© 2008

New Computational Paradigms

Changing Conceptions of What is Computable

  • S. Barry Cooper
  • Benedikt Löwe
  • Andrea Sorbi

Table of contents

  1. Front Matter
    Pages i-xiii
  2. The Turing Model of Computation and its Applications to Logic, Mathematics, Philosophy, and Computer Science

    1. Andrew Hodges
      Pages 3-15
    2. Serikzhan Badaev, Sergey Goncharov
      Pages 19-34
    3. Johan van Benthem
      Pages 35-58
    4. Yiannis N. Moschovakis, Vasilis Paschalis
      Pages 87-118
    5. Viggo Stoltenberg-Hansen, John V. Tucker
      Pages 153-194
    6. Jiří Wiedermann, Dana Pardubská
      Pages 195-209
  3. Logic, Algorithms and Complexity

  4. Models of Computation from Nature

  5. Computable Analysis and Real Computation

    1. Olivier Bournez, Manuel L. Campagnolo
      Pages 383-423
    2. Vasco Brattka, Peter Hertling, Klaus Weihrauch
      Pages 425-491

About this book


In recent years, classical computability has expanded beyond its original scope to address issues related to computability and complexity in algebra, analysis, and physics. The deep interconnection between "computation" and "proof" has originated much of the most significant work in constructive mathematics and mathematical logic of the last 70 years. Moreover, the increasingly compelling necessity to deal with computability in the real world (such as computing on continuous data, biological computing, and physical models) has brought focus to new paradigms of computation that are based on biological and physical models. These models address questions of efficiency in a radically new way and even threaten to move the so-called Turing barrier, i.e. the line between the decidable and the un-decidable.

This book examines new developments in the theory and practice of computation from a mathematical perspective, with topics ranging from classical computability to complexity, from biocomputing to quantum computing. The book opens with an introduction by Andrew Hodges, the Turing biographer, who analyzes the pioneering work that anticipated recent developments concerning computation’s allegedly new paradigms. The remaining material covers traditional topics in computability theory such as relative computability, theory of numberings, and domain theory, in addition to topics on the relationships between proof theory, computability, and complexity theory. New paradigms of computation arising from biology and quantum physics are also discussed, as well as the computability of the real numbers and its related issues.

This book is suitable for researchers and graduate students in mathematics, philosophy, and computer science with a special interest in logic and foundational issues. Most useful to graduate students are the survey papers on computable analysis and biological computing. Logicians and theoretical physicists will also benefit from this book.


Analysis algorithms complexity complexity theory computability theory computer computer science information theory logic mathematical logic model theory philosophy proof theory

Editors and affiliations

  • S. Barry Cooper
    • 1
  • Benedikt Löwe
    • 2
  • Andrea Sorbi
    • 3
  1. 1.University of LeedsUK
  2. 2.University of AmsterdamNetherlands
  3. 3.Università di SienaItaly

Bibliographic information


From the reviews:

“It is addressed to researcher and graduate students … . All contributions to the book have been rigorously refereed, and the standards with respect to layout, references … are high. … This is a piece of excellent pedagogical work. The paper is hereby recommended. … I personally find very readable and informative. … I enjoyed reading these papers, and I assume they are all right when we take them for what they are … .”­­­ (Lars Kristiansen, Studia Logica, Vol. 97, 2011)