Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures
Abstract
We show that realvalue approximations of KolmogorovChaitin (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 integervalue programsize complexity. Nevertheless, K_m proves to be a finergrained 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 arXiv:
 arXiv:1211.4891
 Bibcode:
 2012arXiv1211.4891S
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Computational Complexity;
 Computer Science  Formal Languages and Automata Theory
 EPrint:
 22 pages, 8 images. This article draws heavily from arXiv:1211.1302