Recursive functionals
type: Item
instance of: scholarly article
title: Recursive functionals
zbMATH Open document ID: 0806.03034
author: Luis E. Sanchis
published in: Studies in Logic and the Foundations of Mathematics
publication date: 1993-01-23T00:00:00Z
review text: The subject of the book is computability theory of higher order functions, i.e. functionals. Compared with other texts on the subject, e.g. \textit{P. G. Hinman's} book [Recursion-theoretic hierarchies (1978; Zbl 0371.02017)], the work under review has two distinctive features: it extends the standard theory by allowing partial functions as arguments, and it restricts it by disallowing higher order arguments which are not numerical functions. The author's reason for the restriction is simply that ``a good and sufficiently general description of the theory at this level should be in place before considering extensions to higher types'' (p. vii). Concerning the extension (allowing partial function arguments), the theory is made manageable by restricting to monotonic functionals, and the book demonstrates convincingly that a reasonable theory of computation can be developed in this setting. At the same time a fair amount of the classical theory is carried over to the present situation, most notably Gandy's selector theorem, Kleene's theorem on hyperarithmetic predicates and Grilliot's theorem on effectively discontinuous functionals. No attempt is made, however, to establish some relations with the more general theory of higher order computability due to Scott, although the existence of this theory is shortly mentioned in the preface (but no papers of Scott and e.g. Ershov or Plotkin are among the references). This is particularly disappointing since recently some progress has been made in establishing a proper mathematical context for the treatment of totality, a notion central for the book under review; cf. \textit{U. Berger} [Ann. Pure Appl. Logic 60, 91-117 (1993; Zbl 0776.03031)], or the recent book of \textit{E. R. Griffor}, \textit{I. Lindstroom} and \textit{V. Stoltenberg- Hansen} [Mathematical theory of domains (Cambridge University Press) (1994)]. The book develops the theory from scratch and contains full and carefully worked out proofs. A fair number of exe…
Mathematics Subject Classification ID: 03D65
Mathematics Subject Classification ID: 03-01
Mathematics Subject Classification ID: 03-02
zbMATH DE Number: 108619
zbMATH Keywords: higher-order computability
zbMATH Keywords: recursive functionals with partial arguments
reviewed by: Helmut Schwichtenberg
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 108619