Decision Times of Infinite Computations

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 similarly. 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.