On Sub-Propositional Fragments of Modal Logic

TL;DR

This paper investigates the expressive power and computational complexity of various sub-propositional fragments of modal logics like K, T, K4, and S4, revealing a hierarchy and polynomial satisfiability in certain cases.

Contribution

It introduces a hierarchy of sub-propositional fragments based on expressive power and analyzes their satisfiability complexity, highlighting low expressive power and polynomial cases.

Findings

Hierarchy of fragments based on expressive power

Low expressive power of Horn fragments without diamonds

Polynomial satisfiability for certain fragments

Abstract

In this paper, we consider the well-known modal logics $\mathbf{K}$, $\mathbf{T}$, $\mathbf{K4}$, and $\mathbf{S4}$, and we study some of their sub-propositional fragments, namely the classical Horn fragment, the Krom fragment, the so-called core fragment, defined as the intersection of the Horn and the Krom fragments, plus their sub-fragments obtained by limiting the use of boxes and diamonds in clauses. We focus, first, on the relative expressive power of such languages: we introduce a suitable measure of expressive power, and we obtain a complex hierarchy that encompasses all fragments of the considered logics. Then, after observing the low expressive power, in particular, of the Horn fragments without diamonds, we study the computational complexity of their satisfiability problem, proving that, in general, it becomes polynomial.