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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
Polynomial Local Search in the Polynomial Hierarchy and Witnessing in Fragments of Bounded Arithmetic
, 2008
"... The complexity class of Π p kpolynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the Σ p idefinable functions of T k+1 2 are characterized in terms of Π p kPLS problems. These Π p kPLS problems c ..."
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Cited by 8 (3 self)
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The complexity class of Π p kpolynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the Σ p idefinable functions of T k+1 2 are characterized in terms of Π p kPLS problems. These Π p kPLS problems can be defined in a weak base theory such as S1 2, and proved to be total in T k+1 2. Furthermore, the Π p kPLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new ∀Σb 1(α)principle that is conjectured to separate T k 2 (α) and T k+1 2 (α). 1
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 ..."
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Cited by 3 (0 self)
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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
Bounded arithmetic, cryptography, and complexity
 THEORIA
, 1997
"... This survey discusses theories of bounded arithmetic, growth rates of definable functions, natural proofs, interpolation theorems, connections to cryptography, and the difficulty of obtaining independence results. ..."
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Cited by 3 (0 self)
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This survey discusses theories of bounded arithmetic, growth rates of definable functions, natural proofs, interpolation theorems, connections to cryptography, and the difficulty of obtaining independence results.
Translating I∆0(exp) proofs into weaker systems
 Mathematical Logic Quarterly
, 2000
"... The purpose of this paper is to explore the relationship between I∆0+exp and its weaker subtheories. We give a method of translating certain classes of I∆0+exp proofs into weaker systems of arithmetic such as Buss ’ systems S2. We show if IEi(exp) ⊢ A with a proof P of expindrank(P) ≤ n + 1 where ..."
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Cited by 1 (1 self)
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The purpose of this paper is to explore the relationship between I∆0+exp and its weaker subtheories. We give a method of translating certain classes of I∆0+exp proofs into weaker systems of arithmetic such as Buss ’ systems S2. We show if IEi(exp) ⊢ A with a proof P of expindrank(P) ≤ n + 1 where all ( ∀ ≤: right) or ( ∃ ≤: left) have bounding terms not containing function symbols then S i 2 ⊇ IEi,2 ⊢ A n. Here A is not necessarily a bounded formula. For IOpen(exp) we prove a similar result. Using our translations we show IOpen(exp) � I∆0(exp). Here I∆0(exp) is a conservative extension of I∆0+exp obtained by adding to I∆0 a symbol for 2 x to the language as well as defining axioms for it.
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...