**Author**: Gerald E. Sacks

**Publisher:**Cambridge University Press

**ISBN:**1316739465

**Category:**Mathematics

**Page:**N.A

**View:**2448

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# Search Results for: higher-recursion-theory-perspectives-in-logic

**Author**: Gerald E. Sacks

**Publisher:** Cambridge University Press

**ISBN:** 1316739465

**Category:** Mathematics

**Page:** N.A

**View:** 2448

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the second publication in the Perspectives in Logic series, is an almost self-contained introduction to higher recursion theory, in which the reader is only assumed to know the basics of classical recursion theory. The book is divided into four parts: hyperarithmetic sets, metarecursion, α-recursion, and E-recursion. This text is essential reading for all researchers in the field.

**Author**: Jens E. Fenstad

**Publisher:** Cambridge University Press

**ISBN:** 1107168163

**Category:** Mathematics

**Page:** 237

**View:** 7060

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the tenth publication in the Perspectives in Logic series, Jens E. Fenstad takes an axiomatic approach to present a unified and coherent account of the many and various parts of general recursion theory. The main core of the book gives an account of the general theory of computations. The author then moves on to show how computation theories connect with and unify other parts of general recursion theory. Some mathematical maturity is required of the reader, who is assumed to have some acquaintance with recursion theory. This book is ideal for a second course in the subject.

**Author**: William Levine,Georgia Martin

**Publisher:** Springer Science & Business Media

**ISBN:** 1461206359

**Category:** Computers

**Page:** 353

**View:** 5165

One of the major concerns of theoretical computer science is the classifi cation of problems in terms of how hard they are. The natural measure of difficulty of a function is the amount of time needed to compute it (as a function of the length of the input). Other resources, such as space, have also been considered. In recursion theory, by contrast, a function is considered to be easy to compute if there exists some algorithm that computes it. We wish to classify functions that are hard, i.e., not computable, in a quantitative way. We cannot use time or space, since the functions are not even computable. We cannot use Turing degree, since this notion is not quantitative. Hence we need a new notion of complexity-much like time or spac~that is quantitative and yet in some way captures the level of difficulty (such as the Turing degree) of a function.
*Computational Aspects of Definability*

**Author**: Chi Tat Chong,Liang Yu

**Publisher:** Walter de Gruyter GmbH & Co KG

**ISBN:** 3110275643

**Category:** Mathematics

**Page:** 320

**View:** 2726

This monograph presents recursion theory from a generalized and largely global point of view. A major theme is the study of the structures of degrees arising from two key notions of reducibility, the Turing degrees and the hyperdegrees, using ideas and techniques beyond those of classical recursion theory. These include structure theory, hyperarithmetic determinacy and rigidity, basis theorems, independence results on Turing degrees, as well as applications to higher randomness.

**Author**: Noam Greenberg

**Publisher:** American Mathematical Soc.

**ISBN:** 0821838857

**Category:** Mathematics

**Page:** 99

**View:** 5260

When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of $\alpha$-finiteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees.We show that if an admissible ordinal $\alpha$ is effectively close to $\omega$ (where this closeness can be measured by size or by confinality) then such constructions may be performed in the $\alpha$-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of $\alpha$-r.e. degrees for various classes of admissible ordinals $\alpha$. Together with coding work which shows that for some $\alpha$, the theory of the $\alpha$-r.e. degrees is complicated, we get that for every admissible ordinal $\alpha$, the $\alpha$-r.e. degrees and the classical r.e. degrees are not elementarily equivalent.

**Author**: Arnold Oberschelp

**Publisher:** N.A

**ISBN:** 9783411161713

**Category:** Recursion theory

**Page:** 339

**View:** 960

**Author**: N.A

**Publisher:** N.A

**ISBN:** N.A

**Category:** Electronic journals

**Page:** N.A

**View:** 714

**Author**: Antonio Montalbán

**Publisher:** N.A

**ISBN:** N.A

**Category:**

**Page:** 962

**View:** 6787

**Author**: Peter G. Hinman

**Publisher:** A K Peters/CRC Press

**ISBN:** 9781568812625

**Category:** Mathematics

**Page:** 896

**View:** 6382

This introductory graduate text covers modern mathematical logic from propositional, first-order and infinitary logic and Gödel's Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory. Based on the author's more than 35 years of teaching experience, the book develops students' intuition by presenting complex ideas in the simplest context for which they make sense. The book is appropriate for use as a classroom text, for self-study, and as a reference on the state of modern logic.
*Proceedings of a Conference held in Oxford in July 1976*

**Author**: R. O. Gandy,J. M. E. Hyland

**Publisher:** N.A

**ISBN:** 9780720422009

**Category:** Logic, Symbolic and mathematical

**Page:** 612

**View:** 6578

**Author**: Peter Michael Gerdes

**Publisher:** N.A

**ISBN:** N.A

**Category:**

**Page:** 124

**View:** 3662

*The Official Journal of the Mathematical Association of America*

**Author**: N.A

**Publisher:** N.A

**ISBN:** N.A

**Category:** Mathematicians

**Page:** N.A

**View:** 6878

**Author**: Vanderbilt University

**Publisher:** N.A

**ISBN:** 9780897918916

**Category:** Machine learning

**Page:** 338

**View:** 1727

**Author**: Jacques Dubucs,Michel Bourdeau

**Publisher:** Springer

**ISBN:** 9401792178

**Category:** Philosophy

**Page:** 214

**View:** 6111

Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the developments following Turing’s ground-breaking linkage of computation and the machine, the resulting birth of complexity theory, the innovations of Kolmogorov complexity and resolving the dissonances between proof theoretical semantics and canonical proof feasibility. Finally, it explores one of the most fundamental questions concerning the interface between constructivity and computability: whether the theory of recursive functions is needed for a rigorous development of constructive mathematics. This volume contributes to the unity of science by overcoming disunities rather than offering an overarching framework. It posits that computability’s adoption of a classical, ontological point of view kept these imperatives separated. In studying the relationship between the two, it is a vital step forward in overcoming the disagreements and misunderstandings which stand in the way of a unifying view of logic.
*A Study of Computable Functions and Computably Generated Sets*

**Author**: Robert I. Soare

**Publisher:** Springer Science & Business Media

**ISBN:** 9783540152996

**Category:** Mathematics

**Page:** 437

**View:** 3939

..."The book, written by one of the main researchers on the field, gives a complete account of the theory of r.e. degrees. .... The definitions, results and proofs are always clearly motivated and explained before the formal presentation; the proofs are described with remarkable clarity and conciseness. The book is highly recommended to everyone interested in logic. It also provides a useful background to computer scientists, in particular to theoretical computer scientists." Acta Scientiarum Mathematicarum, Ungarn 1988 ..."The main purpose of this book is to introduce the reader to the main results and to the intricacies of the current theory for the recurseively enumerable sets and degrees. The author has managed to give a coherent exposition of a rather complex and messy area of logic, and with this book degree-theory is far more accessible to students and logicians in other fields than it used to be." Zentralblatt für Mathematik, 623.1988
*Klassische Prädikatenlogik*

**Author**: Hans Hermes

**Publisher:** Springer-Verlag

**ISBN:** 3322996425

**Category:** Technology & Engineering

**Page:** 208

**View:** 1667

Das vorliegende, 1963 in erster Auflage erschienene Buch ist aus Vorlesungen hervorgegangen. Es soll eine Einführung in die klassische zweiwertige Prädikaten logik geben. Die Beschränkung auf die klassische Logik soll nicht besagen, daß diese Logik prinzipiell einen Vorzug vor anderen, nichtklassischen Logiken besitzt. Die klassische Logik empfiehlt sich jedoch als Einführung in die Logik wegen ihrer Einfachheit und als Fundament für die Anwendung deshalb, weil sie der klassischen Mathematik und damit den darauf aufgebauten exakten Wissenschaften zugrunde liegt. Das Buch wendet sich primär an Studierende der Mathematik, die in den An fängervorlesungen bereits einige grundlegende mathematische Begriffe, wie den Gruppenbegriff, kennengelernt haben. Der Leser soll dazu geführt werden, daß er die Vorteile einer Formalisierung einsieht. Der übergang von der Umgangssprache zu einer formalisierten Sprache, welcher erfahrungsgemäß gewisse Schwierigkeiten bereitet, wird eingehend besprochen und eingeübt. Die Analyse desmathemati schen Umgangs mit den grundlegenden mathematischen Strukturen führt in zwangloser Weise zum semantisch begründeten Folgerungsbegriff.
*Perspectives East and West*

**Author**: Barry S. Cooper,Sergei S. Goncharov

**Publisher:** Springer Science & Business Media

**ISBN:** 1461507553

**Category:** Mathematics

**Page:** 375

**View:** 4572

Science involves descriptions of the world we live in. It also depends on nature exhibiting what we can best describe as a high aLgorithmic content. The theme running through this collection of papers is that of the interaction between descriptions, in the form of formal theories, and the algorithmic content of what is described, namely of the modeLs of those theories. This appears most explicitly here in a number of valuable, and substantial, contributions to what has until recently been known as 'recursive model theory' - an area in which researchers from the former Soviet Union (in particular Novosibirsk) have been pre-eminent. There are also articles concerned with the computability of aspects of familiar mathematical structures, and - a return to the sort of basic underlying questions considered by Alan Turing in the early days of the subject - an article giving a new perspective on computability in the real world. And, of course, there are also articles concerned with the classical theory of computability, including the first widely available survey of work on quasi-reducibility. The contributors, all internationally recognised experts in their fields, have been associated with the three-year INTAS-RFBR Research Project "Com putability and Models" (Project No. 972-139), and most have participated in one or more of the various international workshops (in Novosibirsk, Heidelberg and Almaty) and otherresearch activities of the network.

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