Classical recursion theory. Vol. II
type: Item
instance of: scholarly article
title: Classical recursion theory. Vol. II
zbMATH Open document ID: 0931.03057
published in: Studies in Logic and the Foundations of Mathematics
publication date: 1999-10-25T00:00:00Z
review text: [The first volume (1989) has been reviewed (see Zbl 0661.03029), the paperback edition of the first volume has been announced (see Zbl 0744.03044).] Ten years after an impressive first volume, Odifreddi has produced a monumental continuation: together they give \textit{the} picture of recursion theory as developed in the 20th century. This volume includes chapters on theories of recursive functions, hierarchies of recursive functions, recursively enumerable sets, recursively enumerable degrees, limit sets, arithmetical sets, arithmetical degrees, and enumeration degrees. Odifreddi's informal style, its self-contained nature, his constant attention on authorship, the broad, up-to-date list of references, the original treatment with emphasis on ideas and insight are features making the book a fundamental contribution to the subject. Needless to say that we enthusiastically recommend it. For the reader's convenience we insert here the publisher's description: Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of theoretical computer science. Besides giving a detailed description of the theories of abstract complexity theory and of inductive inference, it contributes a uniform picture of the most basic complexity classes, ranging from small time and space bounds to the elementary functions, with a particular attention to polynomial time and space computability. It also deals with primitive recursive functions and larger classes, which are of interest to the proof theorist. The second half of the book starts with the classical theory of recursively enumerable sets and degrees, which constitutes the core of recursion or computability theory. Unlike other texts, usually confined to the Turing degrees, the book covers a variety of other…
Mathematics Subject Classification ID: 03Dxx
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03-01
Mathematics Subject Classification ID: 03-00
Mathematics Subject Classification ID: 03D15
Mathematics Subject Classification ID: 03D20
Mathematics Subject Classification ID: 03D25
Mathematics Subject Classification ID: 03D28
Mathematics Subject Classification ID: 03D30
Mathematics Subject Classification ID: 03D55
Mathematics Subject Classification ID: 68Q15
zbMATH DE Number: 1352586
zbMATH Keywords: Classical recursion theory
zbMATH Keywords: classical recursion theory
zbMATH Keywords: recursive functions
zbMATH Keywords: hierarchies
zbMATH Keywords: recursively enumerable sets
zbMATH Keywords: recursively enumerable degrees
zbMATH Keywords: limit sets
zbMATH Keywords: arithmetical sets
zbMATH Keywords: arithmetical degrees
zbMATH Keywords: enumeration degrees
zbMATH Keywords: computable sets
zbMATH Keywords: complexity theory
zbMATH Keywords: inductive inference
zbMATH Keywords: complexity classes
zbMATH Keywords: reducibilities
zbMATH Keywords: computability theory
zbMATH Keywords: recursive sets
author: Piergiorgio Odifreddi
reviewed by: Cristian S. Calude
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1352586