Modal Fragments

TL;DR

This paper surveys systematic approaches to basis-restricted fragments of propositional and modal logics, analyzing how expressive power and complexity depend on allowed operators, and introduces a unified framework for modal fragments.

Contribution

It unifies two lines of investigation on modal fragments, extending complexity classifications and decidability results using Post-lattice methods.

Findings

Modal fragments' complexity depends on operator basis.

Systematic classification of propositional and modal fragments.

Extended results on learnability of fragments.

Abstract

We survey systematic approaches to basis-restricted fragments of propositional logic and modal logics, with an emphasis on how expressive power and computational complexity depend on the allowed operators. The propositional case is well-established and serves as a conceptual template: Post's lattice organizes fragments via Boolean clones and supports complexity classifications for standard reasoning tasks. For modal fragments, we then bring together two historically independent lines of investigation: a general framework where modal fragments are parameterized by a basis of "connectives" defined by arbitrary modal formulas (initially proposed and studied by logicians such as Kuznetsov and Ratsa in the 1970s), and the more tractable class of what we call simple modal fragments parameterized by Boolean functions plus selected modal operators, where Post-lattice methods enable systematic decidability and dichotomy results. Along the way, we collect and extend results on teachability and exact learnability from examples for both propositional fragments and simple modal fragments, and we conclude by identifying several open problems.