Alan Turing: His Work and Impact

903 Pages·2013·11.72 MB·English

Alan Mathison Turing by Max Newman
(BibliographicMemoirsoftheFellowsoftheRoyalSociety,
vol.1(Nov.1955),pp.253–263)
Andrew Hodges Contributes
A COMMENT ON NEWMAN’S
BIOGRAPHICAL MEMOIR
Newman had to comply with official secrecy and said virtually nothing regarding Turing’s work
from1939to1945.Althoughthewords‘ForeignOffice’wouldhaveconveyed‘codesandciphers’
toallbutthemostnaivereaders,nothingwentbeyondthistoconveyscaleorsignificanceorscien-
tificcontent.IndeedNewman’saccountwentfurtherthansuppressioveriandledintoasuggestio
falsi.Theexpression‘mildroutine’probablyreinforcedtheprevalentimpressionofBletchleyPark
astheresortofleisuredtime-wasters.Turing’sworkhadbeenfarfromroutine,involvingreal-time
dayandnightworkontheU-boatmessages,andhair-raisingmissionstoFrance,theUnitedStates,
and Germany. It also required great intellectual originality. Newman could probably have given a
cluetoitscontentbymakingareferencetoI.J.Good’s1950bookProbabilityandtheweighingof
evidence.Buttherewasnosuchhint,andthe1955readercouldneverhaveguessedthatNewman
had headed the section that used the most advanced electronic technology and Turing’s statistical
theorytobreakHitler’smessages.
AmoresurprisingfeatureofNewman’saccountistheclaimthat‘thedesigners’of‘thenewauto-
maticcomputingmachines’hadworkedinignoranceofTuring’suniversalmachine.Thisisanodd
expressionsinceTuringhimselfwasonesuchdesigner,asNewman’sreferenceto‘thefirstplanof
the ACE’ makes clear, and obviously he knew of his own theory. Moreover, this plan was a very
early one submitted to the NPL in March 1946. Newman can therefore only have meant that von
Neumann’sreportofJune1945waswritteninignoranceofTuring’swork.Theoriginofthedigital
computer is a major point of interest in the history of science, and it seems strange that Newman
lenthisauthoritytosuchanobliqueandvaguecommentonit,withanimplicitassertionaboutvon
Neumannthatisatvariancewithotherevidence.Newman’sstatementisalsomisleadinginitsimpli-
cationthatTuringonlyturnedhisattentiontocomputersinthesummerof1945afterlearningofvon
Neumann’sdesign.Asithappens,NewmanhadactuallywrittentovonNeumannon8February1946
withasharplywordedstatementaboutBritishdevelopments,assertingtheirearlystartandintellectual
independence.1AlreadyhewasapplyingtotheRoyalSocietyforalargegranttofundwhatbecame
theManchestercomputer.‘Byabout18monthsago’,hewrote,‘Ihaddecidedtotrymyhandatstart-
ingupamachineunit...ThiswasbeforeIknewanythingoftheAmericanwork...Iamofcoursein
closetouchwithTuring...’Thedateof‘18monthsago’isthatofAugust1944.Inthelightofwhat
wasrevealedover20yearslater,itseemsobviousthatthesuccessoftheelectronicColossusafter
D-DayprompteddiscussionbetweenTuringandNewmanofhowthelogicoftheuniversalmachine
couldbeimplementedinapracticalform.Allthispre-1945historywasobliteratedbyNewman’s
accountin1955.Itisofcourseverypossiblethattheoverpoweringnatureofofficialsecrecydeterred
NewmanfromgivingeventhefaintesthintofhisownandTuring’swartimeexperienceatBletchley
Park.Unfortunatelythisomissioncontributedtoadistortionofthehistoricalrecord.
1 LetterinthevonNeumannarchive,LibraryofCongress,WashingtonD.C.QuotedbyA.HodgesAlanTuring:the
enigma,p.341.
3
4 PartI
BycourtesyoftheNationalPortraitGallery,London
AlanMathisonTuring 5
ALAN MATHISON TURING
1912–1954
The sudden death of Alan Turing on 7 June 1954 deprived mathematics and science of a great
original mind at the height of its power. After some years of scientific indecision, since the end
of the war, Turing had found, in his chemical theory of growth and form, a theme that gave the
fullestscopeforhisrarecombinationofabilities,asamathematicalanalystwithaflairformachine
computing, and a natural philosopher full of bold original ideas. The preliminary report of 1952,
and the account that will appear posthumously, describe only his first rough sketch of this theory,
andtheunfulfilleddesignmustremainapainfulreminderofthelossthathisearlydeathhascaused
toscience.
AlanMathisonTuringwasborninLondonon23June1912,thesonofJuliusMathisonTuring,
oftheIndianCivilService,andofEthelSaraTuring(ne´eStoney).Thename‘Turing’isofScottish,
perhapsultimatelyofNormanorigin,thefinalgbeinganadditionmadebySirWilliamTuring,of
Aberdeenshire, in the reign James VI and I. The Stoneys, an English-Irish family of Yorkshire
origin, produced some distinguished physicists and engineers in the nineteenth century, three of
whombecameFellowsoftheSociety;andEdithA.Stoneywasoneoftheearlywomenequal-to-
wranglersatCambridge(bracketedwith17thWrangler,1893).
AlanTuring’sinterestinsciencebeganearlyandneverwavered.Bothathispreparatoryschools
andlateratSherborne,whichheenteredin1926,thecontrastbetweenhisabsorbedinterestinsci-
enceandmathematics,andhisindifferencetoLatinand‘Englishsubjects’perplexedanddistressed
his teachers, bent on giving him a well-balanced education. Many of the characteristics that were
stronglymarkedinhislaterlifecanalreadybeclearlyseeninrememberedincidentsofthistime:his
particulardelightinproblems,largeorsmall,thatenabledhimtocombinetheorywiththekindof
experimentshecouldcarryoutwithhisownhands,withthehelpofwhateverapparatuswasathand;
his strong preference for working everything out from first principles instead of borrowing from
others—ahabitwhichgavefreshnessandindependencetohiswork,butalsoundoubtedlyslowed
himdown,andlateronmadehimadifficultauthortoread.Atschoolhomemadeexperimentsinhis
studydidnotfitwellintotheroutineofthehouse:aletterfromhishousemastermentions‘Heaven
knows what witches’ brew blazing on a naked wooden window sill’. But before he left school
hisabilities,andhisobviousseriousnessofpurpose,hadwonhimrespectandaffection,andeven
toleranceforhisownpeculiarmethods.
In 1931 he entered King’s College, Cambridge, as a mathematical scholar. A second class in
PartIoftheTriposshowedhimstilldeterminednottospendtimeonsubjectsthatdidnotinterest
∗
him.InPartIIhewasaWrangler,with‘b ’,andhewonaSmith’sPrizein1936.Hewaselecteda
FellowofKing’sin1935,foradissertationontheCentralLimitTheoremofprobability(whichhe
discoveredanew,inignoranceofrecentpreviouswork).
Itwasin1935thathefirstbegantoworkinmathematicallogic,andalmostimmediatelystarted
on the investigation that was to lead to his best known results, on computable numbers and the
RoyalSocietyMemoir(MaxNewman)
ReproducedfromtheBibliographicMemoirsoftheFellowsoftheRoyalSociety,vol.1(November1955)pp.253–263
bykindpermissionoftheRoyalSocietyandofEdwardandWilliamNewman.
6 PartI
‘Turingmachine’.Thepaperattractedattentionassoonasitappearedandtheresultingcorrespon-
denceledtohisspendingthenexttwoyears(1936–8)inPrinceton,workingwithProfessorAlonzo
Church,thesecondofthemasProctorFellow.
In1938TuringreturnedtoCambridge;in1939thewarbrokeout.Forthenextsixyearshewas
fullyoccupiedwithhisdutiesfortheForeignOffice.Theseyearswerehappyenough,perhapsthe
happiest of his life, with full scope for his inventiveness, a mild routine to shape the day, and a
congenialsetoffellow-workers.Butthelosstohisscientificworkoftheyearsbetweentheagesof
27 and 33 was a cruel one. Three remarkable papers written just before the war, on three diverse
mathematicalsubjects,showthequalityoftheworkthatmighthavebeenproducedifhehadsettled
downtoworkonsomebigproblematthatcriticaltime.ForhisworkfortheForeignOfficehewas
awardedtheO.B.E.
Attheendofthewarmanycircumstancescombinedtoturnhisattentiontothenewautomatic
computingmachines.Theywereinprinciplerealizationsofthe‘universalmachine’whichhehad
described in the 1937 paper for the purpose of a logical argument, though their designers did not
yetknowofTuring’swork.Besidesthistheoreticallink,therewasastrongattractioninthemany-
sidednatureofthework,rangingfromelectriccircuitdesigntotheentirelynewfieldororganizing
mathematical problems for a machine. He decided to decline an offer of a Cambridge University
Lectureship,andjointhegroupthatwasbeingformedattheNationalPhysicalLaboratoryforthe
design,constructionanduseofalargeautomaticcomputingmachine.Inthethreeyears(1945–8)
thatthisassociationlastedhemadethefirstplanoftheACE,theN.P.L’sautomaticcomputer,and
didagreatdealofpioneeringworkinthedesignofsub-routines.
In 1948 he was appointed to a Readership in the University of Manchester, where work was
beginning on the construction of a computing machine by F. C. Williams and T. Kilburn. The
expectation was that Turing would lead the mathematical side of the work, and for a few years
he continued to work, first on the design of the sub-routines out of which the larger programmes
for such a machine are built, and then, as this kind work became standardized, on more general
problemsofnumericalanalysis.From1950onwardheturnedbackforawhiletomathematicsand
finallytohisbiologicaltheory.ButheremainedinclosecontactwiththeComputingMachineLab-
oratory, whose members found him ready to tackle the mathematical problems that arose in their
work,andwhatismore,tofindtheanswers,bythatcombinationofpowerfulmathematicalanalysis
andintuitiveshortcutsthatshowedhimatheartmoreofanappliedthanapuremathematician.
HewaselectedtotheFellowshipoftheSocietyin1951.
Forrecreationheturnedmostlytothose‘home-made’projectsandexperiments,self-contained
both in theory and practice, that have already been mentioned: they remained a ruling passion up
to the last hours of his life. The rule of the game was that everything was to be done with the
materialsathand,andworkedoutfromdatatobefoundinthehouse,orinhisownhead.Thissort
ofself-sufficiencystoodhimingoodsteadinstartingonhistheoryof‘morphogenesis’,wherethe
preliminaryreadingwouldhavedrownedoutamoreorthodoxapproach.Ineverydaylifeitledto
acertainfondnessforthegimcrack,forexamplethefamousBletchleybicycle,thechainofwhich
wouldstayoniftheridercountedhispedal-strokesandexecutedacertainmanoeuvreafterevery
seventeenstrokes.
After the war, feeling in need of violent exercise, he took to long distance running, and found
thathewasverysuccessfulatit.Hewonthe3milesand10mileschampionshipsofhisclub(the
Walton Athletic Club), both in record time, and was placed fifth in the A.A.A. Marathon race in
1947.Hethoughtitquitenaturaltoputthisaccomplishmenttopracticalusefromtimetotime,for
examplebyrunningsomeninemilesfromTeddingtontoatechnicalconferenceatthePostOffice
ResearchStationinNorthLondon,whenthepublictransportprovedtedious.
Inconversationhehadagiftforcomicalbutbrilliantlyaptanalogies,whichfounditsfullscope
in the discussions on ‘brains v. machines’ of the late 1940’s. He delighted in confounding those
who, as he thought, too easily assumed that the two things are separated by an impassable gulf,
bychallengingthemtoproduceanexaminationpaperthatcouldbepassedbyaman,butnotbya
AlanMathisonTuring 7
machine.Theunexpectedelementinhumanbehaviourheproposed,halfseriously,toimitatebya
randomelement,orroulette-wheel,inthemachine.This,hesaid,wouldenableproudownerstosay
‘Mymachine’(insteadof‘Mylittleboy’)‘saidsuchafunnythingthismorning’.
ThosewhoknewTuringwillremembermostvividlytheenthusiasmandexcitementwithwhich
hewouldpursueanyideathatcaughthisinterest,fromaconversationalharetoadifficultscientific
problem.Norwasitonlythepleasureofthechasethatinspiredhim.Hewouldtakethegreatestpains
over services, large or small, to his friends. His colleagues in the computing machine laboratory
foundhimstillasreadyaseverwithhishelpfortheirproblemswhenhisowninterestswerefully
engaged with his bio-chemical theory; and, as another instance, he gave an immense amount of
thoughtandcaretotheselectionofthepresentswhichhegavetohisfriendsandtheirchildrenat
Christmas.
Hisdeath,atatimewhenhewasfullyabsorbedinhisscientificwork,wasagreatandsadloss
tohisfriends,aswellastothewiderworldofscience.
Scientificwork
ThevariedtitlesofTuring’spublishedworkdisguiseitsunityofpurpose.Thecentralproblemwith
whichhestarted,andtowhichheconstantlyreturned,istheextentandthelimitationsofmechanistic
explanationsofnature.Allhiswork,exceptforthreepapersinpuremathematics(1935b,1938aand
b)grewnaturallyoutofthetechnicalproblemsencounteredintheseinquiries.Hiswayoftackling
theproblemwasnotbyphilosophicaldiscussionofgeneralprinciples,butbymathematicalproof
ofcertainlimitedresults:inthefirstinstancetheimpossibilityofthetoosanguineprogrammefor
thecompletemechanizationofmathematics,andinhisfinalwork,thepossibilityof,atanyrate,a
partialexplanationofthephenomenaoforganicgrowthbythe‘blind’operationofchemicallaws.
1.Mathematicallogic
The Hilbert decision-programme of the 1920’s and 30’s had for its objective the discovery of a
general process, applicable to any mathematical theorem expressed in fully symbolical form, for
decidingthetruthorfalsehoodofthetheorem.Afirstblowwasdealtattheprospectsoffindingthis
newphilosopher’sstonebyGo¨del’sincompletenesstheorem(1931),whichmadeitclearthattruth
orfalsehoodofAcouldnotbeequatedtoprovabilityofAornot-Ainanyfinitelybasedlogic,chosen
onceforall;buttherestillremainedinprinciplethepossibilityoffindingamechanicalprocessfor
deciding whether A, or not-A, or neither, was formally provable in a given system. Many were
convinced that no such process was possible, but Turing set out to demonstrate the impossibility
rigorously.Thefirststepwasevidentlytogiveadefinitionof‘decisionprocess’sufficientlyexact
toformthebasisofamathematicalproofofimpossibility.Tothequestion‘Whatisa“mechanical”
process?’Turingreturnedthecharacteristicanswer‘Somethingthatcanbedonebyamachine,and
heembarkedonthehighlycongenialtaskofanalyzingthegeneralnotionofacomputingmachine.
It is difficult to-day to realize how bold an innovation it was to introduce talk about paper tapes
andpatternspunchedinthem,intodiscussionsofthefoundationsofmathematics.Itisworthwhile
quotingfromhispaper(1937a)theparagraphinwhichthecomputingmachineisfirstintroduced,
bothforthesakeofitscontentandtogivetheflavourofTuring’swritings.
‘1.Computingmachines
‘We have said that the computable numbers are those whose decimals are calculable by
finite means. This requires rather more explicit definition. No real attempt will be made
to justify the definitions given until we reach §9. For the present I shall only say that the
justificationliesinthefactthatthehumanmemoryisnecessarilylimited.
8 PartI
‘Wemaycompareamanintheprocessofcomputingarealnumbertoamachinewhich
is only capable of a finite number of conditions q , q , ..., q which will be called “m-
1 2 R
configurations”. The machine is supplied with a “tape” (the analogue of paper) running
through it, and divided into sections (called “squares”) each capable of bearing a “sym-
bol”. At any moment there is just one square, say the r-th, bearing the symbol S(r) which
is “in the machine”. We may call this square the “scanned square”. The symbol on the
scanned square may be called the “scanned symbol”. The “scanned symbol” is the only
one of which the machine is, so to speak, “directly aware”. However, by altering its m-
configurationthemachinecaneffectivelyremembersomeofthesymbolswhichithas“seen”
(scanned) previously. The possible behaviour of the machine at any moment is determined
bythem-configurationq andthescannedsymbolS(r).Thispairq ,S(r)willbecalledthe
n n
“configuration”:thustheconfigurationdeterminesthepossiblebehaviourofthemachine.In
some of the configurations in which the scanned square is blank (i.e. bears no symbol) the
machinewritesdownanewsymbolonthescannedsquare:inotherconfigurationsiterases
thescannedsymbol.Themachinemayalsochangethesquarewhichisbeingscanned,but
only by shifting it one place to right or left. In addition to any of these operations the m-
configurationmaybechanged.Someofthesymbolswrittendownwillformthesequenceof
figureswhichisthedecimaloftherealnumberwhichisbeingcomputed.Theothersarejust
roughnotesto“assistthememory”.Itwillonlybetheseroughnoteswhichwillbeliableto
erasure.
‘Itismycontentionthattheseoperationsincludeallthosewhichareusedinthecomputation
ofanumber.Thedefenceofthiscontentionwillbeeasierwhenthetheoryofthemachines
isfamiliartothereader.’
In succeeding paragraphs he gave arguments for believing that a machine of this kind could
be made to do any piece of work which could be done by a human computer obeying explicit
instructions given to him before the work starts. A machine of the kind he had described could
be made for computing the successive digits of π, another for computing the successive prime
numbers,andsoforth.Suchamachineiscompletelyspecifiedbyatable,whichstateshowitmoves
fromeachofthefinitesetsofpossible‘configurations’toanother.Inthecomputationsmentioned
above, of π and of the successive primes, the machine may be supposed to be designed for its
specialpurpose.Itissuppliedwithablanktapeandstartedoff.Butwemayalsoimagineamachine
supplied with a tape already bearing a pattern which will influence its subsequent behaviour, and
this pattern might be the table, suitably encoded, of a particular computing machine, X. It could
be arranged that this tape would cause the machine, M, into which it was inserted to behave like
machineX.Turingprovedthefundamentalresultthatthereisa‘universal’machine,U (ofwhich
hegavethetable),whichcanbemadetodotheworkofanyassignedspecial-purposemachine,that
istosaytocarryoutanypieceofcomputing,ifatapebearingsuitable‘instructions’isinsertedinto
it.ThemachineU issoconstructedthat,presentedwithatapebearinganyarbitrarypatternitwill
move through a determinate, in general endless, succession of configurations; and it may or may
notprintatleastonedigit,0or1.Ifitdoes,thepatternis‘circle-free’.Itisthereforeaproblem,for
which a decisionprocess might be sought,to determine from inspectionof a tape, whetheror not
it is circle-free. By means of a Cantor diagonal argument, Turing showed that no instruction-tape
will cause the machine U to solve this problem, i.e. no pattern P is such that U, when presented
with P followed by an arbitrary pattern ϒ, will print 0 if ϒ is ‘circle-free’, and 1 if it is not. If
Turing’s thesis is accepted, that the existence of a method for solving such a problem means the
existence of a machine (or an instruction-tape for the universal machine U) that will solve it, it
follows that the discovery of a process for discriminating between circle-free and other tapes is
an insoluble problem, in an absolute and inescapable sense. From this basic insoluble problem it
AlanMathisonTuring 9
wasnotdifficulttoinferthattheHilbertprogrammeoffindingadecisionmethodfortheaxiomatic
system,Z,ofelementarynumber-theory,isalsoimpossible.
In the application he had principally in mind, namely, the breaking down of the Hilbert pro-
gramme,TuringwasunluckilyanticipatedbyafewmonthsbyChurch,whoprovedthesameresult
by means of his ‘λ-calculus’. An offprint arrived in Cambridge just as Turing was ready to send
off his manuscript. But it was soon realised that Turing’s ‘machine’ had a significance going far
beyondthisparticularapplication.ItwasshownbyTuring(1937b)andothersthatthedefinitionsof
‘generalrecursive’(byGo¨delin1931andKleenein1935),‘λ-definable’(byChurchin1936)and
‘computable’ (Turing, 1937a) have exactly the same scope, a fact which greatly strengthened the
belief that they describe a fundamentally important body of functions. Turing’s treatment has the
meritofmakingaparticularlyconvincingcasefortheacceptanceoftheseandnootherprocesses,
as genuinely constructive; and it turned out to be well adapted for use in finding other insoluble
problems,e.g.,inthetheoryofgroupsandsemi-groups.
Turing’sothermajorcontributiontothispartofmathematicallogic,thepaper(1939)onsystems
logic based on ordinals, has received less attention than (1937a), perhaps owing to its difficulty.
The method of Go¨del for constructing an undecidable sentence in any finitely based logic, L, i.e.
a sentence expressible, but neither provable nor disprovable, in L, has led to the consideration of
infinite families of ‘logics’, Lα, one for each ordinal α, where Lα+1 is formed from Lα by the
adjunction as an axiom of a sentence undecidable for Lα, if such exist, and Lα for limit ordinals
α has as ‘provable formulae’, the union of the sets Pβ(β<α), where Pβ is the set of provable
formulaeinLβ.Theprocessmustterminateforsomeγ <ω ,sincethetotalsetofformulae(which
1
does not change) is countable. This procedure opens up the possibility of finding a logic that is
complete, without violating Go¨del’s principle, since Lα may not be finitely based if α is a limit
ordinal. Rosser investigated this possibility in 1937, using the ‘classical’ non-constructive theory
ofordinals.Turingtookuptheproposal,butwiththeprovisothat,althoughsomenon-constructive
stepsmustbemadeifacompletelogicistobeattained,astrictwatchshouldbekeptonthem.He
firstintroducedanewtheoryofconstructiveordinals,orratherofformulae(ofChurch’sλ-calculus)
representingordinals;andheshowedthattheproblemofdecidingwhetheraformularepresentsan
ordinal (in a plausible sense) is insoluble, in the sense of his earlier paper. A formula L of the
λ-calculus is a logic if it gives a means of establishing the truth of number-theoretic theorems;
formally,ifL(A)conv.2impliesthatA(n)conv.2foreachnrepresentinganaturalnumber.The
extentofListhesetofA’ssuchthatL(A)conv.2,i.e.,roughlyspeaking,thesetofA’sforwhichL
provesA(n)istrueforalln.Anordinallogicisnowdefinedtobeaformula(cid:2),suchthat(cid:2)((cid:3))isa
logicwhenever(cid:3)representsanordinal;and(cid:2)iscompleteifeverynumber-theoretictheoremthatis
true,isprobablein(cid:2)((cid:3))forsome(cid:3),i.e.ifgivenAsuchthatA(n)conv.2foreachnrepresenting
anaturalnumber,(cid:2)((cid:3),A)conv.2forsome(cid:3)(dependingonA).Itisnextshown,byanexample,
that formulae (cid:3) , (cid:3) may represent the same ordinal, but yet make (cid:2)((cid:3) ) and (cid:2)((cid:3) ) different
1 2 1 2
logics,inthesensethattheyhavedifferentextents.Anordinallogicforwhichthiscannothappen
isinvariant.Itisonlyininvariantlogicsthatthe‘depth’ofatheoremcanbemeasuredbythesize
oftheordinalrequiredforitsproof.Themaintheoremsofthepaperstate(1)thatcompleteordinal
logicsandinvariantordinallogicsexist,(2)thatnocompleteandinvariantordinallogicexists.
Thispaperisfullofinterestingsuggestionsandideas.In§4Turingconsiders,asasystemwith
the minimal departure from constructiveness, one in which number-theoretic problems of some
classarearbitrarilyassumedtobesoluble:asheputsit,‘Letussupposethatwearesuppliedwith
some unspecified means of solving number-theoretic problems; a kind of oracle, as it were.’ The
availabilityoftheoracleisthe‘infinite’ingredientnecessarytoescapetheGo¨delprinciple.Italso
obviously resembles the stages in the construction of a proof by a mathematician where he ‘has
an idea’, as distinct from making mechanical use of a method. The discussion of this and related
mattersin§11(‘Thepurposeofordinallogics’)throwsmuchlightonTuring’sviewsontheplace
ofintuitioninmathematicalproof.Inthefinalratherdifficult§12theideaadumbratedbyHilbertin
1922ofrecursivedefinitionsoforder-typesotherthanωreceiveditsfirstdetailedexposition.
10 PartI
Besides these two pioneering works, and the papers (1937b, c), arising directly out of
them,Turingpublishedfourpapersofpredominantlylogicalinterest.(A)Thepaper(1942a),with
M. H. A. Newman, on a formal question in Church’s theory of types. (B) A ‘practical form of
type-theory’(1948b)isintendedtogiveRussell’stheoryoftypesaformthatcouldbeusedinordi-
nary mathematics. Since the more flexible Zermelo-von Neumann set-theory has been generally
preferred to type-theory by mathematicians, this paper has received little attention. It contains a
numberofinterestingideas,inparticularadefinitionof‘equivalence’betweenlogicalsystems(p.
89). (C) The use of dots as brackets (1942b), an elaborate discussion of punctuation in symbolic
logic. Finally, (D) contains the proof (1950a) of the insolubility of the word-problem for semi-
groups with cancellation. A finitely generated semi-group without cancellation is determined by
choosing a finite set of pairs of words, (A, B)(i=1 ...k) of some alphabet, and declaring two
i i 1
wordstobe‘equivalent’iftheycanbeprovedsobytheuseoftheequationsPAQ=PBQ,where
i i
P and Q can be arbitrary words (possibly empty). The word-problem for such a semi-group is to
find a process which will decide whether or not two given words are equivalent. The insolubility
of this problem can be brought into relation with the fundamental insoluble machine-tape prob-
lem.Thetableofacomputingmachinestates,foreachconfiguration,whatistheconfigurationthat
followsit.Sinceaconfigurationcanbedenotedbya‘word’,inlettersrepresentingtheinternalcon-
figurationsandtape-symbols,thistablegivesasetofpairswordswhich,whensuitablymodified,
determineasemi-groupwithinsolublewordproblem.SomuchwasprovedbyE.L.Postin1947.
Thequestionbecomesmuchmoredifficultifthesemi-groupisrequiredtosatisfythecancellation
laws,‘AC=BCimpliesA=B’and‘CA=CBimpliesA=B’sincenowaconditionisimposedon
accountofitsmathematicalinterest,andnotbecauseitarisesnaturallyfromthemachineinterpre-
tation.ThiswasthesteptakenbyTuringin(1950a).(Forahelpfuldiscussionandanalysisofthis
difficultpaperseethelongreviewbyW.W.Boone,J.SymbolicLogic,17(1952)74.)
2.Threemathematicalpapers
ShortlybeforethewarTuringmadehisonlycontributionstomathematicsproper.
Thepaper1938acontainsaninterestingtheoremontheapproximationofLiegroupsbyfinite
groups:ifa(connected)Liegroup,L,canforarbitraryε>0beε-approximatedbyafinitegroup
whose multiplication law is an ε-approximation to that of L, in the sense that the two products of
anytwoelementsarewithinεofeachother,thenLmustbebothcompactandabelian.Thetheory
ofrepresentationsoftopologicalgroupsisusedtoapplyJordan’stheoremontheabelianinvariant
subgroupsoffinitegroupsoflineartransformations.
Paper(1938b)liesinthedomainofclassicalgrouptheory.ResultsofR.Baerontheextensions
ofagrouparere-provedbyamoreunifiedandsimplermethod.
Paper(1943)—submittedin1939,butdelayedfouryearsbywar-timedifficulties—showsthat
Turing’s interest in practical computing goes back at least to this time. A method is given for the
calculationoftheRiemannzeta-function,suitableforvaluesoftinarangenotwellcoveredbythe
previousworkofSiegelandTitchmarsh.Thepaperisastraightforwardbuthighlyskilledpieceof
classical analysis. (The post-war paper (1953a) describes an attempt to apply a modified form of
thisprocess,whichfailedowingtomachinetrouble.)
3.Computingmachines
Apartfromthepractical‘programmer’shandbook’,onlytwopublishedpapers(1948aand1950b)
resultedfromTuring’sworkonmachines.Whenbinaryfractionsoffixedlengthareused(asthey
mustbeonacomputingmachine)forcalculationsinvolvingaverylargenumberofmultiplications
and divisions, the necessary rounding-off of exact products introduces cumulative errors, which
graduallyconsumethetrustworthydigitsasthecomputationproceeds.Thepaper(1948a)investi-
gatesquestionsofthefollowingtype:howmanyfiguresoftheansweraretrustworthyifk figures
AlanMathisonTuring 11
are retained in solving n linear equations in n unknowns? The answer depends on the method of
solution,andanumberofdifferentonesareconsidered.Inparticularitisshownthattheordinary
methodofsuccessiveeliminationofthevariablesdoesnotleadtotheverylargeerrorsthathadbeen
predicted,saveinexceptionalcaseswhichcanbespecified.
Theotherpaper(1950b)arisingoutofhisinterestincomputingmachinesisofaverydifferent
nature.Thispaper,oncomputingmachinesandintelligence,containsTuring’sviewsonsomeques-
tionsaboutwhichhehadthoughtagreatdeal.Hereheelaborateshisnotionofan‘examination’to
testmachinesagainstmen,andheexaminessystematicallyaseriesofargumentscommonlyputfor-
wardagainsttheviewthatmachinesmightbesaidtothink.Sincethepaperiseasilyaccessibleand
highlyreadable,itwouldbepointlesstosummarizeit.Theconversationalstyleallowsthenatural
clarityofTuring’sthoughttoprevail,andthepaperisamasterpieceofclearandvividexposition.
Theproposals(1953b)formakingacomputingmachineplaychessareamusing,anddidinfact
produce a defence lasting 30 moves when the method was tried against a weak player; but it is
possiblethatTuringunderestimatedthegapthatseparatescombinatoryfrompositionplay.
4.Chemicaltheoryofmorphogenesis
For the following account of Turing’s final work I am indebted to Dr N. E. Hoskin, who with
DrB.Richardsispreparinganeditionofthematerialforpublication.
The work falls into two parts. In the first part, published (1952) in his lifetime, he set out to
showthatthephenomenaofmorphogenesis(growthandformoflivingthings)couldbeexplained
by consideration of a system of chemical substances whose concentrations varied only by means
of chemical reactions, and by diffusion through the containing medium. If these substances are
considered as form-producers (or ‘morphogens’ as Turing called them) they may be adequate
to determine the formation and growth of an organism, if they result in localized accumulations
of form-producing substances. According to Turing the laws of physical chemistry are sufficient
to account for many of the facts of morphogenesis (a view similar to that expressed by D’Arcy
ThompsoninGrowthandform).
Turingarrivedatdifferentialequationsoftheform
∂X
i =f(X ...,X )+μ∇2X, (i=1,...,n)
∂t i 1 n i
forndifferentmorphogensincontinuoustissue;wheref isthereactionfunctiongivingtherateof
i
growthofX,and∇2X istherateofdiffusionofX.Healsoconsideredthecorrespondingequations
i i i
forasetofdiscreteceils.Thefunctionf involvestheconcentrations,andinhis1952paperTuring
i
considered the X’s as variations from a homogeneous equilibrium. If, then, there are only small
i
departures from equilibrium, it is permissible to linearize the f’s, and so linearize the differential
i
equations. In this way he was able to arrive at the conditions governing the onset of instability.
Assuminginitiallyastateofhomogeneousequilibriumdisturbedbyrandomdisturbancesatt=0,
hediscussedthevariousformsinstabilitycouldtake,onacontinuousringoftissue.Oftheforms
discussedthemostimportantwasthatwhicheventuallyreachedapatternofstationarywaves.The
botanical situation corresponding to this would be an accumulation of the relevant morphogen in
several evenly distributed regions around the ring, and would result in the main growth taking
place at these points. (The examples cited are the tentacles Hydra and whorled leaves.) He also
testedthetheorybyobtainingnumericalsolutionsoftheequations,usingtheelectroniccomputer
atManchester.Inthenumericalexample,inwhichtwomorphogensweresupposedtobepresentin
a ring of twenty cells, he found that a three or four lobed pattern would result. In other examples
hefoundtwo-dimensionalpatterns,suggestiveofdappling;andasystemonaspheregaveresults
indicativeofgastrulation.Healsosuggestedthatstationarywavesintwodimensionscouldaccount
forthephenomenaofphyllotaxis.
In his later work (as yet unpublished) he considered quadratic terms in the reaction functions
in order to take account of larger departures from the state of homogeneous equilibrium. He was
12 PartI
attemptingtosolvetheequationsintwodimensionsonthecomputeratthetimeofhisdeath.The
work is in existence, but unfortunately is in a form that makes it extremely difficult to discover
theresultsheobtained.However,B.Richards,usingthesameequations,investigatedtheproblem
in the case where the organism forms a spherical shell and also obtained numerical results on the
computer. These were compared with the structure of Radiolaria, which have spikes on a basic
spherical shell, and the agreement was strikingly good. The rest of this part of Turing’s work is
incomplete,andlittleelsecanbeobtainedfromit.However,fromRichards’sresultsitseemsthat
considerationofquadratictermsissufficienttodeterminepracticalsolutions,whereaslinearterms
arereallyonlysufficienttodiscusstheonsetofinstability.
The second part of the work is a mathematical discussion of the geometry of phyllotaxis (i.e.
ofmaturebotanicalstructures).Turingdiscussedmanywaysofclassifyingphyllotaxispatternsand
suggested various parameters by which a phyllotactic lattice may be described. In particular, he
showed that if a phyllotactic system is Fibonacci in character, it will change, if at all, to a system
whichhasalsoFibonaccicharacter.Thisisinaccordancewithobservation.However,mostofthis
section was intended merely as a description preparatory to his morphogenetic theory, to account
forthefactsofphyllotaxis;anditisclearthatTuringdidnotintendittostandalone.
The wide range of Turing’s work and interests have made the writer of this notice more than
ordinarily dependent on the help of others. Among many who have given valuable information I
wishtothankparticularlyMrR.Gandy,MrJ.H.Wilkinson,DrB.RichardsandDrN.E.Hoskin;
andMrsTuring,AlanTuring’smother,forconstanthelpwithbiographicalmaterial.
BIBLIOGRAPHY
1935a.OntheGaussianErrorFunction(King’sCollegeFellowshipDissertation)
1935b.Equivalenceofleftandrightalmostperiodicity.J.Lond.Math.Soc.10,284.
1937a.Oncomputablenumbers,withanapplicationtotheEntscheidungsproblem.Proc.Lond.Math.Soc.(2),
42,230.
1937b.Computabilityandλ-definability.J.SymbolicLogic,2,153.
1937c.Thep-functioninλ-K-conversion.J.SymbolicLogic,2,164.
1937d.Correctionto1937a.Proc.Lond.Math.Soc.(2),43,544.
1938a.FiniteapproximationstoLiegroups.Ann.Math.Princeton,39,105.
1938b.Theextensionsofagroup.Comp.Math.5,357.
1939.Systemsoflogicbasedonordinals.Proc.Lond.Math.Soc.(2),45,161.
1942a.(WithM.H.A.Newman)AformaltheoreminChurch’stheoryoftypes.J.SymbolicLogic,7,28.
1942b.TheuseofdotsasbracketsinChurch’ssystem,J.SymbolicLogic,7,146.
1943*. Amethodforthecalculationofthezeta-function.Proc.Lond.Math.Soc.(2),48,180.
1948a.Rounding-offerrorsinmatrixprocesses.Quart.J.Mech.App.Math.1,287.
1948b.Practicalformsoftype-theory.J.SymbolicLogic,13,80.
1950a.Thewordprobleminsemi-groupswithcancellation.Ann.Math. 52,491.
1950b.Computingmachineryandintelligence.Mind59,433.
1950c.Programmers’HandbookfortheManchesterelectroniccomputer
1952.Thechemicalbasisofmorphogenesis.Phil.Trans.B237,37.
1953a.SomecalculationsoftheRiemannzeta-function.Proc.Lond.Math.Soc.(3),3,99.
1953b. Digital computers applied to games: chess, pp. 288–295 of Faster than thought, ed. B. V. Bowden.
Pitman,London.
1954.Solvableandunsolvableproblems.Sci.News31,7.
[A second paper on morphogenesis is being prepared for publication by N. E. Hoskin and B. Richards,
basedonworkleftbyTuring.]
*Receivedfouryearsearlier(7March1939).