Iterated multiplication in VTC^0

11/05/2020 ∙ by Emil Jeřábek, et al. ∙ 0

We show that VTC^0, the basic theory of bounded arithmetic corresponding to the complexity class TC^0, proves the IMUL axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the TC^0 iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, VTC^0 can also prove the integer division axiom, and (by results of Jeřábek) the RSUV-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories Δ^b_1-CR and C^0_2. As a side result, we also prove that there is a well-behaved Δ_0 definition of modular powering in IΔ_0+WPHP(Δ_0).

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