A Circuit Complexity Formulation of Algorithmic Information Theory

TL;DR

This paper introduces a circuit complexity-based prior inspired by Solomonoff's inductive inference, offering a universal, halting-problem-free measure for learning Boolean functions in binary classification tasks.

Contribution

It proposes a novel prior based on circuit complexity that is independent of UTM choice and free of halting problem issues, suitable for machine learning applications.

Findings

Circuit complexity provides a universal, UTM-independent measure.

The model enables recursive calculation of circuit outputs.

It is well-suited for learning Boolean functions from partial data.

Abstract

Inspired by Solomonoffs theory of inductive inference, we propose a prior based on circuit complexity. There are several advantages to this approach. First, it relies on a complexity measure that does not depend on the choice of UTM. There is one universal definition for Boolean circuits involving an universal operation such as nand with simple conversions to alternative definitions such as and, or, and not. Second, there is no analogue of the halting problem. The output value of a circuit can be calculated recursively by computer in time proportional to the number of gates, while a short program may run for a very long time. Our prior assumes that a Boolean function, or equivalently, Boolean string of fixed length, is generated by some Bayesian mixture of circuits. This model is appropriate for learning Boolean functions from partial information, a problem often encountered within machine learning as "binary classification." We argue that an inductive bias towards simple explanations as measured by circuit complexity is appropriate for this problem.