Classical Recursion Theory, Volume 2
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
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THEORIES OF RECURSIVE FUNCTIONS
HIERARCHIES OF RECURSIVE FUNCTIONS
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analogue argument arithmetical arithmetical sets automorphism bounded characterization class of sets coding coinfinite column complexity classes computable functions consider construction contains converges Corollary countable defined definition deterministic e-splitting element elementarily equivalent elementary ensure Exercises exists EXPSPACE extension first-order First-Order Arithmetic Fixed-Point Theorem follows function g given hence Hierarchy Theorem Hint hyperhypersimple hypersimple incomparable induction hypothesis infinite initial segment injury input isomorphic iteration Jockusch Lachlan lattice Lemma length LOGSPACE m-degrees minimal cover minimal degree minimal pair NEXP nondeterministic nondeterministic Turing machine nonrecursive r.e. nontrivial notion obtain one-one oracle ordinal otherwise parameters partial recursive functions particular polynomial time computable primitive recursive functions proof properties Proposition proved PSPACE quantifiers r-maximal r.e. degrees r.e. set recursive enumeration recursive sets relativization result satisfied Section Slaman smallest space stage strings subset Suppose theory total recursive functions tree Turing degrees Turing machine uniformly