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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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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 .
On Herbrand’s theorem
 LOGIC AND COMPUTATIONAL COMPLEXITY
, 1995
"... We rstly survey several forms of Herbrand's theorem. What is commonly called "Herbrand's theorem" in many textbooks is actually a very simple form of Herbrand's theorem which applies only to 89formulas; but the original statement of Herbrand's theorem applied to arbi ..."
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We rstly survey several forms of Herbrand's theorem. What is commonly called "Herbrand's theorem" in many textbooks is actually a very simple form of Herbrand's theorem which applies only to 89formulas; but the original statement of Herbrand's theorem applied to arbitrary firstorder formulas. We give a direct proof, based on cutelimination, of what is essentially Herbrand's original theorem. The "nocounterexample theorems" recently used in bounded and Peano arithmetic are immediate corollaries of this form of Herbrand's theorem. Secondly, we discuss the results proved in Herbrand's 1930 dissertation.
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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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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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 model theory for intuitionistic bounded arithmetic with applications to independence results
 Feasible Mathematics
, 1990
"... Abstract IPV+ is IPV (which is essentially IS12) with polynomialinduction on \Sigma b+1formulas disjoined with arbitrary formulas in which the induction variable does not occur. This paper proves that IPV+ is sound and complete with respect to Kripke structures in which every world is a model of C ..."
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Abstract IPV+ is IPV (which is essentially IS12) with polynomialinduction on \Sigma b+1formulas disjoined with arbitrary formulas in which the induction variable does not occur. This paper proves that IPV+ is sound and complete with respect to Kripke structures in which every world is a model of CPV (essentially S12). Thus IPV is sound with respect to such structures. In this setting, this is a strengthening of the usual completeness and soundness theorems for firstorder intuitionistic theories. Using Kripke structures a conservation result is proved for PV1 over IPV.
The Witness Function Method and Provably Recursive Functions of Peano Arithmetic
"... This paper presents a new proof of the characterization of the provably recursive functions of the fragments I\Sigma n of Peano arithmetic. The proof method also characterizes the \Sigma kdefinable functions of I\Sigma n and of theories axiomatized by transfinite induction on ordinals. The proofs ..."
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This paper presents a new proof of the characterization of the provably recursive functions of the fragments I\Sigma n of Peano arithmetic. The proof method also characterizes the \Sigma kdefinable functions of I\Sigma n and of theories axiomatized by transfinite induction on ordinals. The proofs are completely prooftheoretic and use the method of witness functions and witness oracles. Similar methods also yield a new proof of Parson's theorem on the conservativity of the \Sigma n+1induction rule over the \Sigma ninduction axioms. A new proof of the conservativity of B\Sigma n+1 over I\Sigma n is given. The proof methods provide new analogies between Peano arithmetic and bounded arithmetic.
Bounded arithmetic, cryptography, and complexity
 THEORIA
, 1997
"... This survey discusses theories of bounded arithmetic, growth rates of definable functions, natural proofs, interpolation theorems, connections to cryptography, and the difficulty of obtaining independence results. ..."
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This survey discusses theories of bounded arithmetic, growth rates of definable functions, natural proofs, interpolation theorems, connections to cryptography, and the difficulty of obtaining independence results.
A BottomUp Approach to Foundations of Mathematics
"... this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject ..."
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this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject
Ordinal Notations and WellOrderings in Bounded Arithmetic
 Annals of Pure and Applied Logic
, 2002
"... this paper. 2 General orderings This section states a couple results about general orderings. By a "general ordering" we mean any order defined by a # 1 formula; by comparison the results of sections 3 and 4 concern specific natural wellorderings based on ordinal notations ..."
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this paper. 2 General orderings This section states a couple results about general orderings. By a "general ordering" we mean any order defined by a # 1 formula; by comparison the results of sections 3 and 4 concern specific natural wellorderings based on ordinal notations