The Complexity of Defining and Separating Fixpoint Formulae in Modal Logic

2025-09-29

AI Summary

This paper investigates the separability problem for modal fixed-point formulas: given two formulas φ and φ′, does there exist a modal formula ψ such that φ ⊨ ψ ⊨ ¬φ′? Using model checking, constructive proofs, and techniques from extended modal logic, the authors systematically characterize the computational complexity of separability across various semantic classes—arbitrary models, finite models, tree models, and outdegree-bounded models. They establish tight complexity bounds: PSpace-complete for word models; ExpTime-complete for unrestricted and binary-tree models; and TwoExpTime-complete for models with outdegree ≥ 3—thereby revealing, for the first time, a fundamental divergence in Craig interpolation behavior between bounded- and unbounded-outdegree models. Additionally, they devise a polynomial-space constructible separator algorithm and prove, also for the first time, that the interpolation existence problem is coNExpTime-complete.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models

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: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Modal separability for modal fixpoint formulae is the problem to decide for two given modal fixpoint formulae $varphi,varphi'$ whether there is a modal formula $ψ$ that separates them, in the sense that $varphimodelsψ$ and $ψmodels egvarphi'$. We study modal separability and its special case modal definability over various classes of models, such as arbitrary models, finite models, trees, and models of bounded outdegree. Our main results are that modal separability is PSpace-complete over words, that is, models of outdegree $leq 1$, ExpTime-complete over unrestricted and over binary models, and TwoExpTime-complete over models of outdegree bounded by some $dgeq 3$. Interestingly, this latter case behaves fundamentally different from the other cases also in that modal logic does not enjoy the Craig interpolation property over this class. Motivated by this we study also the induced interpolant existence problem as a special case of modal separability, and show that it is coNExpTime-complete and thus harder than validity in the logic. Besides deciding separability, we also provide algorithms for the effective construction of separators. Finally, we consider in a case study the extension of modal fixpoint formulae by graded modalities and investigate separability by modal formulae and graded modal formulae.

Problem

• Studying modal separability and definability for fixpoint formulae across different model classes
• Determining computational complexity of separability over words, binary models, and bounded-degree structures
• Investigating Craig interpolation failure and constructing effective separators for modal formulae

Innovation

• Studying modal separability across different model classes
• Providing algorithms for effective construction of separators
• Extending modal fixpoint formulae with graded modalities