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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 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.

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  1. Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Praha 1, Czech Republic

    Emil Jeřábek

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  1. Emil Jeřábek
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Jeřábek, E. Open induction in a bounded arithmetic for TC0 . Arch. Math. Logic 54, 359–394 (2015). https://doi.org/10.1007/s00153-014-0414-7

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  • DOI: https://doi.org/10.1007/s00153-014-0414-7

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