A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences. (21st December 2017)

• Record Type: Journal Article
• Authors: Soler-Toscano, Fernando; Zenil, Hector
• Other Names: Innocenti Giacomo Academic Editor.
• Is Part Of: Complexity. Volume 2017(2017)
• Publication Date: 2017-12-21
• Subjects: Chaotic behavior in systems -- Periodicals; Complexity (Philosophy) -- Periodicals; 003
• DOI: 10.1155/2017/7208216
• Languages: English
• ISSNs: 1076-2787

Abstract

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 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 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.