We prove PSPACE-completeness of two classic types of Chess problems when generalized to n-by-n boards. A "retrograde" problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is "valid" or "legal" or "reachable". Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A "helpmate" problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle.

中文翻译：

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逆行和帮友棋问题的复杂性：即使是合作棋也很难

当推广到 n×n 棋盘时，我们证明了两种经典类型的国际象棋问题的 PSPACE 完备性。“逆行”问题询问是否有可能从自然起始位置到达某个位置，即该位置是否“有效”或“合法”或“可达”。大多数现实世界的逆行国际象棋问题都要求这样一个序列的最后几步；我们分析了存在指数长移动序列的决策问题。一个“帮手”问题询问玩家是否有可能被来自给定位置的任何移动序列所击倒。帮手问题本质上是一种合作形式的国际象棋，双方玩家共同努力使特定玩家获胜；它也出现在常规的国际象棋游戏中，用完时间（标记）的玩家只有在他们可能从当前位置被将死时才会输（即，帮助问题有解决方案）。我们的 PSPACE 硬度降低来自名为 Subway Shuffle 的益智游戏的变体。