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Exponential Lower Bounds for AC0-Frege Imply Superpolynomial Frege Lower Bounds

  • Yuval Filmus
  • Toniann Pitassi
  • Rahul Santhanam
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

We give a general transformation which turns polynomial-size Frege proofs to subexponential-size AC0-Frege proofs. This indicates that proving exponential lower bounds for AC0-Frege is hard, since it is a longstanding open problem to prove super-polynomial lower bounds for Frege. Our construction is optimal for tree-like proofs.

As a consequence of our main result, we are able to shed some light on the question of weak automatizability for bounded-depth Frege systems. First, we present a simpler proof of the results of Bonet et al. [5] showing that under cryptographic assumptions, bounded-depth Frege proofs are not weakly automatizable. Secondly, we show that because our proof is more general, under the right cryptographic assumptions, it could resolve the weak automatizability question for lower depth Frege systems.

Keywords

Proof System Double Sequent Pigeonhole Principle Logical Depth Proof Complexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Allender, E., Hellerstein, L., McCabe, P., Pitassi, T., Saks, M.: Minimizing disjunctive normal form formulas and AC0 circuits given a truth table. SIAM Journal on Computing 38(1), 63–84 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atserias, A., Maneva, E.: Mean-payoff games and propositional proofs. Inf. and Comp. 209(4), 664–691 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beame, P.W., Impagliazzo, R., Krajíček, J., Pitassi, T., Pudlák, P., Woods, A.: Exponential lower bounds for the pigeonhole principle. In: Proceedings of the Twenty-Fourth Annual ACM Symposium on Computing, Victoria, B.C., Canada, pp. 200–220 (May 1992)Google Scholar
  4. 4.
    Bonet, M.L., Buss, S.R., Pitassi, T.: Are there hard examples for Frege systems? In: Feasible Mathematics II, pp. 30–56. Birkhäuser, Basel (1995)CrossRefGoogle Scholar
  5. 5.
    Bonet, M.L., Domingo, C., Gavaldà, R., Maciel, A., Pitassi, T.: Non-automatizability of bounded-depth Frege proofs. Computational Complexity 13(1-2), 47–68 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonet, M.L., Pitassi, T., Raz, R.: On interpolation and automatization for Frege systems. SIAM Journal on Computing 29(6), 1939–1967 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buss, S.R.: Polynomial size proofs of the pigeonhole principle. Journal of Symbolic Logic 57, 916–927 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. Journal of Symbolic Logic 44(1), 36–50 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Haken, A.: The intractability of resolution. Theoretical Computer Science 39, 297–305 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Krajíček, J.: Lower bounds to the size of constant-depth propositional proofs. Journal of Symbolic Logic 59(1), 73–86 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Krajíček, J.: Bounded arithmetic, propositional logic, and complexity theory. Cambridge University Press, New York (1995)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Yuval Filmus
    • 1
  • Toniann Pitassi
    • 1
  • Rahul Santhanam
    • 2
  1. 1.Univeristy of TorontoCanada
  2. 2.University of EdinburghScotland, UK

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