Enumerability Decidability Computability
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Enumerability Decidability Computability
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Author : Hans Hermes
language : en
Publisher: Springer
Release Date : 2013-03-14
Enumerability Decidability Computability written by Hans Hermes and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.
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.
Enumerability Decidability Computability
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Author : Hans Hermes
language : en
Publisher:
Release Date : 1969
Enumerability Decidability Computability written by Hans Hermes and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1969 with categories.
Enumerability Decidability Computability
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Author : Hans Hermes
language : en
Publisher:
Release Date : 2014-01-15
Enumerability Decidability Computability written by Hans Hermes and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.
Computability Theory
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Author : Neil D. Jones
language : en
Publisher: Academic Press
Release Date : 2014-06-20
Computability Theory written by Neil D. Jones and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-06-20 with Mathematics categories.
Computability Theory: An Introduction provides information pertinent to the major concepts, constructions, and theorems of the elementary theory of computability of recursive functions. This book provides mathematical evidence for the validity of the Church–Turing thesis. Organized into six chapters, this book begins with an overview of the concept of effective process so that a clear understanding of the effective computability of partial and total functions is obtained. This text then introduces a formal development of the equivalence of Turing machine computability, enumerability, and decidability with other formulations. Other chapters consider the formulas of the predicate calculus, systems of recursion equations, and Post's production systems. This book discusses as well the fundamental properties of the partial recursive functions and the recursively enumerable sets. The final chapter deals with different formulations of the basic ideas of computability that are equivalent to Turing-computability. This book is a valuable resource for undergraduate or graduate students.
Enumerability Decidability Computability
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Author : Hans Hermes
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Enumerability Decidability Computability written by Hans Hermes and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
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.
Computability And Decidability
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Author : J. Loeckx
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Computability And Decidability written by J. Loeckx and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Computers categories.
The present Lecture Notes evolved from a course given at the Technische Hogeschool Eindhoven and later at the Technische Hogeschool Twente. They are intended for computer science students; more specifically, their goal is to introduce the notions of computability and decidability, and to prepare for the study of automata theory, formal language theory and the theory of computing. Except for a general mathematical background no preliminary knowledge is presupposed, but some experience in programming may be helpful. While classical treatises on computability and decidability are oriented towards the foundation of mathematics or mathematical logic, the present notes try to relate the subject to computer science. Therefore, the expose is based on the use of strings rather than on that of natural numbers; the notations are similar to those in use in automata theory; in addition, according to a common usage in formal language theory, most of the proofs of computability are reduced to the semi-formal description of a procedure the constructivity of which is apparent to anybody having some programming experience. Notwithstanding these facts the subject is treated with mathematical rigor; a great number of informal comments are inserted in order to allow a good intuitive understanding. I am indebted to all those who drew my attention to some errors and ambiguities in a preliminary version of these Notes. I want also to thank Miss L.A. Krukerink for her diligence in typing the manuscript.
Computable Functions
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Author : Alexander Shen
language : en
Publisher: American Mathematical Soc.
Release Date : 2003
Computable Functions written by Alexander Shen and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Mathematics categories.
This lively and concise book is based on the lectures for undergraduates given by the authors at the Moscow State University Mathematics Department and covers the basic notions of the general theory of computation. It begins with the definition of a computable function and an algorithm and discusses decidability, enumerability, universal functions, numberings and their properties, $m$-completeness, the fixed point theorem, arithmetical hierarchy, oracle computations, and degrees of unsolvability. The authors also cover specific computational models, such as Turing machines and recursive functions. The intended audience includes undergraduate students majoring in mathematics or computer science, and all mathematicians and programmers who would like to learn the basics of the general theory of computation.
Computability Theory
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Author : S. Barry Cooper
language : en
Publisher: CRC Press
Release Date : 2003-11-17
Computability Theory written by S. Barry Cooper and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-11-17 with Mathematics categories.
Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way.
Computability Enumerability Unsolvability
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Author : S. B. Cooper
language : en
Publisher: Cambridge University Press
Release Date : 1996-01-11
Computability Enumerability Unsolvability written by S. B. Cooper and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-01-11 with Mathematics categories.
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.
The Foundations Of Computability Theory
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Author : Borut Robič
language : en
Publisher: Springer
Release Date : 2015-09-14
The Foundations Of Computability Theory written by Borut Robič and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-09-14 with Computers categories.
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.