Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess is Hard

Josh Brunner; Erik D. Demaine; Dylan Hendrickson; Julian Wellman

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arXiv · 2010.09271

Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess is Hard

Josh Brunner·Erik D. Demaine·Dylan Hendrickson·Julian Wellman

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Abstract

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.

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Josh Brunner, Erik D. Demaine, Dylan Hendrickson, Julian Wellman. 2020-10-19. Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess is Hard. https://arxiv.org/abs/2010.09271

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