Higher-Order Computability

587 Pages·2015·5.196 MB·English

by  John Longley, Dag Normann (auth.)

John Longley · Dag Normann
Higher-Order
Computability
Theory and Applications of Computability
In cooperation with the association Computability in Europe
Series Editors
Prof. P. Bonizzoni
Università degli Studi di Milano-Bicocca
Dipartimento di Informatica Sistemistica e Comunicazione (DISCo)
Milan
Italy
[e-mail]

Prof. V. Brattka
Universität der Bundeswehr München
Fakultät für Informatik
Neubiberg
Germany
[e-mail]

Prof. S.B. Cooper
University of Leeds
Department of Pure Mathematics
Leeds
UK
[e-mail]

Prof. E. Mayordomo
Universidad de Zaragoza
Departamento de Informática e Ingeniería de Sistemas
Zaragoza
Spain
[e-mail]
Prof. P. Panangaden
McGill University
School of Computer Science
Montréal
Canada
[e-mail]
More information about this series at http://www.springer.com/series/8819
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 real-world 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.

Series Advisory Board

Samson Abramsky, University of Oxford
Eric Allender, Rutgers, The State University of New Jersey
Klaus Ambos-Spies, Universität Heidelberg
Giorgio Ausiello, Università di Roma, “La Sapienza”
Jeremy Avigad, Carnegie Mellon University
Samuel R. Buss, University of California, San Diego
Rodney G. Downey, Victoria University of Wellington
Sergei S. Goncharov, Novosibirsk State University
Peter Jeavons, University of Oxford
Nataša Jonoska, University of South Florida, Tampa
Ulrich Kohlenbach, Technische Universität Darmstadt
Ming Li, University of Waterloo
Wolfgang Maass, Technische Universität Graz
Grzegorz Rozenberg, Leiden University and University of Colorado, Boulder
Alan Selman, University at Buffalo, The State University of New York
Wilfried Sieg, Carnegie Mellon University
Jan van Leeuwen, Universiteit Utrecht
Klaus Weihrauch, FernUniversität Hagen
Philip Welch, University of Bristol
John Longley • Dag Normann
Higher-Order Computability
John Longley Dag Normann
School of Informatics Department of Mathematics
The University of Edinburgh The University of Oslo
Edinburgh, UK Oslo, Norway
ISSN 2190-619X ISSN 2190-6203 (electronic)
Theory and Applications of Computability
ISBN 978-3-662-47991-9 ISBN 978-3-662-47992-6 (eBook)
DOI 10.1007/978-3-662-47992-6

Library of Congress Control Number: 2015951088

Springer Heidelberg New York Dordrecht London
© Springer-Verlag Berlin Heidelberg 2015
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or
information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt
from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained herein
or for any errors or omissions that may have been made.

Printed on acid-free paper

Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media (www.springer.com)
ToCarolineandSvanhild

Preface
This book serves as an introduction to an area of computability theory that origi-
natedinthe1950s,andsincethenhasfannedoutinmanydifferentdirectionsunder
theinfluenceofworkersfrombothmathematicallogicandtheoreticalcomputersci-
ence.WhereastheworkofChurchandTuringfromthe1930sprovidedadefinitive
conceptofcomputabilityfornaturalnumbersandsimilardiscretedata,ourpresent
inquiry begins by asking what ‘computability’ might mean for data of more com-
plexkinds.Inparticular,canonedevelopagoodtheoryofcomputabilityinsettings
where ‘computable operations’ may themselves be passed as inputs to other com-
putableoperations?Whatdoesitmeanto‘computewith’(forexample)afunction
whoseargumentsarethemselvesfunctions?
Conceptsofcomputabilityinhigher-ordersettingsareofinterestforavarietyof
reasons within both logic and computer science. For example, from a metamathe-
matical point of view, questions of computability arise naturally in the attempt to
elucidateandstudynotionsofconstructivityinmathematics.Inordertogiveacon-
structiveinterpretationforsomemathematicaltheoryinvolvingfunctions,realnum-
bers, operators on spaces of real-valued functions, or whatever, one needs, first of
all, a clear idea of what is meant by a ‘constructive presentation’ of these math-
ematical objects, and secondly, an understanding of what ways of manipulating
thesepresentationsareacceptedaslegitimateconstructions—inotherwords,ano-
tionofcomputabilityfortheobjectsinquestion.Fromacomputersciencepointof
view,conceptsofhigher-ordercomputabilitybearnaturallyonthequestionofwhat
can and cannot be computed (in principle) within various kinds of programming
languages—particularlylanguagesthatmanipulatehigher-orderdatawhichcannot
bereducedtonaturalnumbersintherequisitesense.Therearealsoapplicationsin
whichthelogicandcomputersciencestrandsintertwineclosely,forinstanceinthe
extractionofcomputerprogramsfrommathematicalproofs.
A particular emphasis of the present book will be the way in which both logic
and computer science have had a deep and formative influence on the theory of
higher-ordercomputability.Thisdualheritageistosomeextentreflectedinthecol-
laborationthathasgivenrisetothisbook,withthesecondauthorrepresentingthe
older mathematical logic tradition, and the first author the more modern computer
vii
viii Preface
scienceone.Asweshallsee,thetheoryinitspresentformconsistsofideasdevel-
opedsometimesjointly,thoughoftenindependently,byavarietyofresearchcom-
munitieseachwithitsowndistinctivemotivations—however,thethesisofthebook
is that these can all be seen, with hindsight, as contributions to a single coherent
subject.
In contrast to the situation for the natural numbers, where the Church–Turing
thesis encapsulates the idea that there is just a single reasonable notion of effec-
tive computability, it turns out that at higher types there are a variety of possible
notions competing for our attention. Indeed, in the early decades of the subject, a
profusionofdifferentconceptsofcomputabilitywereexplored,eachbroadlygiving
risetoitsownstrandofresearch:Kleene’s‘S1–S9’computabilityinaset-theoretic
setting, computability for the total continuous functionals of Kleene and Kreisel,
thepartialcomputablefunctionalsofScottandErshov,thesequentiallycomputable
functionalsasrepresentedbyPlotkin’sprogramminglanguagePCF,thesequential
algorithms of Berry and Curien, and others besides. All of this has given rise to a
largeandbewilderingliterature,withwhichbothauthorshavepreviouslyattempted
to grapple in their respective survey papers [178, 215]. More recently, however,
many of these strands have started to re-converge, owing largely to increased in-
teractionamongtheresearchcommunitiesinvolved,andaunifyingperspectivehas
emergedwhichallowsallofthemtobefittedintoacoherentandconceptuallysatis-
fyingpicture,revealingnotonlytherangeofpossiblecomputabilitynotionsbutthe
relationships between them. The purpose of this book is to present this integrated
viewofthesubjectinasystematicandself-containedway,accessible(forinstance)
toabeginninggraduatestudentwithareasonablegroundinginlogicandthetheory
ofcomputation.
Ouremphasiswillbeonthe‘puretheory’ofcomputabilityinhigher-orderset-
tings:roughlyspeaking,foreachnotionofcomputabilitythatweconsider,weshall
developabodyofdefinitions,theoremsandexamplesbroadlyanalogoustothaten-
countered in a typical introductory course on classical (first-order) computability.
Rather little space will be devoted to applications of these ideas in other areas of
logicorcomputerscience,althoughweshallsometimesalludetosuchapplications
asmotivationsforourstudy.
Thebookisdividedintotwoparts.InPartI(Chapters1–4)weoutlineourvision
of the subject as a whole, introducing the main ideas and reviewing their history,
thendevelopingthegeneralmathematicalframeworkwithinwhichweconductour
investigation. In Part II (Chapters 5–13) we consider, in turn, a succession of par-
ticularcomputabilitynotions(approximatelyoneperchapter)thathaveemergedas
conceptuallynaturalandmathematicallyinteresting.Ouradvicetothereaderisto
beginwithChapter1foraninformalintroductiontothemainconceptsandtheintu-
itionsandmotivationsbehindthem,thenproceedtoChapter3forthegeneraltheory
along with some key examples that will play a pervasive role later on. Thereafter,
theremainingchaptersmaybetackledinmoreorlessanyorderwithonlyamoder-
ateamountofcross-referral.Furtherdetailsofthebook’sstructureandthelogical
dependenciesbetweenchaptersmaybefoundinSection1.3.
Preface ix
The idea of writing this book matured during the second author’s visit to the
University of Edinburgh in the spring of 2009. Whilst the second author’s earlier
book [207] is often cited as a reference work in the area, it only covers one part
ofthe story, andthatpart only upto1980, andwebothfelt thatthetimewas ripe
for a more comprehensive work mapping out the current state of the field for the
benefit of future researchers. In 2009 we agreed on an outline for the contents of
such a book, but our writing began in earnest in 2011, with each of us producing
initialdraftsfordesignatedpartsofthevolume.Inordertoensurecoherenceatthe
leveloflinguisticstyleanduseofterminology,thefirstauthorthenpreparedafull
versionofthemanuscript,whichwasthenrevisedinthelightofthesecondauthor’s
comments.Theauthorshavebeeninsteadycontactwitheachotherthroughoutthe
wholeprocess,andtheyacceptequalresponsibilityforthefinalproduct.1
Ithasbeenourprivilegeovermanyyearstohavedevelopedacloseacquaintance
withthispartofthemathematicallandscapethroughourvarioustraversalsthereof,
andourjourneyshavebeenimmeasurablyenrichedbythecompanyofthenumer-
ous fellow travellers from whom we have learned along the way and with whom
we have shared ideas. Our thanks go to Samson Abramsky, Andrej Bauer, Ulrich
Berger, Jan Bergstra, Chantal Berline, Antonio Bucciarelli, Pierre-Louis Curien,
Thomas Ehrhard, Yuri Ershov, Mart´ın Escardo´, Solomon Feferman, Mike Four-
man,PieterHofstra,MartinHyland,AchimJung,JimLaird,GuyMcCusker,Paul-
Andre´ Mellie`s,JohanMoldestad,YiannisMoschovakis,HannoNickau,LukeOng,
JaapvanOosten,GordonPlotkin,JimRoyer,VladimirSazonov,MatthiasSchro¨der,
Helmut Schwichtenberg, Dana Scott, Alex Simpson, Viggo Stoltenberg-Hansen,
ThomasStreicher andStan Wainer,amongmany others.We alsowish topay par-
ticulartributetothememoryofRobinGandy,whoseinfluencecannotbeoveresti-
mated.
WeareverygratefultoBarryCooperandtheComputabilityinEuropeeditorial
board for the opportunity to publish this work under their auspices, and for their
supportandencouragementthroughouttheproject.WealsothankRonanNugentof
Springer-Verlag for his guidance and assistance with the publication process, and
thecopyeditorfordetectingnumerousminorerrors.
Our academic host organizations, the School of Informatics at the University
of Edinburgh and the Department of Mathematics at the University of Oslo, have
formanyyearsprovidedcongenialandstimulatingenvironmentsforourwork,and
we have each benefited from the hospitality of the other’s institution. Finally, our
heartfeltthanksgotoourrespectivewives,CarolineandSvanhild,fortheirpatience
andlovethroughoutthewritingofthebook,andfortheircheerfultoleranceofthe
levelsofmentalpreoccupationthatsuchundertakingsseeminvariablytoengender.
Edinburgh,Oslo JohnLongley
October2014 DagNormann
1 Alistofknownerrataandotherupdatesrelatingtothecontentofthebookwillbemaintained
onlineathttp://homepages.inf.ed.ac.uk/jrl/HOC updates.pdf.