arXiv:1302.4377math.LO
Transfinite game values in infinite chess
Early discovery. Linked by one identifiable commenter. This paper is outside the main ranking; a mention is not an endorsement.
Abstract
We investigate the transfinite game values arising in infinite chess, providing both upper and lower bounds on the supremum of these values---the omega one of chess---with two senses depending on whether one considers only finite positions or also positions with infinitely many pieces. For lower bounds, we present specific infinite positions with transfinite game values of omega, omega^2, omega^2 times k, and omega^3. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, true omega one.
If you're like me and got very excited by teasers like this: > Another interesting thing we noticed is that there is a computable position in infinite chess, such that ... white has a computable strategy defeating any computable strategy of black—but in the category of arbitrary…
cvossView on HNParent context