A uniform reduction theorem extending a result of J. Grollmann and A. Selman

  • Kenneth W. Regan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 226)


We derive a recursion-theoretic result telling when a family of reductions to a class
can be replaced by a single oracle Turing machine. The theorem is a close analogue of the Uniform Boundedness Theorem of functional analysis, specializing it to the Cantor-set topology on ℙ(Σ*). This generalizes one of the main theorems of J. Grollmann and A. Selman [FOCS '84], namely that NP-hardness implies uniform NP-hardness for ‘promise problems’. We investigate other consequences and problems arising from the theorem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A-S85]
    K. Ambos-Spies. Sublattices of the polynomial-time degrees. Information and Control 65, No. 1, April 1985, pp 63–84.Google Scholar
  2. [Ang80]
    D. Angluin. Counting problems and the polynomial-time hierarchy. Theoretical Computer Science 12, No. 2, October 1980.Google Scholar
  3. [Cu80]
    N. Cutland. Computability. (Cambridge: Camb. University Press, 1980.)Google Scholar
  4. [Dowd82]
    M. Dowd. Forcing and the P hierarchy. Preprint, Rutgers Univ., 1982.Google Scholar
  5. [Dum77]
    M. Dummett. Elements of Intuitionism. (Oxford: Clarendon Press, 1977.)Google Scholar
  6. [EY80]
    S. Even and Y. Yacobi. Cryptography and NP-completeness. Proc. ICALP '80, Springer LNCS 80, pp. 195–207.Google Scholar
  7. [Grl84]
    J. Grollmann. Ph.D dissertation, Univ. of Dortmund, W. Germany, 1984.Google Scholar
  8. [GS84]
    J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. Proc. 25th FOCS, Oct. 1984.Google Scholar
  9. [GS85]
    Ibid. Iowa State Univ. Technical Report TR 85-31, November 1985.Google Scholar
  10. [Hom82]
    S. Homer. Minimal degrees for polynomial reducibilities. Draft, Boston University, 1982.Google Scholar
  11. [KzMt80]
    D. Kozen and M. Machtey. On relative diagonals. TR RC 8184 (#35583), IBM Thomas J. Watson Research Center, Yorktown Hts., NY 10598 USA, 1980.Google Scholar
  12. [Lad75]
    R. Ladner. On the structure of polynomial-time reducibility. J. ACM 22, 1975, pp. 155–171.Google Scholar
  13. [Mel73]
    K. Melhorn. On the size of sets of computable functions. Proc. 14th Symposium on Switching and Automata Theory (now STOC), 1973, pp 190–196.Google Scholar
  14. [Rog67]
    H. Rogers. Theory of Recursive Functions and Effective Computability. (New York: McGraw-Hill, 1967).Google Scholar
  15. [Roy63]
    H. Royden. Real Analysis. (New York: The MacMillan Company, 1963).Google Scholar
  16. [Rud74]
    W. Rudin. Real and Complex Analysis (2nd. edition). (New York: McGraw-Hill, 1974.)Google Scholar
  17. [Sel82]
    A. Selman. Reductions on NP and P-selective sets. Theoretical Computer Science 19, 1982, pp 287–304.Google Scholar
  18. [SY82]
    A. Selman and Y. Yacobi. The complexity of promise problems. Proc. ICALP '82, Springer LNCS 140, 1982, pp. 502–509.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Kenneth W. Regan
    • 1
  1. 1.Merton CollegeOxfordEngland

Personalised recommendations