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Nimber-preserving reduction: Game secrets and homomorphic Sprague-Grundy theorem.

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Title: Nimber-preserving reduction: Game secrets and homomorphic Sprague-Grundy theorem.
Authors: Burke, Kyle1 (AUTHOR) kburke@flsouthern.edu, Ferland, Matthew1,2 (AUTHOR) mferland@usc.edu, Teng, Shang-hua1,2 (AUTHOR) shanghua@usc.edu
Source: Theoretical Computer Science. Jul2024, Vol. 1005, pN.PAG-N.PAG. 1p.
Subjects: Polynomial time algorithms, Homomorphisms, Cryptography, Game theory, Games
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 G 1 , G 2 in I P , a Generalized Geography game G satisfying: nimber (G) = nimber (G 1) ⊕ nimber (G 2). [ABSTRACT FROM AUTHOR]
Copyright of Theoretical Computer Science is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 178022034
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  Data: Nimber-preserving reduction: Game secrets and homomorphic Sprague-Grundy theorem.
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  Data: <searchLink fieldCode="AR" term="%22Burke%2C+Kyle%22">Burke, Kyle</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> kburke@flsouthern.edu</i><br /><searchLink fieldCode="AR" term="%22Ferland%2C+Matthew%22">Ferland, Matthew</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> mferland@usc.edu</i><br /><searchLink fieldCode="AR" term="%22Teng%2C+Shang-hua%22">Teng, Shang-hua</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> shanghua@usc.edu</i>
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  Data: <searchLink fieldCode="JN" term="%22Theoretical+Computer+Science%22">Theoretical Computer Science</searchLink>. Jul2024, Vol. 1005, pN.PAG-N.PAG. 1p.
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  Data: <searchLink fieldCode="DE" term="%22Polynomial+time+algorithms%22">Polynomial time algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Homomorphisms%22">Homomorphisms</searchLink><br /><searchLink fieldCode="DE" term="%22Cryptography%22">Cryptography</searchLink><br /><searchLink fieldCode="DE" term="%22Game+theory%22">Game theory</searchLink><br /><searchLink fieldCode="DE" term="%22Games%22">Games</searchLink>
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  Data: 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 G 1 , G 2 in I P , a Generalized Geography game G satisfying: nimber (G) = nimber (G 1) ⊕ nimber (G 2). [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Theoretical Computer Science is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.tcs.2024.114636
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      – Code: eng
        Text: English
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      – SubjectFull: Polynomial time algorithms
        Type: general
      – SubjectFull: Homomorphisms
        Type: general
      – SubjectFull: Cryptography
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      – SubjectFull: Game theory
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      – SubjectFull: Games
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            NameFull: Ferland, Matthew
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              M: 07
              Text: Jul2024
              Type: published
              Y: 2024
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