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


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).


Approximation Method Computational Mathematic Problem Complexity Algorithm Analysis Constant Number 
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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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