Classifying Recursive Functions

This chapter presents the study of the interplay between recursion in higher types and transfinite recursion. The chapter introduces the theory of partial continuous functionals, based on Scott's notion of an information system. The partial continuous functionals are central for any the analysis of higher type computability, which is based on the rather natural assumption that any computation ought to be finite. The reason is simply that they form the mathematically appropriate domain of a computable functional. The chapter also presents the collapsing results and discusses the extended Grzegorczyk hierarchy. An introduction to partial continuous functional and some general material concerning computability in higher types is also presented in the chapter. The chapter discusses bounded fixed point operators and concerns elimination of detours through higher types by transfinite recursion.

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