Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess Is Hard
We prove PSPACE-completeness of two classic types of Chess problems when generalized to n × 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.
hardness
board games
PSPACE
Theory of computation~Problems, reductions and completeness
17:1-17:14
Regular Paper
The full version of this paper is available at https://arxiv.org/abs/2010.09271.
This work was initiated during open problem solving in the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.892) in Spring 2019. We thank the other participants of that class - in particular, John Urschel - for related discussions and providing an inspiring atmosphere.
Josh
Brunner
Josh Brunner
Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA
Erik D.
Demaine
Erik D. Demaine
Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA
http://erikdemaine.org/
https://orcid.org/0000-0003-3803-5703
Dylan
Hendrickson
Dylan Hendrickson
Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA
https://orcid.org/0000-0002-9967-8799
Julian
Wellman
Julian Wellman
Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA
10.4230/LIPIcs.ISAAC.2020.17
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Josh Brunner, Erik D. Demaine, Dylan Hendrickson, and Julian Wellman
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