Every Sequence Is Decompressible from a Random One

  • David Doty
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)

Abstract

Kučera and Gács independently showed that every infinite sequence is Turing reducible to a Martin-Löf random sequence. We extend this result to show that every infinite sequence S is Turing reducible to a Martin-Löf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. We show that this is the optimal ratio of query bits to computed bits achievable with Turing reductions. As an application of this result, we give a new characterization of constructive dimension in terms of Turing reduction compression ratios.

Keywords

Constructive dimension Kolmogorov complexity Turing reduction compression martingale random sequence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AHLM04]
    Athreya, K.B., Hitchcock, J.M., Lutz, J.H., Mayordomo, E.: Effective strong dimension, algorithmic information, and computational complexity. SIAM Journal on Computing (2004); Preliminary version appeared in: Proceedings of the 21st International Symposium on Theoretical Aspects of Computer Science, pp. 632–643 (to appear)Google Scholar
  2. [Bar68]
    Barzdin′, Y.M.: Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set. Soviet Mathematics Doklady 9, 1251–1254 (1968)MATHGoogle Scholar
  3. [Cha75]
    Chaitin, G.J.: A theory of program size formally identical to information theory. Journal of the Association for Computing Machinery 22, 329–340 (1975)MathSciNetCrossRefMATHGoogle Scholar
  4. [Edg04]
    Edgar, G.A.: Classics on Fractals. Westview Press, Oxford (2004)MATHGoogle Scholar
  5. [Fal90]
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. John Wiley, Chichester (1990)MATHGoogle Scholar
  6. [Fen02]
    Fenner, S.A.: Gales and supergales are equivalent for defining constructive Hausdorff dimension. Technical Report cs.CC/0208044, Computing Research Repository (2002)Google Scholar
  7. [Gác86]
    Gács, P.: Every sequence is reducible to a random one. Information and Control 70, 186–192 (1986)MathSciNetCrossRefMATHGoogle Scholar
  8. [Hau19]
    Hausdorff, F.: Dimension und äusseres Mass. Mathematische Annalen 79, 157–179 (1919); English version appears in: [Edg04], pp. 75–99CrossRefMATHGoogle Scholar
  9. [Hit03]
    Hitchcock, J.M.: Gales suffice for constructive dimension. Information Processing Letters 86(1), 9–12 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. [Huf59]
    Huffman, D.A.: Canonical forms for information-lossless finite-state logical machines. IRE Trans. Circuit Theory CT-6 (Special Supplement) , pp. 41–59 (1959), Also available in: Moore, E.F., (ed.) Sequential Machine: Selected Papers, pp. 866–871. Addison-Wesley (1964)Google Scholar
  11. [Kuč85]
    Kučera, A.: Measure, \({\rm \Pi}_1^0\)-classes and complete extensions of PA. In: Recursion Theory Week. Lecture Notes in Mathematics, vol. 1141, pp. 245–259 (1985)Google Scholar
  12. [Kuč89]
    Kučera, A.: On the use of diagonally nonrecursive functions. In: Studies in Logic and the Foundations of Mathematics, vol. 129, pp. 219–239. North-Holland, Amsterdam (1989)Google Scholar
  13. [Lut03a]
    Lutz, J.H.: Dimension in complexity classes. SIAM Journal on Computing 32, 1236–1259 (2003); Preliminary version appeared in: Proceedings of the Fifteenth Annual IEEE Conference on Computational Complexity, pp. 158–169 (2000)Google Scholar
  14. [Lut03b]
    Lutz, J.H.: The dimensions of individual strings and sequences. Information and Computation 187, 49–79 (2003); Preliminary version appeared in: Proceedings of the 27th International Colloquium on Automata, Languages, and Programming, pp. 902–913 (2000)Google Scholar
  15. [Lut05]
    Lutz, J.H.: Effective fractal dimensions. Mathematical Logic Quarterly 51, 62–72 (2003) (Invited lecture at the International Conference on Computability and Complexity in Analysis, Cincinnati, OH) (August 2003)MathSciNetCrossRefMATHGoogle Scholar
  16. [LV97]
    Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and its Applications, 2nd edn. Springer, Berlin (1997)CrossRefMATHGoogle Scholar
  17. [Mar66]
    Martin-Löf, P.: The definition of random sequences. Information and Control 9, 602–619 (1966)MathSciNetCrossRefMATHGoogle Scholar
  18. [May02]
    Mayordomo, E.: A Kolmogorov complexity characterization of constructive Hausdorff dimension. Information Processing Letters 84(1), 1–3 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. [MM04]
    Merkle, W., Mihailović, N.: On the construction of effective random sets. Journal of Symbolic Logic, 862–878 (2004)Google Scholar
  20. [MN05]
    Miller, J.S., Nies, A.: Randomness and computability: Open questions. Technical report, University of Auckland (2005)Google Scholar
  21. [Rya86]
    Ya Ryabko, B.: Noiseless coding of combinatorial sources. Problems of Information Transmission 22, 170–179 (1986)MATHGoogle Scholar
  22. [Sch71]
    Schnorr, C.P.: A unified approach to the definition of random sequences. Mathematical Systems Theory 5, 246–258 (1971)MathSciNetCrossRefMATHGoogle Scholar
  23. [Soa87]
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)CrossRefMATHGoogle Scholar
  24. [Sul84]
    Sullivan, D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Mathematica 153, 259–277 (1984)MathSciNetCrossRefMATHGoogle Scholar
  25. [Tri82]
    Tricot, C.: Two definitions of fractional dimension. Mathematical Proceedings of the Cambridge Philosophical Society 91, 57–74 (1982)MathSciNetCrossRefMATHGoogle Scholar
  26. [ZL78]
    Ziv, J., Lempel, A.: Compression of individual sequences via variable-rate coding. IEEE Transaction on Information Theory 24, 530–536 (1978)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • David Doty
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

Personalised recommendations