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, 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 I P is Sprague-Grundy-complete for 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 I P , there exists a polynomial-time algorithm to construct, for any pair of games G 1 , G 2 in I P , a prime game (i.e. a game that cannot be written as a sum) G of I P , satisfying: nimber(G) = nimber(G 1 ) ⊕ nimber(G 2 ).