Magic: the Gathering is a popular and famously complicated card game about magical combat. Recently, several authors including Chatterjee and Ibsen-Jensen (2016) and Churchill, Biderman, and Herrick (2019) have investigated the computational complexity of playing Magic optimally. In this paper we show that the ``mate-in-$n$'' problem for Magic is $\Delta^0_n$-hard and that optimal play in two-player Magic is non-arithmetic in general. These results apply to how real Magic is played, can be achieved using standard-size tournament legal decks, and do not rely on stochasticity or hidden information. Our paper builds upon the construction that Churchill, Biderman, and Herrick (2019) used to show that this problem was at least as hard as the halting problem.

中文翻译：

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魔术：聚会和算术一样难

Magic: the Gathering 是一款流行的、以复杂的魔法战斗着称的纸牌游戏。最近，包括 Chatterjee 和 Ibsen-Jensen（2016 年）以及 Churchill、Biderman 和 Herrick（2019 年）在内的几位作者研究了以最佳方式玩万智牌的计算复杂性。在这篇论文中，我们展示了万智牌的“mate-in-$n$”问题是 $\Delta^0_n$-hard 并且两人万智牌中的最佳打法通常是非算术的。这些结果适用于真正的万智牌是如何玩的，可以使用标准大小的锦标赛合法套牌来实现，并且不依赖于随机性或隐藏信息。我们的论文建立在 Churchill、Biderman 和 Herrick（2019）用来表明这个问题至少与停机问题一样困难的结构之上。