Complexity of Łukasiewicz Modal Probabilistic Logics

2025-11-27

AI Summary

This paper investigates the computational complexity of Łukasiewicz multi-valued modal probabilistic logic, targeting fine-grained probabilistic reasoning—such as reasoning about probability bounds—in modal contexts involving knowledge, belief, time, and action. Methodologically, it establishes a formal framework integrating modal, multi-valued, and probabilistic semantics based on Łukasiewicz logic, and defines two variants of the local entailment (consequence) problem. The main contribution is the first proof that both entailment problems are PSPACE-complete, thereby providing an exact characterization of the logic’s computational complexity. This result furnishes a rigorous theoretical foundation for automated reasoning, model checking, and formal verification in modal probabilistic settings, and extends the applicability of multi-valued modal logics to uncertainty reasoning.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Large-scale security measurements

Abstract

Modal probabilistic logics provide a framework for reasoning about probability in modal contexts, involving notions such as knowledge, belief, time, and action. In this paper, we study a particular family of these logics, extending the modal Łukasiewicz many-valued logic. These logics are shown to be capable of expressing nuanced probabilistic concepts, including upper and lower probabilities. Our main contribution is a PSPACE-completeness result for two variants of the local consequence problem, providing a precise computational characterisation.

Problem

• Studying computational complexity of modal probabilistic logics
• Extending modal Łukasiewicz logic for nuanced probabilistic reasoning
• Providing PSPACE-completeness for local consequence problem variants

Innovation

• Extends modal Łukasiewicz many-valued logic
• Expresses upper and lower probabilistic concepts
• Provides PSPACE-completeness for consequence problems