A Model-Theoretic Approach to Belief Change in Answer Set Programming

Abstract

We address the problem of belief change in (nonmonotonic) logic programming under answer set semantics. Our formal techniques are analogous to those of distance-based belief revision in propositional logic. In particular, we build upon the model theory of logic programs furnished by SE interpretations, where an SE interpretation is a model of a logic program in the same way that a classical interpretation is a model of a propositional formula. Hence we extend techniques from the area of belief revision based on distance between models to belief change in logic programs.We first consider belief revision: for logic programs P and Q, the goal is to determine a program R that corresponds to the revision of P by Q, denoted P * Q. We investigate several operators, including (logic program) expansion and two revision operators based on the distance between the SE models of logic programs. It proves to be the case that expansion is an interesting operator in its own right, unlike in classical belief revision where it is relatively uninteresting. Expansion and revision are shown to satisfy a suite of interesting properties; in particular, our revision operators satisfy all or nearly all of the AGM postulates for revision.We next consider approaches for merging a set of logic programs, P 1 , . . . , Pn. Again, our formal techniques are based on notions of relative distance between the SE models of the logic programs. Two approaches are examined. The first informally selects for each program P i those models of P i that vary the least from models of the other programs. The second approach informally selects those models of a program P 0 that are closest to the models of programs P 1 , . . . , Pn. In this case, P 0 can be thought of as a set of database integrity constraints. We examine these operators with regards to how they satisfy relevant postulate sets.Last, we present encodings for computing the revision as well as the merging of logic programs within the same logic programming framework. This gives rise to a direct implementation of our approach in terms of off-the-shelf answer set solvers. These encodings also reflect the fact that our change operators do not increase the complexity of the base formalism.

AGM-Style Belief Revision of Logic Programs under Answer Set Semantics

“…Other work uses logic programs under a variant of the stable semantics to specify database revision, i.e., the revision of knowledge bases given as sets of atomic facts [22]. [12] addresses specific revision (and belief merging) operators based on distances defined in terms of the SE models of the underlying programs. As well, [9] considers the extent to which logic programs per se, are compatible with the AGM approach to revision.…”

“…The postulates and semantic construction of Section 2 are easily adapted to logic programs; for this, we draw on material from [12,10]. To begin, the expansion of logic programs P and Q, P + Q, can be defined as a logic program R where SE (P ) ∩ SE (Q) = SE (R).…”

“…[14,23]. Recently however, specific revision operators based on the monotonic concept of SE-models [28] (which underlies the answer-set semantics of logic programs [21]) have been proposed [12] together with a suitable variant of the AGM postulates; see also [27] for a variant thereof. In recent work [19], the notion of SE-models (and similar concepts) has been put in connection to postulates for belief base change different from AGM.…”

“…We give representation theorems for the AGM-style postulates proposed in [12] by exploiting, first, the recent techniques for Horn revision due to [10] and, second, the properties that program classes enjoy in terms of SE-models [13]. This allows us to give representation theorems for the important classes of disjunctive (generalized and ordinary), normal, positive, and Horn logic programs.…”

“…This allows us to give representation theorems for the important classes of disjunctive (generalized and ordinary), normal, positive, and Horn logic programs. This work is interesting because, on the one hand it shows how the AGM approach can be extended to a seemingly nonmonotonic framework, while on the other hand the formal characterization may provide guiding principles for the development of specific revision operators beyond the ones suggested in [12].…”