The concept of nimbers--a.k.a. Grundy-values or nim-values--is fundamental to combinatorial game theory. Nimbers provide a complete characterization of strategic interactions among impartial games in their disjunctive sums as well as the winnability. In this paper, we initiate a study of nimber-preserving reductions among impartial games. These reductions enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, $\cal{I}^P$ , of polynomially-short impartial rulesets under nimber-preserving reductions, a property we refer to as Sprague-Grundy-complete. In contrast, we also show that not every PSPACE-complete ruleset in $\cal{I}^P$ is Sprague-Grundy-complete for $\cal{I}^P$ . By considering every impartial game as an encoding of its nimber, our technical result establishes the following striking cryptography-inspired homomorphic theorem: Despite the PSPACE-completeness of nimber computation for $\cal{I}^P$ , there exists a polynomial-time algorithm to construct, for any pair of games $G_1$, $G_2$ of $\cal{I}^P$ , a prime game (i.e. a game that cannot be written as a sum) $H$ of $\cal{I}^P$ , satisfying: nimber($H$) = nimber($G_1$) $\oplus$ nimber($G_2$).

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Nimber-Preserving Reductions 和同态 Sprague-Grundy 游戏编码

nimbers 的概念——又名 Grundy 值或 nim 值——是组合博弈论的基础。Nimbers 提供了公平游戏之间战略互动的完整特征，包括它们的分离总和以及可赢性。在本文中，我们启动了一项关于公平游戏中的保数减少的研究。这些减少增强了组合游戏的传统计算特征中的可赢性减少减少。我们证明了广义地理对于自然类 $\cal{I}^P$ 是完备的，它是多项式短公正规则集在保数约简下，我们称之为 Sprague-Grundy-complete 的性质。相比之下，我们还表明，并非 $\cal{I}^P$ 中的每个 PSPACE-complete 规则集都是 $\cal{I}^P$ 的 Sprague-Grundy-complete。