs by searching the clase space using MonteCarlo tree search [8] 6.1 Input To execute the learning algorithms, prepare a Prolog file divided in five parts • preamble • background knowledge, i.e., knowledge valid for all interpretations 24 • LPAD/CPL-program for you which you want to learn the parameters (optional) • language bias information • example interpretations The preamble must come first, the order of the other parts can be changed. For example, consider the Bongard problems of [7]. bongard.pl and bongardkeys.pl represent a Bongard problem for SLIPCOVER. bongard.pl and bongardkeys.pl represent a Bongard problem for LEMUR. 6.1.1 Preamble In the preamble, the SLIPCOVER library is loaded with (see bongard.pl): :- use_module(library(slipcover)). Now you can initialize SLIPCOVER with :- sc. At this point you can start setting parameters for SLIPCOVER such as for example ::::- set_sc(megaex_bottom,20). set_sc(max_iter,2). set_sc(max_iter_structure,5). set_sc(verbosity,1). We will see later the list of available parameters. In the preamble, the LEMUR library is loaded with (see bongard.pl): :- use_module(library(lemur)). Now you can initialize LEMUR with :- lemur. At this point you can start setting parameters for LEMUR such as for example :- set_lm(verbosity,1). A parameter that is particularly important for both SLIPCOVER and LEMUR is verbosity: if set to 1, nothing is printed and learning is fastest, if set to 3 much information is printed and learning is slowest, 2 is in between. This ends the preamble. 25 6.1.2 Background and Initial LPAD/CPL-program Now you can specify the background knowledge with a fact of the form bg( ). where the clauses must currently be deterministic. Alternatively, you can specify a set of clauses by including them in a section between :- begin_bg. and :- end_bg. For example :- begin_bg. replaceable(gear). replaceable(wheel). replaceable(chain). not_replaceable(engine). not_replaceable(control_unit). component(C):replaceable(C). component(C):not_replaceable(C). :- end_bg. from the mach.pl example. If you specify both a bg/1 fact and a section, the clauses of the two will be combined. Moreover, you can specify an initial program with a fact of the form in(

). The initial program is used in parameter learning for providing the structure. The indicated parameters do not matter as they are first randomized. Remember to enclose each clause in parentheses because :- has the highest precedence. For example, bongard.pl has the initial program in([(pos:0.197575 :circle(A), in(B,A)), (pos:0.000303421 :circle(A), triangle(B)), (pos:0.000448807 :triangle(A), circle(B))]). Alternatively, you can specify an input program in a section between :- begin_in. and :- end_in., as for example :- begin_in. pos:0.197575 :- 26 circle(A), in(B,A). pos:0.000303421 :circle(A), triangle(B). pos:0.000448807 :triangle(A), circle(B). :- end_in. If you specify both a in/1 fact and a section, the clauses of the two will be combined. 6.1.3 Language Bias The language bias part contains the declarations of the input and output predicates. Output predicates are declared as output(

/ ). and indicate the predicate whose atoms you want to predict. Derivations for the atoms for this predicates in the input data are built by the system. These are the predicates for which new clauses are generated. Input predicates are those whose atoms you are not interested in predicting. You can declare closed world input predicates with input_cw( / ). For these predicates, the only true atoms are those in the interpretations and those derivable from them using the background knowledge, the clauses in the input/hypothesized program are not used to derive atoms for these predicates. Moreover, clauses of the background knowledge that define closed world input predicates and that call an output predicate in the body will not be used for deriving examples. Open world input predicates are declared with input( / ). In this case, if a subgoal for such a predicate is encountered when deriving a subgoal for the output predicates, both the facts in the interpretations, those derivable from them and the background knowledge, the background clauses and the clauses of the input program are used. Then, you have to specify the language bias by means of mode declarations in the style of Progol. modeh( , ( ,...)). specifies the atoms that can appear in the head of clauses, while modeb( , ( ,...)). 27 specifies the atoms that can appear in the body of clauses. can be an integer or *. indicates how many atoms for the predicate specification are retained in the bottom clause during a saturation step. * stands for all those that are found. Otherwise the indicated number is randomly chosen. For SLIPCOVER, two specialization modes are available: bottom and mode. In the first, a bottom clause is built and the literals to be added during refinement are taken from it. In the latter, no bottom clause is built and the literals to be added during refinement are generated directly from the mode declarations. LEMUR has only specialization mode. Arguments of the form + specifies that the argument should be an input variable of type , i.e., a variable replacing a + argument in the head or a - argument in a preceding literal in the current hypothesized clause. Another argument form is - for specifying that the argument should be a output variable of type . Any variable can replace this argument, either input or output. The only constraint on output variables is that those in the head of the current hypothesized clause must appear as output variables in an atom of the body. Other forms are # for specifying an argument which should be replaced by a constant of type in the bottom clause but should not be used for replacing input variables of the following literals when building the bottom clause or -# for specifying an argument which should be replaced by a constant of type in the bottom clause and that should be used for replacing input variables of the following literals when building the bottom clause. for specifying a constant. Note that arguments of the form # -# are not available in specialization mode mode, if you want constants to appear in the literals you have to indicate them one by one in the mode declarations. An example of language bias for the Bongard domain is output(pos/0). input_cw(triangle/1). 28 input_cw(square/1). input_cw(circle/1). input_cw(in/2). input_cw(config/2). modeh(*,pos). modeb(*,triangle(-obj)). modeb(*,square(-obj)). modeb(*,circle(-obj)). modeb(*,in(+obj,-obj)). modeb(*,in(-obj,+obj)). modeb(*,config(+obj,-#dir)). SLIPCOVER and LEMUR also require facts for the determination/2 Aleph-style predicate that indicate which predicates can appear in the body of clauses. For example determination(pos/0,triangle/1). determination(pos/0,square/1). determination(pos/0,circle/1). determination(pos/0,in/2). determination(pos/0,config/2). state that triangle/1 can appear in the body of clauses for pos/0. SLIPCOVER and LEMUR also allow mode declarations of the form modeh( ,[ ,..., ],[ ,..., ],[ ,..., ]). These mode declarations are used to generate clauses with more than two head atoms. In them, ,..., are schemas, ,..., are atoms such that is obtained from by replacing placemarkers with variables, are the predicates admitted in the body. ,..., are used to indicate which variables should be shared by the atoms in the head. An example of such a mode declaration (from uwcselearn.pl) is modeh(*, [advisedby(+person,+person),tempadvisedby(+person,+person)], [advisedby(A,B),tempadvisedby(A,B)], [professor/1,student/1,hasposition/2,inphase/2, publication/2,taughtby/3,ta/3,courselevel/2,yearsinprogram/2]). If you want to specify negative literals for addition in the body of clauses, you should define a new predicate in the background as in not_worn(C):component(C), \+ worn(C). one_worn:worn(_). 29 none_worn:\+ one_worn. from mach.pl and add the new predicate in a modeb/2 fact modeb(*,not_worn(-comp)). modeb(*,none_worn). Note that successful negative literals do not instantiate the variables, so if you want a variable appearing in a negative literal to be an output variable you must instantiate before calling the negative literals. The new predicates must also be declared as input input_cw(not_worn/1). input_cw(none_worn/0). Lookahead can also be specified with facts of the form lookahead( , ). In this case when a literal matching

is added to the body of clause during refinement, then also the literals matching will be added. An example of such declaration (from muta.pl) is lookahead(logp(_),[(_=_))]). Note that

is copied with copy_term/2 before matching, so variables in common between

and may not be in common in the refined clause. It is also possible to specify that a literal can only be added together with other literals with facts of the form lookahead_cons(

, ). In this case

is added to the body of clause during refinement only together with literals matching . An example of such declaration is lookahead_cons(logp(_),[(_=_))]). Also here

is copied with copy_term/2 before matching, so variables in common between

and may not be in common in the refined clause. Moreover, we can specify lookahead with lookahead_cons_var(

, ). In this case

is added to the body of clause during refinement only together with literals matching and

is not copied before matching, so variables in common between

and are in common also in the refined clause. This is allowed only with specialization set to bottom. An example of such declaration is lookahead_cons_var(logp(B),[(B=_))]). 30 6.1.4 Example Interpretations The last part of the file contains the data. You can specify data with two modalities: models and keys. In the models type, you specify an example model (or interpretation or megaexample) as a list of Prolog facts initiated by begin(model(

)). and terminated by end(model( )). as in begin(model(2)). pos. triangle(o5). config(o5,up). square(o4). in(o4,o5). circle(o3). triangle(o2). config(o2,up). in(o2,o3). triangle(o1). config(o1,up). end(model(2)). The interpretations may contain a fact of the form prob(0.3). assigning a probability (0.3 in this case) to the interpretations. If this is omitted, the probability of each interpretation is considered equal to 1/n where n is the total number of interpretations. prob/1 can be used to set a different multiplicity for the interpretations. The facts in the interpretation are loaded in SWI-Prolog database by adding an extra initial argument equal to the name of the model. After each interpretation is loaded, a fact of the form int( ) is asserted, where id is the name of the interpretation. This can be used in order to retrieve the list of interpretations. Alternatively, with the keys modality, you can directly write the facts and the first argument will be interpreted as a model identifier. The above interpretation in the keys modality is pos(2). triangle(2,o5). config(2,o5,up). square(2,o4). in(2,o4,o5). circle(2,o3). triangle(2,o2). config(2,o2,up). in(2,o2,o3). triangle(2,o1). config(2,o1,up). 31 which is contained in the bongardkeys.pl This is also how model 2 above is stored in SWI-Prolog database. The two modalities, models and keys, can be mixed in the same file. Facts for int/1 are not asserted for interpretations in the key modality but can be added by the user explicitly. Note that you can add background knowledge that is not probabilistic directly to the file writing the clauses taking into account the model argument. For example (carc.pl) contains connected(_M,Ring1,Ring2):Ring1 \= Ring2, member(A,Ring1), member(A,Ring2), !. symbond(Mod,A,B,T):- bond(Mod,A,B,T). symbond(Mod,A,B,T):- bond(Mod,B,A,T). where the first argument of all the atoms is the model. Example registration.pl contains for example party(M,P):participant(M,_, _, P, _). that defines intensionally the target predicate party/1. Here M is the model and participant/4 is defined in the interpretations. You can also define intensionally the negative examples with neg(party(M,yes)):- party(M,no). neg(party(M,no)):- party(M,yes). Then you must indicate how the examples are divided in folds with facts of the form: fold( , ), as for example fold(train,[2,3,...]). fold(test,[490,491,...]). As the input file is a Prolog program, you can define intensionally the folds as in fold(all,F):findall(I,int(I),F). fold/2 is dynamic so you can also write (registration.pl) :- fold(all,F), sample(4,F,FTr,FTe), assert(fold(rand_train,FTr)), assert(fold(rand_test,FTe)). which however must be inserted after the input interpretations otherwise the facts for int/1 will not be available and the fold all would be empty. This command uses sample(N,List,Sampled,Rest) exported from slipcover that samples N elements from List and returns the sampled elements in Sampled and the rest in Rest. If List has N elements or less, Sampled is equal to List and Rest is empty. 32 6.2 Commands 6.2.1 Parameter Learning To execute EMBLEM, prepare an input file in the editor panel as indicated above and call ?- induce_par(

,P). where

is a list of the folds for training and P will contain the input program with updated parameters. For example bongard.pl, you can perform parameter learning on the train fold with ?- induce_par([train],P). 6.2.2 Structure Learning To execute SLIPCOVER, prepare an input file in the editor panel as indicated above and call ?- induce(

,P). where

is a list of the folds for training and P will contain the learned program. For example bongard.pl, you can perform structure learning on the train fold with ?- induce([train],P). A program can also be tested on a test set with test/7 or test_prob/6 as described below. Between two executions of induce/2 you should exit SWI-Prolog to have a clean database. To execute LEMUR, prepare an input file in the editor panel as indicated above and call ?- induce_lm(

,P). where

is a list of the folds for training and P will contain the learned program. For example bongard.pl, you can perform structure learning on the train fold with ?- induce_lm([train],P). Between two executions of induce_lm/2 you should exit SWI-Prolog to have a clean database. 33 6.2.3 Testing A program can also be tested on a test set in SLIPCOVER and LEMUR with ?- test(

, ,LL,AUCROC,ROC,AUCPR,PR). or ?- test_prob(

, ,NPos,NNeg,LL,ExampleList). where

is a list of terms representing clauses and is a list of folds. test/7 returns the log likelihood of the test examples in LL, the Area Under the ROC curve in AUCROC, a dictionary containing the list of points (in the form of Prolog pairs x-y) of the ROC curve in ROC, the Area Under the PR curve in AUCPR, a dictionary containing the list of points of the PR curve in PR. test_prob/6 returns the log likelihood of the test examples in LL, the numbers of positive and negative examples in NPos and NNeg and the list ExampleList containing couples Prob-Ex where Ex is a for a a positive example and \+(a) for a a negative example and Prob is the probability of example a. Then you can draw the curves in cplint on SWISH using C3.js using compute_areas_diagrams(+ExampleList:list,-AUCROC:float,-ROC:dict,-AUCPR:float,-PR:dict) is de (from pack auc) that takes as input a list ExampleList of pairs probability-literal of the form that is returned by test_prob/6. For example, to test on fold test the program learned on fold train you can run the query ?- induce_par([train],P), test(P,[test],LL,AUCROC,ROC,AUCPR,PR). Or you can test the input program on the fold test with ?- in(P), test(P,[test],LL,AUCROC,ROC,AUCPR,PR). In cplint on SWISH, by including :- use_rendering(c3). :- use_rendering(lpad). in the code before :- sc. the curves will be shown as graphs using C3.js and the output program will be pretty printed. You can also draw the curves in cplint on SWISH using R by loading library cplint r with :- use_module(library(cplint_r)). and using 34 ?- test_r(

, ,LL,AUCROC,AUCPR). or predicate compute_areas_diagrams_r(+ExampleList:list,-AUCROC:float,-AUCPR:float) is det that takes as input a list ExampleList of pairs probability-literal of the form that is returned by test_prob/6. 6.3 Parameters Parameters are set with commands of the form :- set_sc(

, ). The available parameters are: • specialization: (values: {bottom,mode}, default value: bottom, valid for SLIPCOVER) specialization mode. • depth_bound: (values: {true,false}, default value: true) if true, the depth of the derivation of the goal is limited to the value of the depth parameter. • depth (values: integer, default value: 2): depth of derivations if depth_bound is set to true • single_var (values: {true,false}, default value: false): if set to true, there is a random variable for each clause, instead of a different random variable for each grounding of each clause • epsilon_em (values: real, default value: 0.1): if the difference in the log likelihood in two successive parameter EM iteration is smaller than epsilon_em, then EM stops • epsilon_em_fraction (values: real, default value: 0.01): if the difference in the log likelihood in two successive parameter EM iteration is smaller than epsilon_em_fraction*(-current log likelihood), then EM stops • iter (values: integer, defualt value: 1): maximum number of iteration of EM parameter learning. If set to -1, no maximum number of iterations is imposed • iterREF (values: integer, defualt value: 1, valid for SLIPCOVER and LEMUR): maximum number of iteration of EM parameter learning for refinements. If set to -1, no maximum number of iterations is imposed. • random_restarts_number (values: integer, default value: 1, valid for EMBLEM, SLIPCOVER and LEMUR): number of random restarts of parameter EM learning 35 • random_restarts_REFnumber (values: integer, default value: 1, valid for SLIPCOVER and LEMUR): number of random restarts of parameter EM learning for refinements • seed (values: seed(integer) or seed(random), default value seed(3032)): seed for the Prolog random functions, see SWI-Prolog manual • c_seed (values: unsigned integer, default value 21344)): seed for the C random functions • logzero (values: negative real, default value log(0.000001)): value assigned to log 0 • max_iter (values: integer, default value: 10, valid for SLIPCOVER): number of interations of beam search • max_var (values: integer, default value: 4, valid for SLIPCOVER and LEMUR): maximum number of distinct variables in a clause • beamsize (values: integer, default value: 100, valid for SLIPCOVER): size of the beam • megaex_bottom (values: integer, default value: 1, valid for SLIPCOVER): number of mega-examples on which to build the bottom clauses • initial_clauses_per_megaex (values: integer, default value: 1, valid for SLIPCOVER): number of bottom clauses to build for each mega-example (or model or interpretation) • d (values: integer, default value: 1, valid for SLIPCOVER): number of saturation steps when building the bottom clause • mcts_beamsize (values: integer, default value: 3, valid for LEMUR): size of the Monte-Carlo tree search beam • mcts_visits (values: integer, default value: +1e20, valid for LEMUR): maximum number of visits • max_iter_structure (values: integer, default value: 10000, valid for SLIPCOVER): maximum number of theory search iterations • background_clauses (values: integer, default value: 50, valid for SLIPCOVER): maximum numbers of background clauses • maxdepth_var (values: integer, default value: 2, valid for SLIPCOVER and LEMUR): maximum depth of variables in clauses (as defined in [5]). • mcts_max_depth (values: integer, default value: 8, valid for LEMUR): maximum depth of default policy search 36 • mcts_c (values: real, default value: 0.7, valid for LEMUR): value of parameter C in the computation of UCT • mcts_iter (values: integer, default value: 20, valid for LEMUR): number of Monte-Carlo tree search iterations • mcts_maxrestarts (values: integer, default value: 20, valid for LEMUR): maximum number of Monte-Carlo tree search restarts • neg_ex (values: given, cw, default value: cw): if set to given, the negative examples in testing are taken from the test folds interpretations, i.e., those examples ex stored as neg(ex); if set to cw, the negative examples are generated according to the closed world assumption, i.e., all atoms for target predicates that are not positive examples. The set of all atoms is obtained by collecting the set of constants for each type of the arguments of the target predicate. • verbosity (values: integer in [1,3], default value: 1): level of verbosity of the algorithms. 6.4 Files The pack/cplint/prolog/examples folder in SWI-Prolog home contains some example programs. The subfolder learning contains some learning examples. The pack/cplint/doc folder in SWI-Prolog home contains this manual in latex, html and pdf. 7 License cplint follows the Artistic License 2.0 that you can find in cplint root folder. The copyright is by Fabrizio Riguzzi. The library CUDD for manipulating BDDs has the following license: Copyright (c) 1995-2004, Regents of the University of Colorado All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: • Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. • Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. • Neither the name of the University of Colorado nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. 37 THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ”AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. References [1] Elena Bellodi and Fabrizio Riguzzi. EM over binary decision diagrams for probabilistic logic programs. In Proceedings of the 26th Italian Conference on Computational Logic (CILC2011), Pescara, Italy, 31 August 31-2 September, 2011, 2011. [2] Elena Bellodi and Fabrizio Riguzzi. EM over binary decision diagrams for probabilistic logic programs. Technical Report CS-2011-01, Dipartimento di Ingegneria, Università di Ferrara, Italy, 2011. [3] Elena Bellodi and Fabrizio Riguzzi. Expectation Maximization over binary decision diagrams for probabilistic logic programs. Intelligent Data Analysis, 17(2):343–363, 2013. [4] Elena Bellodi and Fabrizio Riguzzi. Structure learning of probabilistic logic programs by searching the clause space. Theory and Practice of Logic Programming, 15(2):169–212, 2015. [5] William W. Cohen. Pac-learning non-recursive prolog clauses. Artif. Intell., 79(1):1–38, 1995. [6] L. De Raedt, A. Kimmig, and H. Toivonen. ProbLog: A probabilistic Prolog and its application in link discovery. In International Joint Conference on Artificial Intelligence, pages 2462–2467, 2007. [7] L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 6th Conference on Algorithmic Learning Theory (ALT 1995), volume 997 of LNAI, pages 80–94, Fukuoka, Japan, 1995. Springer. 38 [8] Nicola Di Mauro, Elena Bellodi, and Fabrizio Riguzzi. Bandit-based Monte-Carlo structure learning of probabilistic logic programs. Mach. Learn., 100(1):127–156, July 2015. [9] Robert M Fung and Kuo-Chu Chang. Weighing and integrating evidence for stochastic simulation in bayesian networks. In Fifth Annual Conference on Uncertainty in Artificial Intelligence, pages 209–220. North-Holland Publishing Co., 1990. [10] Muhammad Asiful Islam, CR Ramakrishnan, and IV Ramakrishnan. Inference in probabilistic logic programs with continuous random variables. 12:505–523, 7 2012. [11] D. Koller and N. Friedman. Probabilistic Graphical Models: Principles and Techniques. Adaptive computation and machine learning. MIT Press, Cambridge, MA, 2009. [12] Arun Nampally and CR Ramakrishnan. Adaptive mcmc-based inference in probabilistic logic programs. arXiv preprint arXiv:1403.6036, 2014. [13] Davide Nitti, Tinne De Laet, and Luc De Raedt. Probabilistic logic programming for hybrid relational domains. Mach. Learn., 103(3):407–449, 2016. [14] J. Pearl. Causality. Cambridge University Press, 2000. [15] David Poole. The independent choice logic for modelling multiple agents under uncertainty. Artificial Intelligence, 94(1-2):7–56, 1997. [16] Fabrizio Riguzzi. MCINTYRE: A Monte Carlo system for probabilistic logic programming. Fundamenta Informaticae, 124(4):521–541, 2013. [17] Fabrizio Riguzzi. The distribution semantics is well-defined for all normal programs. In Fabrizio Riguzzi and Joost Vennekens, editors, Proceedings of the 2nd International Workshop on Probabilistic Logic Programming (PLP), volume 1413 of CEUR Workshop Proceedings, pages 69–84, Aachen, Germany, 2015. Sun SITE Central Europe. [18] Fabrizio Riguzzi and Terrance Swift. Tabling and Answer Subsumption for Reasoning on Logic Programs with Annotated Disjunctions. In Technical Communications of the International Conference on Logic Programming, volume 7 of Leibniz International Proceedings in Informatics (LIPIcs), pages 162–171. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2010. [19] Taisuke Sato and Yoshitaka Kameya. Prism: A language for symbolic-statistical modeling. In International Joint Conference on Artificial Intelligence, pages 1330– 1339, 1997. [20] Taisuke Sato and Yoshitaka Kameya. Parameter learning of logic programs for symbolic-statistical modeling. J. Artif. Intell. Res., 15:391–454, 2001. 39 [21] J. Vennekens, M. Denecker, and M. Bruynooghe. Representing causal information about a probabilistic process. In Proceedings of the 10th European Conference on Logics in Artificial Intelligence, LNAI. Springer, September 2006. [22] J. Vennekens, Marc Denecker, and Maurice Bruynooghe. CP-logic: A language of causal probabilistic events and its relation to logic programming. Theory Pract. Log. Program., 9(3):245–308, 2009. [23] J. Vennekens and S. Verbaeten. Logic programs with annotated disjunctions. Technical Report CW386, K. U. Leuven, 2003. [24] J. Vennekens, S. Verbaeten, and M. Bruynooghe. Logic programs with annotated disjunctions. In International Conference on Logic Programming, volume 3131 of LNCS, pages 195–209. Springer, 2004. [25] John Von Neumann. Various techniques used in connection with random digits. Nat. Bureau Stand. Appl. Math. Ser., 12:36–38, 1951. 40

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