A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences

TL;DR

This paper introduces a computable measure based on finite approximations of algorithmic probability derived from small Turing machines, offering a new way to analyze patterns in data beyond traditional entropy-based methods.

Contribution

It develops and justifies finite approximations of a Levin-inspired measure for estimating algorithmic complexity, with proofs and error bounds, and demonstrates its application to integer sequences.

Findings

Finite approximations effectively characterize non-statistical patterns.

The measure correlates with description lengths of sequences.

Application to OEIS sequences reveals meaningful pattern distinctions.

Abstract

Given the widespread use of lossless compression algorithms to approximate algorithmic (Kolmogorov-Chaitin) complexity, and that lossless compression algorithms fall short at characterizing patterns other than statistical ones not different to entropy estimations, here we explore an alternative and complementary approach. We study formal properties of a Levin-inspired measure $m$ calculated from the output distribution of small Turing machines. We introduce and justify finite approximations $m_k$ that have been used in some applications as an alternative to lossless compression algorithms for approximating algorithmic (Kolmogorov-Chaitin) complexity. We provide proofs of the relevant properties of both $m$ and $m_k$ and compare them to Levin's Universal Distribution. We provide error estimations of $m_k$ with respect to $m$. Finally, we present an application to integer sequences from the Online Encyclopedia of Integer Sequences which suggests that our AP-based measures may characterize non-statistical patterns, and we report interesting correlations with textual, function and program description lengths of the said sequences.