On the computational content of the Lawson topology

Abstract

An element of an effectively given domain is computable iff its basic Scott open neighbourhoods are recursively enumerable. We thus refer to computable elements as Scott computable and define an element to be Lawson computable if its basic Lawson open neighbourhoods are recursively enumerable. Since the Lawson topology is finer than the Scott topology, a stronger notion of computability is obtained. For example, in the powerset of the natural numbers with its standard effective presentation, an element is Scott computable iff it is a recursively enumerable set, and it is Lawson computable iff it is a recursive set. Among other examples, we consider the upper powerdomain of Euclidean space, for which we prove that Scott and Lawson computability coincide with two notions of computability for compact sets recently proposed by Brattka and Weihrauch in the framework of type-two recursion theory.