Iterated multiplication in $VTC^0$

Abstract

We show that $VTC^0$, the basic theory of bounded arithmetic corresponding to the complexity class $\mathrm{TC}^0$, proves the $IMUL$ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the $\mathrm{TC}^0$ iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, $VTC^0$ can also prove the integer division axiom, and (by our previous results) the RSUV-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories $\Delta^b_1$-$CR$ and $C^0_2$. As a side result, we also prove that there is a well-behaved $\Delta_0$ definition of modular powering in $I\Delta_0+WPHP(\Delta_0)$.