FLP answer set semantics without circular justifications for general logic programs

Abstract

The answer set semantics presented by Faber et al. [27] has been widely used to define so called FLP answer sets for different types of logic programs. However, it was recently observed that when being extended from normal to more general classes of logic programs, this approach may produce answer sets with circular justifications that are caused by selfsupporting loops. The main reason for this behavior is that the FLP answer set semantics is not fully constructive by a bottom up construction of answer sets. In this paper, we overcome this problem by enhancing the FLP answer set semantics with a level mapping formalism such that every answer set I can be built by fixpoint iteration of a one-step provability operator (more precisely, an extended van Emden-Kowalski operator for the FLP reduct f Π I ). This is inspired by the fact that under the standard answer set semantics, each answer set I of a normal logic program Π is obtainable by fixpoint iteration of the standard van Emden-Kowalski one-step provability operator for the Gelfond-Lifschitz reduct Π I , which induces a level mapping. The enhanced FLP answer sets, which we call well-justified FLP answer sets, are thanks to the level mapping free of circular justifications. As a general framework, the well-justified FLP answer set semantics applies to logic programs with first-order formulas, logic programs with aggregates, description logic programs, hex-programs etc., provided that the rule satisfaction is properly extended to such general logic programs. We study in depth the computational complexity of FLP and well-justified FLP answer sets for general classes of logic programs. Our results show that the level mapping does not increase the worst-case complexity of FLP answer sets. Furthermore, we describe an implementation of the well-justified FLP answer set semantics, and report about an experimental evaluation, which indicates a potential for performance improvements by the level mapping in practice.

selp: A Single-Shot Epistemic Logic Program Solver

“…Other ASP Semantics. Apart from the original semantics for ASP (called stable model semantics, (Gelfond and Lifschitz 1988;Gelfond and Lifschitz 1991)), several different semantics have been proposed that investigate how to interpret more advanced constructs in ASP, like double negation, aggregates, optimization, etc (Lifschitz et al 1999;Pearce 2006;Pelov et al 2007;Ferraris et al 2011;Faber et al 2011;Shen et al 2014). Epistemic reducts may contain double negation, and we have opted to use the FLP semantics by Faber et al (2011), as used by Shen and Eiter (2011), to interpret this.…”

Evaluating Answer Set Programming with Non-Convex Recursive Aggregates

“…Several other stable model semantics were proposed for interpreting logic programs with aggregates. Many of these semantics rely on stability checks that are not based on minimality [30,31,33], and therefore the rewritings presented by [4] and recalled in Section 3 cannot be used for these semantics. A more recent proposal is based on a stability check that essentially eliminates aggregates from program reducts [23], and therefore the rewritings by [4] cannot help also in this case.…”

“…Mainstream ASP solvers [15,20] almost agree on the semantics of aggregates [14,17], here referred to as F-stable model semantics, even if several valid alternatives were also considered in the literature [23,30,31,33]. It is interesting to observe that F-stable model semantics was proposed more than a decade ago, providing a reasonable semantics for aggregates also in the recursive case.…”

On the Properties of Atom Definability and Well-Supportedness in Logic Programming

“…For instance, the unique FLP-stable model of (1) is ∅. Apart from minimality, another property that has been recently considered by Shen et al in [10] is the extension of Fages' well-supportedness [11], originally defined for normal logic programs, to rules with a more general syntax like, for instance, allowing Boolean formulas in the head or the body. Intuitively, a model M is said to be well-supported if its true atoms can be assigned a derivation ordering (via modus ponens) from the positive part of the program, while the interpretation of negated atoms is fixed with respect to M , acting like an assumption a priori.…”