Theory of Computing Systems

, Volume 45, Issue 4, pp 740–755 | Cite as

Constructive Dimension and Turing Degrees



This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim H(S) and constructive packing dimension dim P(S) is Turing equivalent to a sequence R with dim H(R)≥(dim H(S)/dim P(S))−ε, for arbitrary ε>0. Furthermore, if dim P(S)>0, then dim P(R)≥1−ε. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension.

A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dim H(S)/dim P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim H(S)=dim P(S)) such that dim H(S)>0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1.

Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.


Constructive dimension Turing Extractor Degree Randomness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Athreya, K., Hitchcock, J., Lutz, J.H., Mayordomo, E.: Effective strong dimension, algorithmic information and computational complexity. SIAM J. Comput. 37, 671–705 (2007) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bienvenu, L., Doty, D., Stephan, F.: Constructive dimension and weak truth-table degrees. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) Computation and Logic in the Real World—Third Conference of Computability in Europe (CiE 2007), Siena, Italy, June 18–23, 2007, Proceedings. Lecture Notes in Computer Science, vol. 4497, pp. 63–72. Springer, Berlin (2007) Google Scholar
  3. 3.
    Calude, C.S., Chaitin, G.J.: Randomness everywhere. Nature 400, 319–320 (1999) CrossRefGoogle Scholar
  4. 4.
    Doty, D.: Every sequence is decompressible from a random one. In: Logical Approaches to Computational Barriers, Proceedings of the Second Conference on Computability in Europe, Swansea, UK, July 2006. Lecture Notes in Computer Science, vol. 3988, pp. 153–162. Springer, Berlin (2006) Google Scholar
  5. 5.
    Doty, D.: Dimension extractors and optimal decompression. Theory Comput. Syst. 43(3), 425–463 (2008). Special issue of invited papers from Computability in Europe 2006 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fortnow, L., Hitchcock, J.M., Aduri, P., Vinodchandran, N.V., Wang, F.: Extracting Kolmogorov complexity with applications to dimension zero-one laws. In: Proceedings of the 33rd International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 4051, pp. 335–345. Springer, Berlin (2006) CrossRefGoogle Scholar
  7. 7.
    Friedberg, R., Rogers, H.: Reducibilities and completeness for sets of integers. Z. Math. Log. Grundl. Math. 5, 117–125 (1959) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hausdorff, F.: Dimension und äusseres Mass. Math. Ann. 79, 157–179 (1919) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hemaspaandra, L., Hempel, H., Vogel, J.: Optimal separations for parallel versus sequential self-checking: parallelism can exponentially increase self-checking cost. Technical Report TR 691, Department of Computer Science, University of Rochester, May 1998 Google Scholar
  10. 10.
    Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and its Applications, 2nd edn. Springer, Berlin (1997) MATHGoogle Scholar
  11. 11.
    Lutz, J.H.: Dimension in complexity classes. SIAM J. Comput. 32, 1236–1259 (2003) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lutz, J.H.: The dimensions of individual strings and sequences. Inf. Comput. 187, 49–79 (2003) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lutz, J.H.: Effective fractal dimensions. Math. Log. Q. 51, 62–72 (2005). (Invited lecture at the International Conference on Computability and Complexity in Analysis, Cincinnati, OH, August 28–30, 2003) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mayordomo, E.: A Kolmogorov complexity characterization of constructive Hausdorff dimension. Inf. Process. Lett. 84(1), 1–3 (2002) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Miller, J.: Extracting information is hard. Technical Report (2008).
  16. 16.
    Nies, A., Reimann, J.: A lower cone in the wtt degrees of non-integral effective dimension. In: Proceedings of IMS workshop on Computational Prospects of Infinity, Singapore (2009, to appear). Earlier version appeared as Technical Report 63, Workgroup Mathematical Logic and Theoretical Computer Science, University of Heidelberg (2005) Google Scholar
  17. 17.
    Odifreddi, P.: Classical Recursion Theory. Studies in Logic and the Foundations of Mathematics, vol. 125. North-Holland, Amsterdam (1989) MATHGoogle Scholar
  18. 18.
    Post, E.L.: Recursively enumerable sets of positive integers and their decision problems. Bull. Am. Math. Soc. 50, 284–316 (1944) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Reimann, J.: Computability and fractal dimension. Doctoral thesis, Heidelberg (2005) Google Scholar
  20. 20.
    Reimann, J., Slaman, T.: Randomness, Entropy and Reducibility. Manuscript (2005) Google Scholar
  21. 21.
    Ryabko, B.Ya.: Coding of combinatorial sources and Hausdorff dimension. Sov. Math. Dokl. 30, 219–222 (1984) MATHGoogle Scholar
  22. 22.
    Ryabko, B.Ya.: Noiseless coding of combinatorial sources. Probl. Inf. Transm. 22, 170–179 (1986) MATHMathSciNetGoogle Scholar
  23. 23.
    Shaltiel, R.: Recent developments in explicit constructions of extractors. Bull. EATCS 77, 67–95 (2002) MATHMathSciNetGoogle Scholar
  24. 24.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987) Google Scholar
  25. 25.
    Stephan, F.: Hausdorff-dimension and weak truth-table reducibility. Technical Report TR52/05, School of Computing, National University of Singapore (2005) Google Scholar
  26. 26.
    Sullivan, D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153, 259–277 (1984) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Tricot, C.: Two definitions of fractional dimension. Math. Proc. Camb. Philos. Soc. 91, 57–74 (1982) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Turing, A.M.: Systems of logic based on ordinals. Proc. Lond. Math. Soc. 45, 161–228 (1939) MATHCrossRefGoogle Scholar
  29. 29.
    Zimand, M.: Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences. Technical Report 0705.4658, Computing Research Repository (2007) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Laboratoire d’Informatique Fondamentale de MarseilleUniversité de ProvenceMarseille Cedex 13France
  2. 2.Department of Computer ScienceIowa State UniversityAmesUSA
  3. 3.School of Computing and Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore

Personalised recommendations