UC Berkeley

A split of an r.e. set A is a pair of disjoint r.e. sets whose union is A. We investigate information theoretic properties of r.e. sets and properties of their enumerations, and whether these properties are preserved under splittings. The first part is involved with dynamic notions. An r.e. set is speedable if for every computable function, there exists a finite algorithm enumerating membership faster, by the desired computable factor, on infinitely many integers. Remmel (1986) asked whether every speedable set could be split into two speedable sets. Jahn published a proof, answering their question positively. However, his proof was seen to be incorrect and we construct a speedable set that cannot be split into speedable sets, answering their question negatively. We also prove additional splitting results related to speedability.

The second part is involved with effectively closed sets. We introduce the notion of being recursively avoiding and prove several results.