2009/08/24 by Alan Guo, Ezra Miller, Guo, Alan +1 · 1 citation
Computer Science · Mathematics · #05A15 (Primary) #05E40 #06F05 #20M14 #52B20 #68W30 (Secondary) #91A05 #91A46 #Advanced Optimization Algorithms Research #Artificial Intelligence in Games #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.AC #math.CO #msc:05A15 #msc:05E40 #msc:06F05 #msc:20M14 #msc:52B20 #msc:68W30 #msc:91A05 #msc:91A46
paper · pdf · doi:10.48550/arxiv.0908.3473
18 pages, no figures
arxiv created 2009/08/24 · openalex publication_date 2009/08/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these lattice games can be made particularly efficient for octal games, which we generalize to squarefree games. These additionally encompass all heap games in a natural setting, in which the Sprague-Grundy theorem for normal play manifests itself geometrically. We provide an algorithm to compute normal play strategies. The setting of lattice games naturally allows for mis`ere play, where 0 is declared a losing position. Lattice games also allow situations where larger finite sets of positions are declared losing. Generating functions for sets of winning positions provide data structures for strategies of lattice games. We conjecture that every lattice game has a rational strategy: a rational generating function for its winning positions. Additionally, we conjecture that every lattice game has an affine stratification: a partition of its set of winning positions into a finite disjoint union of finitely generated modules for affine semigroups.