Nimber-Preserving Reductions and Homomorphic Sprague-Grundy Game Encodings
(2109.05622)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, $\cal{I}P$ , 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 $G1$, $G2$ 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($G1$) $\oplus$ nimber($G2$).
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