Reasoning Under the Principle of Maximum Entropy for Modal Logics K45, KD45, and S5

TL;DR

This paper introduces modal Markov logic, extending propositional Markov logic to reason under maximum entropy principles for modal logics K45, KD45, and S5, enabling probabilistic reasoning over epistemic states.

Contribution

It develops a novel framework that defines probability distributions over non-equivalent epistemic states using weighted modal formulas, with scalable inference algorithms.

Findings

Provides an algorithm scaling exponentially with knowledge base size

Defines probability over non-equivalent epistemic states

Discusses extension to infinite propositions

Abstract

We propose modal Markov logic as an extension of propositional Markov logic to reason under the principle of maximum entropy for modal logics K45, KD45, and S5. Analogous to propositional Markov logic, the knowledge base consists of weighted formulas, whose weights are learned from data. However, in contrast to Markov logic, in our framework we use the knowledge base to define a probability distribution over non-equivalent epistemic situations (pointed Kripke structures) rather than over atoms, and use this distribution to assign probabilities to modal formulas. As in all probabilistic representations, the central task in our framework is inference. Although the size of the state space grows doubly exponentially in the number of propositions in the domain, we provide an algorithm that scales only exponentially in the size of the knowledge base. Finally, we briefly discuss the case of languages with an infinite number of propositions.