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34
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.
Relating the Bounded Arithmetic and Polynomial Time Hierarchies
 Annals of Pure and Applied Logic
, 1994
"... The bounded arithmetic theory S 2 is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T 2 equals S then T 2 is equal to S 2 and proves that the polynomial time hierarchy collapses to # , and, in fact, to the Boolean hierarchy over # and to # i+1 / ..."
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Cited by 27 (1 self)
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The bounded arithmetic theory S 2 is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T 2 equals S then T 2 is equal to S 2 and proves that the polynomial time hierarchy collapses to # , and, in fact, to the Boolean hierarchy over # and to # i+1 /poly .
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
Relating the PSPACE reasoning power of Boolean Programs and Quantified Boolean Formulas
, 2000
"... We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantifie ..."
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Cited by 13 (9 self)
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We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantified Boolean formula (QBF) characterization of
PSPACE due to Stockmeyer and Meyer. We conclude with a discussion of some closely
related open problems and their implications.
Theories With SelfApplication and Computational Complexity
 Information and Computation
, 2002
"... Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not ne ..."
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Cited by 12 (9 self)
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Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not necessarily total. It has turned out that theories with selfapplication provide a natural setting for studying notions of abstract computability, especially from a prooftheoretic perspective.
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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Cited by 10 (0 self)
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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.
The strength of replacement in weak arithmetic
 Proceedings of the Nineteenth Annual IEEE Symposium on Logic in Computer Science
, 2004
"... The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<a  ∃x
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Cited by 10 (3 self)
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The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<a  ∃x<aφ(i,x) → ∃w ∀i<aφ(i,[w]i) proves the scheme where φ is in the class Γ of formulas. The theory S1 2 BB(Σb 1), and thus in S1 2 every Σb1 formula is equivalent to a strict Σb1 formula (in which all nonsharplybounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S1 2 do not prove either BB(Σb1) or BB(Σb0). We show (unconditionally) that V 0 does not prove BB(ΣB 0), where V 0 (essentially IΣ 1,b 0) is the twosorted theory associated with the complexity class AC0. We show that PV does not prove BB(Σb 0), assuming
End Extensions of Models of Linearly Bounded Arithmetic
, 1994
"... We show that every model of I \Delta 0 has an end extension to a model of a theory (extending Buss' S 0 2 ) where logspace computable function are formalizable. We also show the existence of an isomorphism between models of I \Delta 0 and models of linear arithmetic LA (i.e., secondorder Presburger ..."
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Cited by 10 (0 self)
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We show that every model of I \Delta 0 has an end extension to a model of a theory (extending Buss' S 0 2 ) where logspace computable function are formalizable. We also show the existence of an isomorphism between models of I \Delta 0 and models of linear arithmetic LA (i.e., secondorder Presburger arithmetic with finite comprehension for bounded formulas). 0 Introduction. In the last two decades the research on bounded arithmetic has been focussed mainly on the theories of I \Delta 0 and I \Delta 0+\Omega 1 and on their fragments. The interest in these theories is in part motivated by complexity theoretical considerations. It is well known that the provably recursive function of I \Delta 0+\Omega 1 are those computable by algorithms in the polynomial time hierarchy while the provably recursive functions of I \Delta 0 correspond to the linear time hierarchy. Some open problems in bounded arithmetic are known to be equivalent to problems of complexity theory. More precisely, to the s...
Theory for TC 0 and Other Small Complexity Classes
 Logical Methods in Computer Science
, 2005
"... Abstract We present a general method for introducing finitely axiomatizable "minimal " secondorder theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the firstorder theory \Delta b ..."
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Cited by 9 (4 self)
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Abstract We present a general method for introducing finitely axiomatizable "minimal " secondorder theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the firstorder theory \Delta b