License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FUN.2022.10
URN: urn:nbn:de:0030-drops-159808
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15980/
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Burke, Kyle W. ; Ferland, Matthew ; Teng, Shang-Hua

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

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Abstract

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).

BibTeX - Entry

@InProceedings{burke_et_al:LIPIcs.FUN.2022.10,
  author =	{Burke, Kyle W. and Ferland, Matthew and Teng, Shang-Hua},
  title =	{{Nimber-Preserving Reduction: Game Secrets And Homomorphic Sprague-Grundy Theorem}},
  booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-232-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{226},
  editor =	{Fraigniaud, Pierre and Uno, Yushi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15980},
  URN =		{urn:nbn:de:0030-drops-159808},
  doi =		{10.4230/LIPIcs.FUN.2022.10},
  annote =	{Keywords: Combinatorial Games, Nim, Generalized Geography, Sprague-Grundy Theory, Grundy value, Computational Complexity, Functional-Preserving Reductions}
}

Keywords: Combinatorial Games, Nim, Generalized Geography, Sprague-Grundy Theory, Grundy value, Computational Complexity, Functional-Preserving Reductions
Collection: 11th International Conference on Fun with Algorithms (FUN 2022)
Issue Date: 2022
Date of publication: 23.05.2022

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