Does Treewidth Help in Modal Satisfiability?

TL;DR

This paper explores whether the treewidth of incidence graphs can be used to efficiently determine the satisfiability of modal logic formulas, revealing complexity results across different model classes.

Contribution

It introduces a new incidence graph concept for modal logic and analyzes the parameterized complexity of modal satisfiability based on treewidth and modal depth.

Findings

Modal satisfiability in general models is FPT.

It is W[1]-hard in transitive models.

Modal satisfiability in transitive and Euclidean models is FPT.

Abstract

Many tractable algorithms for solving the Constraint Satisfaction Problem (CSP) have been developed using the notion of the treewidth of some graph derived from the input CSP instance. In particular, the incidence graph of the CSP instance is one such graph. We introduce the notion of an incidence graph for modal logic formulae in a certain normal form. We investigate the parameterized complexity of modal satisfiability with the modal depth of the formula and the treewidth of the incidence graph as parameters. For various combinations of Euclidean, reflexive, symmetric and transitive models, we show either that modal satisfiability is FPT, or that it is W[1]-hard. In particular, modal satisfiability in general models is FPT, while it is W[1]-hard in transitive models. As might be expected, modal satisfiability in transitive and Euclidean models is FPT.