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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 the Bounded Version of Hilbert's Tenth Problem
, 2002
"... Hilbert's Tenth problem concerned the decidability of Diophantine equations over the integers. Its negative solution, the MRDP theorem, amounted to showing that the class of formula of the form (9~y)P (~x; ~y) = Q(~x; ~y) over the natural numbers where P; Q are polynomials is equivalent to the class ..."
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Hilbert's Tenth problem concerned the decidability of Diophantine equations over the integers. Its negative solution, the MRDP theorem, amounted to showing that the class of formula of the form (9~y)P (~x; ~y) = Q(~x; ~y) over the natural numbers where P; Q are polynomials is equivalent to the class of recursively enumerable sets. Provability of this theorem in weak fragments of arithmetic is known to imply NP=coNP. The bounded form of Hilbert's Tenth problem is whether the NPpredicates are the class D of predicates given by formulas of the form (9~y)[( X j y j 2 j P i x i j k ) ^ P (~x; ~y) = Q(~x; ~y)] where P; Q are polynomials with coecients in N. This problem is related to the average case completeness of certain NPproblems. In this paper we give lower bounds on the provability of both these problems in weak fragments of arithmetic and we show certain closure properties of the Dpredicates. We show the theory I 5 E1 can not prove D =NP. Here I m E1 has a nite ...