Probabilistic Theorem Proving

TL;DR

This paper introduces a novel probabilistic theorem proving method that unifies logical inference with probabilistic reasoning, outperforming previous approaches especially when logical structure is involved.

Contribution

It presents the first method combining graphical model inference with first-order theorem proving, including an efficient algorithm and an approximate version.

Findings

Outperforms lifted variable elimination on structured problems

Proves correctness and properties of the new algorithm

Approximate method surpasses lifted belief propagation in accuracy

Abstract

Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable elimination and belief propagation, neither of which take logical structure into account. We propose the first method that has the full power of both graphical model inference and first-order theorem proving (in finite domains with Herbrand interpretations). We first define probabilistic theorem proving, their generalization, as the problem of computing the probability of a logical formula given the probabilities or weights of a set of formulas. We then show how this can be reduced to the problem of lifted weighted model counting, and develop an efficient algorithm for the latter. We prove the correctness of this algorithm, investigate its properties, and show how it generalizes previous approaches. Experiments show that it greatly outperforms lifted variable elimination when logical structure is present. Finally, we propose an algorithm for approximate probabilistic theorem proving, and show that it can greatly outperform lifted belief propagation.