Turing’s Revolution : The Impact of His Ideas about Computability This book provides an overview of the confluence of ideas in Turing’s era and work and examines the impact of his work on mathematical logic and theoretical computer science. It combines contributions by well known scientists on the history and philosophy of computability theory as well as on generalised Turing computability. By looking at the roots and at the philosophical and technical influence of Turing’s work it is possible to gather new perspectives and new research topics which might be considered as a continuation of Turing’s working ideas well into the 21st century Preface (Martin Davis). I Introduction (Sommaruga Strahm). II Turing and the history of computability theory 1. Conceptual Confluence in 1936: Post & Turing Martin Davis and Wilfried Sieg 2. Algorithms: From Al Khwarizmi to Turing and Beyond Wolfgang Thomas 3. The Stored Program Universal Computer: Did Zuse Anticipate Turing and von Neumann Jack Copeland and Giovanni Sommaruga III Generalizing Turing computability theory 1. Theses for Computation and Recursion on Concrete and Abstract Structures Solomon Feferman 2. Generalizing Computability Theory to Abstract Algebras John V. Tucker and Jeffrey Zucker 3. Discrete Transfinite Computation Philip Welch 4. Semantics to Syntax Analyses of Algorithms Yuri Gurevich 5. The Information Content of Typical Reals George Barmpalias and Andy Lewis Pye 6. Proof theoretic Analysis by Iterated Reflection Lev Beklemishev. IV Philosophical reflections 1. Alan Turing and the Foundation of Computer Science Juraj Hromkovic 2. Proving Things about the Informal Stewart Shapiro 3. Why Turing’s Thesis is Not a Thesis Robert Soare 4. Incomputability Emergent and Higher Type Computation S. Barry Cooper