Universality of Infinite Chess
Matthew Bolan, Andreas Tsevas
TL;DR
The work proves that infinite chess on $Z^2$ with infinitely many pieces is as expressive as any open Gale-Stewart game with draws, via explicit computable embeddings. It provides two construction schemes: a bishop-tree embedding that simulates arbitrary open games, and a pawn-based construction that realizes all countable ordinals with only kings and pawns. Consequently, every countable ordinal appears as a game value in infinite chess, and the omega-one of computable positions equals the Church-Kleene ordinal $oldsymbol{ ext{ω}_1^{ ext{CK}}}$; hyperarithmetic phenomena transfer to chess strategies as well. A robust variant shows that even under short-range movement and restricted material, all countable ordinals still arise as game values, demonstrating the deep, unavoidable role of non-local forcing (zugzwang) in achieving transfinite values.
Abstract
We prove that chess played on the infinite chessboard $\mathbb{Z}^2$ with infinitely many pieces is as powerful as it could possibly be, by showing that every open Gale-Stewart game with draws is strategically equivalent to some infinite chess position and vice versa. As our construction is computable and open Gale-Stewart games are well understood, this allows us to resolve many open questions about the complexity of infinite chess with infinitely many pieces. In particular, all countable ordinals arise as the game value of some such chess position. We also give an alternate construction that realizes all countable ordinals as game values, with the pleasing property that it consists only of the king pair and pawns.