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 ω 1 is the maximal decision time of sets of reals. Our main results determine the supremum of countable decision times as σ and that of countable semidecision times as τ, where σ and τ denote he suprema of Σ1and Σ2- definable ordinals, respectively, over Lω1. We further compute analogous suprema for singletons.