Advertisement

Ramsey's theorem in bounded arithmetic

  • Pavel Pudlák
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 533)

Abstract

We shall show that the finite Ramsey theorem as a Δ0 schema is provable in 01. As a consequence we get that propositional formulas expressing the finite Ramsey theorem have polynomial-size bounded-depth Frege proofs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    M.Ajtai, The complexity of the pigeonhole principle, 29-th Symp. on Foundations of Comp. Sci. (1988), pp. 346–355.Google Scholar
  2. [B1]
    S. Buss, Bounded Arithmetic, Bibliopolis, 1986.Google Scholar
  3. [CR]
    S.A. Cook, R.A. Reckhow, The relative efficiency of propositional proof systems, Journ. Symb. Logic 44, (1979), pp.36–50.Google Scholar
  4. [FW]
    P. Frankl, R.M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1(4), (1981), pp.357–368.Google Scholar
  5. [H]
    A. Haken, The intractability of resolution, Theor. Comp. Sci. 39, (1985), pp.297–308.CrossRefGoogle Scholar
  6. [KPT]
    J.Krajíček, P.Pudlák, G.Takeuti, Bounded arithmetic and the polynomial hierarchy, Annals of Pure and Applied Logic, to appear.Google Scholar
  7. [K]
    B. Krishnamurthy, Short proofs for tricky formulas, Acta Informatica 22, (1985), pp.253–275.CrossRefGoogle Scholar
  8. [PW1]
    J.Paris, A.Wilkie, Δ0 sets and induction, in Proc. Jadwisin Logic Conference, Poland, Leeds Univ. Press, 1981, pp.237–248.Google Scholar
  9. [PW2]
    J. Paris, A. Wilkie, Counting Δ 0 sets, Fundamenta Mathematicae 127, (1987), pp. 67–76.Google Scholar
  10. [PWW]
    J.B. Paris, A,J, Wilkie and A.R. Woods, Provability of the Pigeon Hole Principle and the existence of infinitely many primes, JSL 53/4, (1988), pp. 1235–1244.Google Scholar
  11. [PRS]
    P. Pudlák, V. Rödl, P. Savický, Graph complexity, Acta Informatica 25, (1988), pp.515–535.Google Scholar
  12. [R]
    A.A.Razborov, Formulas of bounded depth in basis {&, ⊕} and some combinatorial problems, in Složnost' algoritmov i prikladnaja matematičeskaja logika, S.I.Adjan editor, 1987.Google Scholar
  13. [T]
    S.Toda, On the computational power of PP and ⊕PP, 30-th Symp. on Foundations of Comp. Sci., (1989), pp.514–519.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Pavel Pudlák
    • 1
  1. 1.Mathematical InstitutePraha 1Czechoslovakia

Personalised recommendations