Nimber-Preserving Reduction: Game Secrets And Homomorphic Sprague-Grundy Theorem

Authors Kyle W. Burke, Matthew Ferland, Shang-Hua Teng

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Author Details

Kyle W. Burke
  • Department of Computer Science, Plymouth State University, NH, USA
Matthew Ferland
  • Department of Computer Science, University of Southern California, Los Angeles, CA, USA
Shang-Hua Teng
  • Department of Computer Science, University of Southern California, Los Angeles, CA, USA

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Kyle W. Burke, Matthew Ferland, and Shang-Hua Teng. Nimber-Preserving Reduction: Game Secrets And Homomorphic Sprague-Grundy Theorem. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The concept of nimbers - a.k.a. Grundy-values or nim-values - is fundamental to combinatorial game theory. Beyond the winnability, nimbers provide a complete characterization of strategic interactions among impartial games in disjunctive sums. In this paper, we consider nimber-preserving reductions among impartial games, which 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 polynomial-time nimber-preserving reductions. We refer to this notion of completeness as Sprague-Grundy-completeness. In contrast, we also show that not every PSPACE-complete ruleset in ℐ^P is Sprague-Grundy-complete for ℐ^P. By viewing every impartial game as an encoding of its nimber - a succinct game secret richer than its winnability alone - 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 G₁, G₂ in ℐ^P, a Generalized Geography game G satisfying: nimber(G) = nimber(G₁) ⊕ nimber(G₂).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Combinatorial Games
  • Nim
  • Generalized Geography
  • Sprague-Grundy Theory
  • Grundy value
  • Computational Complexity
  • Functional-Preserving Reductions


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