The decision time of an infinite time algorithm is the supremum of its halting times over all real inputs. The decision time of a set of reals is the least decision time of an algorithm that decides the set; semidecision times of semidecidable sets are defined similary. It is not hard to see that $\omega_1$ is the maximal decision time of sets of reals. Our main results determine the supremum of countable decision times as $\sigma$ and that of countable semidecision times as $\tau$, where $\sigma$ and $\tau$ denote the suprema of $\Sigma_1$- and $\Sigma_2$-definable ordinals, respectively, over $L_{\omega_1}$. We further compute analogous suprema for singletons.
翻译:无限时间算法的确定时间是它对所有实际输入的停顿时间的最优值。 一组真实值的确定时间是决定集的算法的最低决定时间; 半分数组的半决定时间是相似的。 很难看出$\ omga_ 1$是每组真实值的最大决定时间。 我们的主要结果决定了以$gma$计算的可计算决定时间的最优值, 以及以$$计算的可计算半决定时间的最优值。 其中, $\ sigma$ 和$\ tau$ 表示以$\ sigma_ 1$ 和 $\ sigma_ 2$- 可确定值的最优值。 我们进一步计算单吨的类似 suprema 。