Skip to main content
SpringerLink
Log in

On the Autoreducibility of Random Sequences

  • Conference paper
  • First Online:
Mathematical Foundations of Computer Science 2000 (MFCS 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1893))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alon N., Spencer J. The Probabilistic Method. John Wiley and Sons, New York, 1992.

    MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  3. Blahut R. Theory and Practice of Error Control Codes. Addison Wesley, Reading, Ma, 1984.

    Google Scholar 

  4. Book R. On Languages Reducible to Algorithmically Random Languages. SIAM Journal on Computing, Vol. 23, No. 6, 1275–1282, 1984.

    Article  MathSciNet  Google Scholar 

  5. Book R., Lutz J., Wagner K. W. An Observation on Probability Versus Randomness. Math. Systems Theory, 27, 201–209, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  6. Brualdi R. Introductory Combinatorics. North-Holland, NY, 1988.

    Google Scholar 

  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. Cutland N. Introduction to Computability. Cambridge Univ. Press, 1992.

    Google Scholar 

  9. Ebert T. Applications of Recursive Operators to Randomness and Complexity. Ph.D. Thesis, University of California at Santa Barbara, 1998.

    Google Scholar 

  10. Gács P. Every Sequence is Reducible to a Random One. Information and Control, 70, 186–192, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  11. Kim K.H., Roush F.W. Applied Abstract Algebra. John Wiley and Sons, New York, 1983.

    MATH  Google Scholar 

  12. Li M., Vitanyi P. An Introduction to Kolmogorov Complexity and its Applications. Springer-Verlag, New York, 1993.

    MATH  Google Scholar 

  13. Lutz J. König’s Lemma, Randomness, and the Arithmetical Hierarchy. Unpublished Lecture Notes, Iowa St. Univ., 1993.

    Google Scholar 

  14. Lutz J. A Pseudorandom Oracle Characterization of BPP. SIAM Journal on Computing, 22(5):1075–1086, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  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. Martin-Löf P. On the Definition of Random Sequences. Information and Control, 9, 602–619, 1966.

    Article  MathSciNet  Google Scholar 

  17. Wolfgang Merkle. Personal communication, December 1999.

    Google Scholar 

  18. Rogers H. Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge, 1992.

    Google Scholar 

  19. Shiryaev A.N. Probability. Springer-Verlag, New York, 1995.

    MATH  Google Scholar 

  20. Trakhtenbrot, B.A. On Autoreducibility. Soviet Math dokl., 11:814–817, 1970.

    MATH  Google Scholar 

  21. Wagner K. W., Wechsung G. Computational Complexity. Deutscher Verlag der Wissenschaften, Berlin, 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DoCoMo Communications Laboratories, USA, 250 Cambridge Ave, Palo Alto, CA, 94306

    Todd Ebert

  2. Theoretische Informatik, Universität Würzburg, Am Hubland, 97074, Würzburg, Germany

    Heribert Vollmer

Authors
  1. Todd Ebert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heribert Vollmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Aarhus, Ny Munkegade, Bldg. 540, 8000, Aarhus C, Denmark

    Mogens Nielsen

  2. Department of Computer Sciene, Comenius University, 84248, Bratislava, Slovakia

    Branislav Rovan

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ebert, T., Vollmer, H. (2000). On the Autoreducibility of Random Sequences. In: Nielsen, M., Rovan, B. (eds) Mathematical Foundations of Computer Science 2000. MFCS 2000. Lecture Notes in Computer Science, vol 1893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44612-5_29

Download citation

  • DOI: https://doi.org/10.1007/3-540-44612-5_29

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67901-1

  • Online ISBN: 978-3-540-44612-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation