Theory of Computing Systems

, Volume 56, Issue 3, pp 439–464 | Cite as

Feasible Analysis, Randomness, and Base Invariance

  • Santiago Figueira
  • André Nies


We show that polynomial time randomness of a real number does not depend on the choice of a base for representing it. Our main tool is an ‘almost Lipschitz’ condition that we show for the cumulative distribution function associated to martingales with the savings property. Based on a result of Schnorr, we prove that for any base r, n⋅log2 n-randomness in base r implies normality in base r, and that n 4-randomness in base r implies absolute normality. Our methods yield a construction of an absolutely normal real number which is computable in polynomial time.


Base invariance Polynomial time randomness Analysis Normality Martingales 



We thank Verónica Becher, Pablo Heiber, Jack Lutz, Elvira Mayordomo and Theodore A. Slaman. We also thank the referee for careful reading and mindful suggestions. This research was partially carried out while the authors participated in the Buenos Aires Semester in Computability, Complexity and Randomness, 2013. Nies was supported by the Marsden fund of New Zealand. Figueira was supported by UBA (UBACyT 20020110100025) and ANPCyT (PICT-2011-0365).


  1. 1.
    Ambos-Spies, K., Fleischhack, H., Huwig, H.: Diagonalizations over polynomial time computable sets. Theor. Comput. Sci. 51, 177–204 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ambos-Spies, K., Fleischhack, H., Huwig, H.: Diagonalizing over deterministic polynomial time. In: CSL. Lecture Notes in Computer Science, vol. 329, pp. 1–16 (1987) CrossRefGoogle Scholar
  3. 3.
    Ambos-Spies, K., Terwijn, S., Zheng, X.: Resource bounded randomness and weakly complete problems. Theor. Comput. Sci. 172, 195–207 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Becher, V., Figueira, S.: An example of a computable absolutely normal number. Theor. Comput. Sci. 270, 947–958 (2002). doi: 10.1016/S0304-3975(01)00170-0 CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Becher, V., Figueira, S., Picchi, R.: Turing’s unpublished algorithm for normal numbers. Theor. Comput. Sci. 377(1–3), 126–138 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Becher, V., Heiber, P., Slaman, T.A.: A polynomial-time algorithm for computing absolutely normal numbers. Manuscript (2013) Google Scholar
  7. 7.
    Borel, E.: Les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Mat. Palermo 27, 247–271 (1909) CrossRefzbMATHGoogle Scholar
  8. 8.
    Brattka, V., Miller, J.S., Nies, A.: Randomness and differentiability (2011, to appear) Google Scholar
  9. 9.
    Brown, G., Moran, W., Pearce, C.E.M.: A decomposition theorem for numbers in which the summands have prescribed normality properties. J. Number Theory 24(3), 259–271 (1986). doi: 10.1016/0022-314X(86)90034-X CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Calude, C.S., Jürgensen, H.: Randomness as an invariant for number representations. In: Maurer, J.K.H., Rozenberg, G. (eds.) Results and Trends in Theoretical Computer Science, pp. 44–66. Springer, Berlin (1994) CrossRefGoogle Scholar
  11. 11.
    Champernowne, D.G.: The construction of decimals in the scale of ten. J. Lond. Math. Soc. 8, 254–260 (1933) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Green, B., Tao, T.: The primes contain arbitrarily long arithmetic progressions. Ann. Math. 167(2), 481–547 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Hertling, P., Weihrauch, K.: Randomness space. In: Automata, Languages and Programming, pp. 796–807. Springer, Berlin (1998) CrossRefGoogle Scholar
  14. 14.
    Hitchcock, J., Mayordomo, E.: Base invariance of feasible dimension. Manuscript (2012) Google Scholar
  15. 15.
    Lutz, J.H.: Category and measure in complexity classes. SIAM J. Comput. 19(6), 1100–1131 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lutz, J.H.: Almost everywhere high nonuniform complexity. J. Comput. Syst. Sci. 44(2), 220–258 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lutz, J., Mayordomo, E.: Construction of an absolutely normal real number in polynomial time. Manuscript (2012) Google Scholar
  18. 18.
    Martin-Löf, P.: The definition of random sequences. Inf. Control 9, 602–619 (1966) CrossRefzbMATHGoogle Scholar
  19. 19.
    Nies, A.: Computability and Randomness. Clarendon Press, Oxford (2009) CrossRefzbMATHGoogle Scholar
  20. 20.
    Schmidt, W.M.: On normal numbers. Pac. J. Math. 10, 661–672 (1960) CrossRefzbMATHGoogle Scholar
  21. 21.
    Schnorr, C.P.: A unified approach to the definition of a random sequence. Math. Syst. Theory 5, 246–258 (1971) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Schnorr, C.P.: Zufälligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics, vol. 218 (1971) CrossRefzbMATHGoogle Scholar
  23. 23.
    Sierpinski, W.: Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d’un tel nombre. Bull. Soc. Math. Fr. 45, 127–132 (1917) MathSciNetGoogle Scholar
  24. 24.
    Silveira, J.: Invariancia por cambio de base de la aleatoriedad computable y la aleatoriedad con recursos acotados. Ph.D. thesis, University of Buenos Aires (2011). Advisor: Santiago Figueira. Unpublished Google Scholar
  25. 25.
    Staiger, L.: The Kolmogorov complexity of real numbers. In: Proceedings of the 12th International Symposium on Fundamentals of Computation Theory, FCT ’99, pp. 536–546. Springer, London (1999). CrossRefGoogle Scholar
  26. 26.
    Turing, A.M.: A note on normal numbers. In: Britton, J. (ed.) Collected Works of A.M. Turing: Pure Mathematics, pp. 117–119. North Holland, Amsterdam (1992) Google Scholar
  27. 27.
    Wang, Y.: Randomness and complexity. Ph.D. thesis, University of Heidelberg (1996) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departamento de ComputaciónUniversity of Buenos Aires and CONICETBuenos AiresArgentina
  2. 2.The University of AucklandAucklandNew Zealand

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