Classical recursion theory. Vol. II
[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 strong reducibilities, studying both their individual structures and their mutual relationships. The last chapters extend the theory to limit sets and arithmetical sets. The volume ends with the first textbook treatment of the enumeration degrees, which admit a number of applications from algebra to the lambda calculus. The book is a valuable source of information for anyone interested in complexity and computability theory. The student will appreciate the detailed but informal account of a wide variety of basic topics, while the specialist will find a wealth of material sketched in exercises and asides. A massive bibliography of more than a thousand titles completes the treatment on the historical side. Contents: Chapter headings only. Preface. Introduction. VII. Theories of recursive functions. Measures of complexity. Speed of computations. Complexity classes. Time and space measures. Inductive inference. VIII. Hierarchies of recursive functions. Small time and space bounds. Deterministic polynomial time. Nondeterministic polynomial time. The polynomial time hierarchy. Polynomial space. Exponential time and space. Elementary functions. Primitive recursive functions. 0-recursive functions. IX. Recursively enumerable sets. Global properties of recursive sets. Local properties of r.e. sets. Global properties of r.e. sets. Complexity of r.e. sets. Inductive inference of r.e. sets. X. Recursively enumerable degress. The finite injury priority method. Effective Baire category. The infinite injury priority method. The priority method. Many-one degrees. Turing degrees. Comparison of degree theories. Structure inside degrees. Index sets. XI. Limit sets. Jump classes. 1-generic degrees. Structure theory. Minimal degrees. Global properties. Many-one degrees. XII. Arithmetical sets. Forcing in arithmetic. Applications of forcing. Turing degrees of arithmetical sets. XIII. Arithmetical degrees. The theory of arithmetical degrees. An analogue of r.e. sets. An analogue of Post's problem. An analogue of the jump classes. Comparison with r.e. degrees. The theory of enumeration degrees. Enumeration degrees below \(0_e'\). A model of the lambda calculus. Bibliography. Notation index. Subject index.
scientific article; zbMATH DE number 1352586