A computable measure of algorithmic probability by finite approximations with an application to integer sequences

摘要

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 to entropy evaluations; here we explore an alternative and complementary approach. We study formal properties of a Levin-inspired measure m calculated from the out-put distribution of small Turing machines. We introduce and justify finite approximations mk 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 mk and compare them to Levin's Universal Distribution. We provide error estimations of mk 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.