Nimber-Preserving Reductions and Homomorphic Sprague-Grundy Game Encodings
This work addresses foundational issues in combinatorial game theory by introducing new reduction concepts and completeness results, with potential applications in cryptography and algorithm design, though it is incremental in extending traditional computational characterizations.
The paper tackles the problem of nimber-preserving reductions in impartial combinatorial games, proving that Generalized Geography is Sprague-Grundy-complete for a class of polynomially-short impartial rulesets, and establishes a homomorphic theorem enabling polynomial-time construction of prime games with nimbers equal to the XOR of given games' nimbers.
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($G_2$).