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We give a family of tautologies whose algebraic translations have constant-degree, polynomial size polynomial calculus refutations over Z2, but which require superpolynomial size bounded-depth Frege proofs from Count2 axioms. This gives a superpolynomial size separation of bounded-depth Frege plus mod 2 counting axioms from bounded-depth Frege plus parity gates. Combined with another result of the authors, it gives the first size (as opposed to degree) separation between the polynomial calculus and Nullstellensatz systems.
Date of Conference: 8-11 Oct. 2001