Characterizations of stable model semantics for logic programs with arbitrary constraint atoms

Abstract

This paper studies the stable model semantics of logic programs with (abstract) constraint atoms and their properties. We introduce a succinct abstract representation of these constraint atoms in which a constraint atom is represented compactly. We show two applications. First, under this representation of constraint atoms, we generalize the Gelfond-Lifschitz transformation and apply it to define stable models (also called answer sets) for logic programs with arbitrary constraint atoms. The resulting semantics turns out to coincide with the one defined by Son et al., which is based on a fixpoint approach. One advantage of our approach is that it can be applied, in a natural way, to define stable models for disjunctive logic programs with constraint atoms, which may appear in the disjunctive head as well as in the body of a rule. As a result, our approach to the stable model semantics for logic programs with constraint atoms generalizes a number of previous approaches. Second, we show that our abstract representation of constraint atoms provides a means to characterize dependencies of atoms in a program with constraint atoms, so that some standard characterizations and properties relying on these dependencies in the past for logic programs with ordinary atoms can be extended to logic programs with constraint atoms.

Relating weight constraint and aggregate programs: Semantics and representation

“…For logic programs with abstract constraint atoms, it is often said that all of the major semantics coincide for programs with monotone constraints. For example, this is the case for the semantics proposed in (Faber et al 2004;Marek and Remmel 2004;Ferraris 2005;Marek et al 2007;Liu and Truszczyński 2006;Shen et al 2009;Liu et al 2010). What is unexpected is that this is not the case for the stable model semantics for weight constraint programs.…”

“…Recently, ASP was extended to include constraints to facilitate reasoning with sets of atoms. These constraints include weight constraints (Simons et al 2002), aggregates (Faber et al 2004;Ferraris 2005;Pelov et al 2007;, and abstract constraints (Marek and Remmel 2004;Liu and Truszczyński 2006;Marek et al 2007;You et al 2007;Shen et al 2009;Liu et al 2010). Among them, weight constraints and aggregates are the most widely used constraints in practice.…”

“…Among the semantics for aggregate programs, this semantics is known to be the most conservative, in the sense that any answer set under this semantics is an answer set under others, but the reverse may not hold. The relationships of these semantics have been studied in , Shen et al (2009), and Liu et al (2010). In this paper, we refer to the semantics based on conditional satisfaction as the answer set semantics.…”

Vicious Circle Principle and Formation of Sets in ASP Based Languages

“…This generalizes the standard ASP semantics for disjunctive programs. We conjecture that when we adapt our definition of Slog + semantics to disjunctive constraint programs, it will coincide with that of [33,20]. However, our definition seems to be simpler and is based on clear, VCP related intuition.…”

“…More information about Slog and Slog + can be found in Section 3. Shen et al [33] and Liu et al [20] propose equivalent semantics for disjunctive constraint programs (i.e., programs with rules whose bodies are built from constraint atoms and whose heads are epistemic disjunctions of such atoms). This generalizes the standard ASP semantics for disjunctive programs.…”