Fuzzy Aggregates in Fuzzy Answer Set Programming

04/05/2013 ∙ by Emad Saad, et al. ∙ 0

Fuzzy answer set programming is a declarative framework for representing and reasoning about knowledge in fuzzy environments. However, the unavailability of fuzzy aggregates in disjunctive fuzzy logic programs, DFLP, with fuzzy answer set semantics prohibits the natural and concise representation of many interesting problems. In this paper, we extend DFLP to allow arbitrary fuzzy aggregates. We define fuzzy answer set semantics for DFLP with arbitrary fuzzy aggregates including monotone, antimonotone, and nonmonotone fuzzy aggregates. We show that the proposed fuzzy answer set semantics subsumes both the original fuzzy answer set semantics of DFLP and the classical answer set semantics of classical disjunctive logic programs with classical aggregates, and consequently subsumes the classical answer set semantics of classical disjunctive logic programs. We show that the proposed fuzzy answer sets of DFLP with fuzzy aggregates are minimal fuzzy models and hence incomparable, which is an important property for nonmonotonic fuzzy reasoning.

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1 Introduction

Fuzzy answer set programming [Saad2010, Saad2009, Subrahmanian1994] is a declarative programming framework that has been shown effective for knowledge representation and reasoning in fuzzy environments. These include representing and reasoning about actions with fuzzy effects and fuzzy planning [Saad2009, Saad et al.2009] as well as representing and reasoning about fuzzy preferences [Saad2010]. The fuzzy answer set programming framework includes disjunctive fuzzy logic programs [Saad2010], extended fuzzy logic programs [Saad2009], and normal fuzzy logic programs [Subrahmanian1994] with fuzzy answer set semantics. However, the unavailability of fuzzy aggregates in fuzzy answer set programming [Saad2010, Saad2009, Subrahmanian1994] disallows the natural and concise representation of many new interesting problems.

Example 1

Consider the same company control problem described in [Faber et al.2010]. Assume that a company owns of a company shares, represented by the predicate . If the company owns a total sum of more than of shares of the company directly (through itself) or indirectly (through another company controlled by ), then we say that company controls company . Let denotes that company controls company . Let denotes that company controls of company shares through company , since controls and owns of shares. Assume information about companies shares are represented as facts as described below. This company control problem is represented as a classical disjunctive logic program with classical aggregates, described below, whose answer set describes the intuitive and correct solution to the problem as illustrated in [Faber et al.2010] as:

The above representation of the company control problem as a classical disjunctive logic program with classical aggregates is entirely correct if our knowledge regarding the companies shares are prefect. However this is not always the case. Consider our knowledge regarding the company shares is not perfect. Thus, we cannot absolutely assert that some company controls another company as in the above representation. Instead, we can assert that a company controls another company with a certain degree of beliefs. In the presence of such uncertainties, the above company control problem need to be redefined to deal with imperfect knowledge about companies shares (namely fuzzy company control problem), where the imperfect knowledge about the companies shares are represented as a fuzzy set over companies shares. Consequently, a logical framework different from classical disjunctive logic programs with classical aggregates is needed for representing and reasoning about such fuzzy reasoning problems.

Consider that the fuzzy set over companies shares, presented in Example (1), is described as; company owns of company with grade membership ; company owns of company with grade membership ; company owns of company with grade membership ; and company owns of company with grade membership . Consider also that the same company control strategy as in Example (1) is employed. Thus, this fuzzy company control problem cannot be represented as a classical disjunctive logic program with classical aggregates, since classical disjunctive logic programs with classical aggregates do not allow neither representing and reasoning in the presence of fuzzy uncertainty nor allow aggregation over fuzzy sets. Moreover, this fuzzy company control problem cannot be represented as a disjunctive fuzzy logic program with fuzzy answer set semantics either, since disjunctive fuzzy logic programs with fuzzy answer set semantics do not allow aggregations over fuzzy sets by means of fuzzy aggregates for intuitive and concise representation of the problem.

Therefore, we propose to extend disjunctive fuzzy logic programs with fuzzy answer set semantics [Saad2010], denoted by DFLP, with arbitrary fuzzy aggregates to allow intuitive and concise representation of many real-world problems. To the best of our knowledge, this development is the first that defines semantics for fuzzy aggregates in a fuzzy answer set programming framework.

The contributions of this paper are as follows. We extend the original language of DFLP to allow arbitrary fuzzy annotation function including monotone, antimonotone, and nonmonotone annotation functions. We define the notions of fuzzy aggregates and fuzzy aggregate atoms in DFLP. We develop the fuzzy answer set semantics of DFLP with arbitrary fuzzy aggregates, denoted by DFLP, including monotone, antimonotone, and nonmonotone fuzzy aggregates. We show that the presented fuzzy answer set semantics of DFLP subsumes and generalizes both the original fuzzy answer set semantics of DFLP [Saad2010] and the classical answer set semantics of the classical disjunctive logic programs with classical aggregates, denoted by DLP [Faber et al.2010], and consequently subsumes the classical answer set semantics of classical disjunctive logic programs, denoted by DLP [Gelfond and Lifschitz1991]. We show that the fuzzy answer sets of DFLP are minimal fuzzy models and hence incomparable, which is an important property for nonmonotonic fuzzy reasoning.

The choice of DFLP for extension with fuzzy aggregates is interesting for many reasons. First, DFLP is very expressive form of fuzzy answer set programming that allows disjunctions to appear in the head of rules. It has been shown in [Saad2010] that; (1) DFLP is capable of representing and reasoning with both fuzzy uncertainty and qualitative uncertainty in which fuzzy uncertainly need to be defined over qualitative uncertainty; (2) DFLP is shown to be sophisticated logical framework for representing and reasoning about fuzzy preferences; (3) DFLP is a natural extension to DLP and its fuzzy answer set semantics subsumes the classical answer set semantics of DLP [Gelfond and Lifschitz1991]; (4) DFLP with fuzzy answer set semantics subsumes the fuzzy answer set programming framework of [Subrahmanian1994], which are DFLP programs with only an atom appearing in heads of rules.

2 Dflp : Fuzzy Aggregates Disjunctive Fuzzy Logic Programs

In this section we present the basic language of DFLP, the notions of fuzzy aggregates and fuzzy aggregate atoms, and the syntax of DFLP programs.

2.1 The Basic Language of DFLP

Let denotes an arbitrary first-order language with finitely many predicate symbols, function symbols, constants, and infinitely many variables. A term is a constant, a variable or a function. An atom, , is a predicate in , where is the Herbrand base of . The Herbrand universe of is denoted by . Non-monotonic negation or the negation as failure is denoted by . In fuzzy aggregates disjunctive fuzzy logic programs, DFLP, the grade membership values are assigned to atoms in as values from . The set and the relation form a complete lattice, where the join () operation is defined as and the meet () is defined as .

A fuzzy annotation, , is either a constant in (called fuzzy annotation constant), a variable ranging over (called fuzzy annotation variable), or (called fuzzy annotation function) where is a representation of a monotone, antimonotone, or nonmonotone total or partial function and are fuzzy annotations. If is an atom and is a fuzzy annotation then is called a fuzzy annotated atom.

2.2 Fuzzy Aggregate Atoms

A symbolic fuzzy set is an expression of the form
, where is a variable or a function term and is fuzzy annotation variable or fuzzy annotation function, and is a conjunction of fuzzy annotated atoms. A ground fuzzy set is a set of pairs of the form such that is a constant term and is fuzzy annotation constant, and is a ground conjunction of fuzzy annotated atoms. A symbolic fuzzy set or ground fuzzy set is called a fuzzy set term. Let be a fuzzy aggregate function symbol and be a fuzzy set term, then is said a fuzzy aggregate, where , , , , . If is a fuzzy aggregate and is a constant, a variable or a function term, called guard, then we say is a fuzzy aggregate atom, where .

Example 2

The following are examples for fuzzy aggregate atoms representation in DFLP language.

Definition 1

Let be a fuzzy aggregate. A variable, , is a local variable to if and only if appears in and does not appear in the DFLP rule that contains .

Definition (1) characterizes the local variables for a fuzzy aggregate function. For example, for the first fuzzy aggregate atom in Example (2), the variables and are local variables to the fuzzy aggregate .

Definition 2

A global variable is a variable that is not a local variable.

2.3 Dflp Program Syntax

This section defines the syntax of rules and programs in the language of DFLP.

Definition 3

A DFLP rule is an expression of the form

where are atoms, are atoms or fuzzy aggregate atoms, and are fuzzy annotations.

A DFLP rule means that if for each , where , it is believable that the grade membership value of is at least w.r.t. and for each , where , it is not believable that the grade membership value of is at least w.r.t. , then there exists at least , where , such that the grade membership value of is at least .

Definition 4

A DFLP program, , is a set of DFLP rules.

For the simplicity of the presentation, atoms that appear in DFLP programs without fuzzy annotations are assumed to be associated with the fuzzy annotation constant .

Example 3

The fuzzy company control problem described in Example 1 can be concisely and intuitively represented as DFLP program, , that consists of the DFLP rules:

The last DFLP rule in, , says that if it is at least grade membership value believable that company owns a total sum of more than of shares of the company directly (through itself) or indirectly (through another company controlled by ), then it is grade membership value believable that company controls company .

Definition 5

The ground instantiation of a symbolic fuzzy set

is the set of all ground pairs of the form , where is a substitution of every local variable appearing in to a constant from .

Definition 6

A ground instantiation of a DFLP rule, , is the replacement of each global variable appearing in to a constant from , then followed by the ground instantiation of every symbolic fuzzy set, , appearing in .

The ground instantiation of a DFLP program, , is the set of all possible ground instantiations of every DFLP rule in .

Example 4

A ground instantiation of the DFLP rule

with respect to the DFLP program, , described in Example 3, is given as:

3 Fuzzy Aggregates Semantics

A fuzzy aggregate is an aggregation over a fuzzy set that returns the evaluation of a classical aggregate and the grade membership value of the evaluation of that classical aggregate over a given fuzzy set. The fuzzy aggregates that we consider are , , , , and that find the evaluation of the classical aggregates , , , , and respectively along with the grade membership value of their evaluations. The application of fuzzy aggregates is on ground fuzzy sets which are sets of constants terms along with their associated grade membership values.

3.1 Mappings

Let denotes a set of objects. Then, we use to denote the set of all multisets over elements in . Let denotes the set of all real numbers and denotes the set of all natural numbers, and denotes the Herbrand universe. Let be a symbol that does not occur in . Therefore, the mappings of the fuzzy aggregates are given by:

  • .

  • .

  • .

  • .

  • .

The application of and on the empty multiset return and respectively. The application of on the empty multiset returns . However, the application of and on the empty multiset is undefined.

Definition 7

A fuzzy interpretation of a DFLP program, , is a mapping .

3.2 Semantics of Fuzzy Aggregates

The semantics of fuzzy aggregates is defined with respect to a fuzzy interpretation, which is a representation of fuzzy sets. A fuzzy annotated atom, , is true (satisfied) with respect to a fuzzy interpretation, , if and only if . The negation of a fuzzy annotated atom, , is true (satisfied) with respect to if and only if . The evaluation of a fuzzy aggregate, and hence the truth valuation of a fuzzy aggregate atom, are established with respect to a given fuzzy interpretation, , as presented in the following definitions.

Definition 8

Let be a ground fuzzy aggregate and be a fuzzy interpretation. Then, we define to be the multiset constructed from elements in , where is true w.r.t. .

Definition 9

Let be a ground fuzzy aggregate and be a fuzzy interpretation. Then, the evaluation of with respect to is, , the result of the application of to , where if is not in the domain of and

4 Fuzzy Answer Set Semantics of DFLP

In this section we define the satisfaction, fuzzy models, and the fuzzy answer set semantics of fuzzy aggregates disjunctive fuzzy logic programs, DFLP. Let be a DFLP rule and and
.

Definition 10

Let be a ground DFLP program, be a DFLP rule in , be a fuzzy interpretation for , and . Then,

  1. satisfies in iff .

  2. satisfies in iff and and .

  3. satisfies in iff or and or .

  4. satisfies in iff .

  5. satisfies in iff .

  6. satisfies iff satisfies and satisfies .

  7. satisfies iff such that satisfies .

  8. satisfies iff satisfies whenever satisfies or does not satisfy .

  9. satisfies iff satisfies every DFLP rule in and

    • such that satisfies and satisfies in the .

Example 5

Let be a DFLP program that consists of the DFLP rules:

The ground instantiation of is given by:

Let be a fuzzy interpretation of that assign to , to , and assigns to the remaining atoms in . Thus the evaluation of the fuzzy aggregate atom, in w.r.t. to is given as follows, where

and . Therefore,
, and consequently, the fuzzy annotated fuzzy annotated aggregate atom is not satisfied by . This is because and although . Let be a fuzzy interpretation of that assign to , to , and assigns to the remaining atoms in . Thus, and , hence the fuzzy annotated fuzzy annotated aggregate atom is satisfied by , since and .

Let be a fuzzy annotated atom, or the negation of , denoted by . Let be two fuzzy interpretations. Then, we say that is monotone if such that , it is the case that if satisfies then also satisfies . However, is antimonotone if such that it is the case that if satisfies then also satisfies . But, if is not monotone or not antimonotone, then we say is nonmonotone. A fuzzy annotated atom or a fuzzy annotated fuzzy aggregate atom, , or the negation of fuzzy annotated atom or the negation of a fuzzy annotated fuzzy aggregate atom, , can be monotone, antimonotone or nonmonotone, since their fuzzy annotations are allowed to be arbitrary functions. Moreover, fuzzy aggregate atoms by themselves can be monotone, antimonotone or nonmonotone.

Definition 11

A fuzzy model for a DFLP program, , is a fuzzy interpretation for that satisfies . A fuzzy model for is –minimal iff there does not exist a fuzzy model for such that .

Example 6

It can easily verified that the fuzzy interpretation, , for DFLP program, , described in Example (5), is a minimal fuzzy model for . However, the fuzzy interpretation, , for , described in Example (5), is not a fuzzy model for .

Definition 12

Let be a ground DFLP program, be a DFLP rule in , and be a fuzzy interpretation for . Let denotes satisfies . Then, the fuzzy reduct, , of w.r.t. is a ground DFLP program where

Definition 13

A fuzzy interpretation, , of a ground DFLP program, , is a fuzzy answer set for if is -minimal fuzzy model for .

Observe that the definitions of the fuzzy reduct and the fuzzy answer sets for DFLP programs are generalizations of the fuzzy reduct and the fuzzy answer sets of the original DFLP programs described in [Saad2010].

Example 7

It can be easily verified that the DFLP program described in Example (5) has three fuzzy answer sets , , and presented below, where atoms in that are not appearing in , , and are assumed to be assigned the fuzzy annotation .

Example 8

The DFLP program representation of the fuzzy company control problem, , described in Example (3) has one fuzzy answer set, , which, after omitting the facts and assuming atoms in that do not appear in are assigned the annotation , is

The fuzzy answer set, , implies that no company fuzzy controls another company.

5 Dflp Semantics Properties

In this section we study the semantics properties of DFLP programs and its relationship to the original fuzzy answer set semantics of disjunctive fuzzy logic programs, denoted by DFLP [Saad2010]; the classical answer set semantics of classical disjunctive logic programs with classical aggregates, denoted by DLP [Faber et al.2010]; and the original classical answer set semantics of classical disjunctive logic programs, denoted by DLP [Gelfond and Lifschitz1991].

Theorem 1

Let be a DFLP program. The fuzzy answer sets for are –minimal fuzzy models for .

The following theorem shows that the fuzzy answer set semantics of DFLP subsumes and generalizes the fuzzy answer set semantics of DFLP [Saad2010], which are DFLP programs without fuzzy aggregates atoms and with only monotone fuzzy annotation functions.

Theorem 2

Let be a DFLP program and be a fuzzy interpretation. Then, is a fuzzy answer set for iff is a fuzzy answer set for w.r.t. the fuzzy answer set semantics of [Saad2010].

Now we show that the fuzzy answer set semantics of DFLP programs naturally subsumes and generalizes the classical answer set semantics of the classical disjunctive logic programs with the classical aggregates, DLP [Faber et al.2010], which consequently naturally subsumes the classical answer set semantics of the original classical disjunctive logic programs, DLP [Gelfond and Lifschitz1991].

Any DLP program, , is represented as a DFLP program, , where each DLP rule in of the form

is represented, in , as a DFLP rule of the form

where are atoms and are atoms or fuzzy aggregate atoms whose fuzzy aggregates contain fuzzy sets that involve conjunctions of fuzzy annotated atoms with the fuzzy annotation , where represents the truth value true. We call this class of DFLP programs as DFLP. Any DLP program is represented as a DFLP program by the same way as DLP except that DLP disallows classical aggregate atoms. The following results show that DFLP programs subsume both DLP and DLP programs.

Theorem 3

Let be a DFLP program equivalent to a DLP program . Then, is a fuzzy answer set for iff is a classical answer set for , where iff and iff .

Proposition 1

Let be a DFLP program equivalent to a DLP program . Then, is a fuzzy answer set for iff is a classical answer set for , where iff and iff .

6 Conclusions and Related Work

We presented the syntax and semantics of the fuzzy aggregates disjunctive fuzzy logic programs, DFLP, that extends the original disjunctive fuzzy logic programs, DFLP [Saad2010], with arbitrary fuzzy annotation functions and with arbitrary fuzzy aggregates. We introduced the fuzzy answer set semantics of DFLP programs with arbitrary fuzzy aggregates including monotone, antimonotone, and nonmonotone fuzzy aggregates. We have shown that the fuzzy answer set semantics of DFLP subsumes and generalizes the fuzzy answer set semantics of the original DFLP [Saad2010]. In addition, we proved that the fuzzy answer sets of DFLP are minimal fuzzy models and consequently incomparable, which is an important property for nonmonotonic fuzzy reasoning. We have shown that the fuzzy answer set semantics of DFLP subsumes and generalizes the classical answer set semantics of both the classical aggregates classical disjunctive logic programs and the original classical disjunctive logic programs. To the best of our knowledge, this development is the first to consider fuzzy aggregates in fuzzy logical reasoning in general and in fuzzy answer set programming in particular. However, classical aggregates were extensively investigated in classical answer set programming [Faber et al.2010, Niemelä and Simons2001, Pelov et al.2007, Pelov and Truszczynski2004, Ferraris and Lifschitz2005, Ferraris and Lifschitz2010, Pelov2004]. A comprehensive comparisons among these approaches to classical aggregates in classical answer set programming [Faber et al.2010, Niemelä and Simons2001, Pelov et al.2007, Pelov and Truszczynski2004, Ferraris and Lifschitz2005, Ferraris and Lifschitz2010, Pelov2004] in general and between these approaches and DLP in particular is found in [Faber et al.2010].

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