Decision Times of Infinite Computations

Abstract

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 ${\mathit{\omega }}_1$ 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 the suprema of ${\mathrm{\Sigma }}_1$- and ${\mathrm{\Sigma }}_2$-definable ordinals, respectively, over $L_{{\mathit{\omega }}_1}$. We further compute analogous suprema for singletons.