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

TL;DR

This paper proves that determining the validity of retrograde and helpmate chess problems on large boards is computationally very hard, specifically PSPACE-complete, highlighting the complexity of even cooperative chess scenarios.

Contribution

It establishes PSPACE-completeness for classic retrograde and helpmate chess problems on n-by-n boards, extending the understanding of their computational complexity.

Findings

Retrograde problems are PSPACE-complete.

Helpmate problems are PSPACE-complete.

Complexity results are derived via reductions from Subway Shuffle.

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.