2026-07-26T05:32:01Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15980 2024-03-06T10:57:00Z ddc:004 open_access

Nimber-Preserving Reduction: Game Secrets And Homomorphic Sprague-Grundy Theorem Burke, Kyle W. Ferland, Matthew Teng, Shang-Hua Combinatorial Games Nim Generalized Geography Sprague-Grundy Theory Grundy value Computational Complexity Functional-Preserving Reductions 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). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Kyle W. Burke and Matthew Ferland and Shang-Hua Teng 2022 Is Part Of LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FUN.2022.10 urn:nbn:de:0030-drops-159808 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.10 eng https://creativecommons.org/licenses/by/4.0/legalcode