Computational complexity, speedable and levelable sets

Abstract

One of the most interesting aspects of the theory of computational complexity is the speed-up phenomenon such as the theorem of Blum [6, p. 326] which asserts the existence of a 0, 1-valued total recursive function with arbitrarily large speed-up. Blum and Marques [10] extended the speed-up definitions from total to partial recursive functions, or equivalently, to recursively enumerable (r.e.) sets, and introduced speedable and levelable sets. They classified the effectively speedable sets as the subcreative sets but remarked that “the characterizations we provided for speedable and levelable sets do not seem to bear a close relationship to any already well-studied class of recursively enumerable sets.” The purpose of this paper is to give an “information theoretic” characterization of speedable and levelable sets in terms of index sets resembling the jump operator. From these characterizations we derive numerous consequences about the degrees and structure of speedable and levelable sets.