Admissible representations of effective cpo's

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Abstract

The elements of a cpo (complete partial order) D̄ are ‘abstract’ objects in general.

A concrete machine cannot operate with abstract objects but only with names of objects. In this paper the set F of total functions from N to N is suggested as the set of names. ‘Admissible representations’ δ:F→D for effective cpo's are defined by two axioms, which generalize the axioms for Gödel numberings of the partial recursive functions. Topological and recursion theoretic considerations show that the definition is very natural. It is also proved that Pω, the set of subsets of N, is not as suitable as a set of names.

Finally it is proved that for admissible representations the computable functions over cpo's are represented by the computable extensional functions over the names.

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