@MISC{Krajicek93onfrege, author = {Jan Krajicek}, title = {On Frege and Extended Frege Proof Systems}, year = {1993} }

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Abstract

We propose a framework for proving lower bounds to the size of EF - proofs (equivalently, to the number of proof-steps in F-proofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a particular forcing construction explained also in the paper. It reduces the question of proving a lower bound to the question of constructing a partial boolean algebra and a map of formulas into that algebra with particular properties. We show that in principle one can obtain via this method optimal lower bounds (up to a polynomial increase). Introduction A propositional proof system is any polynomial time function P whose range is exactly the set of tautologies TAUT, cf. [17]. For ø a tautology any string ß such that P (ß) = ø is called a P -proof of ø . Any usual propositional calculus, be it resolution or extended resolution, a Hilbert style system based on finitely many axiom schemes and inf...