Products of Weighted Logic Programs

TL;DR

This paper introduces the PRODUCT transformation for weighted logic programs, enabling the combination of programs to optimize proof scores and derive dynamic programming algorithms, with applications in machine learning and information theory.

Contribution

The paper presents the novel PRODUCT transformation that merges weighted logic programs to facilitate complex scoring functions and dynamic programming algorithm derivation.

Findings

Enables merging of weighted logic programs for scoring

Derives important dynamic programming algorithms

Interprets Kullback-Leibler divergence using PRODUCT

Abstract

Weighted logic programming, a generalization of bottom-up logic programming, is a well-suited framework for specifying dynamic programming algorithms. In this setting, proofs correspond to the algorithm's output space, such as a path through a graph or a grammatical derivation, and are given a real-valued score (often interpreted as a probability) that depends on the real weights of the base axioms used in the proof. The desired output is a function over all possible proofs, such as a sum of scores or an optimal score. We describe the PRODUCT transformation, which can merge two weighted logic programs into a new one. The resulting program optimizes a product of proof scores from the original programs, constituting a scoring function known in machine learning as a ``product of experts.'' Through the addition of intuitive constraining side conditions, we show that several important dynamic programming algorithms can be derived by applying PRODUCT to weighted logic programs corresponding to simpler weighted logic programs. In addition, we show how the computation of Kullback-Leibler divergence, an information-theoretic measure, can be interpreted using PRODUCT.