Archive for Mathematical Logic

, Volume 54, Issue 3–4, pp 359–394 | Cite as

Open induction in a bounded arithmetic for TC0

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Abstract

The elementary arithmetic operations \({+,\cdot,\le}\) on integers are well-known to be computable in the weak complexity class TC0, and it is a basic question what properties of these operations can be proved using only TC0-computable objects, i.e., in a theory of bounded arithmetic corresponding to TC0. We will show that the theory VTC0 extended with an axiom postulating the totality of iterated multiplication (which is computable in TC0) proves induction for quantifier-free formulas in the language \({\langle{+,\cdot,\le}\rangle}\) (IOpen), and more generally, minimization for \({\Sigma_{0}^{b}}\) formulas in the language of Buss’s S2.

Keywords

Bounded arithmetic Open induction Threshold circuits Valued fields Real-closed fields 

Mathematics Subject Classification

03F20 03F30 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of MathematicsAcademy of Sciences of the Czech RepublicPraha 1Czech Republic

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