[Paper Review] Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms

Enrico Pontelli, Tran Cao Son+1 more|arXiv (Cornell University)|Oct 10, 2011

Logic, Reasoning, and Knowledge28 references57 citations

TL;DR

This paper introduces four equivalent answer set semantics for logic programs with arbitrary abstract constraint atoms (c-atoms), generalizing stable model semantics to non-monotonic programs with complex constraints. It proposes two fixpoint-based approaches—answer sets by reduct and by complement—and two well-supported model-based definitions—strongly and weakly well-supported models—proving their equivalence and extending classical answer set semantics to c-atoms while preserving key properties like correspondence with well-supported models.

ABSTRACT

In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negation-as-failure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly well-supported models, are generalizations of the notion of well-supported models of normal logic programs to the case of programs with c-atoms. As for the case of fixpoint-based semantics, the difference between these two definitions is rooted in the treatment of naf atoms. We prove that answer sets by reduct (resp. by complement) are equivalent to weakly (resp. strongly) well-supported models of a program, thus generalizing the theorem on the correspondence between stable models and well-supported models of a normal logic program to the class of programs with c-atoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone c-atoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with c-atoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature.

Motivation & Objective

• To develop a uniform, general framework for answer set semantics applicable to logic programs extended with arbitrary abstract constraint atoms (c-atoms), subsuming aggregates, weight constraints, and cardinality constraints.
• To address limitations in prior semantics for c-atoms, particularly the unintuitive answer sets produced by Marek and Remmel's original approach.
• To generalize both fixpoint-based and well-supported model-based answer set definitions from normal logic programs to programs with arbitrary c-atoms.
• To ensure the new semantics coincide with existing semantics for monotone c-atoms and extend classical answer set semantics for normal logic programs.
• To establish theoretical foundations for building block results (e.g., splitting theorems) applicable across various extensions of logic programming.

Proposed method

• Introduces an immediate consequence operator based on conditional satisfaction of c-atoms with respect to a pair of interpretations (I, M), enabling iterative computation of answer sets.
• Defines two fixpoint-based semantics: answer set by reduct and answer set by complement, differing in how they handle negation-as-failure (naf) atoms.
• Proposes two well-supported model-based semantics: strongly and weakly well-supported models, which generalize the notion of well-supported models to c-atoms.
• Uses a notion of conditional satisfaction of c-atoms w.r.t. interpretations I and M, where I represents the current stage of computation and M the candidate answer set.
• Proves that answer sets by reduct correspond to weakly well-supported models, and answer sets by complement correspond to strongly well-supported models.
• Establishes equivalence between the four semantics for normal logic programs and shows consistency with prior semantics for monotone and convex c-atoms.

Experimental results

Research questions

• RQ1 Can a uniform answer set semantics be defined for logic programs with arbitrary abstract constraint atoms, generalizing classical answer set semantics?
• RQ2 How should negation-as-failure be treated in the context of arbitrary c-atoms to preserve desirable properties like minimality and well-supportedness?
• RQ3 Do the proposed semantics for c-atoms preserve key results such as the correspondence between stable models and well-supported models?
• RQ4 Are the new semantics consistent with previously established semantics for monotone and convex c-atoms?
• RQ5 Can the new semantics be related to and subsume existing semantics for aggregates and weight constraints?

Key findings

• The four proposed answer set semantics—answer sets by reduct, by complement, strongly well-supported models, and weakly well-supported models—are equivalent when applied to normal logic programs.
• Answer sets by reduct are equivalent to weakly well-supported models, and answer sets by complement are equivalent to strongly well-supported models, generalizing the classical correspondence between stable and well-supported models.
• The proposed semantics extend the original answer set semantics for normal logic programs and coincide with prior semantics for monotone c-atoms.
• The semantics are consistent with the FLP and Ferraris semantics for monotone aggregates, as proven via equivalence to the FLP reduct-based answer set definition.
• The immediate consequence operator based on conditional satisfaction of c-atoms enables correct and uniform answer set checking across arbitrary c-atoms.
• The framework supports a uniform treatment of aggregates and constraints, enabling future application of building block results (e.g., splitting theorems) across all extensions of logic programming.