Semantics and Compilation of Answer Set Programming with Generalized Atoms

TL;DR

This paper introduces a new semantics called supportedly stable (SFLP) for answer set programming with generalized atoms, providing more intuitive models for complex constructs and offering a compilation method to transform such programs into standard ASP programs.

Contribution

It proposes the supportedly stable semantics for generalized atoms, proves its equivalence to existing semantics for convex cases, and develops a compilation technique for transforming programs with generalized atoms.

Findings

Supportedly stable models are more intuitive for non-convex generalized atoms.

The complexity of supportedly stable models is on the second level of the polynomial hierarchy.

A compilation method is provided to translate programs with generalized atoms into standard ASP programs.

Abstract

Answer Set Programming (ASP) is logic programming under the stable model or answer set semantics. During the last decade, this paradigm has seen several extensions by generalizing the notion of atom used in these programs. Among these, there are aggregate atoms, HEX atoms, generalized quantifiers, and abstract constraints. In this paper we refer to these constructs collectively as generalized atoms. The idea common to all of these constructs is that their satisfaction depends on the truth values of a set of (non-generalized) atoms, rather than the truth value of a single (non-generalized) atom. Motivated by several examples, we argue that for some of the more intricate generalized atoms, the previously suggested semantics provide unintuitive results and provide an alternative semantics, which we call supportedly stable or SFLP answer sets. We show that it is equivalent to the major previously proposed semantics for programs with convex generalized atoms, and that it in general admits more intended models than other semantics in the presence of non-convex generalized atoms. We show that the complexity of supportedly stable models is on the second level of the polynomial hierarchy, similar to previous proposals and to stable models of disjunctive logic programs. Given these complexity results, we provide a compilation method that compactly transforms programs with generalized atoms in disjunctive normal form to programs without generalized atoms. Variants are given for the new supportedly stable and the existing FLP semantics, for which a similar compilation technique has not been known so far.