Archive for Mathematical Logic

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

Open induction in a bounded arithmetic for TC0

  • Emil Jeřábek


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 VTC 0 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 S 2.


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

Mathematics Subject Classification

03F20 03F30 


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© 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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