无限国际象棋的n人配对问题是可判定的
The mate-in-n problem of infinite chess is decidable
摘要：
无限国际象棋是在无限无边的棋盘上下的国际象棋。熟悉的棋子按照他们通常的国际象棋规则移动，每个玩家都努力将对方的国王将死。无限国际象棋中的n个同伴问题（mate-in-n problem）是确定一个指定的玩家是否能在最多n步的情况下从给定的有限位置获胜的问题。这个问题的一个朴素的公式导致了具有2n个交替量词的高算术复杂性的断言--有一个白的移动，这样对于每个黑的回答，有一个白的反移动，等等。在这样的公式中，这个问题似乎是不可判定的;即使到有限的深度，人们也不能期望搜索无限分支的对策树。然而，本文的主要定理，证实了第一作者和C。D. A. Evans，建立了无限国际象棋的mate-in-n问题是可计算判定的，在位置和n中是一致的。此外，有一个可计算的策略，从这样的队友在n个位置的最佳发挥。证明过程表明，mate-in-n问题可以用我们称之为国际象棋的一阶结构来表达，我们证明（在相关片段中）这是一个自动结构，因此其理论是可判定的。事实上，它在Presburger算术中是可以定义的。不幸的是，这种n人配对问题的解决方案似乎并没有解决更一般的获胜位置问题的可判定性，即确定指定玩家是否具有来自给定位置的获胜策略的问题，因为位置可以允许获胜策略，而无需对所需的移动次数进行任何限制。这个问题与无限国际象棋中的超限博弈值有关，而国际象棋的欧米伽1的确切值是未知的。
英文摘要：
Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-n problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. A naive formulation of this problem leads to assertions of high arithmetic complexity with 2n alternating quantifiers---there is a move for white, such that for every black reply, there is a counter-move for white, and so on. In such a formulation, the problem does not appear to be decidable; and one cannot expect to search an infinitely branching game tree even to finite depth. Nevertheless, the main theorem of this article, confirming a conjecture of the first author and C. D. A. Evans, establishes that the mate-in-n problem of infinite chess is computably decidable, uniformly in the position and in n. Furthermore, there is a computable strategy for optimal play from such mate-in-n positions. The proof proceeds by showing that the mate-in-n problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. Indeed, it is definable in Presburger arithmetic. Unfortunately, this resolution of the mate-in-n problem does not appear to settle the decidability of the more general winning-position problem, the problem of determining whether a designated player has a winning strategy from a given position, since a position may admit a winning strategy without any bound on the number of moves required. This issue is connected with transfinite game values in infinite chess, and the exact value of the omega one of chess is not known.
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。
