$VTC^0$中的迭代乘法
Iterated multiplication in $VTC^0$
摘要：
我们表明，$VTC^0$，有界算术的基本理论对应的复杂性类$\mathrm{TC}^0$，证明了$IMUL$公理表示的总体迭代乘法满足其递归定义，形式化一个合适的版本的$\mathrm{TC}^0$迭代乘法算法的Hesse，Allender，和巴林顿。因此，$VTC^0$也可以证明整数除法公理，并（通过我们以前的结果）RSUV-翻译的归纳法和极小化的急剧有界的公式。类似的结果也适用于相关的理论$Δ^b_1$ - $CR$和$C^0_2$。作为一个副作用，我们还证明了有一个良好的$Δ_0$定义的模块供电$IΔ_0+WPHP(Δ_0)$。
英文摘要：
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 $Δ^b_1$-$CR$ 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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