Reasoning About Probabilities, Actions, and Knowledge in Fuzzy Modal Logic

2026-04-24

AI Summary

This study addresses the challenge of uniformly formalizing complex statements that intertwine probability, action, and knowledge within fuzzy modal logic—such as “after performing action a, agent A knows that proposition p holds with probability 0.25.” To this end, the paper introduces a novel fuzzy modal logic equipped with a formal semantics based on Kripke frames augmented with probability measures. The primary contribution lies in the first unified integration of probability, action, and knowledge into a single fuzzy modal logical framework. Furthermore, the work identifies several logically distinct fragments of varying expressiveness, each admitting a satisfiability problem decidable in polynomial time, thereby establishing an upper bound on the satisfiability complexity of the logic over finitely branching models.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We explore a fuzzy modal logic that can formalise probabilistic reasoning about actions and knowledge. In particular, we deal with contexts involving statements about events expressed via modal formulas, e.g., "after doing $a$, the probability of $A$ knowing that $p$ holds increases / decreases / is equal to $0.25$", "according to $A$, $p$ is equally likely to happen after doing $a$ or $b$", etc. We define the semantics of the logic on Kripke frames equipped with probability measures. We analyse the complexity of deciding the satisfiability of formulas of our logic over finitely branching models, for the full language and its fragments of varying expressivity. In particular, we identify several fragments of our logic where satisfiability is decidable in polynomial time.

Problem

• probabilistic reasoning
• fuzzy modal logic
• actions
• knowledge
• satisfiability

Innovation

• fuzzy modal logic
• probabilistic reasoning
• Kripke frames with probability measures
• satisfiability complexity
• polynomial-time fragments