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computational complexity

, Volume 6, Issue 3, pp 256–298 | Cite as

Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

  • S. Buss
  • R. Impagliazzo
  • J. Krajíček
  • P. Pudlák
  • A. A. Razborov
  • J. Sgall
Article

Abstract

We prove a lower bound of the formNΩ(1) on the degree of polynomials in a Nullstellensatz refutation of theCount q polynomials over ℤ m , whereq is a prime not dividingm. In addition, we give an explicit construction of a degreeN Ω(1) design for theCount q principle over ℤ m . As a corollary, using Beameet al. (1994) we obtain a lower bound of the form 2NΩ(1) for the number of formulas in a constant-depth Frege proof of the modular counting principleCount q N from instances of the counting principleCount m M .

We discuss the polynomial calculus proof system and give a method of converting tree-like polynomial calculus derivations into low degree Nullstellensatz derivations.

Further we shwo that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connectiveMOD p follows from a lower bound on the degree of Nullstellensatz proofs with a constant number of levels of extension axioms, where the extension axioms comprise a formalization of the approximation method of Razborov (1987) and Smolensky (1987) (in fact, these two proof systems are basically equivalent).

Keywords

Approximation Method Computational Mathematic Problem Complexity Algorithm Analysis Constant Number 
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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Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  • S. Buss
    • 1
  • R. Impagliazzo
    • 3
  • J. Krajíček
    • 2
  • P. Pudlák
    • 2
  • A. A. Razborov
    • 4
  • J. Sgall
    • 2
  1. 1.Department of MathematicsUniversity of Californian San DiegoLa JollaUSA
  2. 2.Mathematical InstituteAcademy of SciencesPragueCzech Republic
  3. 3.Computer Science EngineeringUniversity of California, San DiegoLa JollaUSA
  4. 4.Steklov Mathematical InstituteAcademy of SciencesMoscowRussia

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