Complexity of the variable-free fragments of non-normal modal logics (extended version)

TL;DR

This paper proves that the satisfiability problem for variable-free fragments of certain non-normal modal logics is computationally hard, specifically NP-hard or coNP-complete, depending on the logic.

Contribution

It establishes the complexity classification of variable-free fragments in a range of non-normal modal logics, including E, EM, EN, and EMN.

Findings

Variable-free fragments of non-normal modal logics are NP-hard or coNP-complete.

Satisfiability problems for these fragments are computationally complex.

Results apply to logics contained in the weak Grzegorczyk logic.

Abstract

We show that the satisfiability problem for the variable-free fragment of every modal logic containing classical propositional logic and contained in the weak Grzegorczyk logic is NP-hard. In particular, the variable-free fragments of the non-normal modal logics E, EM, EN, and EMN are coNP-complete.