New Computational Paradigms: Changing Conceptions of What is Computable

551 Pages·2008·4.904 MB·English

by  Andrew Hodges (auth.), S. Barry Cooper, Benedikt Löwe, Andrea Sorbi (eds.)

New Computational Paradigms
New
(cid:1)(cid:1) (cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1) (cid:1)
Computational Paradigms
Changing Conceptions
of What is Computable
S. Barry Cooper
Benedikt Lo¨we
Andrea Sorbi
Editors
S.BarryCooper BenediktLo¨we
DepartmentofPureMathematics InstituteforLogic,Languageand
UniversityofLeeds Computation
LeedsLS29JT UniversityofAmsterdam
UnitedKingdom PlantageMuidergracht24
[email protected] 1018TVAmsterdam
TheNetherlands
[email protected]
AndreaSorbi
DepartmentofMathematicsand
ComputerScience“RobertoMagari”
Universita`diSiena
PiandeiMantellini44
53100Siena
Italy
[email protected]
ISBN-13:978-0-387-36033-1 e-ISBN-13:978-0-387-68546-5
DOI:10.1007/978-0-387-68546-5
LibraryofCongressControlNumber:2007935311
MathematicsSubjectClassification(2000):68Q01,68Q05
(cid:1)c2008SpringerScience+BusinessMedia,LLC
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Contents
Preface ......................................................... ix
ListofContributors .............................................. xi
AlanTuring,LogicalandPhysical
AndrewHodges................................................... 3
Part I The Turing Model of Computationand its Applications to Logic,
Mathematics,Philosophy,andComputerScience
ComputabilityandNumberings
SerikzhanBadaev,SergeyGoncharov ................................. 19
ComputationasConversation
JohanvanBenthem................................................ 35
ComputationParadigmsinLightofHilbert’sTenthProblem
YuriMatiyasevich ................................................. 59
ElementaryAlgorithmsandTheirImplementations
YiannisN.Moschovakis,VasilisPaschalis .............................. 87
Applicationsofthe Kleene–KreiselDensity Theoremto Theoretical
ComputerScience
DagNormann ....................................................119
ChurchWithoutDogma:AxiomsforComputability
WilfriedSieg .....................................................139
vi Contents
ComputabilityonTopologicalSpacesviaDomainRepresentations
ViggoStoltenberg-Hansen,JohnV.Tucker..............................153
OnthePowerofBroadcastinginMobileComputing
JirˇíWiedermann,DanaPardubská....................................195
PartII Logic,AlgorithmsandComplexity
TheComputationalPowerofBoundedArithmeticfromthePredicative
Viewpoint
SamuelR.Buss ...................................................213
Effective Uniform Bounds from Proofs in Abstract Functional
Analysis
UlrichKohlenbach ................................................223
EffectiveFractalDimensioninAlgorithmicInformationTheory
ElviraMayordomo ................................................259
MetamathematicalPropertiesofIntuitionisticSetTheorieswithChoice
Principles
MichaelRathjen ..................................................287
NewDevelopmentsinProofsandComputations
HelmutSchwichtenberg ............................................313
PartIII ModelsofComputationfromNature
FromCellsto(Silicon)Computers,andBack
GheorghePa˘un...................................................343
Computer Science,Informatics,andNaturalComputing—Personal
Reflections
GrzegorzRozenberg ...............................................373
PartIV ComputableAnalysisandRealComputation
ASurveyonContinuousTimeComputations
OlivierBournez,ManuelL.Campagnolo...............................383
ATutorialonComputableAnalysis
VascoBrattka,PeterHertling,KlausWeihrauch .........................425
AContinuousDerivativeforReal-ValuedFunctions
AbbasEdalat.....................................................493
Contents vii
InfiniteTimeComputableModelTheory
JoelDavidHamkins,RussellMiller,DanielSeabold,SteveWarner ..........521
Index ............................................................. 559
Preface
June2005sawthecomingtogether,inAmsterdam,ofthefirstmeetingofanewre-
searchcommunity,whichsoughttorenew,inthenewcentury,theground-breaking
legacyof Alan Turing.Computabilityin Europe(CiE) originatedwith a 2003pro-
posal for EU funding but rapidly developed into a Europe-wide network of over
400researchersfrom17countries,around70institutions,andanumberofdifferent
research disciplines (mathematics, computerscience, physics, biology,philosophy,
andlogic).Thisbookofinvited(andrigorouslyrefereed)articlesshowcasesthedi-
versity,excitement,andscientificinnovationofthatfirstmeeting,andthepowerful
multidisciplinaritythatitinjectedintocomputationalresearch.
Many of the contributionsto be foundhere reflect the necessity to deal with com-
putability in the real world—computing on continuous data, biological comput-
ing, physical computing, etc.—which has focused attention on new paradigms of
computation,basedonbiologicalandphysicalmodels.Thisbooklooksatnewdevel-
opmentsinthetheoryandpracticeofcomputationfromamathematicalandpredom-
inantlylogicalperspective,withtopicsrangingfromclassicalcomputabilitytocom-
plexity,biocomputing,andquantumcomputing.Traditionaltopicsincomputability
theoryare also coveredas wellas relationshipsamongprooftheory,computability
andcomplexitytheory,andnewparadigmsofcomputationarisingfrombiologyand
quantumphysicsandissuesrelatedtocomputabilitywith/ontherealnumbers.The
bookisaddressedtoresearchersandgraduatestudentsinmathematics,philosophy,
and computer science with a special interest in foundationalissues. Logicians and
theoreticalphysicistswillalsobenefitfromthisbook.
Sincethatfirstconference,CiEhasbecomemorethanthesumofitsparts,reasserting
anoldertraditionofscientificresearch.Thismorethoughtfulapproachiswhatthis
1944quotationfromEinstein1seemstoreferto:
1A.EinsteintoR.A.Thornton,unpublishedletterdatedDec.7,1944;inEinsteinArchive,
HebrewUniversity,Jerusalem.
x Preface
Somanypeopletoday—andevenprofessionalscientists—seem tomelike
someone who has seen thousands of trees but has never seen a forest.
Aknowledgeofthehistoricalandphilosophicalbackgroundgivesthatkind
ofindependencefromprejudicesofhisgenerationfromwhichmostscien-
tistsaresuffering.Thisindependencecreatedbyphilosophicalinsightis—
inmyopinion—themarkofdistinctionbetweenamereartisanorspecialist
andarealseekeraftertruth.
Thereisaparallelbetweenthecompetitivehyperactivespecialismofpartsofcom-
puterscience(andlogic)andthatofthestringtheorycommunitythatLeeSmolin2
focusesoninhisrecentbook.Hepinpoints:
... a more brash, aggressive, and competitive atmosphere, in which theo-
rists vie to respondquicklyto new developments... andare distrustfulof
philosophicalissues. This style supplanted the more reflective, philosoph-
ical style that characterized Einstein and the inventors of quantum theory,
andittriumphedasthecenterofsciencemovedtoAmericaandtheintellec-
tualfocusmovedfromtheexplorationoffundamentalnewtheoriestotheir
application.
ThisbookembodieswhatisspecialaboutwhatCiEistryingtodointakingcomputa-
tionalresearchbeyondtheconstraintsof“normalscience,”whilebuildingacohesive
researchcommunityaroundfundamentalissuesofcomputability.
Leeds S.BarryCooper
Amsterdam BenediktLöwe
Siena AndreaSorbi
July2007
2L.Smolin,TheTroubleWithPhysics:TheRiseofStringTheory,theFallofaScienceand
WhatComesNext,HoughtonMifflin,2006.
List of Contributors
SerikzhanBadaev SamuelR.Buss
KazakhNationalUniversity,Almaty DepartmentofMathematics,University
050038,Kazakhstan ofCalifornia,SanDiego,LaJolla,CA
[email protected] 92093-0112,U.S.A.
[email protected]
JohanvanBenthem
InstituteforLogic,Language& ManuelL.Campagnolo
DM/ISA, Technical University of
Computation(ILLC), University of
Amsterdam,Amsterdam,1018TV,the Lisbon,Tapadada Ajuda, 1349-017
Lisboa,Portugal
Netherlands
and
[email protected]
and SQIG/ITLisboa
DepartmentofPhilosophy,Stanford [email protected]
University, Stanford, CA 94305,
AbbasEdalat
U.S.A.
DepartmentofComputing,Imperial
[email protected]
CollegeLondon,LondonSW7 2AZ,
UnitedKingdom
OlivierBournez
[email protected]
INRIA LorraineandLORIA (UMR
7503CNRS-INPL-INRIA-Nancy2- SergeyGoncharov
UHP),BP23954506Vandœuvre-Lès-
Institute ofMathematicsof Siberian
Nancy,France
Branch of Russian Academy of
[email protected] Sciences, Novosibirsk 6300090,
Russia
VascoBrattka
[email protected]
Laboratoryof FoundationalAspects
ofComputerScience,Departmentof JoelDavidHamkins
MathematicsandAppliedMathematics, TheCollegeofStatenIslandofTheCity
UniversityofCapeTown,Rondebosch UniversityofNewYork,Mathematics,
7701,SouthAfrica StatenIsland,NY10314,U.S.A.
[email protected] and