Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures

TL;DR

This paper demonstrates that numerical approximations of Kolmogorov-Chaitin complexity align with program size, are finer than compression-based measures, and are stable for small strings, supported by an online calculator implementation.

Contribution

It introduces a numerical approximation method for Kolmogorov complexity using small Turing machines and presents an online tool for practical calculations.

Findings

K_m approximations agree with program size.

K_m is finer-grained than compression-based measures.

No correlation between K_m and Logical Depth.

Abstract

We show that real-value approximations of Kolmogorov-Chaitin (K_m) using the algorithmic Coding theorem as calculated from the output frequency of a large set of small deterministic Turing machines with up to 5 states (and 2 symbols), is in agreement with the number of instructions used by the Turing machines producing s, which is consistent with strict integer-value program-size complexity. Nevertheless, K_m proves to be a finer-grained measure and a potential alternative approach to lossless compression algorithms for small entities, where compression fails. We also show that neither K_m nor the number of instructions used shows any correlation with Bennett's Logical Depth LD(s) other than what's predicted by the theory. The agreement between theory and numerical calculations shows that despite the undecidability of these theoretical measures, approximations are stable and meaningful, even for small programs and for short strings. We also announce a first Beta version of an Online Algorithmic Complexity Calculator (OACC), based on a combination of theoretical concepts, as a numerical implementation of the Coding Theorem Method.