Higher-Order Computability PDF
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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 [email protected] Prof. V. Brattka Universität der Bundeswehr München Fakultät für Informatik Neubiberg Germany [email protected] Prof. S.B. Cooper University of Leeds Department of Pure Mathematics Leeds UK [email protected] Prof. E. Mayordomo Universidad de Zaragoza Departamento de Informática e Ingeniería de Sistemas Zaragoza Spain [email protected] Prof. P. Panangaden McGill University School of Computer Science Montréal Canada [email protected] 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.
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