The Complexity of Poor Man's Logic

TL;DR

This paper explores how restricting modal formulas to literals, conjunction, and modal operators affects the complexity of satisfiability problems across different frame classes, revealing varied computational complexities.

Contribution

It demonstrates that the complexity of modal satisfiability varies with frame restrictions, solving an open problem and providing a comprehensive classification.

Findings

Satisfiability in frames with at least one successor per world drops to P.

Satisfiability in frames with at most two successors per world remains PSPACE-complete.

The paper classifies complexity for all other operator restrictions.

Abstract

Motivated by description logics, we investigate what happens to the complexity of modal satisfiability problems if we only allow formulas built from literals, $\wedge$, $\Diamond$, and $\Box$. Previously, the only known result was that the complexity of the satisfiability problem for K dropped from PSPACE-complete to coNP-complete (Schmidt-Schauss and Smolka, 1991 and Donini et al., 1992). In this paper we show that not all modal logics behave like K. In particular, we show that the complexity of the satisfiability problem with respect to frames in which each world has at least one successor drops from PSPACE-complete to P, but that in contrast the satisfiability problem with respect to the class of frames in which each world has at most two successors remains PSPACE-complete. As a corollary of the latter result, we also solve the open problem from Donini et al.'s complexity classification of description logics (Donini et al., 1997). In the last section, we classify the complexity of the satisfiability problem for K for all other restrictions on the set of operators.