Turing Computability: Theory and Applications

Description

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. Front Matter....Pages i-xxxvi Front Matter....Pages 1-1 Defining Computability....Pages 3-22 Computably Enumerable Sets....Pages 23-50 Turing Reducibility....Pages 51-78 The Arithmetical Hierarchy....Pages 79-105 Classifying C.E. Sets....Pages 107-129 Oracle Constructions and Forcing....Pages 131-146 The Finite Injury Method....Pages 147-162 Front Matter....Pages 163-163 Open and Closed Classes....Pages 165-173 Basis Theorems....Pages 175-182 Peano Arithmetic and \(\Pi_1^0\) -Classes....Pages 183-187 Randomness and \(\Pi_1^0\) -Classes....Pages 189-194 Front Matter....Pages 195-195 Minimal Degrees Below \(\emptyset^{\prime\prime}\) ....Pages 197-202 Minimal Degrees Below \(\emptyset^{\prime}\) ....Pages 203-208 Front Matter....Pages 209-209 Banach-Mazur Games....Pages 211-216 Gale-Stewart Games....Pages 217-219 More Lachlan Games....Pages 221-224 Front Matter....Pages 225-225 History of Computability....Pages 227-249 Back Matter....Pages 251-263