A Strongly Grounded Stable Model Semantics for Full Propositional Language

TL;DR

This paper introduces a new stable model semantics called supported semantics for full propositional logic, ensuring minimality and strong grounding, and compares it with existing semantics like equilibrium models.

Contribution

It proposes a novel supported semantics extending stable model semantics to full propositional language, guaranteeing minimality and strong grounding, and relates it to existing non-monotonic reasoning frameworks.

Findings

Supported semantics coincide with derivability in intuitionistic logic.

Extended well-founded semantics to full propositional language.

Reasoning tasks are PSPACE-complete, more expressive than equilibrium models.

Abstract

Answer set programming is one of the most praised frameworks for declarative programming in general and non-monotonic reasoning in particular. There has been many efforts to extend stable model semantics so that answer set programs can use a more extensive syntax. To such endeavor, the community of non-monotonic reasoning has introduced extensions such as equilibrium models and FLP semantics. However, both of these extensions suffer from two problems: intended models according to such extensions (1) are not guaranteed to be minimal, and (2) more importantly, may have self-justifications (i.e., the justification for pertinence of an atom in an intended model may be its own pertinence). Both of these properties directly violate the spirit of stable model semantics. Therefore, we believe that we need a new extension of stable model semantics that guarantees both minimality and being strongly grounded. This paper introduces one such extension using two different approaches: firstly, by extending the goal-reachability interpretation of logic programs to the full propositional language and, secondly, using derivability in intuitionistic propositional logic. We show that both these approaches give the same semantics, that we call the supported semantics. Moreover, using the first approach, we also extend well-founded semantics to full propositional language. Furthermore, we discuss how our supported models relate to other existing semantics for non-monotonic reasoning including the equilibrium models. Last, but not the least, we discuss the complexity of reasoning about supported models and show that all interesting reasoning tasks (such as brave/cautious reasoning) are PSPACE-complete. Therefore, supported model semantics is a much more expressive semantics then the existing semantics such as equlibrium models (that have reasoning procedures in $\Delta^P_3$).