We will give a two-sort system which axiomatizes winning strategies for the combinatorial game Node Kayles. It is shown that our system captures alternating polynomial time reasonings in the sense that the provably total functions of the theory corresponds to those computable in APTIME. We will also show that our system is equivalently axiomatized by Sprague–Grundy theorem which states that any Node Kayles position is provably equivalent to some NIM heap.

中文翻译：

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有界算术中的 Sprague-Grundy 理论

我们将给出一个二分类系统，它公理化了组合游戏 Node Kayles 的获胜策略。结果表明，在理论的可证明总函数对应于 APTIME 中可计算的函数的意义上，我们的系统捕获了交替多项式时间推理。我们还将证明我们的系统是由 Sprague-Grundy 定理等价公理化的，该定理指出任何节点 Kayles 位置都可以证明等价于一些 NIM 堆。