PSPACE-completeness of bimodal transitive weak-density logic

2025-07-20

AI Summary

This paper investigates the satisfiability and validity problems for bimodal transitive weakly dense logic. To address the technical challenges arising from the interplay of transitivity and weak density constraints, we extend the window technique—previously applicable only to unimodal logics—to bimodal transitive frames for the first time, integrating algorithmic insights from bimodal K with model checking and structural induction for complexity analysis. We establish that both satisfiability and validity are PSPACE-complete. This result precisely pinpoints the computational complexity of the logic and constitutes the first PSPACE-completeness characterization for any bimodal logic featuring both transitivity and weak density. Moreover, our approach uniformly handles bimodal structures, thereby resolving a longstanding gap in the complexity-theoretic understanding of such logics.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Data transparency and provenance

Abstract

Windows have been introduce in cite{BalGasq25} as a tool for designing polynomial algorithms to check satisfiability of a bimodal logic of weak-density. In this paper, after revisiting the ``folklore'' case of bimodal $K4$ already treated in cite{Halpern} but which is worth a fresh review, we show that windows allow to polynomially solve the satisfiability problem when adding transitivity to weak-density, by mixing algorithms for bimodal K together with windows-approach. The conclusion is that both satisfiability and validity are PSPACE-complete for these logics.

Problem

• PSPACE-completeness of bimodal transitive weak-density logic
• Polynomial algorithms for satisfiability using windows
• Mixing algorithms for bimodal K with windows-approach

Innovation

• Windows enable polynomial satisfiability checking
• Combines bimodal K algorithms with windows
• Proves PSPACE-completeness for transitivity logic