The omega one of chess
Infinite chess is chess played on an infinite edgeless chessboard. Since checkmates, when they occur, do so after finitely many moves, this is an open game and therefore subject to the theory of transfinite game values for open games.
Specifically, the game value (for white) of a position in infinite chess is defined by recursion. The positions with value $0$ are precisely those in which white has already won. If a position $p$ has white to move, then the value of $p$ is $\alpha+1$ if and only if $\alpha$ is minimal such that white may legally move from $p$ to a position with value $\alpha$. If a position $p$ has black to play, where black has a legal move from $p$, and every move by black from $p$ has a value, then the value of $p$ is the supremum of these values.
The term omega one of chess refers either to the ordinal $\omega_1^{\mathfrak{Ch}}$ or to $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, depending respectively on whether one is considering only finite positions or also positions with infinitely many pieces.[1]
- The ordinal $\omega_1^{\mathfrak{Ch}}$ is the supremum of the game values of the winning finite positions for white in infinite chess.
- The ordinal $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$ is the supremum of the game values of all the winning positions for white in infinite chess, including positions with infinitely many pieces.
- Similarly, $\omega_1^{\mathfrak{Ch}_3}$ and $\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}$ are the analogous ordinals for infinite three-dimensional chess, as described in .
There is an entire natural hierarchy of intermediate concepts between $\omega_1^{\mathfrak{Ch}}$ and $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, corresponding to the various descriptive-set-theoretic complexities of the positions. For example, we may denote by $\omega_1^{\mathfrak{Ch},c}$ the 'computable' omega one of chess, which is the supremum of the game values of the computable positions of infinite chess. Similarly, one may define $\omega_1^{\mathfrak{Ch},\text{hyp}}$ to be the supremum of the values of the hyperarithmetic positions and $\omega_1^{\mathfrak{Ch},\Delta^1_2}$ to be the supremum of the $\Delta^1_2$ definable positions, and so on.
Since there are only countably many finite positions, whose game values form an initial segment of the ordinals, it follows that $\omega_1^{\mathfrak{Ch}}$ is a countable ordinal. The next theorem shows more, that $\omega_1^{\mathfrak{Ch}}$ is bounded by the Church-Kleene ordinal $\omega_1^{ck}$, the supremum of the computable ordinals. Thus, the game value of any finite position in infinite chess with a game value is a computable ordinal.
- $\omega_1^{\mathfrak{Ch}}\leq\omega_1^{\mathfrak{Ch},c}\leq\omega_1^{\mathfrak{Ch},\text{hyp}}\leq\omega_1^{ck}$. Thus, the game value of any winning finite position in infinite chess, as well as any winning computable position or winning hyperarithmetic position, is a computable ordinal. Furthermore, if a designated player has a winning strategy from a position $p$, then there is such a strategy with complexity hyperarithmetic in $p$.
- $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\leq\omega_1$. The value of a winning position $p$ is a $p$-computable ordinal, and hence less than $\omega_1^{ck,p}$.
- Similarly, $\omega_1^{\mathfrak{Ch}_3}\leq\omega_1^{ck}$.
Evans and Hamkins [1] have proved that the omega one of infinite 3D chess is true $\omega_1$, as large as it can be: $\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}=\omega_1$. Meanwhile, there remains a large gap in the best-known bounds for $\omega_1^{\mathfrak{Ch}}$ and $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, with Evans and Hamkins finding (computable) infinite positions having value $\omega^3$ and somewhat beyond.
References
- Evans, C D A and Hamkins, Joel David. Transfinite game values in infinite chess. (under review) www arχiv bibtex