Algebraic proof systems over formulas

Dima Grigoriev, Edward A. Hirsch

Research output: Contribution to journalConference articlepeer-review

26 Scopus citations

Abstract

We introduce two algebraic propositional proof systems ℱ-script N sign script S sign and ℱ-script P sign script C sign. The main difference of our systems from (customary) Nullstellensatz and polynomial calculus is that the polynomials are represented as arbitrary formulas (rather than sums of monomials). Short proofs of Tseitin's tautologies in the constant-depth version of ℱ-script N sign script S sign provide an exponential separation between this system and Polynomial Calculus. We prove that ℱ-script N sign script S sign (and hence ℱ-script P sign script C sign) polynomially simulates Frege systems, and that the constant-depth version of ℱ-script P sign script C sign over finite field polynomially simulates constant-depth Frege systems with modular counting. We also present a short constant-depth ℱ-script P sign script C sign (in fact, ℱ-script N sign script S sign) proof of the propositional pigeon-hole principle. Finally, we introduce several extensions of our systems and pose numerous open questions.

Original languageEnglish
Pages (from-to)83-102
Number of pages20
JournalTheoretical Computer Science
Volume303
Issue number1
DOIs
StatePublished - 28 Jun 2003
Externally publishedYes
EventLogic and Complexity in Computer Science - Creteil, France
Duration: 3 Sep 20015 Sep 2001

Keywords

  • Algebraic propositional proof systems
  • Frege systems

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