Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization

Eric Allender

doi:10.53733/148

Abstract

We survey recent developments related to the Minimum Circuit Size Problem and time-bounded Kolmogorov Complexity.

Highlights

When listing the many accomplishments of Vaughan Jones, the obituaries did not mention the fact that he is indirectly responsible for much of the modern development of the study of Algorithmic Information Theory, known as Kolmogorov Complexity
Theorems about Minimum Circuit Size Problem (MCSP) and the various MKμP problems are stated not in terms of exactly computing the circuit size or the complexity of a string, but in terms of approximating these values. This is usually presented in terms of two thresholds θ1 < θ2, where the desired solution is to say yes if the complexity of x is less than θ1, and to say no if the complexity of x is greater than θ2, and any answer is allowed if the complexity of x lies in the “gap” between θ1 and θ2
Certain connections between computational learning theory and Kolmogorov complexity were identified soon after computational learning theory emerged as a field

Summary

Introduction

When listing the many accomplishments of Vaughan Jones, the obituaries did not mention the fact that he is indirectly responsible for much of the modern development of the study of Algorithmic Information Theory, known as Kolmogorov Complexity. Theorems about MCSP and the various MKμP problems are stated not in terms of exactly computing the circuit size or the complexity of a string, but in terms of approximating these values. This is usually presented in terms of two thresholds θ1 < θ2, where the desired solution is to say yes if the complexity of x is less than θ1, and to say no if the complexity of x is greater than θ2, and any answer is allowed if the complexity of x lies in the “gap” between θ1 and θ2. It is worth making this investment in defining the various distinct notions

Connections to Learning Theory

Findings

Magnification