Decidability in First-Order Modal Logic with Non-Rigid Constants and Definite Descriptions

2025-09-09

AI Summary

This paper investigates the decidability of first-order modal logics incorporating non-rigid constants and definite descriptions. To address the challenge that enhanced expressivity often leads to undecidability, we systematically analyze key monotonic fragments—specifically, the two-variable fragment (with counting) and the guarded fragment—and develop a reduction technique integrating extended-domain semantics, transitive closure operators for modeling cyclic structures, and constructive model generation to handle non-rigid reference and recursion. Our main contributions are: (i) the first decidability proofs for both fragments over the Kₙ and S5ₙ frame classes; (ii) tight complexity bounds for these logics; and (iii) a resolution of a long-standing boundary problem—namely, establishing Ackermann-hardness for the basic modal logic with transitive closure over finite acyclic frames.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

While modal extensions of decidable fragments of first-order logic are usually undecidable, their monodic counterparts, in which formulas in the scope of modal operators have at most one free variable, are typically decidable. This only holds, however, under the provision that non-rigid constants, definite descriptions and non-trivial counting are not admitted. Indeed, several monodic fragments having at least one of these features are known to be undecidable. We investigate these features systematically and show that fundamental monodic fragments such as the two-variable fragment with counting and the guarded fragment of standard first-order modal logics $mathbf{K}_{n}$ and $mathbf{S5}_{n}$ are decidable. Tight complexity bounds are established as well. Under the expanding-domain semantics, we show decidability of the basic modal logic extended with the transitive closure operator on finite acyclic frames; this logic, however, is Ackermann-hard.

Problem

• Decidability of monodic first-order modal logic fragments
• Impact of non-rigid constants and definite descriptions
• Complexity bounds for guarded and two-variable fragments

Innovation

• Monodic fragments with non-rigid constants
• Decidable two-variable fragment with counting
• Guarded fragment under expanding-domain semantics