Computability In Context

Author : S Barry Cooper,Andrea Sorbi
Publisher : World Scientific
Page : 420 pages
File Size : 42,5 Mb
Release : 2011-02-25
Category : Mathematics
ISBN : 9781908978769

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Computability in Context by S Barry Cooper,Andrea Sorbi Pdf

Computability has played a crucial role in mathematics and computer science, leading to the discovery, understanding and classification of decidable/undecidable problems, paving the way for the modern computer era, and affecting deeply our view of the world. Recent new paradigms of computation, based on biological and physical models, address in a radically new way questions of efficiency and challenge assumptions about the so-called Turing barrier. This volume addresses various aspects of the ways computability and theoretical computer science enable scientists and philosophers to deal with mathematical and real-world issues, covering problems related to logic, mathematics, physical processes, real computation and learning theory. At the same time it will focus on different ways in which computability emerges from the real world, and how this affects our way of thinking about everyday computational issues. Contents:Computation, Information, and the Arrow of Time (P Adriaans & P van Emde Boas)The Isomorphism Conjecture for NP (M Agrawal)The Ershov Hierarchy (M M Arslanov)Complexity and Approximation in Reoptimization (G Ausiello et al.)Definability in the Real Universe (S B Cooper)HF-Computability (Y L Drshov et al.)The Mathematics of Computing Between Logic and Physics (G Longo & T Paul)Liquid State Machines: Motivation, Theory, and Applications (W Maass)Experiments on an Internal Approach to Typed Algorithms in Analysis (D Normann)Recursive Functions: An Archeological Look (P Odifreddi)Reverse Mathematics and Well-Ordering Principles (M Rathjen & A Weiermann)Discrete Transfinite Computation Models (P D Welch) Readership: Researchers in computational mathematics, logic, and theoretical computer science. Keywords:Computability;Logic;Real World;Turing Barrier;Real Computation;Learning Theory

Author : Martin D. Davis,Elaine J. Weyuker
Publisher : Academic Press
Page : 446 pages
File Size : 50,8 Mb
Release : 2014-05-10
Category : Reference
ISBN : 9781483264585

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Computability, Complexity, and Languages by Martin D. Davis,Elaine J. Weyuker Pdf

Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science provides an introduction to the various aspects of theoretical computer science. Theoretical computer science is the mathematical study of models of computation. This text is composed of five parts encompassing 17 chapters, and begins with an introduction to the use of proofs in mathematics and the development of computability theory in the context of an extremely simple abstract programming language. The succeeding parts demonstrate the performance of abstract programming language using a macro expansion technique, along with presentations of the regular and context-free languages. Other parts deal with the aspects of logic that are important for computer science and the important theory of computational complexity, as well as the theory of NP-completeness. The closing part introduces the advanced recursion and polynomial-time computability theories, including the priority constructions for recursively enumerable Turing degrees. This book is intended primarily for undergraduate and graduate mathematics students.

Author : Benedikt Löwe,Dag Normann,Ivan Soskov,Alexandra Soskova
Publisher : Springer
Page : 319 pages
File Size : 41,9 Mb
Release : 2011-09-15
Category : Computers
ISBN : 9783642218750

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Models of Computation in Context by Benedikt Löwe,Dag Normann,Ivan Soskov,Alexandra Soskova Pdf

This book constitutes the refereed proceedings of the 7th Conference on Computability in Europe, CiE 2011, held in Sofia, Bulgaria, in June/July 2011. The 22 revised papers presented together with 11 invited lectures were carefully reviewed and selected with an acceptance rate of under 40%. The papers cover the topics computability in analysis, algebra, and geometry; classical computability theory; natural computing; relations between the physical world and formal models of computability; theory of transfinite computations; and computational linguistics.

Author : John Longley,Dag Normann
Publisher : Springer
Page : 571 pages
File Size : 48,9 Mb
Release : 2015-11-06
Category : Computers
ISBN : 9783662479926

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Higher-Order Computability by John Longley,Dag Normann Pdf

This book offers a self-contained exposition of the theory of computability in a higher-order context, where 'computable operations' may themselves be passed as arguments to other computable operations. The subject originated in the 1950s with the work of Kleene, Kreisel and others, and has since expanded in many different directions under the influence of workers from both mathematical logic and computer science. The ideas of higher-order computability have proved valuable both for elucidating the constructive content of logical systems, and for investigating the expressive power of various higher-order programming languages. In contrast to the well-known situation for first-order functions, it turns out that at higher types there are several different notions of computability competing for our attention, and each of these has given rise to its own strand of research. In this book, the authors offer an integrated treatment that draws together many of these strands within a unifying framework, revealing not only the range of possible computability concepts but the relationships between them. The book will serve as an ideal introduction to the field for beginning graduate students, as well as a reference for advanced researchers

Author : Richard L Epstein,Walter A Carnielli
Publisher : Advanced Reasoning Forum
Page : 382 pages
File Size : 44,7 Mb
Release : 2018-10-24
Category : Mathematics
ISBN : 9780981550732

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Computability by Richard L Epstein,Walter A Carnielli Pdf

Now in a new edition!--the classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents sufficient formal logic to give a full development of Gödel's incompleteness theorems. Part IV considers the significance of the technical work with a discussion of Church's Thesis and readings on the foundations of mathematics. This new edition contains the timeline "Computability and Undecidability" as well as the essay "On mathematics".

Author : A.J. Kfoury,Robert N. Moll,Michael A. Arbib
Publisher : Springer Science & Business Media
Page : 259 pages
File Size : 45,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461257493

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A Programming Approach to Computability by A.J. Kfoury,Robert N. Moll,Michael A. Arbib Pdf

Computability theory is at the heart of theoretical computer science. Yet, ironically, many of its basic results were discovered by mathematical logicians prior to the development of the first stored-program computer. As a result, many texts on computability theory strike today's computer science students as far removed from their concerns. To remedy this, we base our approach to computability on the language of while-programs, a lean subset of PASCAL, and postpone consideration of such classic models as Turing machines, string-rewriting systems, and p. -recursive functions till the final chapter. Moreover, we balance the presentation of un solvability results such as the unsolvability of the Halting Problem with a presentation of the positive results of modern programming methodology, including the use of proof rules, and the denotational semantics of programs. Computer science seeks to provide a scientific basis for the study of information processing, the solution of problems by algorithms, and the design and programming of computers. The last 40 years have seen increasing sophistication in the science, in the microelectronics which has made machines of staggering complexity economically feasible, in the advances in programming methodology which allow immense programs to be designed with increasing speed and reduced error, and in the develop ment of mathematical techniques to allow the rigorous specification of program, process, and machine.

Author : Borut Robič
Publisher : Springer Nature
Page : 422 pages
File Size : 49,8 Mb
Release : 2020-11-13
Category : Computers
ISBN : 9783662624210

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The Foundations of Computability Theory by Borut Robič Pdf

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. Finally, in the new Part IV the author revisits the computability (Church-Turing) thesis in greater detail. He offers a systematic and detailed account of its origins, evolution, and meaning, he describes more powerful, modern versions of the thesis, and he discusses recent speculative proposals for new computing paradigms such as hypercomputing. 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. This new edition is completely revised, with almost one hundred pages of new material. In particular the author applied more up-to-date, more consistent terminology, and he addressed some notational redundancies and minor errors. He developed a glossary relating to computability theory, expanded the bibliographic references with new entries, and added the new part described above and other new sections.

Author : S. Barry Cooper
Publisher : CRC Press
Page : 420 pages
File Size : 40,5 Mb
Release : 2017-09-06
Category : Mathematics
ISBN : 9781351991964

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Computability Theory by S. Barry Cooper Pdf

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.

Author : Dexter C. Kozen
Publisher : Springer
Page : 399 pages
File Size : 51,9 Mb
Release : 2013-11-11
Category : Computers
ISBN : 9783642857065

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Automata and Computability by Dexter C. Kozen Pdf

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.

Author : Martin Davis,Ron Sigal,Elaine J. Weyuker
Publisher : Academic Press
Page : 631 pages
File Size : 52,5 Mb
Release : 1994-02-03
Category : Computers
ISBN : 9780122063824

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Computability, Complexity, and Languages by Martin Davis,Ron Sigal,Elaine J. Weyuker Pdf

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.

Author : Nicholas Pippenger
Publisher : Cambridge University Press
Page : 268 pages
File Size : 55,6 Mb
Release : 1997-05-28
Category : Computers
ISBN : 0521553806

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Theories of Computability by Nicholas Pippenger Pdf

A mathematically sophisticated introduction to Turing's theory, Boolean functions, automata, and formal languages.

Author : Anuradha A. Puntambekar
Publisher : Technical Publications
Page : 308 pages
File Size : 44,8 Mb
Release : 2020-12-01
Category : Computers
ISBN : 9789333223843

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Automata and Computability by Anuradha A. Puntambekar Pdf

The book has been developed to provide comprehensive and consistent coverage of concepts of automata theory, formal languages and computation. This book begins by giving prerequisites for the subject, like strings, languages, types of automata, deterministic and non-deterministic automata. It proceeds forward to discuss advanced concepts like regular expressions, context free grammar and pushdown automata. The text then goes on to give a detailed description of context free and non context free languages and Turing Machine with its complexity. This compact and well-organized book provides a clear understanding of the subject with its emphasis on concepts along with a large number of examples.

Author : Neil D. Jones
Publisher : MIT Press
Page : 494 pages
File Size : 54,7 Mb
Release : 1997
Category : Computers
ISBN : 0262100649

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Computability and Complexity by Neil D. Jones Pdf

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

Author : Robert McNaughton
Publisher : Unknown
Page : 424 pages
File Size : 49,6 Mb
Release : 1993-09
Category : Computers
ISBN : 0962388564

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Elementary Computability, Formal Languages, and Automata by Robert McNaughton Pdf

Author : Arnold Beckmann,Victor Mitrana,Mariya Soskova
Publisher : Springer
Page : 363 pages
File Size : 41,5 Mb
Release : 2015-06-19
Category : Computers
ISBN : 9783319200286

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Evolving Computability by Arnold Beckmann,Victor Mitrana,Mariya Soskova Pdf

This book constitutes the refereed proceedings of the 11th Conference on Computability in Europe, CiE 2015, held in Bucharest, Romania, in June/July 2015. The 26 revised papers presented were carefully reviewed and selected from 64 submissions and included together with 10 invited papers in this proceedings. The conference CiE 2015 has six special sessions: two sessions, Representing Streams and Reverse Mathematics, were introduced for the first time in the conference series. In addition to this, new developments in areas frequently covered in the CiE conference series were addressed in the further special sessions on Automata, Logic and Infinite Games; Bio-inspired Computation; Classical Computability Theory; as well as History and Philosophy of Computing.