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Bounded Arithmetic and Constant Depth Frege Proofs
, 2004
"... We discuss the ParisWilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and nonrelativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0depth for PKproofs that makes the translation from boun ..."
Abstract

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We discuss the ParisWilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and nonrelativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0depth for PKproofs that makes the translation from bounded arithmetic to propositional logic particularlytransparent. Using this, we give new proofs of the witnessing theorems for S12and T 12; namely, new proofs that the \Sigma b1definable functions of S12are polynomial time computable and that the \Sigma b1definable functions of T 12 are in Polynomial Local Search (PLS). Both proofs generalize to \Sigma