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Proof Complexity In Algebraic Systems And Bounded Depth Frege Systems With Modular Counting
"... We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, us ..."
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Cited by 31 (9 self)
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We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 for the number of formulas in a constantdepth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discuss
Lower Bounds for Propositional Proofs and Independence Results in Bounded Arithmetic
 Proceedings of the 23rd ICALP, Lecture Notes in Computer Science
, 1996
"... . We begin with a highly informal discussion of the role played by Bounded Arithmetic and propositional proof systems in the reasoning about the world of feasible computations. Then we survey some known lower bounds on the complexity of proofs in various propositional proof systems, paying special a ..."
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Cited by 27 (10 self)
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. We begin with a highly informal discussion of the role played by Bounded Arithmetic and propositional proof systems in the reasoning about the world of feasible computations. Then we survey some known lower bounds on the complexity of proofs in various propositional proof systems, paying special attention to recent attempts on reducing such bounds to some purely complexity results or assumptions. As one of the main motivations for this research we discuss provability of extremely important propositional formulae that express hardness of explicit Boolean functions with respect to various nonuniform computational models. 1. Propositional proofs as feasible proofs of plain statements Interesting and viable logical theories do not appear as result of sheer speculation. Conversely, they attempt to summarize and capture a certain amount of reasoning of a certain style about a certain class of objects that had existed in the math community before the mathematical logics entered the stage....
Tautologies From PseudoRandom Generators
, 2001
"... We consider tautologies formed from a pseudorandom number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a ..."
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Cited by 16 (0 self)
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We consider tautologies formed from a pseudorandom number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a form of a hardness condition posed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at nonlogicians, of the relation between propositional proof complexity and bounded arithmetic. It is a fundamental problem of mathematical logic to decide if tautologies can be inferred in propositional calculus in substantially fewer steps than it takes to check all possible truth assignments. This is closely related to the famous P/NP problem of Cook [3]. By propositional calculus I mean any textbook system based on a nite number of inference rules and axiom schemes that is sound and complete. The qualication substantially less means that the nu...
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.
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.
Superbits, Demibits, and NP/qpolynatural Proofs
 Journal of Computer and System Sciences
, 1997
"... We introduce the superbit conjecture, which allows the development of a theory generalizing the notion of pseudorandomness so as to fool nondeterministic statistical tests. This new kind of pseudorandomness rules out the existence of NP=polynatural properties that can work against P=poly. This ..."
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We introduce the superbit conjecture, which allows the development of a theory generalizing the notion of pseudorandomness so as to fool nondeterministic statistical tests. This new kind of pseudorandomness rules out the existence of NP=polynatural properties that can work against P=poly. This is an important extension of the original theory of P=polynatural proofs [10]. We also introduce the closely related demibit conjecture which is more intuitive and is the source of interesting open problems. 1 Introduction By exploiting the theory of pseudorandom generators [1, 11, 3, 7], Razborov and Rudich [10] give evidence that the proof techniques used in the last sixteen years of nonuniform, nonmonotone circuit lower bounds will be unable to solve the barrier problems of complexity theory. They argue that known lower bound arguments are all "natural" which means that they exploit some P natural combinatorial property (see Section 3.1). Furthermore, they show that if strong e...
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