A characterization of the leaf language classes 1

Bovet, Crescenzi, and Silvestri ( 1992, 1995), and independently Vereshchagin ( 1994), showed that many complexity classes in the polynomial time setting are leaf language classes, i.e. classes which are determined by two disjoint languages. They gave many examples but they did not characterize the set of leaf language classes. This will be done in &his paper. It will be shown that the set of leaf language classes equals the set of countable complexity classes that are closed downward with respect to polynomial-time many-one reducibility <L and closed under join. Moreover, the set of classes characterizable by two complementary leaf languages will be shown to be equal to the set of complexity classes which, with respect to &, have a complete set and are closed downward. @ 1997 Published by Elsevier Science B.V.