© 2015

Higher-Order Computability

  • Valuable for researchers in mathematical logic and theoretical computer science

  • Consolidates work carried out in this domain since the 1950s

  • Asks what ‘computability’ means for data more complex than natural numbers


Part of the Theory and Applications of Computability book series (THEOAPPLCOM)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Background and General Theory

    1. Front Matter
      Pages 1-2
    2. John Longley, Dag Normann
      Pages 3-34
    3. John Longley, Dag Normann
      Pages 35-50
    4. John Longley, Dag Normann
      Pages 51-114
    5. John Longley, Dag Normann
      Pages 115-164
  3. Particular Models

    1. Front Matter
      Pages 165-166
    2. John Longley, Dag Normann
      Pages 167-210
    3. John Longley, Dag Normann
      Pages 211-278
    4. John Longley, Dag Normann
      Pages 279-348
    5. John Longley, Dag Normann
      Pages 349-430
    6. John Longley, Dag Normann
      Pages 431-462
    7. John Longley, Dag Normann
      Pages 463-488
    8. John Longley, Dag Normann
      Pages 489-504
    9. John Longley, Dag Normann
      Pages 505-534
    10. John Longley, Dag Normann
      Pages 535-546
  4. Back Matter
    Pages 547-571

About this book


This book offers a self-contained exposition of the theory of computability in a higher-order context, where 'computable operations' may themselves be passed as arguments to other computable operations. The subject originated in the 1950s with the work of Kleene, Kreisel and others, and has since expanded in many different directions under the influence of workers from both mathematical logic and computer science. The ideas of higher-order computability have proved valuable both for elucidating the constructive content of logical systems, and for investigating the expressive power of various higher-order programming languages.


In contrast to the well-known situation for first-order functions, it turns out that at higher types there are several different notions of computability competing for our attention, and each of these has given rise to its own strand of research. In this book, the authors offer an integrated treatment that draws together many of these strands within a unifying framework, revealing not only the range of possible computability concepts but the relationships between them.


The book will serve as an ideal introduction to the field for beginning graduate students, as well as a reference for advanced researchers.



Computability Models Computability Theory Intensional Models Kleene Computability Lambda Algebras

Authors and affiliations

  1. 1.Informatics ForumSchool of InformaticsEdinburghUnited Kingdom
  2. 2.Dept. of MathematicsThe University of OsloOsloNorway

Bibliographic information

  • Book Title Higher-Order Computability
  • Authors John Longley
    Dag Normann
  • Series Title Theory and Applications of Computability
  • Series Abbreviated Title Theory Applications Computability
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2015
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Computer Science Computer Science (R0)
  • Hardcover ISBN 978-3-662-47991-9
  • Softcover ISBN 978-3-662-51711-6
  • eBook ISBN 978-3-662-47992-6
  • Series ISSN 2190-619X
  • Series E-ISSN 2190-6203
  • Edition Number 1
  • Number of Pages XVI, 571
  • Number of Illustrations 0 b/w illustrations, 2 illustrations in colour
  • Topics Theory of Computation
    Mathematics of Computing
  • Buy this book on publisher's site
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“This book is a true tour de force. The content is both mathematics and computer science; it is simultaneously an introductory textbook for students, a tutorial for seasoned researchers, a reference work, and a research monograph; ... The references to the literature are thorough. The writing is technical mathematics, and is good by that standard: there is ample intuitive explanation, and it is overall readable.” (Robert S. Lubarsky, Mathematical Reviews, April 2018)