Modal Fragments

2026-03-05

AI Summary

This study investigates the interplay between expressive power and computational complexity in restricted operator fragments of propositional and modal logics. By integrating Post’s lattice theory with modal logic frameworks, the work introduces the notion of “simple modal fragments,” extending Boolean clone theory to the modal setting and establishing a unified parameterized analysis methodology. The paper systematically characterizes the boundaries of these fragments—parameterized by admissible logical operators—with respect to decidability, computational complexity (including dichotomy results), and learnability (encompassing teachability and exact learnability). This approach unifies two long-standing, independent research directions, offering a cohesive theoretical framework for understanding the structure and learnability of logical fragments.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• modal fragments
• propositional logic
• expressive power
• computational complexity
• Boolean clones

Innovation

• modal fragments
• Post's lattice
• Boolean clones
• computational complexity
• exact learnability