Interpreting Localized Computational Effects Using Operators of Higher Type
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
Some reasons for generalising domain theory
Mathematical Structures in Computer Science
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It is an empirical observation arising from the study of highertype computability that a wide range of approaches to defining aclass of (hereditarily) total functionals over ℕ leads inpractice to a relatively small handful of distinct type structures.Among these are the type structure C of KleeneKreisel continuousfunctionals, its effective substructure Ceffand the type structure HEO of the hereditarily effectiveoperations. However, the proofs of the relevant equivalencesare often non-trivial, and it is not immediately clear why theseparticular type structures should arise so ubiquitously.In this paper we present some new results that go some waytowards explaining this phenomenon. Our results show that a largeclass of extensional collapse constructions always give riseto C, Ceff or HEO (as appropriate). We obtainversions of our results for both the standard and modifiedextensional collapse constructions. The proofs make essential useof a technique due to Normann.Many new results, as well as some previously known ones, can beobtained as instances of our theorems, but more importantly, theproofs apply uniformly to a whole family of constructions, andprovide strong evidence that the three type structures underconsideration are highly canonical mathematical objects.