Strong jump-traceability I: The computably enumerable case

Abstract

Recent investigations in algorithmic randomness have lead to the discovery and analysis of the fundamental class K of reals called the K-trivial reals, defined as those whose initial segment complexity is identical with that of the sequence of all 1's. There remain many important open questions concerning this class, such as whether there is a combinatorial characterization of the class and whether it coincides with possibly smaller subclasses, such as the class of reals which are not sufficiently powerful as oracles to cup a Turing incomplete Martin–Löf random real to the halting problem. Hidden here is the question of whether there exist proper natural subclasses of K . We show that the combinatorial class of computably enumerable, strongly jump-traceable reals, defined via the jump operator by Figueira, Nies and Stephan [Santiago Figueira, André Nies, Frank Stephan, Lowness properties and approximations of the jump, Electr. Notes Theor. Comput. Sci. 143 (2006) 45–57], is such a class, and show that like K , it is an ideal in the computably enumerable degrees. This is the first example of a class of reals defined by a “cost function” construction which forms a proper subclass of K . Further, we show that every c.e., strongly jump-traceable set is not Martin–Löf cuppable, thus giving a combinatorial property which implies non-ML cuppability.