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Complexity
Volume 2017, Article ID 7208216, 10 pages
https://doi.org/10.1155/2017/7208216
Research Article

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

1Grupo de Lógica, Lenguaje e Información, Universidad de Sevilla, Sevilla, Spain
2Algorithmic Nature Group, LABORES, Paris, France
3Algorithmic Dynamics Lab, Center for Molecular Medicine, Science for Life Laboratory (SciLifeLab), Department of Medicine, Solna, Karolinska Institute, Stockholm, Sweden
4Group of Structural Biology, Department of Computer Science, University of Oxford, Oxford, UK

Correspondence should be addressed to Hector Zenil; moc.liamg@clinezh

Received 19 February 2017; Revised 22 June 2017; Accepted 7 August 2017; Published 21 December 2017

Academic Editor: Giacomo Innocenti

Copyright © 2017 Fernando Soler-Toscano and Hector Zenil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given the widespread use of lossless compression algorithms to approximate algorithmic (Kolmogorov-Chaitin) complexity and that, usually, generic lossless compression algorithms fall short at characterizing features other than statistical ones not different from entropy evaluations, here we explore an alternative and complementary approach. We study formal properties of a Levin-inspired measure calculated from the output distribution of small Turing machines. We introduce and justify finite approximations 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 and and compare them to Levin’s Universal Distribution. We provide error estimations of with respect to . Finally, we present an application to integer sequences from the On-Line Encyclopedia of Integer Sequences, which suggests that our AP-based measures may characterize nonstatistical patterns, and we report interesting correlations with textual, function, and program description lengths of the said sequences.