Computability And Decidability

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Enumerability · Decidability Computability

Author : Hans Hermes
Publisher : Springer Science & Business Media
ISBN 13 : 3642461786
Total Pages : 260 pages
Book Rating : 4.6/5 (424 download)

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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.

Computability and Decidability

Author : J. Loeckx
Publisher : Springer Science & Business Media
ISBN 13 : 3642806899
Total Pages : 84 pages
Book Rating : 4.6/5 (428 download)

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Book Synopsis Computability and Decidability by : J. Loeckx

Download or read book Computability and Decidability written by J. Loeckx and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Computability & Unsolvability

Author : Martin Davis
Publisher : Courier Corporation
ISBN 13 : 0486614719
Total Pages : 292 pages
Book Rating : 4.4/5 (866 download)

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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.

Computability Theory

Author : Neil D. Jones
Publisher : Academic Press
ISBN 13 : 1483218481
Total Pages : 169 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Computability Theory by : Neil D. Jones

Download or read book Computability Theory written by Neil D. Jones and published by Academic Press. This book was released on 2014-06-20 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Automata and Computability

Author : Dexter C. Kozen
Publisher : Springer
ISBN 13 : 364285706X
Total Pages : 399 pages
Book Rating : 4.6/5 (428 download)

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Book Synopsis Automata and Computability by : Dexter C. Kozen

Download or read book Automata and Computability written by Dexter C. Kozen and published by Springer. This book was released on 2013-11-11 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: These are my lecture notes from CS381/481: Automata and Computability Theory, a one-semester senior-level course I have taught at Cornell Uni versity for many years. I took this course myself in thc fall of 1974 as a first-year Ph.D. student at Cornell from Juris Hartmanis and have been in love with the subject ever sin,:e. The course is required for computer science majors at Cornell. It exists in two forms: CS481, an honors version; and CS381, a somewhat gentler paced version. The syllabus is roughly the same, but CS481 go es deeper into thc subject, covers more material, and is taught at a more abstract level. Students are encouraged to start off in one or the other, then switch within the first few weeks if they find the other version more suitaLle to their level of mathematical skill. The purpose of t.hc course is twofold: to introduce computer science students to the rieh heritage of models and abstractions that have arisen over the years; and to dew!c'p the capacity to form abstractions of their own and reason in terms of them.

Computability and Decidability

Author : Jacques Loeckx
Publisher :
ISBN 13 : 9780387058696
Total Pages : 0 pages
Book Rating : 4.0/5 (586 download)

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Book Synopsis Computability and Decidability by : Jacques Loeckx

Download or read book Computability and Decidability written by Jacques Loeckx and published by . This book was released on 1972 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Enumerability, Decidability, Computability

Author : Hans Hermes
Publisher : Springer
ISBN 13 : 3662116863
Total Pages : 255 pages
Book Rating : 4.6/5 (621 download)

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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.

Mathematics in Computing

Author : Gerard O’Regan
Publisher : Springer Nature
ISBN 13 : 3030342093
Total Pages : 468 pages
Book Rating : 4.0/5 (33 download)

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Book Synopsis Mathematics in Computing by : Gerard O’Regan

Download or read book Mathematics in Computing written by Gerard O’Regan and published by Springer Nature. This book was released on 2020-01-10 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: This illuminating textbook provides a concise review of the core concepts in mathematics essential to computer scientists. Emphasis is placed on the practical computing applications enabled by seemingly abstract mathematical ideas, presented within their historical context. The text spans a broad selection of key topics, ranging from the use of finite field theory to correct code and the role of number theory in cryptography, to the value of graph theory when modelling networks and the importance of formal methods for safety critical systems. This fully updated new edition has been expanded with a more comprehensive treatment of algorithms, logic, automata theory, model checking, software reliability and dependability, algebra, sequences and series, and mathematical induction. Topics and features: includes numerous pedagogical features, such as chapter-opening key topics, chapter introductions and summaries, review questions, and a glossary; describes the historical contributions of such prominent figures as Leibniz, Babbage, Boole, and von Neumann; introduces the fundamental mathematical concepts of sets, relations and functions, along with the basics of number theory, algebra, algorithms, and matrices; explores arithmetic and geometric sequences and series, mathematical induction and recursion, graph theory, computability and decidability, and automata theory; reviews the core issues of coding theory, language theory, software engineering, and software reliability, as well as formal methods and model checking; covers key topics on logic, from ancient Greek contributions to modern applications in AI, and discusses the nature of mathematical proof and theorem proving; presents a short introduction to probability and statistics, complex numbers and quaternions, and calculus. This engaging and easy-to-understand book will appeal to students of computer science wishing for an overview of the mathematics used in computing, and to mathematicians curious about how their subject is applied in the field of computer science. The book will also capture the interest of the motivated general reader.

Decidability of Logical Theories and Their Combination

Author : João Rasga
Publisher : Springer Nature
ISBN 13 : 3030565548
Total Pages : 185 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Decidability of Logical Theories and Their Combination by : João Rasga

Download or read book Decidability of Logical Theories and Their Combination written by João Rasga and published by Springer Nature. This book was released on 2020-10-20 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides a self-contained introduction to decidability of first-order theories and their combination. The technical material is presented in a systematic and universal way and illustrated with plenty of examples and a range of proposed exercises. After an overview of basic first-order logic concepts, the authors discuss some model-theoretic notions like embeddings, diagrams, and elementary substructures. The text then goes on to explore an applicable way to deduce logical consequences from a given theory and presents sufficient conditions for a theory to be decidable. The chapters that follow focus on quantifier elimination, decidability of the combination of first-order theories and the basics of computability theory. The inclusion of a chapter on Gentzen calculus, cut elimination, and Craig interpolation, as well as a chapter on combination of theories and preservation of decidability, help to set this volume apart from similar books in the field. Decidability of Logical Theories and their Combination is ideal for graduate students of Mathematics and is equally suitable for Computer Science, Philosophy and Physics students who are interested in gaining a deeper understanding of the subject. The book is also directed to researchers that intend to get acquainted with first-order theories and their combination.

Computability, Complexity, and Languages

Author : Martin Davis
Publisher : Academic Press
ISBN 13 : 0122063821
Total Pages : 631 pages
Book Rating : 4.1/5 (22 download)

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Book Synopsis Computability, Complexity, and Languages by : Martin Davis

Download or read book Computability, Complexity, and Languages written by Martin Davis and published by Academic Press. This book was released on 1994-02-03 with total page 631 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introductory text covers the key areas of computer science, including recursive function theory, formal languages, and automata. Additions to the second edition include: extended exercise sets, which vary in difficulty; expanded section on recursion theory; new chapters on program verification and logic programming; updated references and examples throughout.

Automata and Computability

Author : Dexter C. Kozen
Publisher : Springer Science & Business Media
ISBN 13 : 1461218446
Total Pages : 407 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Automata and Computability by : Dexter C. Kozen

Download or read book Automata and Computability written by Dexter C. Kozen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides undergraduate students with an introduction to the basic theoretical models of computability, and develops some of the model's rich and varied structure. The first part of the book is devoted to finite automata and their properties. Pushdown automata provide a broader class of models and enable the analysis of context-free languages. In the remaining chapters, Turing machines are introduced and the book culminates in analyses of effective computability, decidability, and Gödel's incompleteness theorems. Students who already have some experience with elementary discrete mathematics will find this a well-paced first course, and a number of supplementary chapters introduce more advanced concepts.

Proofs and Algorithms

Author : Gilles Dowek
Publisher : Springer Science & Business Media
ISBN 13 : 0857291211
Total Pages : 161 pages
Book Rating : 4.8/5 (572 download)

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Book Synopsis Proofs and Algorithms by : Gilles Dowek

Download or read book Proofs and Algorithms written by Gilles Dowek and published by Springer Science & Business Media. This book was released on 2011-01-11 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: Logic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation. Proofs and Algorithms: Introduction to Logic and Computability is an introduction to the fundamental concepts of contemporary logic - those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, - Church's undecidability theorem, Gödel’s incompleteness theorem, the theorem asserting the semi-decidability of provability - that have profoundly changed our vision of reasoning, computation, and finally truth itself. Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.

Computability and Logic

Author : George S. Boolos
Publisher : Cambridge University Press
ISBN 13 : 0521877520
Total Pages : 365 pages
Book Rating : 4.5/5 (218 download)

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Book Synopsis Computability and Logic by : George S. Boolos

Download or read book Computability and Logic written by George S. Boolos and published by Cambridge University Press. This book was released on 2007-09-17 with total page 365 pages. Available in PDF, EPUB and Kindle. Book excerpt: This fifth edition of 'Computability and Logic' covers not just the staple topics of an intermediate logic course such as Godel's incompleteness theorems, but also optional topics that include Turing's theory of computability and Ramsey's theorem.

Enumerability. Decidability. Computability

Author : Hans Hermes
Publisher :
ISBN 13 :
Total Pages : 245 pages
Book Rating : 4.:/5 (7 download)

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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:

Lectures on the Philosophy of Mathematics

Author : Joel David Hamkins
Publisher : MIT Press
ISBN 13 : 0262542234
Total Pages : 350 pages
Book Rating : 4.2/5 (625 download)

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Book Synopsis Lectures on the Philosophy of Mathematics by : Joel David Hamkins

Download or read book Lectures on the Philosophy of Mathematics written by Joel David Hamkins and published by MIT Press. This book was released on 2021-03-09 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.

Limits of Computation

Author : Edna E. Reiter
Publisher : CRC Press
ISBN 13 : 1439882061
Total Pages : 283 pages
Book Rating : 4.4/5 (398 download)

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Book Synopsis Limits of Computation by : Edna E. Reiter

Download or read book Limits of Computation written by Edna E. Reiter and published by CRC Press. This book was released on 2012-10-29 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: Limits of Computation: An Introduction to the Undecidable and the Intractable offers a gentle introduction to the theory of computational complexity. It explains the difficulties of computation, addressing problems that have no algorithm at all and problems that cannot be solved efficiently. The book enables readers to understand: What does it mean for a problem to be unsolvable or to be NP-complete? What is meant by a computation and what is a general model of a computer? What does it mean for an algorithm to exist and what kinds of problems have no algorithm? What problems have algorithms but the algorithm may take centuries to finish? Developed from the authors’ course on computational complexity theory, the text is suitable for advanced undergraduate and beginning graduate students without a strong background in theoretical computer science. Each chapter presents the fundamentals, examples, complete proofs of theorems, and a wide range of exercises.

Computability and Complexity

Author : Neil D. Jones
Publisher : MIT Press
ISBN 13 : 9780262100649
Total Pages : 494 pages
Book Rating : 4.1/5 (6 download)

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Book Synopsis Computability and Complexity by : Neil D. Jones

Download or read book Computability and Complexity written by Neil D. Jones and published by MIT Press. This book was released on 1997 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computability and complexity theory should be of central concern to practitioners as well as theorists. Unfortunately, however, the field is known for its impenetrability. Neil Jones's goal as an educator and author is to build a bridge between computability and complexity theory and other areas of computer science, especially programming. In a shift away from the Turing machine- and G�del number-oriented classical approaches, Jones uses concepts familiar from programming languages to make computability and complexity more accessible to computer scientists and more applicable to practical programming problems. According to Jones, the fields of computability and complexity theory, as well as programming languages and semantics, have a great deal to offer each other. Computability and complexity theory have a breadth, depth, and generality not often seen in programming languages. The programming language community, meanwhile, has a firm grasp of algorithm design, presentation, and implementation. In addition, programming languages sometimes provide computational models that are more realistic in certain crucial aspects than traditional models. New results in the book include a proof that constant time factors do matter for its programming-oriented model of computation. (In contrast, Turing machines have a counterintuitive "constant speedup" property: that almost any program can be made to run faster, by any amount. Its proof involves techniques irrelevant to practice.) Further results include simple characterizations in programming terms of the central complexity classes PTIME and LOGSPACE, and a new approach to complete problems for NLOGSPACE, PTIME, NPTIME, and PSPACE, uniformly based on Boolean programs. Foundations of Computing series