Advertisement

Theory of Computing Systems

, Volume 47, Issue 2, pp 317–341 | Cite as

Non-Uniform Reductions

  • Harry Buhrman
  • Benjamin HescottEmail author
  • Steven Homer
  • Leen Torenvliet
Open Access
Article

Abstract

We study properties of non-uniform reductions and related completeness notions. We strengthen several results of Hitchcock and Pavan (ICALP (1), Lecture Notes in Computer Science, vol. 4051, pp. 465–476, Springer, 2006) and give a trade-off between the amount of advice needed for a reduction and its honesty on NEXP. We construct an oracle relative to which this trade-off is optimal. We show, in a more systematic study of non-uniform reductions, among other things that non-uniformity can be removed at the cost of more queries. In line with Post’s program for complexity theory (Buhrman and Torenvliet in Bulletin of the EATCS 85, pp. 41–51, 2005) we connect such ‘uniformization’ properties to the separation of complexity classes.

Keywords

Non-uniform reductions Reductions with advice Non-uniform complexity NEXP complete set EXP complete set NP complete set 

References

  1. 1.
    Agrawal, M.: Pseudo-random generators and structure of complete degrees. In IEEE Conference on Computational Complexity, pp. 139–147 (2002) Google Scholar
  2. 2.
    Agrawal, M., Biswas, S.: Polynomial isomorphism of 1-L complete sets. In: Proc. Structure in Complexity Theory 7th Annual Conference, San Diego, California, pp. 75–80. IEEE Computer Society, Los Alamitos (1993) Google Scholar
  3. 3.
    Allender, E.: Isomorphisms and 1-L reductions. J. Comput. Syst. Sci. 36(6), 336–350 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Allender, E., Buhrman, H., Koucký, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. In: FOCS, pp. 669–678. IEEE Computer Society, Los Alamitos (2002) Google Scholar
  5. 5.
    Ambos-Spies, K.: p-mitotic sets. In: Börger, E., Hasenjäger, G., Roding, D. (eds.) Logic and Machines. Lecture Notes in Computer Science, vol. 177, pp. 1–23. Springer, Berlin (1984) Google Scholar
  6. 6.
    Balcázar, J., Díaz, J., Gabarró, J.: Structural Complexity I. Springer, Berlin (1988) zbMATHGoogle Scholar
  7. 7.
    Berman, L., Hartmanis, H.: On isomorphisms and density of NP and other complete sets. SIAM J. Comput. 6, 305–322 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Buhrman, H., Mayordomo, E.: An excursion to the Kolmogorov random strings. In: Proceedings Structure in Complexity Theory, 10th Annual Conference (STRUCTURES95), Minneapolis, pp. 197–205. IEEE Computer Society, Los Alamitos (1995) Google Scholar
  9. 9.
    Buhrman, H., Torenvliet, L.: Complicated complementations. In: Proceedings 14th IEE Conference on Computational Complexity, pp. 227–236. IEEE Computer Society, Los Alamitos (1999) Google Scholar
  10. 10.
    Buhrman, H., Torenvliet, L.: Separating complexity classes using structural properties. In: Proceedings 19th IEE Conference on Computational Complexity, pp. 130–138. IEEE Computer Society, Los Alamitos (2004) CrossRefGoogle Scholar
  11. 11.
    Buhrman, H., Torenvliet, L.: A Post’s program for complexity theory. In Bulletin of the EATCS 85, pp. 41–51 (2005) Google Scholar
  12. 12.
    Buhrman, H., Homer, S., Torenvliet, L.: On complete sets for nondeterministic classes. Math. Syst. Theory 24, 179–200 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Buhrman, H., Spaan, E., Torenvliet, L.: Bounded reductions. In: Ambos-Spies, K., Homer, S., Schöning, U. (eds.) Complexity Theory, pp. 83–99. Cambridge University Press, Cambridge (1993) Google Scholar
  14. 14.
    Buhrman, H., Spaan, E., Torenvliet, L.: The relative power of logspace and polynomial time reductions. Comput. Complexity 3(3), 231–244 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Buhrman, H., van Melkebeek, D., Fortnow, L., Torenvliet, L.: Using autoreducibility to separate complexity classes. SIAM J. Comput. 29(5), 1497–1520 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Fenner, S., Fortnow, L., Kurtz, S.A.: The isomorphism conjecture holds relative to an oracle. In: Proc. 33rd IEEE Symposium Foundations of Computer Science, pp. 30–39 (1992) Google Scholar
  17. 17.
    Ganesan, K., Homer, S.: Complete problems and strong polynomial reducibilities. In: Proc. Symposium on Theoretical Aspects of Computer Science. Springer Lecture Notes in Computer Science, vol. 349, pp. 240–250. Springer, Berlin (1988) Google Scholar
  18. 18.
    Glaßer, C., Selman, A.L., Travers, S.D., Zhang, L.: Non-mitotic sets. In: Arvind, V., Prasad, S. (eds.) STTCS. Lecture Notes in Computer Science, vol. 4855, pp. 146–157. Springer, Berlin (2007) Google Scholar
  19. 19.
    Hartmanis, J., Hemachandra, L.: One-way functions and the non-isomorphism of NP-complete sets. Theor. Comput. Sci. 81(1), 155–163 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Homer, S., Kurtz, S., Royer, J.: A note on many-one and 1-truth table complete sets. Theor. Comput. Sci. 115(2), 383–389 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hitchcock, J.M., Pavan, A.: Hardness hypotheses, derandomization, and circuit complexity. In: 24th Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 336–347. Springer, Berlin (2004) CrossRefGoogle Scholar
  22. 22.
    Hitchcock, J.M., Pavan, A.: Comparing reductions to NP-complete sets. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP (1). Lecture Notes in Computer Science, vol. 4051, pp. 465–476. Springer, Berlin (2006) Google Scholar
  23. 23.
    Homer, S., Selman, A.L.: Oracles for structural properties: the isomorphism problem and public-key cryptography. J. Comput. Syst. Sci. 44(2), 287–301 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Homer, S., Selman, A.L.: Computability and Complexity Theory. Springer, New York (2001) zbMATHGoogle Scholar
  25. 25.
    Kurtz, S., Mahaney, S., Royer, J.: The isomorphism conjecture fails relative to a random oracle. In Proc. 21nd Annual ACM Symposium on Theory of Computing, pp. 157–166 (1989) Google Scholar
  26. 26.
    Krentel, M.: The complexity of optimization problem. J. Comput. Syst. Sci. 36, 490–509 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ladner, R., Lynch, N., Selman, A.: A comparison of polynomial time reducibilities. Theor. Comput. Sci. 1, 103–123 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mayordomo, E.: Almost every set in exponential time is p-bi-immune. Theor. Comput. Sci. 136(2), 487–506 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ronneburger, D.: Kolmogorov complexity and derandomization. PhD thesis, Rutgers University, New Brunswick, NJ, October 2004 Google Scholar
  30. 30.
    Watanabe, O.: A comparison of polynomial time completeness notions. Theor. Comput. Sci. 54, 249–265 (1987) zbMATHCrossRefGoogle Scholar
  31. 31.
    Young, P.: Juris Hartmanis: Fundamental contributions to the isomorphism problems. In: Selman, A.L. (ed.) Complexity Theory Retrospective, pp. 108–146. Springer, Berlin (1990) Google Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Harry Buhrman
    • 1
  • Benjamin Hescott
    • 2
    Email author
  • Steven Homer
    • 3
  • Leen Torenvliet
    • 4
  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Computer Science DepartmentTufts UniversityMedfordUSA
  3. 3.Computer Science DepartmentBoston UniversityBostonUSA
  4. 4.ILLCAmsterdamThe Netherlands

Personalised recommendations