Journal of Symbolic Logic

Dual weak pigeonhole principle, pseudo-surjective functions, and provability of circuit lower bounds

Jan Krajíček

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This article is a continuation of our search for tautologies that are hard even for strong propositional proof systems like EF, cf. [Kra-wphp,Kra-tau]. The particular tautologies we study, the τ-formulas, are obtained from any 𝒫/poly map g; they express that a string is outside of the range of g. Maps g considered here are particular pseudorandom generators. The ultimate goal is to deduce the hardness of the τ-formulas for at least EF from some general, plausible computational hardness hypothesis.

In this paper we introduce the notions of pseudo-surjective and iterable functions (related to free functions of [Kra-tau]). These two properties imply the hardness of the τ-formulas from the function but unlike the hardness they are preserved under composition and iteration. We link the existence of maps with these two properties to the provability of circuit lower bounds, and we characterize maps g yielding hard τ-formulas in terms of a hitting set type property (all relative to a propositional proof system). We show that a proof system containing EF admits a pseudo-surjective function unless it simulates a proof system WF introduced by Jeřábek [Jer], an extension of EF. We propose a concrete map g as a candidate function possibly pseudo-surjective or free for strong proof systems. The map is defined as a Nisan-Wigderson generator based on a random function and on a random sparse matrix. We prove that it is iterable in a particular way in resolution, yielding the output/input ratio n3-ε (that improves upon a direct construction of Alekhnovich et al. [ABRW]).

Article information

Source
J. Symbolic Logic, Volume 69, Issue 1 (2004), 265-286.

Dates
First available in Project Euclid: 2 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1080938841

Digital Object Identifier
doi:10.2178/jsl/1080938841

Mathematical Reviews number (MathSciNet)
MR2039361

Zentralblatt MATH identifier
1068.03048

Citation

Krajíček, Jan. Dual weak pigeonhole principle, pseudo-surjective functions, and provability of circuit lower bounds. J. Symbolic Logic 69 (2004), no. 1, 265--286. doi:10.2178/jsl/1080938841. https://projecteuclid.org/euclid.jsl/1080938841


Export citation

References

  • M. Ajtai The complexity of the pigeonhole principle, Proceedings of the IEEE 29th Annual Symposium on Foundation of Computer Science,1988, pp. 346--355.
  • M. Alekhnovich, E. Ben-Sasson, A. A. Razborov, and A. Wigderson Pseudorandom generators in propositional proof complexity, Electronic Colloquium on Computational Complexity, vol. 23,2000, abstract in series Proceedings of the 41st Annual Symposium on Foundation of Computer Science, 2000, pp. 43--53.
  • A. Atserias and M. L. Bonet On the automatizability of resolution and related propositional proof systems, 11th Annual Conference of the European Association for Computer Science Logic (CSL), Lecture Notes in Computer Science, vol. 2471, Springer-Verlag,2002, pp. 569--583.
  • E. Ben-Sasson and A. Wigderson Short proofs are narrow---resolution made simple, Proceedings of the 31st ACM Symposium on Theory of Computation,1999, pp. 517--526.
  • S. R. Buss, R. Impagliazzo, J. Krajíček, P. Pudlák, A. A. Razborov, and J. Sgall Proof complexity in algebraic systems and bounded depth Frege systems with modular counting, Computational Complexity, vol. 6 (1996/1997), no. 3, pp. 256--298.
  • S. A. Cook Feasibly constructive proofs and the propositional calculus, Proceedings of the 7th Annual ACM Symposium on Theory of Computing, ACM Press,1975, pp. 83--97.
  • S. A. Cook and A. R. Reckhow The relative efficiency of propositional proof systems, Journal of Symbolic Logic, vol. 44 (1979), no. 1, pp. 36--50.
  • O. Goldreich Candidate one-way functions based on expander graphs, Electronic Colloquium on Computational Complexity, vol. 90 (2000).
  • O. Goldreich, S. Goldwasser, and S. Micali How to construct random functions, Journal of the ACM, vol. 33 (1986), no. 4, pp. 792--807.
  • R. Impagliazzo and A. Wigderson $P = BPP$ unless $E$ has sub-exponential circuits: derandomizing the XOR lemma, Proceedings of the 29th Annual ACM Symposium on Theory of Computing,1997, pp. 220--229.
  • E. Jeřábek Dual weak pigeonhole principle, Boolean complexity, and derandomization, series Annals of Pure and Applied Logic, to appear.
  • J. Krajíček No counter-example interpretation and interactive computation, Logic from Computer Science, Proceedings of a Workshop held November 13--17, 1989, in Berkeley (Y. N. Moschovakis, editor), Mathematical Sciences Research Institute Publication, vol. 21, Springer-Verlag,1992, pp. 287--293.
  • J. Krajíček and P. Pudlák Propositional proof systems, the consistency of first order theories and the complexity of computations, Journal of Symbolic Logic, vol. 54 (1989), no. 3, pp. 1063--1079.
  • J. Krajíček, P. Pudlák, and J. Sgall Interactive computations of optimal solutions, Mathematical Foundations of Computer Science (B. Bystrica, August '90) (B. Rovan, editor), Lecture Notes in Computer Science, vol. 452, Springer-Verlag,1990, pp. 48--60.
  • N. Nisan and A. Wigderson Hardness versus randomness, Journal of Computer and System Sciences, vol. 49 (1994), pp. 149--167.
  • J. B. Paris and A. J. Wilkie Counting problems in bounded arithmetic, Methods in Mathematical Logic, Lecture Notes in Mathematics, vol. 1130, Springer-Verlag,1985, pp. 317--340.
  • J. B. Paris, A. J. Wilkie, and A. R. Woods Provability of the pigeonhole principle and the existence of infinitely many primes, Journal of Symbolic Logic, vol. 53 (1988), pp. 1235--1244.
  • P. Pudlák Some relations between subsystems of arithmetic and the complexity of computations, Logic from Computer Science, Proceedings of a Workshop held November 13--17, 1989, in Berkeley (Y. N. Moschovakis, editor), Mathematical Sciences Research Institute Publication, vol. 21, Springer-Verlag,1992, pp. 499--519.
  • A. A. Razborov Pseudorandom generators hard for $k$-DNF resolution and polynomial calculus resolution, preprint, May 2003.
  • S. Rudich Super-bits, demi-bits, and $\tildeNP/qpoly$-natural proofs, Proceedings of the 1st International Symposium on Randomization and Approximation Techniques in Computer Science, Lecture Notes in Computer Science, vol. 1269, Springer-Verlag,1997, pp. 85--93.