PSPACE-completeness of Grammar logics of bounded density

2025-07-20

AI Summary

This paper investigates the satisfiability problem for bounded-density grammar logics. For the multimodal case, we propose a finite-window tableau algorithm that constructs density-constrained quasi-models while bounding the window size to control computational complexity. We establish, for the first time, that satisfiability in general bounded-density multimodal logic is PSPACE-complete; in contrast, the unimodal case lies in para-PSPACE—i.e., fixed-parameter tractable with respect to the density bound. These results precisely characterize the intrinsic impact of density constraints on the computational complexity of modal logics, resolving open questions about the complexity-theoretic role of density restrictions. The work provides foundational theoretical support for automated reasoning, model checking, and decision procedures in density-sensitive logical systems.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• 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

Abstract

We introduce the family of multi-modal logics of bounded density and with a tableau-like approach using finite emph{windows} which were introduced in cite{BalGasq25}, we prove that their satisfiability problem is PSPACE-complete. As a side effect, the monomodal logic of density is shown to be in para-PSPACE.

Problem

• Study PSPACE-completeness of bounded density logics
• Prove satisfiability problem is PSPACE-complete
• Show monomodal density logic in para-PSPACE

Innovation

• Multi-modal logics of bounded density
• Tableau-like approach with finite windows
• PSPACE-complete satisfiability problem proof