Comparing Functional Paradigms for Exact Real-Number Computation [chapter]

Andrej Bauer, Martín Hötzel Escardó, Alex Simpson
2002 Lecture Notes in Computer Science  
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 up to second-order types, and we relate this fact to an analogous comparison of type hierarchies over the external and internal real numbers in
more » ... a 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, although it is known that, in the extensional approach, parallel primitives are necessary for programming total first-order functions, we demonstrate that, in the intensional approach, such primitives are not needed for second-order types and below.
doi:10.1007/3-540-45465-9_42 fatcat:2hnypgcb7fal7dvjfqj5yegdoq