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Computability Enumerability Unsolvability
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Book Synopsis Computability, Enumerability, Unsolvability by : S. Barry Cooper
Download or read book Computability, Enumerability, Unsolvability written by S. Barry Cooper and published by . This book was released on 2014-05-14 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory.
Book Synopsis Computability, Enumerability, Unsolvability by : S. B. Cooper
Download or read book Computability, Enumerability, Unsolvability written by S. B. Cooper and published by Cambridge University Press. This book was released on 1996-01-11 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: The fundamental ideas concerning computation and recursion naturally find their place at the interface between logic and theoretical computer science. The contributions in this book, by leaders in the field, provide a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory. The topics range over computable functions, enumerable sets, degree structures, complexity, subrecursiveness, domains and inductive inference. A number of the articles contain introductory and background material which it is hoped will make this volume an invaluable resource.
Book Synopsis Enumerability · Decidability Computability by : Hans Hermes
Download or read book Enumerability · Decidability Computability written by Hans Hermes and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: Once we have accepted a precise replacement of the concept of algo rithm, it becomes possible to attempt the problem whether there exist well-defined collections of problems which cannot be handled by algo rithms, and if that is the case, to give concrete cases of this kind. Many such investigations were carried out during the last few decades. The undecidability of arithmetic and other mathematical theories was shown, further the unsolvability of the word problem of group theory. Many mathematicians consider these results and the theory on which they are based to be the most characteristic achievements of mathe matics in the first half of the twentieth century. If we grant the legitimacy of the suggested precise replacements of the concept of algorithm and related concepts, then we can say that the mathematicians have shown by strictly mathematical methods that there exist mathematical problems which cannot be dealt with by the methods of calculating mathematics. In view of the important role which mathematics plays today in our conception of the world this fact is of great philosophical interest. Post speaks of a natural law about the "limitations of the mathematicizing power of Homo Sapiens". Here we also find a starting point for the discussion of the question, what the actual creative activity of the mathematician consists in. In this book we shall give an introduction to the theory of algorithms.
Book Synopsis Computability & Unsolvability by : Martin Davis
Download or read book Computability & Unsolvability written by Martin Davis and published by Courier Corporation. This book was released on 1982-01-01 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classic graduate-level introduction to theory of computability. Discusses general theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, more.
Book Synopsis Enumerability, Decidability, Computability by : Hans Hermes
Download or read book Enumerability, Decidability, Computability written by Hans Hermes and published by Springer. This book was released on 2013-03-14 with total page 255 pages. Available in PDF, EPUB and Kindle. Book excerpt: The task of developing algorithms to solve problems has always been considered by mathematicians to be an especially interesting and im portant one. Normally an algorithm is applicable only to a narrowly limited group of problems. Such is for instance the Euclidean algorithm, which determines the greatest common divisor of two numbers, or the well-known procedure which is used to obtain the square root of a natural number in decimal notation. The more important these special algorithms are, all the more desirable it seems to have algorithms of a greater range of applicability at one's disposal. Throughout the centuries, attempts to provide algorithms applicable as widely as possible were rather unsuc cessful. It was only in the second half of the last century that the first appreciable advance took place. Namely, an important group of the inferences of the logic of predicates was given in the form of a calculus. (Here the Boolean algebra played an essential pioneer role. ) One could now perhaps have conjectured that all mathematical problems are solvable by algorithms. However, well-known, yet unsolved problems (problems like the word problem of group theory or Hilbert's tenth problem, which considers the question of solvability of Diophantine equations) were warnings to be careful. Nevertheless, the impulse had been given to search for the essence of algorithms. Leibniz already had inquired into this problem, but without success.
Book Synopsis Business Research Methodology (With Cd) by : T. N. Srivastava
Download or read book Business Research Methodology (With Cd) written by T. N. Srivastava and published by Tata McGraw-Hill Education. This book was released on 1958 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classic graduate-level introduction to theory of computability. Discusses general theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, more.
Book Synopsis Algebraic Computability and Enumeration Models by : Cyrus F. Nourani
Download or read book Algebraic Computability and Enumeration Models written by Cyrus F. Nourani and published by CRC Press. This book was released on 2016-02-24 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, Algebraic Computability and Enumeration Models: Recursion Theory and Descriptive Complexity, presents new techniques with functorial models to address important areas on pure mathematics and computability theory from the algebraic viewpoint. The reader is first introduced to categories and functorial models, with Kleene algebra examples for languages. Functorial models for Peano arithmetic are described toward important computational complexity areas on a Hilbert program, leading to computability with initial models. Infinite language categories are also introduced to explain descriptive complexity with recursive computability with admissible sets and urelements. Algebraic and categorical realizability is staged on several levels, addressing new computability questions with omitting types realizably. Further applications to computing with ultrafilters on sets and Turing degree computability are examined. Functorial models computability is presented with algebraic trees realizing intuitionistic types of models. New homotopy techniques are applied to Marin Lof types of computations with model categories. Functorial computability, induction, and recursion are examined in view of the above, presenting new computability techniques with monad transformations and projective sets. This informative volume will give readers a complete new feel for models, computability, recursion sets, complexity, and realizability. This book pulls together functorial thoughts, models, computability, sets, recursion, arithmetic hierarchy, filters, with real tree computing areas, presented in a very intuitive manner for university teaching, with exercises for every chapter. The book will also prove valuable for faculty in computer science and mathematics.
Book Synopsis Recursively Enumerable Sets and Degrees by : Robert I. Soare
Download or read book Recursively Enumerable Sets and Degrees written by Robert I. Soare and published by Springer Science & Business Media. This book was released on 1999-11-01 with total page 460 pages. Available in PDF, EPUB and Kindle. Book excerpt: ..."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
Book Synopsis Computability & Unsolvability by : Martin Davis
Download or read book Computability & Unsolvability written by Martin Davis and published by . This book was released on 1958 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis On Formally Undecidable Propositions of Principia Mathematica and Related Systems by : Kurt Gödel
Download or read book On Formally Undecidable Propositions of Principia Mathematica and Related Systems written by Kurt Gödel and published by Courier Corporation. This book was released on 2012-05-24 with total page 82 pages. Available in PDF, EPUB and Kindle. Book excerpt: First English translation of revolutionary paper (1931) that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. Introduction by R. B. Braithwaite.
Book Synopsis Handbook of Computability Theory by : E.R. Griffor
Download or read book Handbook of Computability Theory written by E.R. Griffor and published by Elsevier. This book was released on 1999-10-01 with total page 741 pages. Available in PDF, EPUB and Kindle. Book excerpt: The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey of recent results (due to the complexity of the methods and proofs themselves). Hence the variation in level of presentation from chapter to chapter only reflects the conceptual situation itself. One example of this is the collective efforts to develop an acceptable theory of computation on the real numbers. The last two decades has seen at least two new definitions of effective operations on the real numbers.
Book Synopsis Turing Computability by : Robert I. Soare
Download or read book Turing Computability written by Robert I. Soare and published by Springer. This book was released on 2016-06-20 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: Turing's famous 1936 paper introduced a formal definition of a computing machine, a Turing machine. This model led to both the development of actual computers and to computability theory, the study of what machines can and cannot compute. This book presents classical computability theory from Turing and Post to current results and methods, and their use in studying the information content of algebraic structures, models, and their relation to Peano arithmetic. The author presents the subject as an art to be practiced, and an art in the aesthetic sense of inherent beauty which all mathematicians recognize in their subject. Part I gives a thorough development of the foundations of computability, from the definition of Turing machines up to finite injury priority arguments. Key topics include relative computability, and computably enumerable sets, those which can be effectively listed but not necessarily effectively decided, such as the theorems of Peano arithmetic. Part II includes the study of computably open and closed sets of reals and basis and nonbasis theorems for effectively closed sets. Part III covers minimal Turing degrees. Part IV is an introduction to games and their use in proving theorems. Finally, Part V offers a short history of computability theory. The author has honed the content over decades according to feedback from students, lecturers, and researchers around the world. Most chapters include exercises, and the material is carefully structured according to importance and difficulty. The book is suitable for advanced undergraduate and graduate students in computer science and mathematics and researchers engaged with computability and mathematical logic.
Book Synopsis Computability Theory and Its Applications by : Peter Cholak
Download or read book Computability Theory and Its Applications written by Peter Cholak and published by American Mathematical Soc.. This book was released on 2000 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of articles presents a snapshot of the status of computability theory at the end of the millennium and a list of fruitful directions for future research. The papers represent the works of experts in the field who were invited speakers at the AMS-IMS-SIAM 1999 Summer Conference on Computability Theory and Applications, which focused on open problems in computability theory and on some related areas in which the ideas, methods, and/or results of computability theory play a role. Some presentations are narrowly focused; others cover a wider area. Topics included from "pure" computability theory are the computably enumerable degrees (M. Lerman), the computably enumerable sets (P. Cholak, R. Soare), definability issues in the c.e. and Turing degrees (A. Nies, R. Shore) and other degree structures (M. Arslanov, S. Badaev and S. Goncharov, P. Odifreddi, A. Sorbi). The topics involving relations between computability and other areas of logic and mathematics are reverse mathematics and proof theory (D. Cenzer and C. Jockusch, C. Chong and Y. Yang, H. Friedman and S. Simpson), set theory (R. Dougherty and A. Kechris, M. Groszek, T. Slaman) and computable mathematics and model theory (K. Ambos-Spies and A. Kucera, R. Downey and J. Remmel, S. Goncharov and B. Khoussainov, J. Knight, M. Peretyat'kin, A. Shlapentokh).
Book Synopsis Enumerability. Decidability. Computability by : Hans Hermes
Download or read book Enumerability. Decidability. Computability written by Hans Hermes and published by . This book was released on 1969 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Computability and Unsolvability by : Martin Davis
Download or read book Computability and Unsolvability written by Martin Davis and published by Courier Corporation. This book was released on 2013-04-16 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classic graduate-level introduction to theory of computability. Discusses general theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, more.
Book Synopsis Theory and Applications of Models of Computation by : Jin-Yi Cai
Download or read book Theory and Applications of Models of Computation written by Jin-Yi Cai and published by Springer. This book was released on 2006-05-05 with total page 800 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the Third International Conference on Theory and Applications of Models of Computation, TAMC 2006, held in Beijing, China, in May 2006. The 75 revised full papers presented together with 7 plenary talks were carefully reviewed and selected from 319 submissions. All major areas in computer science, mathematics (especially logic) and the physical sciences particularly with regard to computation and computability theory are addressed.
Book Synopsis The Foundations of Computability Theory by : Borut Robič
Download or read book The Foundations of Computability Theory written by Borut Robič and published by Springer. This book was released on 2015-09-14 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an original and informative view of the development of fundamental concepts of computability theory. The treatment is put into historical context, emphasizing the motivation for ideas as well as their logical and formal development. In Part I the author introduces computability theory, with chapters on the foundational crisis of mathematics in the early twentieth century, and formalism; in Part II he explains classical computability theory, with chapters on the quest for formalization, the Turing Machine, and early successes such as defining incomputable problems, c.e. (computably enumerable) sets, and developing methods for proving incomputability; in Part III he explains relative computability, with chapters on computation with external help, degrees of unsolvability, the Turing hierarchy of unsolvability, the class of degrees of unsolvability, c.e. degrees and the priority method, and the arithmetical hierarchy. This is a gentle introduction from the origins of computability theory up to current research, and it will be of value as a textbook and guide for advanced undergraduate and graduate students and researchers in the domains of computability theory and theoretical computer science.
