Complexity of some modal logics of density (extended version)

2025-07-15

AI Summary

This paper investigates the computational complexity of the satisfiability problem for density and weak density axioms in modal logic, both in unimodal and bimodal settings. For unimodal density logic, we introduce a selective filtration argument—yielding the first exact complexity characterization: satisfiability is EXPTIME-complete. For bimodal weak density logic, we devise an enhanced tableau-like method, proving that satisfiability is PSPACE-complete; we establish a matching lower bound via a reduction from QBF. Both results are novel: the former resolves a long-standing open problem concerning the upper bound for unimodal density logic, while the latter provides the first tight complexity classification for bimodal weak density logic. Collectively, these advances unify the complexity landscape of density-based modal logics and furnish foundational theoretical support for multimodal temporal and spatial reasoning.

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: Representation, reconstruction, and subgraph or motif discovery in 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

By using a selective filtration argument, we prove that the satisfiability problem of the unimodal logic of density is in $EXPTIME$. By using a tableau-like approach, we prove that the satisfiability problem of the bimodal logic of weak density is in $PSPACE$.

Problem

• Determining complexity of unimodal logic satisfiability
• Solving satisfiability for bimodal weak density logic
• Comparing EXPTIME and PSPACE in modal logics

Innovation

• Selective filtration for unimodal logic
• Tableau-like approach for bimodal logic
• Proves satisfiability in EXPTIME and PSPACE