On the Autoreducibility of Random Sequences

  • Todd Ebert
  • Heribert Vollmer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1893)


A language A⊂- {0, 1}* is called i.o. autoreducible if A is Turing-reducible to itself via a machine M such that, for infinitely many input words w, M does not query its oracle A about w. We examine the question if algorithmically random languages in the sense of Martin-Löf are i.o. autoreducible. We obtain the somewhat counterintuitive result that every algorithmically random language is polynomial-time i.o. auto-reducible where the autoreducing machine poses its queries in a “quasi-nonadaptive” way; however, if in the above definition the “infinitely many” is replaced by “almost all,” then every algorithmically random language is not autoreducible in this stronger sense. Further results obtained give upper and lower bounds on the number of queries of the autoreducing machine M and the number of inputs w for which M does not query the oracle about w.


Random Sequence Turing Machine Binary Sequence Recursive Function Code Word 
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  1. 1.
    Alon N., Spencer J. The Probabilistic Method. John Wiley and Sons, New York, 1992.zbMATHGoogle Scholar
  2. 2.
    Bennett C., Gill J. Relative to a Random Oracle A, P A ≠ NP A co NP A With Probability 1. SIAM Journal on Computing, 10, 96–113, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blahut R. Theory and Practice of Error Control Codes. Addison Wesley, Reading, Ma, 1984.Google Scholar
  4. 4.
    Book R. On Languages Reducible to Algorithmically Random Languages. SIAM Journal on Computing, Vol. 23, No. 6, 1275–1282, 1984.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Book R., Lutz J., Wagner K. W. An Observation on Probability Versus Randomness. Math. Systems Theory, 27, 201–209, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brualdi R. Introductory Combinatorics. North-Holland, NY, 1988.Google Scholar
  7. 7.
    Buhrman H., van Melkebeek D., Regan K.W., Sivakumar D., Strauss M., A generalization of resource-bounded measure, with an application to the BPP vs. EXP problem. Manuscript, 1999. Preliminary versiond appeared in Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science, pp. 161–171, 1998, and as University of Chicago, Department of Computer Science, Technical Report TR-97-04, May 1997.Google Scholar
  8. 8.
    Cutland N. Introduction to Computability. Cambridge Univ. Press, 1992.Google Scholar
  9. 9.
    Ebert T. Applications of Recursive Operators to Randomness and Complexity. Ph.D. Thesis, University of California at Santa Barbara, 1998.Google Scholar
  10. 10.
    Gács P. Every Sequence is Reducible to a Random One. Information and Control, 70, 186–192, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kim K.H., Roush F.W. Applied Abstract Algebra. John Wiley and Sons, New York, 1983.zbMATHGoogle Scholar
  12. 12.
    Li M., Vitanyi P. An Introduction to Kolmogorov Complexity and its Applications. Springer-Verlag, New York, 1993.zbMATHGoogle Scholar
  13. 13.
    Lutz J. König’s Lemma, Randomness, and the Arithmetical Hierarchy. Unpublished Lecture Notes, Iowa St. Univ., 1993.Google Scholar
  14. 14.
    Lutz J. A Pseudorandom Oracle Characterization of BPP. SIAM Journal on Computing, 22(5):1075–1086, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lutz J. The Quantitative Structure of Exponential Time. In: Hemaspaandra L., Selman A., editors, Complexity Theory Restropsective II, Springer Verlag, 1997, 225–260.Google Scholar
  16. 16.
    Martin-Löf P. On the Definition of Random Sequences. Information and Control, 9, 602–619, 1966.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Wolfgang Merkle. Personal communication, December 1999.Google Scholar
  18. 18.
    Rogers H. Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge, 1992.Google Scholar
  19. 19.
    Shiryaev A.N. Probability. Springer-Verlag, New York, 1995.zbMATHGoogle Scholar
  20. 20.
    Trakhtenbrot, B.A. On Autoreducibility. Soviet Math dokl., 11:814–817, 1970.zbMATHGoogle Scholar
  21. 21.
    Wagner K. W., Wechsung G. Computational Complexity. Deutscher Verlag der Wissenschaften, Berlin, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Todd Ebert
    • 1
  • Heribert Vollmer
    • 2
  1. 1.DoCoMo Communications Laboratories, USAPalo Alto
  2. 2.Theoretische InformatikUniversität WürzburgWürzburgGermany

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