Computable one-to-one enumerations of effective domains

Abstract

In this paper we consider ω-algebraic complete partial orders where the compact elements are not maximal in the partial order. Under the assumption that the compact elements admit a one-to-one enumeration such that the restriction of the order to them is completely enumerable, it is shown that the computable domain elements also can be effectively enumerated without repetition. Such computable one-to-one enumerations of the computable domain elements are minimal among all enumerations of these elements with respect to the reducibility of one enumeration to another. The admissible indexings which are usually used in computability studies of continuous complete partial orders are maximal among the computable enumerations. As it is moreover shown, admissible numberings are recursively isomorphic to the directed sum of a computable family of computable one-to-one enumerations. Both results generalize well known theorems by Friedberg and Schinzel, respectively, for the partial recursive functions. The proof uses a priority argument.

On leave from Siemens Corporate Laboratories for Research and Technology, Munich, West Germany. Supported by a grant of the Italian C.N.R. to work at the Computer Science Department of the University of Pisa and by the Siemens Corporate Laboratories for Research and Technology.