With Alex Simpson and MartÃn EscardÃ³.

**Abstract:** We compare the definability of total functionals over the reals in two functional-programming approaches to exact real-number computation: the *extensional* approach, in which one has an abstract datatype of real numbers; and the *intensional* approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide for second-order types, and we relate this fact to an analogous comparison of type hierarchies over the *external* and *internal* real numbers in Dana Scott’s category of equilogical spaces. We do not know whether similar coincidences hold at third-order types. However, we relate this question to a purely topological conjecture about the Kleene-Kreisel continuous functionals over the natural numbers. Finally, we demonstrate that, in the intensional approach to exact real-number computation, parallel primitives are not required for programming second-order total functionals over the reals.

**Published in:** In Proceedings ICALP 2002, Springer LNCS 2380, pp. 488-500, 2002.

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