Advertisement

computational complexity

, Volume 3, Issue 2, pp 186–205 | Cite as

Relativized isomorphisms of NP-complete sets

  • Judy Goldsmith
  • Deborah Joseph
Article

Abstract

In this paper, we present several results about collapsing and non-collapsing degrees ofNP-complete sets. The first, a relativized collapsing result, is interesting because it is the strongest known constructive approximation to a relativization of Berman and Hartmanis' 1977 conjecture that all ≤ m P -complete sets forNP arep-isomorphic. In addition, the collapsing result explores new territory in oracle construction, particularly in combining overlapping and apparently incompatible coding and diagonalizing techniques. We also present non-collapsing results, which are notable for their technical simplicity, and which rely on no unproven assumptions such as one-way functions. The basic technique developed in these non-collapsing constructions is surprisingly robust, and can be combined with many different oracle constructions.

Key words

complexity classes isomorphisms NP-completeness collapsing degrees relativized computation sparse oracles 

Subject classifications

68Q15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K. Ambos-Spies, H. Fleischhack, and H. Huwig, Diagonalizations over polynomial time computable sets,Theoret. Comput. Sci. 54 (1987).Google Scholar
  2. T. Baker, J. Gill andR. Soloway, Relativizations of theP=NP question,SIAM J. Comp. 4 (1975), 431–442.Google Scholar
  3. J. Balcázar, R. Book, andU. Schöning, The polynomial-time hierarchy and sparse oracles,J. Assoc. Comput. Mach. 33 (1986), 603–617.Google Scholar
  4. J. Balcázar andU. Schöning, Bi-immune sets for complexity classes,Math. Sys. Theory 18 (1985), 1–10.Google Scholar
  5. C. Bennet andJ. Gill, Relative to a random oracleA, P A ≠ NP A with probability one,SIAM J. Comp. 10 (1981), 96–113.Google Scholar
  6. L. Berman, Polynomial reducibilities and complete sets, Ph.D. Thesis, Cornell University, 1977.Google Scholar
  7. L. Berman andJ. Hartmanis, On isomorphisms and density ofNP and other complete sets,SIAM J. Comp. 6 (1977), 305–322.Google Scholar
  8. M. Blum and R. Impagliazzo, Generic oracles and oracle classes,Proc. 28th Ann. Symp. Found. Comput. Sci. (1987), 118–126.Google Scholar
  9. M. Dowd, Isomorphism of complete sets,Unpublished manuscript, (1978).Google Scholar
  10. S. Fenner, Notions of resource-bounded category and genericity,University of Chicago Technical Report 90-32 (1990).Google Scholar
  11. S. Fenner, L. Fortnow, and S. Kurtz, An oracle relative to which the isomorphism conjecture holds,Proc. 33nd Ann. Symp. Found. Comput. Sci. (1992), 30–39.Google Scholar
  12. L. Fortnow, H. Karloff, K. Lund, and N. Nisan, The polynomial time hierarchy has interactive proofs (Note),Unpublished manuscript (1989).Google Scholar
  13. L. Fortnow andM. Sipser, Are there interactive protocols forcoNPlanguages?”Information Procession Letters 28 (1988), 249–251.Google Scholar
  14. J. Goldsmith, Polynomial isomorphisms and near-testable sets, PhD. Thesis, University of Wisconsin-Madison (1988) (Also available as University of Wisconsin Technical Report 816 (1989))Google Scholar
  15. J. Goldsmith and D. Joseph, Three results on the polynomial isomorphisms of complete sets,Proc. 27th Ann. Symp. Found. Comput. Sci. (1986), 390–397. (Note that an additional construction, as described in Theorem 2.2, was presented at the conference.)Google Scholar
  16. J. Hartmanis andL. Hemachandra, One-way functions and the non-isomorphism ofNP-complete sets,Theoret. Comput. Sci. 81 (1991), 155–63.Google Scholar
  17. S. Homer and A. Selman, Oracles for structural properties: the isomorphism problem and public-key cryptography,Proc. 4th Ann. Structure in Complexity Theory (1989), 3–14.Google Scholar
  18. K. Ko, T. Long andD. Du, A note on one-way functions and polynomial-time isomorphisms,Theoret. Comput. Sci. 39 (1986b), 225–237 (Journal version ofProc. Twenty-first Ann. ACM Symp. Theor. Comput. (1986), 295–303).Google Scholar
  19. S. Kurtz, A relativized failure of the Berman-Hartmanis conjecture,Unpublished manuscript (1983) (to appear,Theoret. Comput. Sci.)Google Scholar
  20. S. Kurtz, S. Mahaney, and J. Royer, Collapsing degrees (extended abstract),Proc. 27th Ann. Symp. Found. Comput. Sci. (1986), 380–389.Google Scholar
  21. S. Kurtz, S. Mahaney, and J. Royer, Progress on collapsing degrees,Proc. 2nd Ann. Structure in Complexity, IEEE Computer Society (1987), 126–131.Google Scholar
  22. S. Kurtz, S. Mahaney, and J. Royer, Collapsing degrees,J. Comput. System Sci. 37 (1988) (Journal version of Kurtz,et al. 1986).Google Scholar
  23. S. Kurtz, S. Mahaney, and J. Royer, The isomorphism conjecture fails relative to a random oracle,Proc. Twenty-first Ann. ACM Symp. Theor. Comput. (1989), 157–166.Google Scholar
  24. T. Long, One-way functions, isomorphisms, and complete sets (abstract),Abstracts of the Amer. Math. Soc. 9 (1988), 125.Google Scholar
  25. T. Long andA. Selman, Relativizing Complexity Classes with sparse oracles,J. Assoc. Comput. Mach. 33 (1986), 618–627.Google Scholar

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Judy Goldsmith
    • 1
  • Deborah Joseph
    • 2
  1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCANADA
  2. 2.Department of Computer SciencesUniversity of Wisconsin-MadisonMadisonUSA

Personalised recommendations