Series Overview
The book series Theory and Applications of Computability is published by Springer
in cooperation with the Association Computability in Europe.
Books published in this series will be of interest to the research community and graduate students, with a
unique focus on issues of computability. The perspective of the series is multidisciplinary, recapturing the
spirit of Turing by linking theoretical and realworld concerns from computer science, mathematics, biology,
physics, and the philosophy of science.
The series includes research monographs, advanced and graduate texts, and books that offer an original
and informative view of computability and computational paradigms.
Volume 8
Dusko Pavlovic
Programs as Diagrams
From Categorical Computability to Computable Categories
1st Edition., 2023, XVII, 252 p.
Hardcover, ISBN 9783031348266
It is not always clear what computer programs mean in the various languages in which they can be written,
yet a picture can be worth 1000 words, a diagram 1000 instructions.
In this unique textbook/reference, programs are drawn as string diagrams in the language of categories,
which display a universal syntax of mathematics (Computer scientists use them to analyze the program semantics;
programmers to display the syntax of computations). Here, the stringdiagrammatic depictions of computations
are construed as programs in a singleinstruction programming language. Such programs as diagrams show how
functions are packed in boxes and tied by strings. Readers familiar with categories will learn about the
foundations of computability; readers familiar with computability gain access to category theory.
Additionally, readers familiar with both are offered many opportunities to improve the approach.
Topics and features:
 Delivers a ‘crash’ diagrambased course in theory of computation
 Uses singleinstruction diagrammatic programming language
 Offers a practical introduction into categories and string diagrams as computational tools
 Reveals how computability is programmability, rather than an ‘ether’ permeating computers
 Provides a categorical model of intensional computation is unique up to isomorphism
 Serves as a stepping stone into research of computable categories
In addition to its early chapters introducing computability for beginners,
this flexible textbook/resource also contains both middle chapters that expand for suitability
to a graduate course as well as final chapters opening up new research.


Volume 7
Damir D. Dzhafarov,
Carl Mummert
Reverse Mathematics
Problems, Reductions, and Proofs
1st Edition., 2022, X, 480 p. 20 illus.
Hardcover, ISBN 9783031113666
Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems.
Typical questions include: Can we prove this result without first proving that one? Can a computer solve this
problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful
results as well as significant foundational insights.
This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions
and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics.
It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and
other parts of computability that have become integral to research in the field.
Topics and features:
 Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model construction
 Offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results
 Provides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic arguments
 Includes a large number of exercises of varying levels of difficulty, supplementing each chapter
The text will be accessible to students with a standard first year course in mathematical logic.
It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas.


Volume 6
Vasco Brattka,
Peter Hertling (Eds.)
Handbook of Computability and Complexity in Analysis
1st Edition., 2021, XXIV, 426 p. 23 illus.
Hardcover, ISBN 9783030592332
Computable analysis is the modern theory of computability and complexity in analysis that arose out of Turing's seminal work in the 1930s.
This was motivated by questions such as: which real numbers and real number functions are computable,
and which mathematical tasks in analysis can be solved by algorithmic means?
Nowadays this theory has many different facets that embrace topics from computability theory, algorithmic randomness, computational complexity,
dynamical systems, fractals, and analog computers, up to logic, descriptive set theory, constructivism, and reverse mathematics.
In recent decades computable analysis has invaded many branches of analysis, and researchers have studied computability and complexity questions
arising from real and complex analysis, functional analysis, and the theory of differential equations, up to (geometric) measure theory and topology.
This handbook represents the first coherent crosssection through most active research topics on the more theoretical side of the field.
It contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic,
and descriptive complexity. All chapters are written by leading experts working at the cutting edge of the respective topic.
Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions
into many branches of computable analysis, and a wealth of information and references that will help them to navigate the modern research
literature in this field.


Volume 5
S. Barry Cooper,
Mariya Soskova (Eds.)
The Incomputable
Journeys Beyond the Turing Barrier
1st Edition., 2017, X, 292 p. 10 illus.
Hardcover, ISBN 9783319436678
This book questions the relevance of computation to the physical universe. Our theories deliver computational descriptions,
but the gaps and discontinuities in our grasp suggest a need for continued discourse between researchers from different disciplines,
and this book is unique in its focus on the mathematical theory of incomputability and its relevance for the real world.
The core of the book consists of thirteen chapters in five parts on extended models of computation; the search for natural
examples of incomputable objects; mind, matter, and computation; the nature of information, complexity, and randomness;
and the mathematics of emergence and morphogenesis.
This book will be of interest to researchers in the areas of theoretical computer science, mathematical logic, and philosophy.



Editorial Board
Founding Editors: P. Bonizzoni, V. Brattka, S.B. Cooper, E. Mayordomo
Proposals
Formal or informal book proposals can be
sent directly to members of the editorial board
(contact details are on the linked web pages above).
Series Advisory Board
 Samson Abramsky (Oxford, UK)
 Eric Allender (Rutgers, New Jersey, USA)
 Klaus AmbosSpies (Heidelberg, Germany)
 Giorgio Ausiello (Rome, Italy)
 Jeremy Avigad (Carnegie Mellon, Pittsburgh, USA)
 Samuel R. Buss (California, San Diego, USA)
 Rodney G. Downey (Wellington, New Zealand)
 Sergei S. Goncharov (Novosibirsk, Russia)
 Peter Jeavons (Oxford, UK)
 Nataša Jonoska (South Florida, Tampa, USA)
 Ulrich Kohlenbach (Darmstadt, Germany)
 Ming Li (Waterloo, Canada)
 Wolfgang Maass (Graz, Austria)
 Grzegorz Rozenberg (Leiden, The Netherlands and Colorado, USA)
 Alan Selman (Buffalo, New York, USA)
 Wilfried Sieg (Carnegie Mellon, Pittsburgh, USA)
 Jan van Leeuwen (Utrecht, The Netherlands)
 Klaus Weihrauch (Hagen, Germany)
 Philip Welch (Bristol, UK)
Series
Volume 4
Robert I. Soare
Turing Computability
Theory and Applications
1st Edition., 2016, XXXVI, 263 p. 4 illus.
Hardcover, ISBN 9783642319327
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.


Volume 3
Dag Normann,
John Longley
Higher Order Computability
1st Edition., 2015, X, 575 p. 2 illus.
Hardcover, ISBN 9783662479919
This book offers a selfcontained exposition of the theory of computability in a higherorder 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 higherorder computability have proved valuable both for elucidating the constructive content
of logical systems, and for investigating the expressive power of various higherorder programming languages.


Volume 2
Douglas S. Bridges,
Luminiţa Simona Vîţă
Apartness and Uniformity
A Constructive Development
1st Edition., 2011, XIV, 198 p. 3 illus.
Hardcover, ISBN 9783642224140
Largely an exposition of the authors' own research, this is the first
book dealing with the apartness approach to constructive topology, and
is a valuable addition to the literature on constructive mathematics and
on topology in computer science. It is aimed at graduate students and
advanced researchers in theoretical computer science, mathematics, and
logic who are interested in constructive/algorithmic aspects of
topology.


Volume 1
Rodney G. Downey,
Denis R. Hirschfeldt
Algorithmic Randomness and Complexity
1st Edition., 2010, XXVIII, 855 p. 8 illus.
Hardcover, ISBN 9780387955674
This is the first comprehensive treatment of this important field, designed to be both a
reference tool for experts and a guide for newcomers. It surveys a broad section of work
in the area, and presents most of its major results and techniques in depth. It will be of
interest to researchers and students in computability theory, algorithmic information theory,
and theoretical computer science.
This book has received the Shoenfield Prize 2016, which is awarded by the Association for Symbolic
Logic for outstanding expository writing in the field of logic.


