1 Introduction

cplint is a suite of programs for reasoning with LPADs [11, 12] and CP-logic programs [10, 13].

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 lpadsld.pl, lpad.pl and cpl.pl:

• lpadsld.pl: computes the probability of a query using the top-down procedure described in in [7] and [8]. It is based on SLDNF resolution and is an adaptation of the interpreter for ProbLog [4].

  It was proved correct [8] with respect to the semantics of LPADs 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) [9] programs and PRM models [6]: 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.

• lpad.pl: computes the probability of a query using a top-down procedure based on SLG resolution [3]. 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.
• 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.

The modules for answering queries using the definition of the semantics of LPADs and CP-logic are semlpadsld.pl, semlpad.pl and semcpl.pl:

• 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.
• 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.
• 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 [13].

2 Installation

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 http://www.ncc.up.pt/~vsc/Yap/downloads.html .

cplint requires glu (a subpackage of vis) and glib-2.0. You can download glu from http://vlsi.colorado.edu/~vis/getting_VIS_2.1.html You can download glib-2.0 (version ≥ 2.0) 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 suite of tests of the various programs. 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. 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

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 [12] 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 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.

4 Commands

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 .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 one of the modules (such as for example lpad.pl) 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 by using 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 can 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

After having parsed a program, in order to read in a new program you must restart Yap when using semlpadsld.pl and semlpad.pl. With the other modules, you can directly parse a new program.

When using 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 semcpl.pl, you can print the execution process by using the command print. after p(file). Moreover, you can build an execution process given a context by issuing the command parse(file). and then build(context). where context is a list of atoms that are true in the context. semcpl.pl can print “Invalid program” in the case in which no execution process exists.

When using cpl.pl you can print a partial execution model including all the clauses involved in the query issued with print. 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

from the Yap prompt after having loaded the module. The current value can be read with

from the Yap prompt. The available parameters are:

• epsilon_parsing (valid for all six modules): if (1 - the sum of the probabilities of all the head atoms) is smaller than epsilon_parsing then cplint adds the null events to the head. Default value 0.00001
• save_dot (valid for all goal-oriented modules): 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 between variables and clauses can be evinced from the message printed on the screen, such as

  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 &

• ground_body (valid for lpadsld.pl and all semantic modules): determines how non ground clauses are treated: if true, ground clauses are obtained from a non ground clause by replacing each variable with a constant, if 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.

5 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 .uni (for universe) in the same directory where the .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 coin.uni that is a Prolog file containing facts of the form

where var_list is a list of variables names (each must be included in single quotes) and const_list is a list of constants. The semantic modules generate the grounding by instantiating in all possible ways the variables of var_list with the constants of 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 .uni must contain a fact of the form

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 ti must be specified with a fact of the form

The file .uni can contain both universe and mode declaration, the ones to be used depend on the value of the parameter grounding: with value variables, the universe declarations are used, with value modes the mode declarations are used.

With semcpl.pl only mode declarations can be used.

6 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.

When using lpadsld.pl, 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 can be used with 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 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 where the probabilities in the head do not sum up to one so the null event is automatically added to the head. 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.

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 the files:

• testlpadsld_gbtrue.pl, testlpadsld_gbfalse.pl, testlpad.pl, testcpl.pl, testsemlpadsld.pl, testsemlpad.pl testsemcpl.pl: Prolog programs for testing the modules. They are executed when issuing the command make installcheck during the installation. To execute them afterwords, load the file and issue the command t.
• Subdirectory examples:
  • alarm.cpl: representation of the Bayesian network in Figure 2 of [12].
  • coin.cpl: coin example from [12].
  • coin2.cpl: coin example with two coins.
  • dice.cpl: dice example from [12].
  • twosideddice.cpl, threesideddice.cpl game with idealized dice with two or three sides. Used in the experiments in [8].
  • ex.cpl: first example in [8].
  • exapprox.cpl: example showing the problems of approximate inference (see [8]).
  • exrange.cpl: example showing the problems with non range restricted programs (see [8]).
  • female.cpl: example showing the dependence of probabilities in the head from variables in the body (from [12]).
  • mendel.cpl, mendels.cpl: programs describing the Mendelian rules of inheritance, taken from [2].
  • paper_ref.cpl, paper_ref_simple.cpl: paper citations examples, showing reference uncertainty, inspired by [6].
  • 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 [5].
  • win.cpl, light.cpl, trigger.cpl, throws.cpl, hiv.cpl,
    invalid.cpl: programs taken from [13]. invalid.cpl is an example of a program that is invalid but sound.

  The files *.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.

• 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 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.

lpad.pl, semlpad.pl and cpl.pl are based on the SLG system by Weidong Chen and David Scott Warren , Copyright (C) 1993 Southern Methodist University, 1993 SUNY at Stony Brook, see the file COYPRIGHT_SLG for detailed information on this copyright.

References