Results 1  10
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13
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.
Quantified Propositional Calculus and a SecondOrder Theory for NC¹
, 2004
"... Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S ..."
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Cited by 9 (2 self)
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Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce
The provable total search problems of bounded arithmetic
, 2007
"... We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument use ..."
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Cited by 8 (4 self)
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We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument uses a translation of first order proofs into large, uniform propositional proofs in a system in which the soundness of the rules can be witnessed by polynomial time reductions between games. We show that ∀ ˆ Σ b 1(α) conservativity of of T i+1 2 (α) over T i 2(α) already implies ∀ ˆ Σ b 1(α) conservativity of T2(α) over T i 2(α). We translate this into propositional form and give a polylogarithmic width CNF GI3 such that if GI3 has small R(log) refutations then so does any polylogarithmic width CNF which has small constant depth refutations. We prove a resolution lower bound for GI3. We use our characterization to give a sufficient condition for the totality of a relativized NP search problem to be unprovable in T i 2(α) in terms of a nonlogical question about multiparty communication protocols.
A thirdorder bounded arithmetic theory for PSPACE
 of Lecture Notes in Computer Science
, 2004
"... Abstract. We present a novel thirdorder theory W 1 1 of bounded arithmetic suitable for reasoning about PSPACE functions. This theory has the advantages of avoiding the smash function symbol and is otherwise much simpler than previous PSPACE theories. As an example we outline a proof in W 1 1 that ..."
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Cited by 7 (3 self)
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Abstract. We present a novel thirdorder theory W 1 1 of bounded arithmetic suitable for reasoning about PSPACE functions. This theory has the advantages of avoiding the smash function symbol and is otherwise much simpler than previous PSPACE theories. As an example we outline a proof in W 1 1 that from any configuration in the game of Hex, at least one player has a winning strategy. We then exhibit a translation of theorems of W 1 1 into families of propositional tautologies with polynomialsize proofs in BPLK (a recent propositional proof system for PSPACE and an alternative to G). This translation is clearer and more natural in several respects than the analogous ones for previous PSPACE theories. Keywords: Bounded arithmetic, propositional proof complexity, PSPACE, quantified propositional calculus 1
Disjoint NPpairs from propositional proof systems
 In Proc. 3rd Conference on Theory and Applications of Models of Computation
, 2006
"... Abstract. For a proof system P we introduce the complexity class DNPP(P) of all disjoint NPpairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NPpairs of these proof ..."
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Cited by 7 (3 self)
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Abstract. For a proof system P we introduce the complexity class DNPP(P) of all disjoint NPpairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NPpairs of these proof systems hard or complete for DNPP(P). Moreover we demonstrate that nonequivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for DNPP(P) and the reductions between the canonical pairs exist. 1
On an optimal quantified propositional proof system and a complete language for NP . . .
 In Proceedings of the 11th International Symposium on Fundamentals of Computing Theory, LNCS #1279
, 1997
"... . In this paper we prove that if there exists an optimal quantified propositional proof system then there exists a complete language for NP " coNP. 1 Introduction There is a lot of activity around the complexity of propositional proof systems (see [7, 12]). Research into propositional proof system ..."
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. In this paper we prove that if there exists an optimal quantified propositional proof system then there exists a complete language for NP " coNP. 1 Introduction There is a lot of activity around the complexity of propositional proof systems (see [7, 12]). Research into propositional proof systems is motivated by the open problem NP=coNP? Research into quantified propositional proof systems is not so popular. The study of quantified propositional proof systems may be related to the problem NP=PSPACE? Some deep results about connections between quantified propositional proof systems and bounded arithmetic are contained in [8]. We propose to study the problem of the existence of an optimal quantified propositional proof system. The similar problem for propositional proof systems has been studied in [9]. It is not known whether complete languages exist for NP " coNP and Sipser has shown in [10] that there are relativizations so that NP " coNP has no complete languages (see also [4...
Separation Results for the Size of ConstantDepth Propositional Proofs
, 2004
"... This paper proves exponential separations between depth d  LK and depth (d + 2 )  LK for every d 2 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d  LK and depth (d+1)  LK for d N . We investigate the relationship between ..."
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Cited by 5 (3 self)
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This paper proves exponential separations between depth d  LK and depth (d + 2 )  LK for every d 2 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d  LK and depth (d+1)  LK for d N . We investigate the relationship between the sequencesize, treesize and height of depth d  LKderivations for d 2 N , and describe transformations between them.
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
Consistency and Gamesin Search of New Combinatorial Principles
, 2004
"... We show that a semantical interpretation of Herbrand's disjunctions can be used to obtain 2 independent sentences whose nature is more combinatorial than the nature of the usual consistency statements. Then we apply this method to Bounded Arithmetic and present 8 1 combinatorial sentences tha ..."
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Cited by 1 (1 self)
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We show that a semantical interpretation of Herbrand's disjunctions can be used to obtain 2 independent sentences whose nature is more combinatorial than the nature of the usual consistency statements. Then we apply this method to Bounded Arithmetic and present 8 1 combinatorial sentences that characterize all 8 1 sentences provable in S 2 . We use the concept of a two player game to describe these sentences.
A Propositional Proof System for Log Space
"... Abstract. The proof system G ∗ 0 of the quantified propositional calculus corresponds to NC 1, and G ∗ 1 corresponds to P, but no formulabased proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL ∗. We begin by defining a class ΣCNF (2) of ..."
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Abstract. The proof system G ∗ 0 of the quantified propositional calculus corresponds to NC 1, and G ∗ 1 corresponds to P, but no formulabased proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL ∗. We begin by defining a class ΣCNF (2) of quantified formulas that can be evaluated in log space. Then GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF (2) formulas and no cut formula that is not quantifier free contains a nonparameter free variable. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of Σ B 0rec into a family of tautologies that have polynomial size GL ∗ proofs. Σ B 0rec is a theory of bounded arithmetic that is known to correspond to log space. To do the translation, we find an appropriate axiomatization of Σ B 0rec, and put Σ B 0rec proofs into a new normal form. To show that GL ∗ is not too strong, we prove the soundness of GL ∗ in such a way that it can be formalized in Σ B 0rec. This is done by giving a log space algorithm that witnesses GL ∗ proofs. 1