Depth as Randomness Deficiency

Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program in absence of auxiliary information. Variants known as "logical depth" and "computational depth", are expressed in Kolmogorov complexity theory. In this article we derive quantitative relation between logical depth and computational depth and unify the different "depth" notions by relating them to A. Kolmogorov and L. Levin's fruitful notion of "randomness deficiency". Subsequently, we revisit the computational depth of infinite strings, introducing the notion of super deep sequences and relate it with other approaches.

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Luis Antunes 0002, Armando Matos, Andre Souto, Paul M. B. Vitányi
• Keywords: Computation Time, Computational Depth, CORR 2008, Education, Program Length