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14
An application of boolean complexity to separation problems in bounded arithmetic
 Proc. London Math. Society
, 1994
"... We develop a method for establishing the independence of some Zf(a)formulas from S'2(a). In particular, we show that T'2(a) is not VZ*(a)conservative over S'2(a). We characterize the Z^definable functions of T2 as being precisely the functions definable as projections of polynomial local search ( ..."
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Cited by 54 (15 self)
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We develop a method for establishing the independence of some Zf(a)formulas from S'2(a). In particular, we show that T'2(a) is not VZ*(a)conservative over S'2(a). We characterize the Z^definable functions of T2 as being precisely the functions definable as projections of polynomial local search (PLS) problems. Although it is still an open problem whether bounded arithmetic S2 is finitely axiomatizable, considerable progress on this question has been made: S2 +1 is V2f+1conservative over T'2 [3], but it is not V2!f+2conservative unless £f+2 = Ylf+2 [10], and in addition, T2 is not VZf+1conservative over S'2 unless LogSpace s? = Af+1 [8]. In particular, S2 is not finitely axiomatizable provided that the polynomialtime hierarchy does not collapse [10]. For the theory S2(a) these results imply (with some additional arguments) absolute results: S'2 + (a) is V2f+,(a)conservative but not VZf+2(a)conservative over T'2(a), and T'2(a) is not VZf+i(c*)conservative over S'2(a). Here a represents a new uninterpreted predicate symbol adjoined to the language of arithmetic which may be used in induction formulas; from a computer science perspective, a represents an oracle. In this paper we pursue this line of investigation further by showing that T'2(a) is also not V2f(a)conservative over S'2(a). This was known for / = 1, 2 by [9,17] (see also [2]), and our present proof uses a version of the pigeonhole principle similar to the arguments in [2,9]. Perhaps more importantly, we formulate a general method (Theorem 2.6) which can be used to show the unprovability of other 2f(a)formulas from S'2(a). Our methods are analogous in spirit to the proof strategy of [8]: prove a witnessing theorem to show that provability of a Zf+1(a)formula A in S'2(a) implies that it is witnessed by a function of certain complexity and then employ techniques of boolean complexity to construct an oracle a such that the formula A cannot be witnessed by a function of the prescribed complexity. Our formula A shall be 2f(a) and thus we can use the original witnessing theorem of [2]. The boolean complexity used is the same as in [8], namely Hastad's switching lemmas [6].
On TruthTable Reducibility to SAT
, 2002
"... We show that polynomial time truthtable reducibility via Boolean circuits to SAT is the same as logspace truthtable reducibility via Boolean formulas to SAT and the same as logspace Turing reducibility to SAT . In addition, we prove that a constant number of rounds of parallel queries to SAT i ..."
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Cited by 51 (2 self)
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We show that polynomial time truthtable reducibility via Boolean circuits to SAT is the same as logspace truthtable reducibility via Boolean formulas to SAT and the same as logspace Turing reducibility to SAT . In addition, we prove that a constant number of rounds of parallel queries to SAT is equivalent to one round of parallel queries.
FirstOrder Proof Theory of Arithmetic
, 1998
"... this article will deal only with the intensional approach. The reader who wants to see the numeralwise representability approach can consult Smorynski [1977] and any number of textbooks such as Mendelson [1987]. The intensional approach is due to Feferman [1960]. An e#ective unification of the two a ..."
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Cited by 29 (1 self)
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this article will deal only with the intensional approach. The reader who wants to see the numeralwise representability approach can consult Smorynski [1977] and any number of textbooks such as Mendelson [1987]. The intensional approach is due to Feferman [1960]. An e#ective unification of the two approaches can be given using the fact (independently due to Wilkie and to Nelson [1986]) that I# 0 +# 1 and S 2 are interpretable in Q ; since both I# 0 +# 1 and S 2 admit a relatively straightforward intensional arithmetization of metamathematics (see Wilkie and Paris [1987] and Buss [1986]), this allows strong forms of incompleteness obtained via the intensional approach to apply also to the theory Q ; paragraph 2.1.4 below sketches how the interpretation of S 2 in Q can be used to give an intensional arithmetization in Q . The book of Smullyan [1992] gives a modern, indepth treatment of Godel's incompleteness theorems
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 .
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.
Polynomial Local Search in the Polynomial Hierarchy and Witnessing in Fragments of Bounded Arithmetic
, 2008
"... The complexity class of Π p kpolynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the Σ p idefinable functions of T k+1 2 are characterized in terms of Π p kPLS problems. These Π p kPLS problems c ..."
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Cited by 8 (3 self)
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The complexity class of Π p kpolynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the Σ p idefinable functions of T k+1 2 are characterized in terms of Π p kPLS problems. These Π p kPLS problems can be defined in a weak base theory such as S1 2, and proved to be total in T k+1 2. Furthermore, the Π p kPLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new ∀Σb 1(α)principle that is conjectured to separate T k 2 (α) and T k+1 2 (α). 1
The Witness Function Method and Provably Recursive Functions of Peano
 Logic, Methodology and Philosophy of Science IX
, 1994
"... This paper presents a new proof of the characterization of the provably recursive functions of the fragments I# n of Peano arithmetic. The proof method also characterizes the # k definable functions of I# n and of theories axiomatized by transfinite induction on ordinals. The proofs are complete ..."
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Cited by 5 (0 self)
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This paper presents a new proof of the characterization of the provably recursive functions of the fragments I# n of Peano arithmetic. The proof method also characterizes the # k definable functions of I# 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.
Unprovability of Consistency Statements in Fragments of Bounded Arithmetic
, 1995
"... This paper deals with the weak fragments of arithmetic PV and S and their inductionfree fragments PV  and S 1 2 . We improve the bootstrapping of S 2 , which allows us to show that the theory S 2 can be axiomatized by the set of axioms BASIC together with any of the following induction s ..."
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Cited by 4 (2 self)
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This paper deals with the weak fragments of arithmetic PV and S and their inductionfree fragments PV  and S 1 2 . We improve the bootstrapping of S 2 , which allows us to show that the theory S 2 can be axiomatized by the set of axioms BASIC together with any of the following induction schemas: # 1 PIND , # 1 PIND or # 1 LIND .
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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Cited by 3 (0 self)
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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.
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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Cited by 2 (0 self)
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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