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13
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 ..."
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Cited by 57 (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.
Witnessing Functions in Bounded Arithmetic and Search Problems
, 1994
"... We investigate the possibility to characterize (multi)functions that are \Sigma b i definable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems defined over polynomialtime structures. We obtain the following results: 1. A reformulation of known ..."
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Cited by 33 (5 self)
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We investigate the possibility to characterize (multi)functions that are \Sigma b i definable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems defined over polynomialtime structures. We obtain the following results: 1. A reformulation of known characterizations of (multi)functions that are \Sigma b 1  and \Sigma b 2 definable in the theories S 1 2 and T 1 2 . 2. New characterizations of (multi)functions that are \Sigma b 2  and \Sigma b 3  definable in the theory T 2 2 . 3. A new nonconservation result: the theory T 2 2 (ff) is not 8\Sigma b 1 (ff) conservative over the theory S 2 2 (ff). To prove that the theory T 2 2 (ff) is not 8\Sigma b 1 (ff)conservative over the theory S 2 2 (ff), we present two examples of a \Sigma b 1 (ff)principle separating the two theories: (a) the weak pigeonhole principle WPHP (a 2 ; f; g) formalizing that no function f is a bijection between a 2 and a with the inverse...
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 30 (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 .
Satisfiability, Branchwidth and Tseitin Tautologies
, 2002
"... For a CNF , let w b () be the branchwidth of its underlying hypergraph. ..."
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Cited by 30 (2 self)
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For a CNF , let w b () be the branchwidth of its underlying hypergraph.
Dual weak pigeonhole principle, pseudosurjective functions, and provability of circuit lower bounds
"... ..."
Provably Total Functions in Bounded Arithmetic Theories . . .
, 2002
"... This paper investigates the provably total functions of fragments of first and secondorder Bounded Arithmetic. The (strongly) \Sigma bidefinable functions of Si13 and Ri3 are precisely the (strong) FP\Sigma p i13 [wit, logO(1)] functions. The \Sigma 1,bidefinable functions of V i12 and U i2 ..."
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Cited by 18 (4 self)
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This paper investigates the provably total functions of fragments of first and secondorder Bounded Arithmetic. The (strongly) \Sigma bidefinable functions of Si13 and Ri3 are precisely the (strong) FP\Sigma p i13 [wit, logO(1)] functions. The \Sigma 1,bidefinable functions of V i12 and U i2 are the EXPTIME\Sigma 1,p i1[wit, poly] functions and the \Sigma 1,bi definable functions of V i2 are the EXPTIME\Sigma 1,p ifunctions. We give witnessing theorems for these theories and prove conservation results for Ri3 over Si13 and for U i2 over V i12.
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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Cited by 6 (0 self)
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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.
Approximate counting by hashing in bounded arithmetic
, 2008
"... We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the b ..."
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Cited by 4 (0 self)
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We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomialtime hierarchy.
On independence of variants of the weak pigeonhole principle
, 2007
"... The principle sPHP a b (P V (α)) states that no oracle circuit can compute a surjection of a onto b. We show that sPHP ϱ(a) π(a) P (a) (P V (α)) is independent of P V1(α)+sPHP Π(a) (P V (α)) for various choices of the parameters π, Π, ϱ, P. We also improve the known separation of iWPHP(P V) from S 1 ..."
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Cited by 3 (1 self)
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The principle sPHP a b (P V (α)) states that no oracle circuit can compute a surjection of a onto b. We show that sPHP ϱ(a) π(a) P (a) (P V (α)) is independent of P V1(α)+sPHP Π(a) (P V (α)) for various choices of the parameters π, Π, ϱ, P. We also improve the known separation of iWPHP(P V) from S 1 2 + sWPHP(P V) under cryptographic assumptions.
Combinatorics in Bounded Arithmetics
, 2004
"... A basic aim of logic is to consider what axioms are used in proving various theorems of mathematics. This thesis will be concerned with such issues applied to a particular area of mathematics: combinatorics. We will consider two widely known groups of proof methods in combinatorics, namely, probabil ..."
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Cited by 2 (0 self)
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A basic aim of logic is to consider what axioms are used in proving various theorems of mathematics. This thesis will be concerned with such issues applied to a particular area of mathematics: combinatorics. We will consider two widely known groups of proof methods in combinatorics, namely, probabilistic methods and methods using linear algebra. We will consider certain applications of such methods, both of which are significant to Ramsey theory. The systems we choose to work in are various theories of bounded arithmetic. For the probabilistic method, the key point is that we use the weak pigeonhole principle to simulate the probabilistic reasoning. We formalize various applications of the ordinary probabilistic method and linearity of expectations, making partial progress on the Local Lemma. In the case of linearity of expectations, we show how to eliminate the weak pigeonhole principle by simulating the derandomization technique of “conditional probabilities.” We consider linear algebra methods applied to various set system theorems. We formalize some theorems using a linear algebra principle as an extra axiom. We also show how weaker results can be attained by giving alternative proofs that avoid linear algebra, and thus also avoid the extra axiom. We formalize upper and lower Ramsey bounds. For the lower bounds, both the probabilistic methods and the linear algebra methods are used. We provide a stratification of the various Ramsey lower bounds, showing that stronger bounds can be proved in stronger theories. A natural question is whether or not the axioms used are necessary. We provide “reversals” in a few cases, showing that the principle used to prove the theorem is in fact a consequence of the theorem (over some base theory). Thus this work can be seen as a (humble) beginning in the direction of developing the Reverse Mathematics of finite combinatorics.