# Classical Recursion Theory, Volume II

**By**

- P. Odifreddi, University of Turin, Italy

Volume II of *Classical Recursion Theory* describes the universe from a local (bottom-upor synthetical) point of view, and covers the whole spectrum, from therecursive to the arithmetical sets.

The first half of the book provides a detailed picture of the computablesets from the perspective of Theoretical Computer Science. Besides giving adetailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexityclasses, ranging from small time and space bounds to the elementary functions,with a particular attention to polynomial time and space computability. It alsodeals with primitive recursive functions and larger classes, which are ofinterest to the proof theorist.

The second half of the book starts with the classical theory of recursivelyenumerable sets and degrees, which constitutes the core of Recursion orComputability Theory. Unlike other texts, usually confined to the Turingdegrees, the book covers a variety of other strong reducibilities, studyingboth their individual structures and their mutual relationships. The lastchapters extend the theory to limit sets and arithmetical sets. The volumeends with the first textbook treatment of the enumeration degrees, whichadmit a number of applications from algebra to the Lambda Calculus.

The book is a valuable source of information for anyone interested inComplexity and Computability Theory. The student will appreciate the detailedbut informal account of a wide variety of basic topics, while the specialistwill find a wealth of material sketched in exercises and asides. A massivebibliography of more than a thousand titles completes the treatment on thehistorical side.

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### Book information

- Published: September 1999
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-50205-6

### Reviews

The scope of the material is amazing. For instance, I can think of no other book with a treatment of w-REA sets inductive reference, and models of lambda calculus! This is especially gratifying in a world of continuing mathematical fragmentation.

Rodney G. Downey, Mathematical Reviews

### Table of 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. &egr;_{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. **Enumeration degrees.** Enumeration degrees. The theory of enumeration degrees. Enumeration degrees below 0'_{e}. A model of the Lambda calculus. Bibliography. Notation index. Subject index.