Numerical evaluation of algorithmic complexity for short strings: A glance into the innermost structure of randomness

• Published in: Applied mathematics and computation Vol. 219; no. 1; pp. 63 - 77
• Main Authors: Delahaye, Jean-Paul, Zenil, Hector
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.09.2012; Elsevier
• Subjects: Algorithmic (program-size) complexity; Algorithmic probability; Algorithms; Approximation; Artificial Intelligence; Busy Beaver problem; Chaitin’s Ω; Complexity; Compressing; Computation; Computational Engineering, Finance, and Science; Computer Science; Computer Science and Game Theory; General Finance; Halting probability; Kolmogorov–Chaitin complexity; Levin–Chaitin coding theorem; Levin’s universal distribution; Logic in Computer Science; Mathematical analysis; Mathematical models; Multiagent Systems; Quantitative Finance; Strings; Kolmogorov–Chaitin complexity; Algorithmic (program-size) complexity; Levin’s universal distribution; Halting probability; Levin–Chaitin coding theorem; Busy Beaver problem; Algorithmic probability; Chaitin’s Ω
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2011.10.006

We describe an alternative method (to compression) that combines several theoretical and experimental results to numerically approximate the algorithmic Kolmogorov–Chaitin complexity of all ∑n=182n bit strings up to 8 bits long, and for some between 9 and 16 bits long. This is done by an exhaustive execution of all deterministic 2-symbol Turing machines with up to four states for which the halting times are known thanks to the Busy Beaver problem, that is 11019960576 machines. An output frequency distribution is then computed, from which the algorithmic probability is calculated and the algorithmic complexity evaluated by way of the Levin–Chaitin coding theorem.