Generalized Modal Satisfiability

TL;DR

This paper classifies the computational complexity of modal satisfiability across all finite propositional operator sets, revealing a trichotomy of PSPACE-complete, coNP-complete, or polynomial cases, for both formulas and circuits.

Contribution

It provides a complete complexity classification for modal satisfiability with any finite set of propositional operators, extending previous work to an infinite set of problems.

Findings

Modal satisfiability complexity varies with propositional operators

Identifies a trichotomy: PSPACE-complete, coNP-complete, or in P

Results apply to both formulas and modal circuits

Abstract

It is well known that modal satisfiability is PSPACE-complete (Ladner 1977). However, the complexity may decrease if we restrict the set of propositional operators used. Note that there exist an infinite number of propositional operators, since a propositional operator is simply a Boolean function. We completely classify the complexity of modal satisfiability for every finite set of propositional operators, i.e., in contrast to previous work, we classify an infinite number of problems. We show that, depending on the set of propositional operators, modal satisfiability is PSPACE-complete, coNP-complete, or in P. We obtain this trichotomy not only for modal formulas, but also for their more succinct representation using modal circuits. We consider both the uni-modal and the multi-modal case, and study the dual problem of validity as well.