Randomness and Initial Segment Complexity for Probability Measures

Authors André Nies , Frank Stephan

Thumbnail PDF


  • Filesize: 448 kB
  • 14 pages

Document Identifiers

Author Details

André Nies
  • School of Computer Science, The University of Auckland, New Zealand
Frank Stephan
  • Department of Mathematics and Department of Computer Science, National University of Singapore, Singapore

Cite AsGet BibTex

André Nies and Frank Stephan. Randomness and Initial Segment Complexity for Probability Measures. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 55:1-55:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures. We begin with examples, and a robustness property related to Solovay tests. Our main work connects our property to the growth of the initial segment complexity for measures μ; the latter is defined as a μ-average over the complexity of strings of the same length. We show that a maximal growth implies our weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We briefly discuss K-triviality for measures, which means that the growth of initial segment complexity is as slow as possible. We show that full Martin-Löf randomness of a measure implies Martin-Löf absolute continuity; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Quantum information theory
  • algorithmic randomness
  • probability measure on Cantor space
  • Kolmogorov complexity
  • statistical superposition
  • quantum states


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Laurent Bienvenu, Wolfgang Merkle, and Alexander Shen. A simple proof of the Miller-Yu Theorem. Fundamenta Informaticae, 83(1-2):21-24, 2008. Google Scholar
  2. Cristian Calude. Information and randomness. Monographs in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 1994. With forewords by Gregory J. Chaitin and Arto Salomaa. Google Scholar
  3. Gregory J. Chaitin. A theory of program size formally identical to information theory. J. Assoc. Comput. Mach., 22:329-340, 1975. Google Scholar
  4. Quinn Culver. Topics in Algorithmic Randomness and Effective Probability. PhD thesis, PhD thesis, University of Notre Dame, 2015. Google Scholar
  5. Rodney G. Downey and Denis Hirschfeldt. Algorithmic randomness and complexity. Springer-Verlag, Berlin, 2010. 855 pages. Google Scholar
  6. Michael Hochman. Upcrossing inequalities for stationary sequences and applications. Annals of Probability, 37(6):2135-2149, 2009. Google Scholar
  7. Mathieu Hoyrup. The dimension of ergodic random sequences. In Christoph Dürr and Thomas Wilke, editors, STACS, pages 567-576, 2012. Google Scholar
  8. Mathieu Hoyrup and Cristóbal Rojas. Computability of probability measures and Martin-Löf randomness over metric spaces. Information and Computation, 207(7):830-847, 2009. Google Scholar
  9. Leonid A. Levin. The concept of a random sequence. Dokl. Akad. Nauk SSSR, 212:548-550, 1973. Google Scholar
  10. Ming Li and Paul Vitányi. An introduction to Kolmogorov complexity and its applications. Graduate Texts in Computer Science. Springer-Verlag, New York, second edition, 1997. Google Scholar
  11. Per Martin-Löf. The definition of random sequences. Information and Control, 9:602-619, 1966. Google Scholar
  12. Daniel Mauldin and Michael Monticino. Randomly generated distributions. Israel Journal of Mathematics, 91(1-3):215-237, 1995. Google Scholar
  13. Joseph S. Miller. The K-degrees, low for K-degrees, and weakly low for K sets. Notre Dame J. Form. Log., 50(4):381-391, 2009. URL: https://doi.org/10.1215/00294527-2009-017.
  14. Joseph S. Miller and Liang Yu. On initial segment complexity and degrees of randomness. Trans. Amer. Math. Soc., 360:3193-3210, 2008. Google Scholar
  15. Joseph S. Miller and Liang Yu. Oscillation in the initial segment complexity of random reals. Advances in Mathematics, 226(6):4816-4840, 2011. Google Scholar
  16. Kenshi Miyabe, André Nies, and Jing Zhang. Using almost-everywhere theorems from analysis to study randomness. Bull. Symb. Logic, 22:305-331, 2016. URL: http://arxiv.org/abs/1411.0732.
  17. André Nies. Computability and Randomness, volume 51 of Oxford Logic Guides. Oxford University Press, Oxford, 2009. 444 pages. Paperback version 2011. URL: https://doi.org/10.1093/acprof:oso/9780199230761.001.0001.
  18. André Nies and Volkher Scholz. Martin-Löf random quantum states. Journal of Mathematical Physics, 60(9):092201, 2019. Available at doi.org/10.1063/1.5094660. Google Scholar
  19. André Nies, Frank Stephan, and Sebastiaan Terwijn. Randomness, relativization and Turing degrees. J. Symbolic Logic, 70(2):515-535, 2005. Google Scholar
  20. Editor André Nies. Logic Blog 2017. Available at https://arxiv.org/abs/1804.05331, 2017. URL: https://arxiv.org/abs/1804.05331.
  21. Piergiorgio Odifreddi. Classical Recursion Theory (Volume I). North-Holland Publishing Co., Amsterdam, 1989. Google Scholar
  22. Piergiorgio Odifreddi. Classical recursion theory: The theory of functions and sets of natural numbers, volume 125. Elsevier, 1992. Google Scholar
  23. Claus-Peter Schnorr. Process complexity and effective random tests. J. Comput. System Sci., 7:376-388, 1973. Fourth Annual ACM Symposium on the Theory of Computing (Denver, Colo., 1972). Google Scholar
  24. Paul C. Shields. The Ergodic Theory of Discrete Sample Paths. Graduate Studies in Mathematics 13. American Mathematical Society, 1996. Google Scholar
  25. Robert I. Soare. Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic, Omega Series. Springer-Verlag, Heidelberg, 1987. Google Scholar
  26. Frank Stephan. Recursion theory. Lecture notes, School of Computing, 2012. Google Scholar