1 Introduction

cplint is an interpreter for LPADs [10, 11] and CP-logic programs [9, 12]. It is described in [6] and [7]. It is an adaptation of the interpreter for ProbLog [3].

It was proved correct [7] for range restricted acyclic programs [1] without function symbols.

It is also able to deal with extensions of LPADs and CP-logic: the clause bodies can contain setof and 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 setof or bagof operator. These extended features have been introduced in order to represent CLP(BN) [8] programs and PRM models [5]: setof and 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.

These extensions are work in progress: they have been implemented but there is no paper yet that describes the semantics of the extended language.

2 Installation

cplint is distributed in source code in the CVS version of Yap. Download it by following the instruction in http://www.ncc.up.pt/%7Evsc/Yap/downloads.html .

cplint requires glu (a subpackage of vis) and glib. You can download glu from http://vlsi.colorado.edu/%7Evis/getting_VIS_2.1.html You can download glib from http://www.gtk.org/ . This is a standard GNU package so it is easy to install it using the package management software of your Linux or Cygwin distribution.

Install glu:

1. downlad glu-2.1.tar.gz
2. decompress it
3. cd glu-2.1
4. mkdir arch
5. cd arch
6. ../configure
7. make
8. su
9. make install

This will install glu into /usr/local, if you want to install to a different DIR use ../configure --prefix DIR

Install Yap together with cplint: when compiling Yap following the instuction of the INSTALL file in the root of the Yap folder, use

Under Windows, you have to use Cygwin (glu does not compile under MinGW), so

If you installed glu in DIR, use --enable-cplint=DIR

After having performed make install you can do make installcheck that will execute a test of cplint. If no error is reported you have a working installation of cplint.

3 Syntax

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. For example, the CP-logic clause

No parentheses are necessary. The pi are numeric expressions that can involve variables appearing in the body. It is up to the user to ensure that the numeric expressions are legal, i.e. that they sum up to less than one for every instantiation of the clause for which the body is true in an instance.

If the clause has an empty body, it can be represented like this

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.

stands for

The coin example of [11] is represented as (see file coin.cpl)

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 is slightly more probable that it lands 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.

4 Commands

The program must be stored in a text file with extension .cpl. Suppose you have stored the example above in file coin.cpl. In order to answer queries from this program, you have to run yap, load cplint by issuing the command

at the command prompt. Then you must parse the source file coin.cpl with the command

if coin.cpl is in the current directory, or

if coin.cpl is in a different directory. At this point you can pose query to the program. You have to use the predicate 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 head(coin), biased(coin) can be asked with the query

For computing the probability of a conjunction given another conjunction you have to use the predicate 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. For example, the probability of the query heads(coin) given the evidence biased(coin) can be asked with the query

The package contains also a program semantics.pl that computes the probability of queries by using directly the semantics of LPADs (i.e. by generating all the ground instances and then testing the query with each of them). This is provided for testing purposes only. After having compiled the program, you can use the same commands of cplint.pl. semantics.pl requires an extra file in the directory where coin.cpl is: a file with extension .uni (for universe) that contains, for each variable, the list of constants to which the variable can be instantiated. For example, in our case the current directory will contain a file coin.uni that is a Prolog file containing facts of the form

where VarList is a list of variables names (each must be included in single quotes) and ConstList is a list of constants. semantics.pl generates the grounding by instantiating in all possible ways the variables of VarList with the constants of ConstList. 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.

You can test your installation of cplint by using the program test.pl: compile cplint.pl, compile test.pl and exectue the query t. A number of queries are executed against the example programs and the returned probabilities are checked: if t. succeeds, then cplint is working.

5 Extensions

In this section we will present the extensions to the syntax of LPADs and CP-logic programs that cplint can handle.

The first is the use of some standard Prolog predicates. The bodies can contain the built-in predicates:

The bodies can also contain the following library predicates:

plus the predicate

that, given a list of numbers, computes its arithmetic mean.

Moreover, the bodies can contain the predicates setof/3 and 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 Z such that there exists an instantiation of X and Y for which foo(X,Y,Z) is true.

An example of the use of setof and bagof is in the file female.cpl:

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 P is 0).

Another use of setof and 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 student.cpl) you can find the following clauses:

where registr_stu(R,S) expresses that registration R refers to student S and registr_gr(R,G) expresses that registration R reports grade 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 has been introduced 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 paper_ref_simple.cpl):

The first clauses states that the paper P cited in a citation 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

in the head, where Atom must contain Variable. The meaning is the following: the set of all the atoms obtained by instantiating Variable of Atom with a term taken from 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 List.

A more elaborate example is present in file 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:

This is an example of a CP-logic program, because the probabilities in the head do not sum up to one. The first clause states that, if the topic of a paper X is theory and of paper Y is theory, there is a probability of 0.005 that there is a citation from X to Y. The other clauses consider the remaining cases for the topics.

6 Parameters

The proof procedure has two parameters that can be set with the command

from the Yap prompt after having compiled cplint.pl. The current value can be read with

from the Yap prompt. Available parameters:

• epsilon_parsing: if (1 - the sum of the probabilities of all the head atoms) is smaller than epsilon_parsing then the clause is considered as an LPAD clause, otherwise it is considered as a CP-logic clause. Default value 0.00001
• savedot: if true a graph representing the BDD is saved in the file cpl.dot in the current directory in dot format. The variables names are of the form Xn_m where n is the number of the multivalued variable and m is the number of the binary variable. The correspondence of variables to clauses can be evinced from the list printed on the screen of the form

  Variables: [(2,[X=2,X1=1]),(2,[X=1,X1=0]),(1,[])]

  where the first element of each couple is the clause number of the input file (starting from 1). In the example above variable X0 corresponds to clause 2 with the substitutions X=2,X1=1, variable X1 corresponds to clause 2 with the substitutions X=1,X1=0 and variable X2 corresponds to clause 1 with the empty substitution. You can view the graph with graphviz ( www.graphviz.org ) using the command

  dotty cpl.dot &

7 Additional Files

In the directory where Yap keeps the library files (usually /usr/local/share/Yap) you can find the directory cplint that contains additional files. cplint contains

• semantics.pl: Prolog program for computing the probability according to the semantics.
• test.pl: Prolog program for testing the system.
• Subdirectory examples:
  • alarm.cpl: representation of the Bayesian network in Figure 2 of [11].
  • coin.cpl: coin example from [11].
  • coin2.cpl: coin example with two coins.
  • dice.cpl: dice example from [11].
  • twosideddice.cpl,threesideddice.cpl game with idealized dice with two or three sides. Used in the experiments in [7].
  • es.cpl: first example in [7].
  • esapprox.cpl: example showing the problems of approximate inference (see [7]).
  • esrange.cpl: example showing the problems with non range restricted programs (see [7]).
  • female.cpl: example showing the dependence of probabilities in the head from variables in the body (from [11]).
  • mendel.cpl: program describing the Mendelian rules of inheritance, taken from [2].
  • paper_ref.cpl,paper_ref_simple.cpl: paper citations examples, showing reference uncertainty, inspired by [5].
  • paper_ref_not.cpl: paper citations example showing that negation can be used also for predicates defined by clauses with uniform in the head.
  • school.cpl: example inspired by the example school_32.yap from the source distribution of Yap in the CLPBN directory.
  • school_simple.cpl: simplified version of school.cpl.
  • student.cpl: student example from Figure 1.3 of [4].

  The files *.uni that are present for some of the examples are used by semantics.pl. Some of the example files contain in an initial comment some queries together with their result.

• Subdirectory doc: contains this manual in latex, html and pdf.

8 License

cplint, as Yap, follows the Artistic License 2.0 that you can find in Yap CVS root dir. The copyright is by Fabrizio Riguzzi.

The program uses the library CUDD for manipulating BDDs that is included in glu. For the use of CUDD, the following license must be accepted:

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.

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