Depth as Randomness Deficiency

  • Authors:

  • Luís Antunes;Armando Matos;André Souto;Paul Vitányi

  • Affiliations:

  • Universidade do Porto, Faculdade de Ciências, Rua Campo Alegre, 1021/1055, 4169007, Porto, Portugal and Instituto de Telecomunicações, Rua Campo Alegre, 1021, 4169007, Porto, ...;Universidade do Porto, Faculdade de Ciências, Rua Campo Alegre, 1021/1055, 4169007, Porto, Portugal and LIACC, Rua Campo Alegre, 1021, 4169007, Porto, Portugal;Universidade do Porto, Faculdade de Ciências, Rua Campo Alegre, 1021/1055, 4169007, Porto, Portugal and Instituto de Telecomunicações, Rua Campo Alegre, 1021, 4169007, Porto, ...;University of Amsterdam, Computer Science Department, Amsterdam, The Netherlands

  • Venue:
  • Theory of Computing Systems - Special Issue: Computation and Logic in the Real World; Guest Editors: S. Barry Cooper, Elvira Mayordomo and Andrea Sorbi
  • Year:
  • 2009

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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.