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10
On Interpolation and Automatization for Frege Systems
, 2000
"... The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the ..."
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Cited by 51 (7 self)
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The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexitytheoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC 0 Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC 0 Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial size TC 0 Frege. As a corollary, we obtain that TC 0 Frege as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integ...
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.
No Feasible Interpolation for TC 0 Frege Proofs
, 1997
"... The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the ..."
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Cited by 9 (3 self)
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The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexitytheoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC 0 Frege systems. More specifically, we show that unless factoring is feasible, neither Frege nor TC 0 Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomialsize TC 0 Frege. In particular, we show how to carry out the proof for the Chinese Remainder Theorem, which may be of independent interest. As a corollary, we obtain that TC 0 Frege as well as any proof system...
On the Computational Content of Intuitionistic Propositional Proofs
, 2000
"... this paper is to show that the constructive character of intuitionistic logic manifests itself not only on the level of computability but, in case of the propositional fragment, also on the level of polynomial time computability. ..."
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Cited by 7 (0 self)
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this paper is to show that the constructive character of intuitionistic logic manifests itself not only on the level of computability but, in case of the propositional fragment, also on the level of polynomial time computability.
Foundations of Instance Level Updates in Expressive Description Logics
, 2011
"... In description logic (DL), ABoxes are used for describing the state of affairs in an application domain. We consider the problem of updating ABoxes when the state changes, assuming that update information is described at an atomic level, i.e., in terms of possibly negated ABox assertions that involv ..."
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Cited by 2 (0 self)
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In description logic (DL), ABoxes are used for describing the state of affairs in an application domain. We consider the problem of updating ABoxes when the state changes, assuming that update information is described at an atomic level, i.e., in terms of possibly negated ABox assertions that involve only atomic concepts and roles. We analyze such basic ABox updates in several standard DLs, in particular addressing questions of expressibility and succinctness: can updated ABoxes always be expressed in the DL in which the original ABox was formulated and, if so, what is the size of the updated ABox? It turns out that DLs have to include nominals and the ‘@’ constructor of hybrid logic for updated ABoxes to be expressible, and that this still holds when updated ABoxes are approximated. Moreover, the size of updated ABoxes is exponential in the role depth of the original ABox and the size of the update. We also show that this situation improves when updated ABoxes are allowed to contain additional auxiliary symbols. Then, DLs only need to include nominals for updated ABoxes to exist, and the size of updated ABoxes is polynomial in the size of both the original ABox and the update.
Propositional proof complexity — an introduction
 In Ulrich Berger and Helmut Schwichtenberg, editors, Computational Proof Theory
, 1997
"... ..."
Descriptive Complexity and Finite Models
"... This paper introduces algebraic proof systems for the propositional calculus. We present new results concerning the relative efficiency of these systems, and also survey what is currently known. Many open problems are presented. 1 Introduction A fundamental problem in logic and computer science is ..."
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This paper introduces algebraic proof systems for the propositional calculus. We present new results concerning the relative efficiency of these systems, and also survey what is currently known. Many open problems are presented. 1 Introduction A fundamental problem in logic and computer science is understanding the efficiency of propositional proof systems. It has been known for a long time that NP = coNP if and only if there exists an efficient propositional proof system, but despite 25 years of research, this problem is still not resolved. (See [46] for an excellent survey of this area.) The intention of the present article is to introduce a new algebraic approach to this problem. Our proof systems are simpler than classical proof systems, and purely algebraic. It is our hope that by studying proof complexity in this light, that new upper and lower bound techniques may emerge. The use of the Nullstellensatz for propositional refutations may have been first suggested in a paper by Lo...
On the Computational Content of Intuitionistic
"... this paper is to show that the constructive character of intuitionistic logic manifests itself not only on the level of computability but, in case of the propositional fragment, also on the level of polynomial time computability ..."
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this paper is to show that the constructive character of intuitionistic logic manifests itself not only on the level of computability but, in case of the propositional fragment, also on the level of polynomial time computability
INTRODUCTION TO THE COMBINATORICS AND COMPLEXITY OF CUT ELIMINATION
"... Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted ..."
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Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted