An MML-based tool for evaluating the complexity of (stochastic) logic theories

TL;DR

This paper introduces a novel MML-based method and software tool for evaluating the complexity of stochastic logic theories expressed in Turing-complete languages, addressing the challenge of approximating Kolmogorov complexity.

Contribution

It presents the first general MML coding scheme for logic programs and a software implementation for theory evaluation in probabilistic and non-probabilistic contexts.

Findings

Effective coding and evaluation of logic theories demonstrated

Applicable to both probabilistic and non-probabilistic scenarios

Provides a new approach to approximate Kolmogorov complexity in logic programming

Abstract

Theory evaluation is a key problem in many areas: machine learning, scientific discovery, inverse engineering, decision making, software engineering, design, human sciences, etc. If we have a set of theories that are able to explain the same set of phenomena, we need a criterion to choose which one is best. There are, of course, many possible criteria. Model simplicity is one of the most common criteria in theory evaluation. The Minimum Message Length (MML) is a solid approach to evaluate theories relative to a given evidence or data. Theories can be expressed in specific or general (Turing-complete) languages. First-order logic, and logic programming in particular, is a Turing-complete language. Evaluating the simplicity of a theory or program described in a Turing-complete language is much more difficult than just counting the number of lines or bits. It is, in fact, the problem of calculating its Kolmogorov complexity, which is uncomputable. Few works in the literature have been able to present accurate and effective approximations for a Turing-complete language. In this work, we present the first general MML coding scheme for logic programs. With this scheme, we can quantify the bits of information required to code (or send) a theory, a set of data or the same data given the theory. As a realization of the above-mentioned schemes, we present a software tool which is able to code and evaluate a set of alternative (stochastic) theories (programs) against a set of examples. We illustrate the application of the tool to a variety of non-probabilistic and probabilistic scenarios.