For any class C und closed under NC1 reductions, it is shown that all sets complete for C under first-order (equivalently, Dlogtimeuniform AC0) reductions are isomorphic under first-order computable isomorphisms.


Primitive Generator Random Restriction Probabilistic Construction Length Encoder Circuit Family 
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  1. AAI+97.
    M. Agrawal, E. Allender, R. Impagliazzio, T. Pitassi, and S. Rudich. Reducing the complexity of reductions. In Proceedings of Annual ACM Symposium on the Theory of Computing, pages 730–738, 1997.Google Scholar
  2. AAR98.
    M. Agrawal, E. Allender, and S. Rudich. Reductions in circuit complexity: An isomorphism theorem and a gap theorem. J. Comput. Sys. Sci., 57:127–143, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  3. ABI93.
    E. Allender, J. Balcázar, and N. Immerman. A first-order isomorphism theorem. In Proceedings of the Symposium on Theoretical Aspects of Computer Science, 1993.Google Scholar
  4. AG91.
    E. Allender and V. Gore. Rudimentary reductions revisited. Information Processing Letters, 40:89–95, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  5. AGHP90.
    N. Alon, O. Goldreich, J. Hastad, and R. Peralta. Simple constructions of almost k-wise independent random variables. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science, pages 544–553, 1990.Google Scholar
  6. Agr96.
    M. Agrawal. On the isomorphism problem for weak reducibilities. J. Comput. Sys. Sci., 53(2):267–282, 1996.zbMATHCrossRefGoogle Scholar
  7. Agr01.
    M. Agrawal. Towards uniform AC0 isomorphisms. In Proceedings of the Conference on Computational Complexity, 2001. to be presented.Google Scholar
  8. BDG88.
    J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.Google Scholar
  9. BH77.
    L. Berman and J. Hartmanis. On isomorphism and density of NP and other complete sets. SIAM Journal on Computing, 1:305–322, 1977.CrossRefMathSciNetGoogle Scholar
  10. BIS90.
    D. Barrington, N. Immerman, and H. Straubing. On uniformity within NC1. J. Comput. Sys. Sci., 74:274–306, 1990.CrossRefMathSciNetGoogle Scholar
  11. CSV84.
    A. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13:423–439, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  12. ER60.
    P. Erdös and R. Rado. Intersection theorems for systems of sets. J. London Math. Soc., 35:85–90, 1960.zbMATHCrossRefMathSciNetGoogle Scholar
  13. FFK92.
    S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science, pages 30–39, 1992. To appear in SIAM J. Comput.Google Scholar
  14. FSS84.
    M. Furst, J. Saxe, and M. Sipser. Parity, circuits, and the polynomial hierarchy. Mathematical Systems Theory, 17:13–27, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  15. IL95.
    N. Immerman and S. Landau. The complexity of iterated multiplication. Information and Computation, 116:103–116, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  16. Imm87.
    N. Immerman. Languages that capture complexity classes. SIAM Journal on Computing, 16:760–778, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Jon75.
    N. Jones. Space-bounded reducibility among combinatorial problems. J. Comput. Sys. Sci., 11:68–85, 1975.zbMATHCrossRefGoogle Scholar
  18. KMR88.
    S. Kurtz, S. Mahaney, and J. Royer. The structure of complete degrees. In A. Selman, editor, Complexity Theory Retrospective, pages 108–146. Springer-Verlag, 1988.Google Scholar
  19. KMR89.
    S. Kurtz, S. Mahaney, and J. Royer. The isomorphism conjecture fails relative to a random oracle. In Proceedings of Annual ACM Symposium on the Theory of Computing, pages 157–166, 1989.Google Scholar
  20. Lin92.
    S. Lindell. A purely logical characterization of circuit complexity. In Proceedings of the Structure in Complexity Theory Conference, pages 185–192, 1992.Google Scholar
  21. NN90.
    J. Naor and M. Naor. Small-bias probability spaces: Efficient constructions and applications. In Proceedings of Annual ACM Symposium on the Theory of Computing, pages 213–223, 1990.Google Scholar
  22. NW94.
    N. Nisan and A. Wigderson. Hardness vs. randomness. J. Comput. Sys. Sci., 49(2):149–167, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  23. Sip83.
    M. Sipser. Borel sets and circuit complexity. In Proceedings of Annual ACM Symposium on the Theory of Computing, pages 61–69, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Manindra Agrawal
    • 1
  1. 1.Department of Computer ScienceIIT KanpurKanpurIndia

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