The Inverse Method Implements the Automata Approach for Modal Satisfiability

TL;DR

This paper demonstrates that Voronkov's inverse method for modal logic K can be viewed as an on-the-fly automata-based satisfiability check, providing both practical efficiency and theoretical insights.

Contribution

It shows that the inverse method is an optimized on-the-fly automata approach for modal logic K and extends to handle global axioms, maintaining optimal complexity.

Findings

Inverse method aligns with automata-based emptiness testing.

Implementation is an optimized on-the-fly automata procedure.

Extends to global axioms with preserved complexity.

Abstract

Tableaux-based decision procedures for satisfiability of modal and description logics behave quite well in practice, but it is sometimes hard to obtain exact worst-case complexity results using these approaches, especially for EXPTIME-complete logics. In contrast, automata-based approaches often yield algorithms for which optimal worst-case complexity can easily be proved. However, the algorithms obtained this way are usually not only worst-case, but also best-case exponential: they first construct an automaton that is always exponential in the size of the input, and then apply the (polynomial) emptiness test to this large automaton. To overcome this problem, one must try to construct the automaton "on-the-fly" while performing the emptiness test. In this paper we will show that Voronkov's inverse method for the modal logic K can be seen as an on-the-fly realization of the emptiness test done by the automata approach for K. The benefits of this result are two-fold. First, it shows that Voronkov's implementation of the inverse method, which behaves quite well in practice, is an optimized on-the-fly implementation of the automata-based satisfiability procedure for K. Second, it can be used to give a simpler proof of the fact that Voronkov's optimizations do not destroy completeness of the procedure. We will also show that the inverse method can easily be extended to handle global axioms, and that the correspondence to the automata approach still holds in this setting. In particular, the inverse method yields an EXPTIME-algorithm for satisfiability in K w.r.t. global axioms.