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Theories for Complexity Classes and their Propositional Translations
 Complexity of computations and proofs
, 2004
"... We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus. ..."
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Cited by 30 (7 self)
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We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.
Structure and Definability in General Bounded Arithmetic Theories
, 1999
"... This paper is motivated by the questions: what are the \Sigma ..."
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Cited by 18 (6 self)
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This paper is motivated by the questions: what are the \Sigma
A Propositional Proof System for. . .
"... . In this paper we introduce Gentzenstyle quantied propositional proof systems L i for the theories R i 2 . We formalize the systems L i within the bounded arithmetic theory R 1 2 and we show that for i 1, R i 2 can prove the validity of a sequent derived by an L i proof. This stateme ..."
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. In this paper we introduce Gentzenstyle quantied propositional proof systems L i for the theories R i 2 . We formalize the systems L i within the bounded arithmetic theory R 1 2 and we show that for i 1, R i 2 can prove the validity of a sequent derived by an L i proof. This statement is formally called iRFN(L i ). We show if R i 2 ` 8xA(x) where A 2 b i , then for each integer n there is a translation of the formula A into quantied propositional logic such that R i 2 proves there is an L i proof of this translated formula. Using the proofs of these two facts we show that L i is in some sense the strongest system for which R i 2 can prove iRFN and we show for i j 2 that the 8 b j consequences of R i 2 are nitely axiomatized. 1. Introduction Propositional proof systems and bounded arithmetic are closely connected. Cook [10] introduced the equational arithmetic theory PV of polynomial time computable functions and showed PV co...