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

TL;DR

This paper introduces a fuzzy modal logic framework for formalising probabilistic reasoning about actions and knowledge, with semantics based on Kripke frames and probability measures.

Contribution

It defines the semantics of a fuzzy modal logic for probabilistic reasoning, analyzes its complexity, and identifies decidable fragments with polynomial-time satisfiability.

Findings

Decidability of satisfiability varies across logic fragments.

Several fragments have polynomial-time satisfiability decision procedures.

The semantics are based on Kripke frames with probability measures.

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.