Descriptive Complexity of Computable Sequences

  • Bruno Durand
  • Alexander Shen
  • Nikolai Vereshagin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1563)


Our goal is to study the complexity of infinite binary recursive sequences. We introduce several measures of the quantity of information they contain. Some measures are based on size of programs that generate the sequence, the others are based on the Kolmogorov complexity of its finite prefixes. The relations between these complexity measures are established. The most surprising among them are obtained using a specific two-players game.


Complexity Measure Binary String Computable Function Special Node Kolmogorov Complexity 
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  1. 1.
    G.J. Chaitin. “On the length of programs for computing finite binary sequences: statistical considerations,” J. of ACM, 16:145–159, 1969.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    G.J. Chaitin. “Information-theoretic characterizations of recursive infinite strings,” Theor. Comp. Sci., 2:45–48, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    B. Durand, S. Porrot. Comparison between the complexity of a function and the complexity of its graph. In MFCS’98. Lecture Notes in Computer Science, 1998.Google Scholar
  4. 4.
    A.N. Kolmogorov. “Three approaches to the quantitative definition of information.” Problems of Information Transmission, 1(1):1–7, 1965.MathSciNetGoogle Scholar
  5. 5.
    M. Li, P. Vitányi. An Introduction to Kolmogorov Complexity and its Applications. Second edition. Springer Verlag, 1997.Google Scholar
  6. 6.
    D.W. Loveland. “A variant of Kolmogorov concept of Complexity”, Information and Control, 15:510–526, 1969.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R.J. Solomonoff. “A formal theory of inductive inference, part 1 and part 2,” Information and Control, 7:1–22, 224–254, 1964.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    V.A. Uspensky, A.Kh. Shen’. “Relations between varieties of Kolmogorov complexities,” Math. Systems Theory, 29:271–292, 1996.zbMATHMathSciNetGoogle Scholar
  9. 9.
    A.K. Zvonkin, L.A. Levin. “The complexity of finite objects and the development of the concepts of information and randomness by means of theory of algorithms.” Russian Math. Surveys, 25(6):83–124, 1970.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Bruno Durand
    • 1
  • Alexander Shen
    • 2
  • Nikolai Vereshagin
    • 3
  1. 1.LIP, Ecole Normale Supérieure de LyonLyon Cedex 07France
  2. 2.Institute of Problems of Information TransmissionMoscowRussia
  3. 3.Dept. of Mathematical Logic and Theory of AlgorithmsMoscow State UniversityVorobjevy Gory, MoscowRussia

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