The Characterization of the Leaf Language Classes

The results of this paper were obtained independently by the two authors and appeared in their Ph.D. theses [7, 20]. A preliminary version with the title was presented by the rst author at the Conference of Structure in Complexity Theory 1994 [6]. Im Neuenheimer Feld 294, 69120 Heidelberg, Germany, email: [email protected] Dipartimento di Scienze dell'Informazione, Via Salaria 113, 00198 Roma, Italy, email: [email protected] Bovet, Crescenzi, and Silvestri [8, 9], and independently Vereshchagin [22], 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 this paper. It will be shown that the set of leaf language classes equals the set of countable complexity classes which, with respect to polynomial-time many-one reducibility , are closed downward and closed under join. Moreover, the set of classes characterized 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.