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
Classical Recursion Theory, Volume II, 1st Edition
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.
Quotes and reviews
@from:Rodney G. Downey
@qu: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.