Modal Fragments

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.

💡 Research Summary

The paper “Modal Fragments” surveys systematic approaches to basis‑restricted fragments of propositional logic and modal logics, focusing on how the choice of operators influences expressive power, computational complexity, and learnability. It begins by recalling Emil Post’s lattice, a complete classification of Boolean clones (sets of Boolean functions closed under composition and containing projections). Post’s lattice is finite‑generated, well‑quasi‑ordered, and its membership and inclusion problems are decidable (NC¹ for truth‑table representations, coNP or Θ₂ᴾ for circuit/formula representations). This algebraic framework underlies the study of propositional fragments: each fragment corresponds to a clone generated by its allowed connectives, and complexity classifications for satisfiability, model‑checking, and related tasks can be derived from the clone’s position in the lattice.

The authors then describe two historically independent lines of research on modal fragments. The first, originating with Kuznetsov and Raţă in the 1970s, defines a fragment by a finite set of arbitrary modal formulas taken as “connectives”. The fragment consists of all formulas obtained by uniform substitution from this set. While this framework is extremely general, many meta‑problems (expressive completeness, expressive containment) turn out to be undecidable or only decidable for locally tabular logics such as S5. The second line, revived in the 1990s by Jeavons, Creignou, and others, restricts attention to “simple modal fragments”. Here a fragment is specified by a set B of Boolean functions and a separate set M of modal operators (□, ◇, etc.). The Boolean part is analyzed via Post’s lattice, while the modal part is treated independently. This yields a rich collection of dichotomy theorems: for each Boolean clone B, the satisfiability or model‑checking problem for the fragment (B, M) falls into one of a few complexity classes (AC⁰, AC⁰