Nimber-Preserving Reductions and Homomorphic Sprague-Grundy Game Encodings

Abstract: 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, $\cal{I}<sup>P$ , 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 $\cal{I}<sup>P$ is Sprague-Grundy-complete for $\cal{I}<sup>P$. 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 $\cal{I}<sup>P$ , there exists a polynomial-time algorithm to construct, for any pair of games $G_1$, $G_2$ of $\cal{I}<sup>P$ , a prime game (i.e. a game that cannot be written as a sum) $H$ of $\cal{I}<sup>P$ , satisfying: nimber($H$) = nimber($G_1$) $\oplus$ nimber($G_2$).