The settling-time reducibility ordering

Abstract

AbstractTo each computable enumerable (c.e.) setAwith a particular enumeration {As}s∈ωthere is associated a settling functionmA(x), wheremA(x) is the last stage when a number less than or equal toxwas enumerated intoA. One c.e. setAis settling time dominated by another setB(B>stA) if for every computable functionf, for all but finitely manyx, mB(x) >f(mA(x)). This settling-time ordering, which is a natural extension to an ordering of the idea of domination, was first introduced by Nabutovsky and Weinberger in [3] and Soare [6]. They desired a sequence of sets descending in this relationship to give results in differential geometry. In this paper we examine properties of the <stordering. We show that it is not invariant under computable isomorphism, that any countable partial ordering embeds into it. that there are maximal and minimal sets, and that two c.e. sets need not have an inf or sup in the ordering. We also examine a related ordering, the strong settling-time ordering where we require for all computablefandg, for almost allx, mB(x) >f(mA(g(x))).