We study algorithmic complexity of solving subtraction games in a~fixed dimension with a finite difference set. We prove that there exists a game in this class such that any algorithm solving the game runs in exponential time. Also we prove an existence of a game in this class such that solving the game is PSPACE-hard. The results are based on the construction introduced by Larsson and W\"astlund. It relates subtraction games and cellular automata.

中文翻译：

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多维减法游戏的计算难度

我们研究了用有限差分集在固定维度上解决减法游戏的算法复杂性。我们证明在这个类中存在一个游戏，使得任何解决游戏的算法都在指数时间内运行。我们还证明了此类游戏的存在性，使得解决该游戏是 PSPACE 难的。结果基于 Larsson 和 W\"astlund 引入的构造。它涉及减法游戏和元胞自动机。