[Paper Review] Modal Fragments

Nick Bezhanishvili, Balder ten Cate+2 more|arXiv (Cornell University)|Mar 5, 2026

Logic, Reasoning, and Knowledge0 citations

TL;DR

A survey unifying two traditions of studying fragments of propositional and modal logic, using Post’s lattice to classify expressive power and complexity, and reporting learnability results.

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.

Motivation & Objective

• Bridge two historically independent lines of research on logic fragments by presenting a unified framework.
• Survey key results on expressive power and complexity for propositional and modal fragments.
• Discuss learnability and teachability from examples for both propositional and simple modal fragments.
• Identify open problems and directions for future work in the study of logical fragments.

Proposed method

• Review a general framework where modal fragments are parameterized by a basis of connectives defined by arbitrary modal formulas.
• Exhibit and analyze the simpler, more tractable class of simple modal fragments parameterized by Boolean functions plus selected modal operators.
• Leverage Post’s lattice (Boolean clones) to classify expressivity and complexity of fragments.
• Present and extend results on teachability and exact learnability from examples for propositional and modal fragments.
• Connect results to broader logical frameworks and outline future research directions.

Experimental results

Research questions

• RQ1 How do expressive power and computational complexity of fragments depend on the allowed set of operators?
• RQ2 What is the correspondence between Boolean clones (Post lattice) and syntactic/propositional fragments?
• RQ3 What dichotomy or classification results exist for the decidability and complexity of modal fragments under different bases?
• RQ4 What learnability or teachability results can be transferred to or extended for propositional and simple modal fragments?

Key findings

• Two historically independent research lines on modal fragments are brought together into a unified narrative.
• The literature yields dichotomy-type and classification results for the complexity of reasoning tasks under various fragments.
• There are results on teachability and exact learnability from examples for both propositional and simple modal fragments.
• Boolean-clone (Post lattice) methods provide systematic analyses of expressivity and complexity across fragments.
• The survey identifies open problems and avenues for further development in basis-restricted fragment research.