Turing Computability

Robert Soare is the Paul Snowden Russell Distinguished Service Professor Emeritus of Mathematics and Computer Science at the University of Chicago. He was the founding chairman of the Department of Computer Science in 1983. He has supervised the dissertations of nineteen Ph.D. students using the content of this book. He wrote the primary reference on computability theory for students and researchers: Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets (Springer, 1987). He is the author of numerous papers on computability theory and mathematical logic. His 1974 Annals of Mathematics paper on automorphisms of computably enumerable sets was selected in the 2003 book by Gerald Sacks as one of the most important in mathematical logic in the twentieth century. He has been an invited speaker at the International Congress of Mathematicians, and a plenary speaker at the International Congress of Logic, Methodology, and Philosophy of Science, the Association of Symbolic Logic Centennial in 2000, the British Mathematical Colloquium in 2012, the Royal Society Meeting on the Incomputable in 2012, and Computability in Europe (CiE) in 2007 and 2012. He was the winner of the 2011 University of Chicago Award for Excellence in Graduate Teaching and is a Fellow of the American Mathematical Society.

Part I Foundations of Computability.- Chap. 1 Defining Computability.- Chap. 2 Computably Enumerable Sets.- Chap. 3 Turing Reducibility.- Chap. 4 The Arithmetical Hierarchy.- Chap. 5 Classifying C.E. Sets.- Chap. 6 Oracle Constructions and Forcing.- Chap. 7 The Finite Injury Method.- Part II Trees and 01 Classes.- Chap. 8 Open and Closed Classes.- Chap. 9 Basis Theorems.- Chap. 10 Peano Arithmetic and 01-Classes.- Chap. 11 Randomness and 01-Classes.- Part III Minimal Degrees.- Chap. 12 Minimal Degrees Below Ø .- Chap. 13 Minimal Degrees Below Ø .- Part IV Games in Computability Theory.- Chap. 14 Banach-Mazur Games.- Chap. 15 Gale-Stewart Games.- Chap. 16 More Lachlan Games.- Part V History of Computability.- Chap. 17 History of Computability.- References.- Index.

"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)

“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)