Effective Operators and Continuity Revisited

In programming language semantics different kinds of semantical domains are used, among them Scott domains and metric spaces. D. Scott raised the problem of finding a suitable class of spac~ which should include Scott domains and metric spaces such that effective mappings between these spaces are continuous. It is well known that between spaces like effectively given Scott domains or constructive metric spaces such operators are effectively continuous and vice versa. But, as an example of Friedberg shows, effective mappings from metric spaces into Scott domains are not continuous in general. In a joint paper P. Young and the author presented a condition which under fairly general effectivity assumptions forces effective mappings between separable countable topological T0-spaces to be effectively continuous. In this paper the condition is weakened. Moreover, a large class of separable countable T0-spaces is given, and it is proved that a mapping between spaces of the class is effectively continuous, iff it is effective and satisfies the condition. A modification of Friedberg's example shows that the result is false without the extra condition. Among others the class of spaces contains all reeursively separable recursive metric spaces in which one can effectively pass from convergent normed recursive Cauehy sequences to their limits and all Scott domains that can be obtained via product and function space constructions from fiat domains with at least three elements. The topology of the spaces in this class is effectively equivalent to the topology generated by those elements in the distributive lattice of all completely enumerable subsets of the space which possess a pseudocomplement and are regular with respect to this operation. 1 I n t r o d u c t i o n In programming language semantics different kinds of semantical domains are used, among them Scott domains and metric spaces (cf. e.g., America and Rutten 1988, de Bakker and Zucker 1982, Nivat 1979, Reed and Roscoe 1988, Scott 1972, 1973, 1976, 1982, Scott and Strachey 1971). In his Logic Colloquium '83 talk D. Scott considered the problem of finding a suitable class of spaces which should include Scott domains and metric spaces such that effective mappings between these spaces are continuous. Continuity is an essential property of mappings which appear as the meaning of program constructs such as procedures: since each converging computation can use only a finite amount of information about its input, it follows that if the value of a computable map with respect to a given input can be found, it must be determined by some finite approximation of the input. If one studies the behaviour of procedures in a system where execution is based on rewriting, then the meaning of a procedure is a map that transforms program code. Maps between semantic domains that are determined by computable operations on the (syntactic) code are called effective. There has been a long interest in logic and constructive mathematics in the question of whether effective maps are continuous: Myhill and Shepherdson (1955) showed that on the set of all partial recursive functions each effective operator is effectively continuous and vice versa. Kreisel, Lacombe and Shoenfield (1959) obtained an analogous result with respect to the set of all total recursive functions. The first result has been lifted to effectively given Scott domains by various authors (cf. Egli and Constable 1976, Er~ov 1977, Weihrauch and Deil 1980) and