Download or read book Algorithms for Constructing Computably Enumerable Sets written by Kenneth J. Supowit and published by Springer Nature. This book was released on 2023-05-23 with total page 191 pages. Available in PDF, EPUB and Kindle. Book excerpt: Logicians have developed beautiful algorithmic techniques for the construction of computably enumerable sets. This textbook presents these techniques in a unified way that should appeal to computer scientists. Specifically, the book explains, organizes, and compares various algorithmic techniques used in computability theory (which was formerly called "classical recursion theory"). This area of study has produced some of the most beautiful and subtle algorithms ever developed for any problems. These algorithms are little-known outside of a niche within the mathematical logic community. By presenting them in a style familiar to computer scientists, the intent is to greatly broaden their influence and appeal. Topics and features: · All other books in this field focus on the mathematical results, rather than on the algorithms. · There are many exercises here, most of which relate to details of the algorithms. · The proofs involving priority trees are written here in greater detail, and with more intuition, than can be found elsewhere in the literature. · The algorithms are presented in a pseudocode very similar to that used in textbooks (such as that by Cormen, Leiserson, Rivest, and Stein) on concrete algorithms. · In addition to their aesthetic value, the algorithmic ideas developed for these abstract problems might find applications in more practical areas. Graduate students in computer science or in mathematical logic constitute the primary audience. Furthermore, when the author taught a one-semester graduate course based on this material, a number of advanced undergraduates, majoring in computer science or mathematics or both, took the course and flourished in it. Kenneth J. Supowit is an Associate Professor Emeritus, Department of Computer Science & Engineering, Ohio State University, Columbus, Ohio, US.
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.
Download or read book Information and Randomness written by Cristian S. Calude and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first edition of the monograph Information and Randomness: An Algorithmic Perspective by Crist ian Calude was published in 1994. In my Foreword I said: "The research in algorithmic information theory is already some 30 years old. However, only the recent years have witnessed a really vigorous growth in this area. . . . The present book by Calude fits very well in our series. Much original research is presented. . . making the approach richer in consequences than the classical one. Remarkably, however, the text is so self-contained and coherent that the book may also serve as a textbook. All proofs are given in the book and, thus, it is not necessary to consult other sources for classroom instruction. " The vigorous growth in the study of algorithmic information theory has continued during the past few years, which is clearly visible in the present second edition. Many new results, examples, exercises and open prob lems have been added. The additions include two entirely new chapters: "Computably Enumerable Random Reals" and "Randomness and Incom pleteness". The really comprehensive new bibliography makes the book very valuable for a researcher. The new results about the characterization of computably enumerable random reals, as well as the fascinating Omega Numbers, should contribute much to the value of the book as a textbook. The author has been directly involved in these results that have appeared in the prestigious journals Nature, New Scientist and Pour la Science.
Download or read book Computability Theory written by S. Barry Cooper and published by CRC Press. This book was released on 2017-09-06 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Algorithmic Randomness and Complexity written by Rodney G. Downey and published by Springer Science & Business Media. This book was released on 2010-10-29 with total page 883 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computability and complexity theory are two central areas of research in theoretical computer science. This book provides a systematic, technical development of "algorithmic randomness" and complexity for scientists from diverse fields.
Download or read book Logic and Theory of Algorithms written by Arnold Beckmann and published by Springer Science & Business Media. This book was released on 2008-06-03 with total page 613 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 4th International Conference on Computability in Europe, CiE 2008, held in Athens, Greece, in June 2008. The 36 revised full papers presented together with 25 invited tutorials and lectures were carefully reviewed and selected from 108 submissions. Among them are papers of 6 special sessions entitled algorithms in the history of mathematics, formalising mathematics and extracting algorithms from proofs, higher-type recursion and applications, algorithmic game theory, quantum algorithms and complexity, and biology and computation.
Download or read book Problems in the Constructive Trend in Mathematics IV written by V. P. Orevkov and published by American Mathematical Soc.. This book was released on 1970 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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
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 289 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.
Download or read book Computability and Models written by Barry S. Cooper and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Problems in the Constructive Trend in Mathematics V written by V. P. Orevkov and published by American Mathematical Soc.. This book was released on 1972 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Matemati eskaja Logika Teorija Algoritmov i Teorija Mno estv written by S. I. Adi︠a︡n and published by American Mathematical Soc.. This book was released on 1977 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: Papers celebrating Petr Sergeevič Novikov and his work in descriptive set theory and algorithmic problems of algebra.
Download or read book The Theory of Computability written by R. Sommerhalder and published by Addison Wesley Publishing Company. This book was released on 1988 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Handbook of the History and Philosophy of Mathematical Practice written by Bharath Sriraman and published by Springer Nature. This book was released on with total page 3221 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Computability and Complexity written by Adam Day and published by Springer. This book was released on 2016-11-30 with total page 788 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Festschrift is published in honor of Rodney G. Downey, eminent logician and computer scientist, surfer and Scottish country dancer, on the occasion of his 60th birthday. The Festschrift contains papers and laudations that showcase the broad and important scientific, leadership and mentoring contributions made by Rod during his distinguished career. The volume contains 42 papers presenting original unpublished research, or expository and survey results in Turing degrees, computably enumerable sets, computable algebra, computable model theory, algorithmic randomness, reverse mathematics, and parameterized complexity, all areas in which Rod Downey has had significant interests and influence. The volume contains several surveys that make the various areas accessible to non-specialists while also including some proofs that illustrate the flavor of the fields.
Download or read book Indivisible Characteristics of Recursively Enumerable Sets written by Ellen S. Chih and published by . This book was released on 2015 with total page 54 pages. Available in PDF, EPUB and Kindle. Book excerpt: A split of an r.e. set A is a pair of disjoint r.e. sets whose union is A. We investigate information theoretic properties of r.e. sets and properties of their enumerations, and whether these properties are preserved under splittings. The first part is involved with dynamic notions. An r.e. set is speedable if for every computable function, there exists a finite algorithm enumerating membership faster, by the desired computable factor, on infinitely many integers. Remmel (1986) asked whether every speedable set could be split into two speedable sets. Jahn published a proof, answering their question positively. However, his proof was seen to be incorrect and we construct a speedable set that cannot be split into speedable sets, answering their question negatively. We also prove additional splitting results related to speedability. The second part is involved with effectively closed sets. We introduce the notion of being recursively avoiding and prove several results.
Download or read book Finite Semigroups And Universal Algebra written by Jorge Almeida and published by World Scientific. This book was released on 1995-01-27 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: Motivated by applications in theoretical computer science, the theory of finite semigroups has emerged in recent years as an autonomous area of mathematics. It fruitfully combines methods, ideas and constructions from algebra, combinatorics, logic and topology. In simple terms, the theory aims at a classification of finite semigroups in certain classes called “pseudovarieties”. The classifying characteristics have both structural and syntactical aspects, the general connection between them being part of universal algebra. Besides providing a foundational study of the theory in the setting of arbitrary abstract finite algebras, this book stresses the syntactical approach to finite semigroups. This involves studying (relatively) free and profinite free semigroups and their presentations. The techniques used are illustrated in a systematic study of various operators on pseudovarieties of semigroups.
