Abstract
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.
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, study the notion of super deep sequences and relate it with other approaches.
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The authors from University of Porto are partially supported by KCrypt (POSC/EIA/60819/2004) and funds granted to LIACC through the Programa de Financiamento Plurianual, Fundação para a Ciência e Tecnologia and Programa POSI.
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Antunes, L., Matos, A., Souto, A. et al. Depth as Randomness Deficiency. Theory Comput Syst 45, 724–739 (2009). https://doi.org/10.1007/s00224-009-9171-0
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DOI: https://doi.org/10.1007/s00224-009-9171-0