Modal Logic S5 Satisfiability in Answer Set Programming

TL;DR

This paper introduces an Answer Set Programming approach to improve the efficiency of S5 modal logic satisfiability checking, reducing formula size and enhancing performance compared to traditional Skolemisation-based methods.

Contribution

It proposes novel ASP encodings leveraging reachability and entailment relations to optimize S5 satisfiability evaluation, offering an alternative to SAT-based solvers.

Findings

Reachability-based encodings are highly effective.

Performance is comparable to state-of-the-art SAT-based solvers.

Entailment relations may introduce overhead and are less effective.

Abstract

Modal logic S5 has attracted significant attention and has led to several practical applications, owing to its simplified approach to dealing with nesting modal operators. Efficient implementations for evaluating satisfiability of S5 formulas commonly rely on Skolemisation to convert them into propositional logic formulas, essentially by introducing copies of propositional atoms for each set of interpretations (possible worlds). This approach is simple, but often results into large formulas that are too difficult to process, and therefore more parsimonious constructions are required. In this work, we propose to use Answer Set Programming for implementing such constructions, and in particular for identifying the propositional atoms that are relevant in every world by means of a reachability relation. The proposed encodings are designed to take advantage of other properties such as entailment relations of subformulas rooted by modal operators. An empirical assessment of the proposed encodings shows that the reachability relation is very effective and leads to comparable performance to a state-of-the-art S5 solver based on SAT, while entailment relations are possibly too expensive to reason about and may result in overhead. This paper is under consideration for acceptance in TPLP.