无限国际象棋中的超限博弈值
Transfinite game values in infinite chess
摘要：
我们调查超限游戏中产生的无限国际象棋的价值观，提供上界和下界上确界的这些价值观-欧米茄一个国际象棋-有两个意义上取决于是否考虑只有有限的位置或无限多件的位置。对于下界，我们给出了超限博弈值为ω、ω ^2、ω ^2乘以k和ω ^3的特定无限位置。通过将树嵌入到国际象棋中，我们证明了存在一个可计算的无限国际象棋位置，如果要求玩家根据确定性可计算策略进行游戏，则该位置是白色的胜利，但如果没有该限制，则该位置是平局。最后，我们证明了在无限三维象棋中，每一个可数序数都是一个位置的博弈值，因此无限三维象棋的Ω 1是尽可能大的，即真Ω 1。
英文摘要：
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.
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