Turing Computability: Theory and Applications (Theory and Applications of Computability) - Hardcover

Soare, Robert I.

Hardcover

ISBN 10: 3642319327 ISBN 13: 9783642319327

Verlag: Springer, 2016

Inhaltsangabe

<P>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.&NBSP;</P><P>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.</P><P>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.</P><P></P>

Über diesen Titel

Críticas

“At a time when computability theory is enjoying remarkable activity and fruitfulness, and benefiting from having a large number of students and young researchers, there is no question that the subject is ready for a new standard introductory text. The present book shares all the features that helped its predecessor become such a standard thirty years ago, and at the same time, it is modern, and it is relevant to today’s state of the field. The subject will be well-served by it.” (Damir D. Dzhafarov, Bulletin of Symbolic Logic, Vol. 23 (1), March, 2017)

Reseña del editor

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.

Details zur Ausgabe

• Verlag: Springer
• Erscheinungsdatum: 2016
• ISBN 10: 3642319327
• ISBN 13: 9783642319327
• Einband: Tapa dura
• Sprache: Englisch
• Auflage: 1
• Anzahl der Seiten: 300