\texttt{cplint} is a suite of programs for reasoning with LPADs \cite{VenVer03-TR,VenVer04-ICLP04-IC} and CP-logic programs \cite{VenDenBru-JELIA06,CP-logic-unp}.
It consists of three Prolog modules for answering queries using goal-oriented procedures plus
three
Prolog modules for answering queries using the definition of the semantics of LPADs and CP-logic.
The modules for answering queries using using goal-oriented procedures are \texttt{lpadsld.pl}, \texttt{lpad.pl} and
\texttt{cpl.pl}:
\begin{itemize}
\item\texttt{lpadsld.pl}: computes the probability of a query using the top-down procedure described in
in \cite{Rig-AIIA07-IC} and \cite{Rig-RCRA07-IC}. It is based on SLDNF resolution and is an adaptation of the interpreter for ProbLog \cite{DBLP:conf/ijcai/RaedtKT07}.
It was proved correct \cite{Rig-RCRA07-IC} with respect to the semantics of LPADs for range restricted acyclic programs \cite{DBLP:journals/ngc/AptB91} without function symbols.
It is also able to deal with extensions of LPADs and CP-logic: the clause bodies can contain \texttt{setof} and \texttt{bagof}, the probabilities in the head may be depend on variables in the body and it is possible to specify a uniform distribution in the head with reference to a \texttt{setof} or \texttt{bagof} operator. These extended features have been introduced in order to represent CLP(BN) \cite{SanPagQaz03-UAI-IC} programs and PRM models \cite{Getoor+al:JMLR02}:
\texttt{setof} and \texttt{bagof} allow to express dependency of an attribute from an aggregate function of another attribute, as in CLP(BN) and PRM, while the possibility of specifying a uniform distribution allows the use of the reference uncertainty feature of PRM.
\item\texttt{lpad.pl}: computes the probability of a query using a top-down procedure based on SLG resolution \cite{DBLP:journals/jacm/ChenW96}. As a consequence, it works for any sound LPADs, i.e., any LPAD such that each of its instances has a two valued well founded model.
\item\texttt{cpl.pl}: computes the probability of a query using a top-down procedure based on SLG resolution and moreover checks that the CP-logic program is valid, i.e., that it has at least an execution model.
\item\texttt{semlpadsld.pl}: given an LPAD $P$, it generates all the instances of $P$. The probability of a query $Q$ is computed by identifying all the instances where $Q$ is derivable by SLDNF resolution.
\item\texttt{semlpad.pl}: given an LPAD $P$, it generates all the instances of $P$. The probability of a query $Q$ is computed by identifying all the instances where $Q$ is derivable by SLG resolution.
\item\texttt{semlcpl.pl}: given an LPAD $P$, it builds an execution model of $P$, i.e., a probabilistic process that satisfy the principles of universal causation, sufficient causation, independent causation, no deus ex machina events and temporal precedence. It uses the definition of the semantics given in \cite{CP-logic-unp}.
%For program with function symbols, the semantics of LPADs and CP-logic are not defined. However, the interpreter accepts programs with function symbols and, if it does not go into a loop, it returns an answer. What is the meaning of this answer is subject of current study.
\texttt{cplint} is distributed in source code in the CVS version of Yap. It includes Prolog and C files. Download it by following the instruction in \href{http://www.ncc.up.pt/~vsc/Yap/downloads.html}{http://www.ncc.up.pt/$\sim$vsc/Yap/downloads.html}.
After having performed \texttt{make install} you can do \texttt{make installcheck} that will execute a suite of tests of the various programs. If no error is reported you have a working installation of \texttt{cplint}.
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.
No parentheses are necessary. The \texttt{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.
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.
All six modules accept the same commands for reading in files and answering queries.
The LPAD or CP-logic program must be stored in a text file with extension \texttt{.cpl}. Suppose you have stored the example above in file \texttt{coin.cpl}.
In order to answer queries from this program, you have to run Yap,
load one of the modules (such as for example \texttt{lpad.pl}) by issuing the command
At this point you can pose query to the program by using the predicate \texttt{s/2} (for solve) that takes as its first argument a conjunction of goals in the form of a list and returns the computed probability as its second argument. For example, the probability of the conjunction \texttt{head(coin),biased(coin)} can be asked with the query
For computing the probability of a conjunction given another conjunction you can use the predicate \texttt{sc/3} (for solve conditional) that take takes as input the query conjunction as its first argument, the evidence conjunction as its second argument and returns the probability in its third argument.
After having parsed a program, in order to read in a new program you must restart Yap when using
\texttt{semlpadsld.pl} and \texttt{semlpad.pl}. With the other modules, you can directly parse a new program.
When using \texttt{lpad.pl}, the system can print the message ``Uunsound program'' in the case in which an instance with a three valued well founded model is found. Moreover, it can print the message ``It requires the choice of a head atom from a non ground head'': in this case, in order to answer the query, all the groundings of the culprit clause must be generated, which may be impossible for programs with function symbols.
When using \texttt{semcpl.pl}, you can print the execution process by using the command \texttt{print.}
after \texttt{p(file).} Moreover, you can build an execution process given a context by issuing the command \texttt{parse(file)}. and then
\texttt{build(context).} where \texttt{context} is a list of atoms that are true in the context.
\texttt{semcpl.pl} can print ``Invalid program'' in the case in which no execution process exists.
When using \texttt{cpl.pl} you can print a partial execution model including all the clauses involved in the query issued with \texttt{print.}\texttt{cpl.pl} can print the messages ``Uunsound program'', ``It requires the choice of a head atom from a non ground head'' and ``Invalid program''.
The modules make use of a number of parameters in order to control their behavior. They that can be set with the command
\begin{verbatim}
set(parameter,value).
\end{verbatim}
from the Yap prompt after having loaded the module.
The current value can be read with
\begin{verbatim}
setting(parameter,Value).
\end{verbatim}
from the Yap prompt.
The available parameters are:
\begin{itemize}
\item
\verb|epsilon_parsing| (valid for all six modules): if (1 - the sum of the probabilities of all the head atoms) is smaller than
\verb|epsilon_parsing|
then \texttt{cplint} adds the null events to the head. Default value 0.00001
\item\verb|save_dot| (valid for all goal-oriented modules): if \texttt{true} a graph representing the BDD is saved in the file \texttt{cpl.dot} in the current directory in dot format.
The variables names are of the form \verb|Xn_m| where \texttt{n} is the number of the multivalued
variable and \texttt{m} is the number of the binary variable. The correspondence between variables and
clauses can be evinced from the message printed on the screen, such as
\begin{verbatim}
Variables: [(2,[X=2,X1=1]),(2,[X=1,X1=0]),(1,[])]
\end{verbatim}
where the first element of each couple is the clause number of the input file (starting from 1).
In the example above variable \texttt{X0} corresponds to clause \texttt{2} with the substitutions \texttt{X=2,X1=1},
variable \texttt{X1} corresponds to clause \texttt{2} with the substitutions \texttt{X=1,X1=0} and
variable \texttt{X2} corresponds to clause \texttt{1} with the empty substitution.
You can view the graph with \texttt{graphviz} (\href{www.graphviz.org}{www.graphviz.org}) using the
command
\begin{verbatim}
dotty cpl.dot &
\end{verbatim}
\item\verb|ground_body| (valid for \texttt{lpadsld.pl} and all semantic modules): determines how non ground clauses are treated: if \texttt{true}, ground clauses are obtained from a non ground clause by replacing each variable with a constant, if \texttt{false}, ground clauses are obtained by replacing only variables in the head with a constant. In the case where the body contains variables not in the head, setting it to false means that the body represents an existential event.
\end{itemize}
\section{Semantic Modules}
The three semantic modules need to produce a grounding of the program in order to compute the semantics.
They require an extra file with extension \texttt{.uni} (for universe) in the same directory where the \texttt{.cpl} file is.
There are two ways to specify how to ground a program. The first consists in providing the list of constants to which each variable can be instantiated. For example, in our case the current directory will contain a file \texttt{coin.uni} that is a Prolog file containing facts of the form
where \verb|var_list| is a list of variables names (each must be included in single quotes) and \verb|const_list| is a list of constants. The semantic modules generate the grounding by instantiating in all possible ways the variables of \verb|var_list| with the constants of \verb|const_list|. Note that the variables are identified by name, so a variable with the same name in two different clauses will be instantiated with the same constants.
The other way to specify how to ground a program consists in using mode and type information. For each predicate, the file \texttt{.uni} must contain a fact of the form
\begin{verbatim}
mode(predicate(t1,...,tn)).
\end{verbatim}
that specifies the number and types of each argument of the predicate. Then, the list of constants that
are in the domain of each type \texttt{ti} must be specified with a fact of the form
\begin{verbatim}
type(ti,list_of_constants).
\end{verbatim}
The file \texttt{.uni} can contain both universe and mode declaration, the ones to be used depend on the value of the parameter \texttt{grounding}: with value \texttt{variables}, the universe declarations are used, with value \texttt{modes} the mode declarations are used.
With \texttt{semcpl.pl} only mode declarations can be used.
When using \texttt{lpadsld.pl}, the bodies can contain the predicates \texttt{setof/3} and \texttt{bagof/3} with the same meaning as in Prolog. Existential quantifiers are allowed in both, so for example the query
returns all the instantiations of \texttt{Z} such that there exists an instantiation of \texttt{X} and \texttt{Y} for which \texttt{foo(X,Y,Z)} is true.
An example of the use of \texttt{setof} and \texttt{bagof} is in the file \texttt{female.cpl}:
\begin{verbatim}
male(C):M/P ; female(C):F/P:-
person(C),
setof(Male,known_male(Male),LM),
length(LM,M),
setof(Female,known_female(Female),LF),
length(LF,F),
P is F+M.
person(f).
known_female(a).
known_female(b).
known_female(c).
known_male(d).
known_male(e).
\end{verbatim}
The disjunctive rule expresses the probability of a person of unknown sex of being male or female depending on the number of males and females that are known.
This is an example of the use of expressions in the probabilities in the head that depend on variables in the body. The probabilities are well defined because they always sum to 1 (unless \texttt{P} is 0).
Another use of \texttt{setof} and \texttt{bagof} is to have an attribute depend on an aggregate function of another attribute, similarly to what is done in PRM and CLP(BN).
So, in the classical school example (available in \texttt{student.cpl}) you can find the following
clauses:
\begin{verbatim}
student_rank(S,h):0.6 ; student_rank(S,l):0.4:-
bagof(G,R^(registr_stu(R,S),registr_gr(R,G)),L),
average(L,Av),Av>1.5.
student_rank(S,h):0.4 ; student_rank(S,l):0.6:-
bagof(G,R^(registr_stu(R,S),registr_gr(R,G)),L),
average(L,Av),Av =< 1.5.
\end{verbatim}
where \verb|registr_stu(R,S)| expresses that registration \texttt{R} refers to student \texttt{S} and \verb|registr_gr(R,G)| expresses that registration \texttt{R} reports grade \texttt{G} which is a natural number. The two clauses express a dependency of the rank of the student from the average of her grades.
Another extension can be used with \texttt{lpadsld.pl} in order to be able to represent reference uncertainty of PRMs. Reference uncertainty means that the link structure of a relational model is not fixed but is uncertain: this is represented by having the instance referenced in a relationship be chosen uniformly from a set. For example, consider a domain modeling scientific papers: you have a single entity, paper, and a relationship, cites, between paper and itself that connects the citing paper to the cited paper. To represent the fact that the cited paper and the citing paper are selected uniformly from certain sets, the following clauses can be used (see file \verb|paper_ref_simple.cpl|):
The first clauses states that the paper \texttt{P} cited in a citation \texttt{C} is selected uniformly from the set of all papers with topic theory.
The second clauses expresses that the citing paper is selected uniformly from the papers with
topic ai.
These clauses make use of the predicate
\begin{verbatim}
uniform(Atom,Variable,List)
\end{verbatim}
in the head, where \texttt{Atom} must contain \texttt{Variable}. The meaning is the following: the set of all the atoms obtained by instantiating \texttt{Variable} of \texttt{Atom} with a term taken from \texttt{List} is generated and the head is obtained by having a disjunct for each instantiation with probability $1/N$ where $N$ is the length of \texttt{List}.
A more elaborate example is present in file \verb|paper_ref.cpl|:
where the cited paper depends on the topic of the citing paper. In particular, if the topic is theory, the cited paper is selected uniformly from the papers about theory with probability 0.9 and from the papers about ai with probability 0.1. if the topic is ai, the cited paper is selected uniformly from the papers about theory with probability 0.01 and from the papers about ai with probability 0.99.
PRMs take into account as well existence uncertainty, where the existence of instances is also probabilistic. For example, in the paper domain, the total number of citations may be unknown and a citation between any two paper may have a probability of existing. For example, a citation between two paper may be more probable if they are about the same topic:
The first clause states that, if the topic of a paper \texttt{X} is theory and of paper \texttt{Y} is theory, there is a probability of 0.005 that there is a citation from \texttt{X} to \texttt{Y}. The other clauses consider the remaining cases for the topics.
In the directory where Yap keeps the library files (usually \texttt{/usr/local/share/ Yap}) you can find the directory \texttt{cplint} that contains the files:
\item\verb|testlpadsld_gbtrue.pl, testlpadsld_gbfalse.pl, testlpad.pl,| \verb|testcpl.pl, testsemcpl.pl|: Prolog programs for testing the modules. They are executed when issuing the command \texttt{make installcheck} during the installation. To execute them afterwords, load the file and issue the command \texttt{t.}
\item\verb|paper_ref_not.cpl|: paper citations example showing that negation can be used also for predicates defined by clauses with \texttt{uniform} in the head.
\item\texttt{school.cpl}: example inspired by the example \verb|school_32.yap| from the
source distribution of Yap in the \texttt{CLPBN} directory.
\item\verb|school_simple.cpl|: simplified version of \texttt{school.cpl}.
\item\verb|student.cpl|: student example from Figure 1.3 of \cite{GetFri01-BC}.
\item\texttt{win.cpl, light.cpl, trigger.cpl, throws.cpl, hiv.cpl,}\\\texttt{ invalid.cpl}: programs taken from \cite{CP-logic-unp}. \texttt{invalid.cpl} is an example of a program that is invalid but sound.
The files \texttt{*.uni} that are present for some of the examples are used by the semantical modules. Some of the example files contain in an initial comment some queries together with their result.
