# Introduction

cplint is a suite of programs for reasoning with LPADs/CP-logic programs , , , . 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.

# 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). 

# Syntax

cplint permits the definition of discrete probability distributions and continuous probability densities.

## 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

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  is represented as (file coin.pl)

heads(Coin):1/2 ; tails(Coin):1/2 :-
toss(Coin),\+biased(Coin).

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)

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 containing 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 containing variable Var and D is a list of values each taking the same probability (1 over the length of D).

### ProbLog Syntax

You can also use ProbLog  syntax, so a general clause takes the form

p1::h1 ; ... ; pn::hn :- Body

where the pi are numeric expressions.

### PRISM Syntax

You can also use PRISM  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 usually 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. This behavior can be changed with the setting prism_memoization: if set to true then memoization is performed. Its default value is false, i.e., no memoization.

For example, the coin example above in PRISM syntax becomes (coinmsw.pl)

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).

## Continuous Probability Densities

cplint handles continuous or integer 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 $$\alpha$$ 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 (rate).

• binomial(Var,N,P) Var follows a binomial distribution with parameters N (number of trials) and P (success probability).

• geometric(Var,P) Var follows a geometric distribution with parameter P (success probability).

• exponential(Var,Lambda) Var follows an exponential distribution with parameter Lambda (rate, or inverse scale).

• pascal(Var,R,P) Var follows an exponential distribution with parameters R (number of failures) and P (success probability).

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

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 $\left[\begin{array}{rr} 1&0\\ 0&1 \end{array}\right]$.

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) :- 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

val(I,X) :-
mean(M),
val(I,M,X).
mean(M): gaussian(M,1.0, 5.0).
val(_,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 Gaussian 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).
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 of 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.

Moreover, user defined distributions are allowed with the syntax:

A:user(Var,Density):-Body.

where Var appears in A and will contain the sampled value and Density is an atom of the form predicate(Parameters). If predicate in predicate(Parameters) has arity n, then the user has to define predicate predicate/n+1 such that

predicate(Parameters,Var)

called with Parameters instantiated, returns in Var a value sampled from the user defined density. The definition of predicate/n+1 should appear after :- end_lpad.

If likelihood weighting or particle filtering will be used for inference, then the user has to define also predicate predicate/n+2 such that predicate(Parameters,Var,Dens), when called with Parameters and Var instantiated, returns in Dens the value of the probability density of Var. The definition of predicate/n+2 should appear after :- end_lpad. Moreover, if the density is discrete, the program must include the fact

disc(predicate).

also after :- end_lpad.

See for example gauss_mean_est_user.pl and binomial_user.pl.

### Distributional Clauses Syntax

You can also use the syntax of Distributional Clauses (DC) . 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 without 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
% 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

# 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 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 [15,18,20].

For the semantics of programs with continuous random variables, see  that defines the probability space for $$N$$ continuous random variables by considering the Borel $$\sigma$$-algebra over $$\mathbb{R}^N$$ 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,  defines the semantics of distributional clauses by resorting to a stochastic $$Tp$$ 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.

# Inference

cplint answers queries using the module pita or mcintyre. The first performs the program transformation technique of . 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 . 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.
toss(Coin),\+biased(Coin).

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)).
:- mc.
toss(Coin),\+biased(Coin).

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.
sampled_male(X):0.5:-
db(male(X)).
male(john).
male(david).

You can also use findall/3 on subgoals defined by database clauses (persons.pl)

:- use_module(library(pita)).
:- pita.
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.
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 $$2^n$$). The Yap version of cplint includes reasoning algorithms that 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.

## Unconditional Queries

The unconditional probability of an atom can be asked using pita with the predicate

prob(:Query:atom,-Probability:float) is nondet

as in

?- prob(heads(coin),P).

If the query is non-ground, prob/2 returns in backtracking the successful instantiations together with their probability.

When using mcintyre, the predicate for querying is

mc_prob(:Query:atom,-Probability:float,+Options:list) is det

where 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.

For example

?- mc_prob(heads(coin),P,[]).

You can also use

mc_prob(:Query:atom,-Probability:float) is det

which is equivalent to mc_prob/3 with an empty option list. In general, all the predicates that admit a list of options as an argument have a corresponding version without the list of options that is equivalente to calling the first with an empty option list.

With mcintyre, you can also take a given number of samples with

mc_sample(:Query:atom,+Samples:int,-Probability:float,
Options:list) is det

where Options is a list of options, the following are recognised by mc_sample/4:

• successes(-Successes:int) Number of successes

• failures(-Failures:int) Number of failures

• 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.

For example (coinmc.pl)

?- mc_sample(heads(coin),1000,P,[successes(S),failures(F)]).

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). As another example, the call

?- mc_sample(heads(coin),1000,Prob).

samples heads(coin) 1000 times and returns the estimated probability that a sample is true.

You can also sample using Gibbs sampling with

mc_gibbs_sample(:Query:atom,+Samples:int,-Probability:float,
+Options:list) is det

where Options is a list of options, the following are recognised by mc_gibbs_sample/4:

• block(+Block:int) Perform blocked Gibbs: Block variables are sampled together, default value 1

• mix(+Mix:int) The first Mix samples are discarded (mixing time), default value 0

• successes(-Successes:int) Number of successes

• failures(-Failures:int) Number of failures

mc_gibbs_sample/3 is equivalent to mc_gibbs_sample/4 with an empty option list.

Moreover, you can sample arguments of queries with

mc_sample_arg(:Query:atom,+Samples:int,?Arg:var,
-Values:list,+Options:list) is det

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 . For example pcfglr.pl and plcg.pl satisfy this constraint while markov_chain.pl and var_obj.pl doesn’t.

Options is a list of options, the following are recognised by mc_sample_arg/5:

• successes(-Successes:int) Number of successes

• failures(-Failures:int) Number of failures

• bar(-BarChar:dict) BarChart is a dict for rendering with c3 as a bar chart with with a bar for each possible value of L, the list of values of Arg for which the query succeeds in a world sampled at random. The size of the bar is the number of samples returning that list of values.

An example of use of mc_sample_arg/4 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) is det

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

mc_sample_arg_first(:Query:atom,+Samples:int,?Arg:var,
-Values:list,+Options:list) is det

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 has a bar for each value of Arg returned as a first answer for the query in a world sampled at random. The size of the bar is the number of samples that returned that value.

The predicate

mc_sample_arg_one(:Query:atom,+Samples:int,?Arg:var,
-Values:list,+Options:list) is det

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 has a bar for each value of Arg returned by sampling with uniform probability one answer from those returned by the query in a world sampled at random. The size of the bar is the number of samples.

The predicate

mc_gibbs_sample_arg(:Query:atom,+Samples:int,?Arg:var,-
Values:list,+Options:list) is det

samples an argument of the query using Gibbs sampling. The same options as those of mc_gibbs_sample/4 are recognized.

Finally, you can compute expectations with

mc_expectation(:Query:atom,+N:int,?Arg:var,-Exp:float) is det

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.

The predicate

mc_gibbs_expectation(:Query:atom,+N:int,?Arg:var,
-Exp:float) is det

computes an expectation with Gibbs sampling.

### Drawing BDDs

With pita, you can obtain the BDD for a query with the predicates

bdd_dot_file(:Query:atom,+FileName:string,-Var:list) is nondet
bdd_dot_string(:Query:atom,-DotString:string,-Var:list) is nondet

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 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 by including

:- use_rendering(graphviz).

before :- pita.

For example (coin.pl)

?- bdd_dot_string(heads(coin),BDD,Var).

returns the BDD for the query heads(coin) and the list of associations between rule groundings and multivalued variables.

## 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) is nondet

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).

You also have

prob(:Query:atom,:Evidence:atom,-Probability:float,
+Options:list) is nondet

where Options is a list of options, the following are recognised by prob/4:

• 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.

as in

?- prob(heads(coin),biased(coin),P,[bar(Chart)}).

prob/3 is equivalent to prob/4 with an empty option list.

When using mcintyre, you can ask conditional queries with rejection sampling, Metropolis-Hastings Markov Chain Monte Carlo or Gibbs sampling. In rejection sampling , 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  (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 regular MCINTYRE. The sample is accepted with a probability of $$\min\{1,\frac{N_0}{N_1}\}$$ where $$N_0$$ is the number of choices sampled in the previous sample and $$N_1$$ is the number of choices sampled in the current sample. In  the lag is always 1 but the proof in  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.

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) is det

where Options is a list of options, the following are recognised by mc_sample_arg/5:

• successes(-Successes:int) Number of successes

• failures(-Failures:int) Number of failures

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) is det

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 successes

• failures(-Failures:int) Number of failures

• 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).

The predicate

mc_gibbs_sample(:Query:atom,:Evidence:atom,+Samples:int,
-Probability:float,+Options:list) is det

performs Gibbs sampling. Options is a list of options, the following are recognised by mc_gibbs_sample/5:

• block(+Block:int) Perform blocked Gibbs: Block variables are sampled together, default value 1

• mix(+Mix:int) The first Mix samples are discarded (mixing time), default value 0

• successes(-Successes:int) Number of successes

• failures(-Failures:int) Number of failures

Moreover, you can sample arguments of queries with rejection sampling, Metropolis-Hastings MCMC or Gibbs sampling using

mc_rejection_sample_arg(:Query:atom,:Evidence:atom,
+Samples:int,?Arg:var,-Values:list,+Options:list) is det
mc_mh_sample_arg(:Query:atom,:Evidence:atom,
+Samples:int,?Arg:var,-Values:list,+Options:list) is det
mc_gibbs_sample_arg(:Query:atom,+Samples:int,
?Arg:var,-Values:list,+Options:list) is det

that return the distribution of values for Arg in Query in Samples of Query given that Evidence is true. Options is a list of options, the following are recognised:

• mix(+Mix:int) The first Mix samples are discarded (mixing time), default value 0 (only MH and GIbbs)

• lag(+Lag:int) lag between each sample, Lag sampled choices are forgotten, default value 1 (only MH)

• block(+Block:int) Perform blocked Gibbs: Block variables are sampled together, default value 1 (only Gibbs)

• bar(-BarChar:dict) BarChart is a bar chart of the possible values

The predicates return 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.

mc_gibbs_sample_arg(:Query:atom,+Samples:int,
?Arg:var,-Values:list,+Options:list) is det

An example of use of the above predicates is

?- mc_mh_sample_arg(eval(2,Y),eval(1,3),1000,Y,V,[]).

To compute conditional expectations, use

mc_rejection_expectation(:Query:atom,:Evidence:atom,+N:int,
?Arg:var,-Exp:float) is det
mc_mh_expectation(:Query:atom,:Evidence:atom,+N:int,
?Arg:var,-Exp:float,+Options:list) is det
mc_gibbs_expectation(:Query:atom,:Evidence:atom,+N:int,
?Arg:var,-Exp:float,+Options:list) is det

where Options is a list of options, the same as those of the predicates for conditional argument sampling are recognised. For example

?- mc_mh_expectation(eval(2,Y),eval(1,3),1000,Y,E,[]).

of arithm.pl 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.

Note that conditional inference is not allowed for PRISM programs with the setting prism_memoization set to false, as sampled values are not stored in that case and conditioning would have no effect.

## 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(val(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 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) is det

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,P).

from indian_gpa.pl samples 1000 times 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) is det

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(val(0,X),(val(1,9),val(2,8)),100,X,L).

from gauss_mean_est.pl samples 100 values for X in val(0,X) given that val(1,9) and val(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) is det

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) is det

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 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) is det

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) is det

that computes the expected value of Arg in Query given that Evidence is true. It uses N particles.

## Causal Inference

pita and mcintyre support causal reasoning, i.e., computing the effect of actions using the do-calculus .

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.

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.

## 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.

### Using C3.js

You can draw the probability of a query being true and being false as a bar chart using the predicates

bar1(+Probability:float,-Chart:dict) is det
bar(+Probability:float,-Chart:dict) is det
bar(+Successes:int,+Failures:int,-Chart:dict) is det
argbar(+Values:list,-Chart:dict) is det

They return a dict for rendering with C3.js as a bar chart: the first returns bar chart with a single bar for the probability, the second a chart with bar for the probability and a bar for one minus the probability, the third a chart with a bar for the number of successes and a bar for the number of failures, and the fourth a chart with a for bar each value, where Values is a list of couples V-N where V is the value and N is the number of samples returning that value.

To render C3.js charts you have to include

:- use_rendering(c3).

before :- pita.

You can also use the bar(-Chart:dict) option of many predicates as in

?- prob(heads(coin),biased(coin),P,[bar(Chart)]).

P will be instantiated with a chart with a bar for the probability of heads(coin) true and a bar for the probability of heads(coin) false, given that biased(coin) is true.

Another example is

?- mc_prob(heads(coin),P,[bar(Chart)]).

that returns a chart representation of the probability.

?- mc_sample(heads(coin),1000,P,[bar(Chart)]).

returns in Chart a diagram with one bar for the number of successes and one bar for the number of failures.

The options of mc_sample_arg/5, mc_sample_arg_first/5, mc_mh_sample_arg/6, mc_rejection_sample_arg/6, can be used for visualizing the results of sampling arguments.

An example is

?- mc_sample_arg(reach(s0,0,S),50,S,ValList,[bar(Chart)]).

The same result can be achieved with

?- mc_sample_arg(reach(s0,0,S),50,S,ValList),argbar(ValList,Chart)

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,-Chart:dict,+Options:list) is det

draws a histogram of the samples in List that 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,-Chart:dict,+Options:list) is det

draws a line chart of the density of the samples in List that must take the same form as for histogram/3.

In histogram/3 and density/3 Options is a list of options, the following are recognised:

• 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

• nbins(+NBins:int) the number of bins for dividing the domain, default value 40

In this way you can specify the limits and the number of intervals of the $$X$$.

The predicate

densities(+PriorList:list,+PostList:list,-Chart:dict,
+Options:list) is det

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 same options as for histogram/3 and density/3 are recognized.

For example, the query

?-  mc_sample_arg(value(0,X),1000,X,L0,[]),
histogram(L0,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.

?- 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,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.

Predicates histogram/3, density/3 and densities/4 each have a version with one argument less that is equivalent to the predicate called with an empty option list.

### Using R

You have to load library cplint_r (a SWI-Prolog pack) with

:- use_module(library(cplint_r)).

Then you can use predicates

bar_r/1
bar_r/2
argbar_r/1

that work as their C3.js counterpart but do not return the graph as an argument as the graph is printed with a different mechanism.

You also have

histogram_r(+List:list,+Options:list) is det

that works as histogram/3.

density_r(+List:list) is det

is like density/3 with the number of bins is determined by R.

densities_r(+PriorList:list,+PostList:list) is det

is like densities/3 with the number of bins is determined by R.

See gauss_mean_est_R.pl for an example of use of these predicates.

ml

## Parameters

The inference modules have a number of parameters in order to control their behavior. They can be set with the directive

:- set_pita(<parameter>,<value>).

or

:- set_mc(<parameter>,<value>).

after initialization (:-pita. or :-mc.) but outside :-begin/end_lpad. The current value can be read with

?- setting_pita(<parameter>,Value).

or

?- setting_mc(<parameter>,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.

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.

• prism_memoization: false: original prism semantics, true: semantics with memoization

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.

• prism_memoization: false: original prism semantics, true: semantics with memoization

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.

## 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)).
:- pita.
:- table path/2.
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.

# 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 , , 

• 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 

• LEMUR (LEarning with a Monte carlo Upgrade of tRee search): an algorithm for learning the structure of programs by searching the clase space using Monte-Carlo tree search 

## Input

To execute the learning algorithms, prepare a Prolog file divided in five parts

• preamble

• background knowledge, i.e., knowledge valid for all interpretations

• 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 . bongard.pl and bongardkeys.pl represent a Bongard problem for SLIPCOVER. bongard.pl and bongardkeys.pl represent a Bongard problem for LEMUR.

### 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.

Now you can specify the background knowledge with a fact of the form

bg(<list of terms representing clauses>).

where the clauses must 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(<list of terms representing clauses>).

The initial program is used in parameter learning for providing the structure. 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 :-
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.

The annotations of the head atoms of the initial program can be probabilities, as in the example above, in this case the parameters do not matter as they are first randomized. The type of randomization depends on the setting alpha. If it takes value 0, a truncated Dirichlet process is used to initialize the parameters: the probability of being true of each Boolean random variable used to represent multivalued random variables is sampled and independently uniformly in [0,1].

If it takes a value $$\geq 0$$, the parameters are sampled from a symmetric Dirichlet distribution, i.e. a Dirichlet distribution with vector of parameters $$(\alpha,\ldots,\alpha)$$.

The annotations of the head atoms of the initial program can also be p(<prob>) with <prob> a probability, in this case the parameter is fixed so it is not tuned by learning, as in

in([(pos:0.5 :-
circle(A),
in(B,A)),
(pos:p(0.5) :-
circle(A),
triangle(B))]).

The annotations of the head atoms of the initial program can also be t(<prob>,<args>) with <prob> either a probability, in this case it is the initial value of the parameter, or a variable, in this case the parameter is initially randomized, and <args> a tuple of variables that also appear in the clause. In this case a different parameter is learned for every grounding of <args> that make the body true.

For example, we can set the initial value of the parameter of the second clause to 0.9 with

in([(pos:0.5 :-
circle(A),
in(B,A)),
(pos:t(0.9) :-
circle(A),
triangle(B))]).

With the program below we learn a different parameter for every instantiation of C in the second clause:

in([(pos:0.5 :-
circle(A),
in(_B,A)),
(pos:t(_,C) :-
triangle(A),
config(A,C))]).

### Language Bias

The language bias part contains the declarations of the input and output predicates. Output predicates are declared as

output(<predicate>/<arity>).

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(<predicate>/<arity>).

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 or in the 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(<predicate>/<arity>).

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(<recall>,<predicate>(<arg1>,...)).

specifies the atoms that can appear in the head of clauses, while

modeb(<recall>,<predicate>(<arg1>,...)).

specifies the atoms that can appear in the body of clauses. <recall> can be an integer or *. <recall> 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

+<type>

specifies that the argument should be an input variable of type <type>, i.e., a variable replacing a +<type> argument in the head or a -<type> argument in a preceding literal in the current hypothesized clause.

Another argument form is

-<type>

for specifying that the argument should be a output variable of type <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

#<type>

for specifying an argument which should be replaced by a constant of type <type> in the bottom clause but should not be used for replacing input variables of the following literals when building the bottom clause or

-#<type>

for specifying an argument which should be replaced by a constant of type <type> in the bottom clause and that should be used for replacing input variables of the following literals when building the bottom clause.

<constant>

for specifying a constant.

Note that arguments of the form #<type> -#<type> 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).
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(<r>,[<s1>,...,<sn>],[<a1>,...,<an>],[<P1/Ar1>,...,<Pk/Ark>]). 

These mode declarations are used to generate clauses with more than two head atoms. In them, <s1>,...,<sn> are schemas, <a1>,...,<an> are atoms such that <ai> is obtained from $$\verb|<si>|$$ by replacing placemarkers with variables, <Pi/Ari> are the predicates admitted in the body. <a1>,...,<an> 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(*,
[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(_).
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(<literal>,<list of literals>).

In this case when a literal matching <literal> is added to the body of clause during refinement, then also the literals matching <list of literals> will be added. An example of such declaration (from muta.pl) is

lookahead(logp(_),[(_=_))]).

Note that <list of literals> is copied with copy_term/2 before matching, so variables in common between <literal> and <list of literals> 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(<literal>,<list of literals>).

In this case <literal> is added to the body of clause during refinement only together with literals matching <list of literals>. An example of such declaration is

lookahead_cons(logp(_),[(_=_))]).

Also here <list of literals> is copied with copy_term/2 before matching, so variables in common between <literal> and <list of literals> may not be in common in the refined clause.

Moreover, we can specify lookahead with

lookahead_cons_var(<literal>,<list of literals>).

In this case <literal> is added to the body of clause during refinement only together with literals matching <list of literals> and <list of literals> is not copied before matching, so variables in common between <literal> and <list of literals> 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=_))]).

### 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(<name>)). and terminated by end(model(<name>)). 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(<id>) 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).

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(<fold_name>,<list of model identifiers>), 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.

## Commands

### Parameter Learning

To execute EMBLEM, prepare an input file in the editor panel as indicated above and call

?- induce_par(<list of folds>,P).

where <list of folds> 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).

### Structure Learning

To execute SLIPCOVER, prepare an input file in the editor panel as indicated above and call

induce(+List_of_folds:list,-P:list) is det

where List_of_folds 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(+List_of_folds:list,-P:list) is det

where List_of_folds 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.

### Testing

A program can also be tested on a test set in SLIPCOVER and LEMUR with

test(+Program:list,+List_of_folds:list,-LL:float,
-AUCROC:float,-ROC:list,-AUCPR:float,-PR:list) is det

or

test(+Program:list,+List_of_folds:list,-NPos:int,-NNeg:int,
-LL:float,-ExampleList:list) is det

where Program is a list of terms representing clauses and List_of_folds 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 det

(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

test_r(+Program:list,+List_of_folds:list,-LL:float,
-AUCROC:float,-AUCPR:float) is det

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.

## Parameters

Parameters are set with commands of the form

:- set_sc(<parameter>,<value>).

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

• 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 ).

• mcts_max_depth (values: integer, default value: 8, valid for LEMUR): maximum depth of default policy search

• 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.

• alpha (values: floating point $$\geq 0$$, default value: 0): parameter of the symmetric Dirichlet distribution used to initialize the parameters. If it takes value 0, a truncated Dirichlet process is used to sample parameters: the probability of being true of each Boolean random variable used to represent multivalued random variables is sampled uniformly and independently in [0,1]. If it takes a value $$\geq 0$$, the parameters are sampled from a symmetric Dirichlet distribution, i.e. a Dirichlet distribution with vector of parameters $$(\alpha,\ldots,\alpha)$$.

• verbosity (values: integer in [1,3], default value: 1): level of verbosity of the algorithms.

## 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.

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:

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.

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 CAU-SED
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.

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