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55
The Complexity Of Propositional Proofs
 Bulletin of Symbolic Logic
, 1995
"... This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on ..."
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Cited by 105 (2 self)
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This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on
Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic
"... A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We ..."
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Cited by 86 (2 self)
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A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: 1. Feasible interpolation theorems for the following proof systems: (a) resolution. (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (ff). (c) linear equational calculus. (d) cutting planes. 2. New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]). (b) for the cutting planes proof system with coefficients written in unary ([4]). 3. An alternative proof of the independence result of [43] concerning the provability of circuitsize lower bounds ...
Predicative Recursion and Computational Complexity
, 1992
"... The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct r ..."
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Cited by 45 (3 self)
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The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct reference to polynomials, time, or even computation. Complexity classes characterized in this way include polynomial time, the functional polytime hierarchy, the logspace decidable problems, and NC. After developing these "resource free" definitions, we apply them to redeveloping the feasible logical system of Cook and Urquhart, and show how this firstorder system relates to the secondorder system of Leivant. The connection is an interesting one since the systems were defined independently and have what appear to be very different rules for the principle of induction. Furthermore it is interesting to see, albeit in a very specific context, how to retract a second order statement, ("inducti...
Witnessing Functions in Bounded Arithmetic and Search Problems
, 1994
"... We investigate the possibility to characterize (multi)functions that are \Sigma b i definable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems defined over polynomialtime structures. We obtain the following results: 1. A reformulation of known ..."
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Cited by 35 (4 self)
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We investigate the possibility to characterize (multi)functions that are \Sigma b i definable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems defined over polynomialtime structures. We obtain the following results: 1. A reformulation of known characterizations of (multi)functions that are \Sigma b 1  and \Sigma b 2 definable in the theories S 1 2 and T 1 2 . 2. New characterizations of (multi)functions that are \Sigma b 2  and \Sigma b 3  definable in the theory T 2 2 . 3. A new nonconservation result: the theory T 2 2 (ff) is not 8\Sigma b 1 (ff) conservative over the theory S 2 2 (ff). To prove that the theory T 2 2 (ff) is not 8\Sigma b 1 (ff)conservative over the theory S 2 2 (ff), we present two examples of a \Sigma b 1 (ff)principle separating the two theories: (a) the weak pigeonhole principle WPHP (a 2 ; f; g) formalizing that no function f is a bijection between a 2 and a with the inverse...
Mathematically Strong Subsystems of Analysis With Low Rate of Growth of Provably Recursive Functionals
, 1995
"... This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of ..."
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Cited by 34 (21 self)
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This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifierfree choice ACqf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non standard' axiom F  which does not hold in the full settheoretic model but in the strongly majorizable functionals): From a proof GnA # +ACqf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full settheoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +ACqf+F  proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +ACqf+# for suitable #. 1
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.
An Equivalence between Second Order Bounded Domain Bounded Arithmetic and First Order Bounded Arithmetic
, 1993
"... We introduce a bounded domain version V 2 (BD) of Buss's second order theory V 2 of bounded arithmetic and show that this version is equivalent to the rst order theory S 3 : More precisely, we construct two natural interpretations V 3 and S 2 (BD) which are inverse to each other and pr ..."
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Cited by 28 (4 self)
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We introduce a bounded domain version V 2 (BD) of Buss's second order theory V 2 of bounded arithmetic and show that this version is equivalent to the rst order theory S 3 : More precisely, we construct two natural interpretations V 3 and S 2 (BD) which are inverse to each other and preserve the syntactic structure of bounded formulae. As a corollary, for the bounded domain case we obtain Buss's result concerning 1 expressibility in V 2 as a direct consequence of his main result for rst order theories. Using only plain corollaries of the cut elimination theorem, we show that V 2 (BD) prove the same formulae where 8 stand for rst order quanti ers. Combined with the above mentioned result this gives an alternative proof of Buss's characterization of 2 functions. All this readily extends to the case V k (BD) vs. S k+1 (i; k 1).
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 .
Logics for Reasoning about Cryptographic Constructions
 In Proc. 44th IEEE Symposium on Foundations of Computer Science
, 2003
"... We present two logical systems for reasoning about cryptographic constructions which are sound with respect to standard cryptographic definitions of security. Soundness of the first system is proved using techniques from nonstandard models of arithmetic. Soundness of the second system is proved by ..."
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Cited by 25 (1 self)
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We present two logical systems for reasoning about cryptographic constructions which are sound with respect to standard cryptographic definitions of security. Soundness of the first system is proved using techniques from nonstandard models of arithmetic. Soundness of the second system is proved by an interpretation into the first system. We also present examples of how these systems may be used to formally prove the correctness of some elementary cryptographic constructions.