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The strength of replacement in weak arithmetic
 Proceedings of the Nineteenth Annual IEEE Symposium on Logic in Computer Science
, 2004
"... The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<a  ∃x
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The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<a  ∃x<aφ(i,x) → ∃w ∀i<aφ(i,[w]i) proves the scheme where φ is in the class Γ of formulas. The theory S1 2 BB(Σb 1), and thus in S1 2 every Σb1 formula is equivalent to a strict Σb1 formula (in which all nonsharplybounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S1 2 do not prove either BB(Σb1) or BB(Σb0). We show (unconditionally) that V 0 does not prove BB(ΣB 0), where V 0 (essentially IΣ 1,b 0) is the twosorted theory associated with the complexity class AC0. We show that PV does not prove BB(Σb 0), assuming
Theory for TC 0 and Other Small Complexity Classes
 Logical Methods in Computer Science
, 2005
"... Abstract We present a general method for introducing finitely axiomatizable "minimal " secondorder theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the firstorder theory \Delta b ..."
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Abstract We present a general method for introducing finitely axiomatizable "minimal " secondorder theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the firstorder theory \Delta b
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
The Riemann Integral in Weak Systems of Analysis
"... Abstract: Taking as a starting point (a modification of) a weak theory of arithmetic of Jan Johannsen and Chris Pollett (connected with the hierarchy of counting functions), we introduce successively stronger theories of bounded arithmetic in order to set up a system for analysis (TCA 2). The extend ..."
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Abstract: Taking as a starting point (a modification of) a weak theory of arithmetic of Jan Johannsen and Chris Pollett (connected with the hierarchy of counting functions), we introduce successively stronger theories of bounded arithmetic in order to set up a system for analysis (TCA 2). The extended theories preserve the connection with the counting hierarchy in the sense that the algorithms which the systems prove to halt are exactly the ones in the hierarchy. We show that TCA 2 has the exact strength to develop Riemannian integration for functions with a modulus of uniform continuity.
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
Root finding with threshold circuits
, 2011
"... We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials—given by a list of coefficients in binary—can be computed to a given accuracy by a uniform TC 0 algorithm (a uniform family of constantdepth polynomialsize threshold circuits). The bas ..."
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We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials—given by a list of coefficients in binary—can be computed to a given accuracy by a uniform TC 0 algorithm (a uniform family of constantdepth polynomialsize threshold circuits). The basic idea is to compute the inverse function of the polynomial by a power series. We also discuss an application to the theory VTC 0 of bounded arithmetic.
Open induction in a TC 0 arithmetic Emil Jeˇrábek
"... Correspondence of theories of bounded arithmetic T and computational complexity classes C: Provably total computable functions of T are Cfunctions T can do reasoning using Cpredicates (comprehension, induction,...) Feasible reasoning: Given a natural concept P ∈ C, what can we prove about P using ..."
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Correspondence of theories of bounded arithmetic T and computational complexity classes C: Provably total computable functions of T are Cfunctions T can do reasoning using Cpredicates (comprehension, induction,...) Feasible reasoning: Given a natural concept P ∈ C, what can we prove about P using only concepts from C? That is: what does T prove about P? Our P: elementary integer arithmetic operations +, ·, ≤