从大量的匹配到可计算性的极限
From heaps of matches to the limits of computability
摘要：
我们研究所谓的不变的游戏玩一个固定数量的$d$堆的比赛。一个博弈由一个长度为$d$的整数向量的有限列表$\mathcal{M}$来描述，该列表指定了合法的移动。一步棋是通过增加$\mathcal{M}$中的一个向量来改变当前的游戏状态，前提是得到的向量的所有元素都是非负的。例如，在一个双堆游戏中，向量$(1,-2)$意味着将一个匹配添加到第一个堆中，并从第二个堆中删除两个匹配。如果为$(1,-2) \in \mathcal{M}$，则允许这样的移动，前提是第二个堆中至少有两个匹配项。两个玩家轮流，一个玩家不能移动就输了。我们表明，这些游戏拥抱计算的普遍性，因此，一些关于他们的基本问题是算法不可判定的。特别是，我们证明，有没有算法，两个游戏$\mathcal{M}$和$\mathcal{M}'$（具有相同数量的堆）作为输入，并确定它们是否是等价的意义上说，每一个开始的位置，这是第一个玩家赢的游戏之一是第一个玩家赢的其他。
英文摘要：
We study so-called invariant games played with a fixed number $d$ of heaps of matches. A game is described by a finite list $\mathcal{M}$ of integer vectors of length $d$ specifying the legal moves. A move consists in changing the current game-state by adding one of the vectors in $\mathcal{M}$, provided all elements of the resulting vector are nonnegative. For instance, in a two-heap game, the vector $(1,-2)$ would mean adding one match to the first heap and removing two matches from the second heap. If $(1,-2) \in \mathcal{M}$, such a move would be permitted provided there are at least two matches in the second heap. Two players take turns, and a player unable to make a move loses. We show that these games embrace computational universality, and that therefore a number of basic questions about them are algorithmically undecidable. In particular, we prove that there is no algorithm that takes two games $\mathcal{M}$ and $\mathcal{M}'$ (with the same number of heaps) as input, and determines whether or not they are equivalent in the sense that every starting-position which is a first player win in one of the games is a first player win in the other.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
