Decision Times of Infinite Computations
Merlin Carl, Philipp Schlicht, PHILIP D. WELCH · Notre Dame Journal of Formal Logic · 2022
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 ω1 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 Σ1- and Σ2-definable ordinals, respectively, over Lω1. We further compute analogous suprema for singletons.