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A ModelTheoretic Property of Sharply Bounded Formulae, with some Applications
, 1998
"... We define a property of substructures of models of arithmetic, that of being lengthinitial , and show that sharply bounded formulae are absolute between a model and its lengthinitial submodels. We use this to prove independence results for some weak fragments of bounded arithmetic by construct ..."
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We define a property of substructures of models of arithmetic, that of being lengthinitial , and show that sharply bounded formulae are absolute between a model and its lengthinitial submodels. We use this to prove independence results for some weak fragments of bounded arithmetic by constructing appropriate models as lengthinitial submodels of some given model. Mathematics Subject Classification: 03F30, 03H15 Introduction First we review the definitions of the theories S i 2 and T i 2 of Bounded Arithmetic introduced by S. Buss [2]: The language of these theories is the language of Peano Arithmetic extended by symbols for the functions b 1 2 xc, jxj := dlog 2 (x + 1)e and x#y := 2 jxj\Deltajyj . A quantifier of the form 8x t , 9x t with x not occurring in t is called a bounded quantifier. Furthermore, a quantifier of the form 8x jtj, 9x jtj is called sharply bounded. A formula is called sharply bounded if all quantifiers in it are sharply bounded. The class of sh...
On the b 1 bitcomprehension rule
 Logic Colloquium 98
, 2000
"... Summary. The theory � b 1CR of Bounded Arithmetic axiomatized by the � b 1bitcomprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1definable functions of � b 1CR are those in uniform TC 0, and the � b 2definable functions are computable by counte ..."
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Summary. The theory � b 1CR of Bounded Arithmetic axiomatized by the � b 1bitcomprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1definable functions of � b 1CR are those in uniform TC 0, and the � b 2definable functions are computable by counterexample computations using TC 0functions. The latter is used to show that a collapse of stronger theories to � b 1CR implies that NP is contained in nonuniform TC 0. 1
Divide and Conquer in Parallel Complexity and Proof Theory
, 1992
"... Copyright Stephen Austin Bloch, 1992 All rights reserved. The dissertation of Stephen Bloch is approved, and it is acceptable in quality and form for publication on micro ..."
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Copyright Stephen Austin Bloch, 1992 All rights reserved. The dissertation of Stephen Bloch is approved, and it is acceptable in quality and form for publication on micro
On the $\Delta^b_1$BitComprehension Rule
, 2000
"... Introduction The \Delta b 1 bitcomprehension rule roughly states the following: Given n and a predicate A(x) that has been proven to be \Delta b 1 , i.e., equivalent to both an NP  (\Sigma b 1 ) and a coNP  (\Pi b 1 ) predicate, there is a number w of length n such that for every i ..."
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Introduction The \Delta b 1 bitcomprehension rule roughly states the following: Given n and a predicate A(x) that has been proven to be \Delta b 1 , i.e., equivalent to both an NP  (\Sigma b 1 ) and a coNP  (\Pi b 1 ) predicate, there is a number w of length n such that for every i ! n, the ith bit of w is set if and only if A(i) holds. One can think of w as coding the set of small i such that A(i) holds. We consider the theory of Bounded Arithmetic \Delta b 1 CR that has this rule as its main axiom. This theory is re