On the Existence of modulo p Cardinality Functions

  • Miklós Ajtai
Conference paper
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 13)

Abstract

Suppose that A = 〈A, …〉 is a first-order structure with a finite number of finitary relation and function symbols, and p is a prime number. We say that a function μ with mod p values is a mod p cardinality function if it is defined on the first-order definable subsets of A, A 2, … and it satisfies the following basic properties of the usual notion of cardinality. It is invariant under one-to-one first-order definable maps, it is additive with respect to disjoint union, it is multiplicative with respect to direct product, and the cardinality of singletons is 1. We show that if there is a first-order definable ordering of the universe then the existence of a mod p cardinality function is equivalent to the following statement: There are no two first-order definable equivalence relations Ф and Ψ on a (first-order definable) subset X of A i for some i = 1,2,… with the following properties: (1) each class of Ф contains exactly p elements, and (2) each class of Ψ with one exception contains exactly p elements, the exceptional class contains 1 element. (We will refer to this statement as the modulo p counting principle.)

Keywords

Fermat Almaden 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Ajtai. \(\Sigma_1^1\)-formulae on finite structures. Annals of Pure and Applied Logic, 24:1 – 48, 1983.CrossRefGoogle Scholar
  2. [2]
    M. Ajtai. The complexity of the pigeon hole principle. In Proceedings of IEEE 29th Annual Symposium on Foundations of Computer Science, 1988. pp. 346 – 355. (Submitted to Combinatorica.)Google Scholar
  3. [3]
    M. Ajtai. Parity and the pigeonhole principle. In S.R. Buss and P.J. Scott, editors, Feasible Mathematics, pages 1–24. Birkhäuser, 1990.CrossRefGoogle Scholar
  4. [4]
    M. Ajtai. The independence of the modulo pcounting principles. IBM Research Report, RJ9422 (83781), November, 1993.Google Scholar
  5. [5]
    M. Ajtai. Symmetric systems of linear equations mod p. IBM Research Report, RJ9442 (82674), July 1993.Google Scholar
  6. [6]
    M. Ajtai and A. Wigderson. Deterministic simulation of probabilistic constant depth circuits Advances in Computing Research, Volume 5, pages 199 – 222, 1989.Google Scholar
  7. [7]
    P. Beame, R. Impagliazzo, J. Krajíček, T. Pitassi, P. Pudlák, and A. Woods. Exponential lower bounds for the pigeonhole principle. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, Victoria, 1992.Google Scholar
  8. [8]
    P. Beame and T. Pitassi. An exponential separation between the matching principle and the pigeonhole principle. In Proceedings of IEEE 8th Annual Symposium on Logic in Computer Science, pages 308–319, 1993.Google Scholar
  9. [9]
    S. Bellantoni, T. Pitassi, and A. Urquhart. Approximation and small depth Frege proofs. SIAM Journal of Computing, 1161 – 1179, Dec. 1992.Google Scholar
  10. [10]
    M. Furst, J. B. Saxe, and M. Sipser. Parity circuits and the polynomial time hierarchy. Mathematical Systems Theory, 17:13–27, 1984. Preliminary version in Proceedings of the 22nd IEEE Symposium on Foundations of Computer Science, 1981.CrossRefGoogle Scholar
  11. [11]
    J. Krajíček, P. Pudlák, and A. Woods. Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Submitted, 1991.Google Scholar
  12. [12]
    T. Pitassi, P. Beame, and R. Impagliazzo. Exponential lower bounds for the pigeonhole principle. Computational Complexity, 3(2) 97 – 140 (1993).CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Miklós Ajtai
    • 1
  1. 1.K/53IBM Almaden Research CenterSan JoseUSA

Personalised recommendations