Comparing C.E. Sets Based on Their Settling Times

Abstract

To each computable enumerable (c.e.) set A with a particular enumeration {A _s }_s ∈ ω, there is associated a settling function m _A (x), where m _A (x) is the last stage when a number less than or equal to x was enumerated into A. In [7], R.W. Robinson classified the complexity of c.e. sets into two groups of complexity based on whether or not the settling function was dominant. An extension of this idea to a more refined ordering of c.e. sets was first introduced by Nabutovsky and Weinberger in [6] and Soare [9], for application to differential geometry. There they defined one c.e. set A to settling time dominate another c.e. set B (B > _st A) if for every computable function f , for all but finitely many x, m _B (x) > f(m _A (x)). In [4] Csima and Soare introduced a stronger ordering, where B > _sst A if for all computable f and g, for almost all x, m _B (x) > f(m _A (g(x))). We give a survey of the known results about these orderings, make some observations, and outline the open questions.