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Bounded Arithmetic and Propositional Proof Complexity
 in Logic of Computation
, 1995
"... This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of t ..."
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This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cutfree proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.
On Herbrand's Theorem
 In Logic and Computational Complexity
, 1995
"... We firstly survey several forms of Herbrand's theorem. What is commonly called "Herbrand's theorem" in many textbooks is actually a very simple form of Herbrand's theorem which applies only to ##formulas; but the original statement of Herbrand's theorem applied to ..."
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We firstly survey several forms of Herbrand's theorem. What is commonly called "Herbrand's theorem" in many textbooks is actually a very simple form of Herbrand's theorem which applies only to ##formulas; but the original statement of Herbrand's theorem applied to arbitrary firstorder formulas. We give a direct proof, based on cutelimination, of what is essentially Herbrand's original theorem. The "nocounterexample theorems" recently used in bounded and Peano arithmetic are immediate corollaries of this form of Herbrand's theorem.