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. (shrink)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

We extend an independence result proved in our earlier paper "Solovay's Theorem Cannot Be Simplified" (Annals of Pure and Applied Logic 112 (2001)). Our method uses the Barwise.

In this paper we show that there are various pseudo-jump operators definable over inadmissible $J_{\beta}$ that relate to the failure of admissiblity and to non-regularity. We will use these ideas to construct some intermediate degrees.

The current literature standardly conceives of voting power in terms of decisiveness: the ability to change the voting outcome by unilaterally changing one’s vote. I argue that this classic conception of voting power, which fails to account for partial decisiveness or efficacy, produces erroneous results because it saddles the concept of voting power with implausible microfoundations. This failure in the measure of voting power in turn reflects a philosophical mistake about the concept of social power in general: a failure to (...) recognize that an agent can exercise individual social power with others’ assistance, in virtue of the group’s collective power, sometimes even when she could not unilaterally scuttle the group’s collective power. I therefore develop a conception of efficacy that admits of degrees and defend a Recursive Measure of voting power that takes partial efficacy into account. (shrink)

This volume contains papers from the recursion theory session of the Kazan Workshop on Recursion and Complexity Theory. Recursion theory, the study of computability, is an area of mathematical logic that has traditionally been particularly strong in the United States and the former Soviet Union. This was the first workshop ever to bring together about 50 international experts in recursion theory from the United States, the former Soviet Union and Western Europe.

This graduate-level_text by a master in the field builds a function theory of the rational field that combines aspects of classical and intuitionist analysis. Topics include recursive convergence, recursive and relative continuity, recursive and relative differentiability, the relative integral, elementary functions, and transfinite ordinals. 1961 edition.