A Note on the Computational Complexity of Selfmate and Reflexmate Chess Problems

TL;DR

This paper proves that determining selfmate, reflexmate, and semi-reflexmate chess problems is computationally very complex, specifically EXPTIME-complete, by adapting existing complexity reductions.

Contribution

It establishes the EXPTIME-completeness of selfmate, reflexmate, and semi-reflexmate chess problems through a modified reduction approach.

Findings

Selfmate, reflexmate, and semi-reflexmate are all EXPTIME-complete.

The complexity reduction is adapted from existing EXPTIME-hardness proofs.

These problems are computationally very challenging to solve in general.

Abstract

A selfmate is a Chess problem in which White, moving first, needs to force Black to checkmate within a specified number of moves. The reflexmate is a derivative of the selfmate in which White compels Black to checkmate with the added condition that if either player can checkmate, they must do that (when this condition applies only to Black, it is a semi-reflexmate). We slightly modify the reduction of EXPTIME-hardness of Chess and apply the reduction to these Chess problems. It is proved that selfmate, reflexmate, and semi-reflexmate are all EXPTIME-complete.