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Open induction in a bounded arithmetic for TC^0 (1404.7435v2)
Published 29 Apr 2014 in cs.LO and math.LO
Abstract: The elementary arithmetic operations $+,\cdot,\le$ on integers are well-known to be computable in the weak complexity class $\mathrm{TC}0$, and it is a basic question what properties of these operations can be proved using only $\mathrm{TC}0$-computable objects, i.e., in a theory of bounded arithmetic corresponding to $\mathrm{TC}0$. We will show that the theory $\mathit{VTC}0$ extended with an axiom postulating the totality of iterated multiplication (which is computable in $\mathrm{TC}0$) proves induction for quantifier-free formulas in the language $\langle +,\cdot,\le \rangle$ (IOpen), and more generally, minimization for $\Sigmab_0$ formulas in the language of Buss's $S_2$.