Open answer set programming with guarded programs

Abstract

Open answer set programming (OASP) is an extension of answer set programming where one may ground a program with an arbitrary superset of the program's constants. We define a fixed point logic (FPL) extension of Clark's completion such that open answer sets correspond to models of FPL formulas and identify a syntactic subclass of programs, called (loosely) guarded programs. Whereas reasoning with general programs in OASP is undecidable, the FPL translation of (loosely) guarded programs falls in the decidable (loosely) guarded fixed point logic (µ(L)GF). Moreover, we reduce normal closed ASP to loosely guarded OASP, enabling for the first time, a characterization of an answer set semantics by µLGF formulas.We further extend the open answer set semantics for programs with generalized literals. Such generalized programs (gPs) have interesting properties, e.g., the ability to express infinity axioms. We restrict the syntax of gPs such that both rules and generalized literals are guarded. Via a translation to guarded fixed point logic, we deduce 2-exptime-completeness of satisfiability checking in such guarded gPs (GgPs). Bound GgPs are restricted GgPs with exptime-complete satisfiability checking, but still sufficiently expressive to optimally simulate computation tree logic (CTL). We translate Datalog lite programs to GgPs, establishing equivalence of GgPs under an open answer set semantics, alternation-free µGF, and Datalog lite. This is a revised and extended version of [Heymans et al. 2005a] and [Heymans et al. 2006a].

FO(ID) as an Extension of DL with Rules

“…[3], [14] and [10] both provide strong semantic integrations for a more general kind of rules, with [10] also defining a guarded fragment that, like ours, can be mapped into guarded fixpoint logic. The main feature that distinguishes our approach from these is of a KR nature.…”

Quantified Equilibrium Logic and Hybrid Rules

“…Two variations of this semantics, the open [8] and generalised open answer set [9] semantics, consider open domains, thereby relaxing the PNA. An extended Herbrand structure is a first-order structure based on a universe U = (D, id), where D ⊇ C. In the open answer set semantics the UNA applies.…”

“…An extended Herbrand structure is a first-order structure based on a universe U = (D, id), where D ⊇ C. In the open answer set semantics the UNA applies. Note that every answer set of a program is also an open answer set [8], but the converse does not hold in general:…”

A Revised Concept of Safety for General Answer Set Programs

“…The latter definition of answer set has been further studied in [10,11] where the basis of a first-oder programming language, RASPL-1, is described. An alternative approach to a first-order ASP language is developed in [8,7].…”