Advertisement

Lower bounds by recursion theoretic arguments

  • Uwe Schöning
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 226)

Abstract

Using methods and notions stemming from recursion theory, new lower bounds on the "distance" between certain intractable sets (like NP-complete or EXPTIME-complete sets) and the sets in P are obtained. Here, the distance of two sets A and B is a function on natural numbers that, for each n, gives the number of strings of size n on which A and B differ. Yesha [6] has shown that each NP-complete set has a distance of at least O(log log n) from each set in P, assuming P ≠ NP. Similarly, whithout an additional assumption, each EXPTIME-complete set has a distance of at least O(log log n) from each set in P.

In this paper the following will be shown:

  1. 1.

    Assuming P ≠ NP, no NP-complete set that is a (weak) p-cylinder can be within a distance of q(n) to any set in P where q is any polynomial. (Note that all "naturally known" NP-complete sets have been shown to be p-cylinders [3]).

     
  2. 2.

    No EXPTIME-complete set can be within a distance of 2nc to any set in P for some constant c>0.

     

The second result improves Yesha's by at least two exponentials.

Keywords

Polynomial Time SIAM Journal Recursive Function Random Oracle Recursion Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.L. Balcázar and U. Schöning, Bi-immune sets for complexity classes, Mathematical Systems Theory 18 (1985), 1–10.Google Scholar
  2. 2.
    C.H. Bennett and J. Gill, Relative to a random oracle A, pA≠NPA≠coNPA with probability 1, SIAM Journal on Computing 10 (1981), 96–113.Google Scholar
  3. 3.
    L. Berman and J. Hartmanis, On isomorphism and density of NP and other complete sets, SIAM Journal on Computing 6 (1977), 305–327.Google Scholar
  4. 4.
    H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.Google Scholar
  5. 5.
    A.C. Yao, Theory and applications of trapdoor functions, 23rd IEEE Symp. Foundations of Computer Science 1982, 80–91.Google Scholar
  6. 6.
    Y. Yesha, On certain polynomial-time truth-table reducibilities of complete sets to sparse sets, SIAM Journal on Computing 12 (1983), 411–425.Google Scholar
  7. 7.
    P. Young, Some structural properties of polynomial reducibilities and sets in NP, Proc. 15th Ann. ACM Symp. Theory of Computing, 1983, 392–401.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Uwe Schöning
    • 1
  1. 1.EWH Koblenz, InformatikKoblenzWest Germany

Personalised recommendations