Algebraic proof systems over formulas

We introduce two algebraic propositional proof systems F - NS and F - PC . 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 tautolo...

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Bibliographic Details
Published inTheoretical computer science Vol. 303; no. 1; pp. 83 - 102
Main Authors Grigoriev, Dima, Hirsch, Edward A.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 28.06.2003
Elsevier
Subjects
Algebraic propositional proof systems
Frege systems
Mathematics
Frege systems
Algebraic propositional proof systems
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Summary:We introduce two algebraic propositional proof systems F - NS and F - PC . 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 - NS provide an exponential separation between this system and Polynomial Calculus. We prove that F - NS (and hence F - PC ) polynomially simulates Frege systems, and that the constant-depth version of F - PC over finite field polynomially simulates constant-depth Frege systems with modular counting. We also present a short constant-depth F - PC (in fact, F - NS ) proof of the propositional pigeon-hole principle. Finally, we introduce several extensions of our systems and pose numerous open questions.
Bibliography:ObjectType-Article-2
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ISSN:0304-3975
1879-2294
DOI:10.1016/S0304-3975(02)00446-2