Towards an Arithmetic for Partial Computable Functionals

158 Pages·2014·0.91 MB·English

by  Basil Karadais

Towards an Arithmetic for
Partial Computable
Functionals
Basil A. Kara´dais
Munich 2013

Towards an Arithmetic for
Partial Computable
Functionals
Basil A. Kara´dais
Dissertation
an der Fakulta¨t fu¨r Mathematik, Informatik und Statistik
der Ludwig–Maximilians–Universita¨t
Mu¨nchen
vorgelegt von
Basil A. Kara´dais
aus Thessaloniki
Mu¨nchen, den 24. Juni 2013
Erstgutachter: Prof.Dr.HelmutSchwichtenberg
Zweitgutachter: Prof.GiovanniSambin
Tagdermu¨ndlichenPru¨fung: 12. August2013
Eidesstattliche Versicherung
Hiermit erkla¨re ich an Eidesstatt, dass die Dissertation von mir selbststa¨ndig, ohne
unerlaubteBeihilfeangefertigtist.
VasileiosKara´dais
Mu¨nchen,den21. Juni2013

Abstract
InEnglish.Thethesisconcernsitselfwithnon-flatScottinformationsystemsasanap-
propriatedenotationalsemanticsfortheproposedtheoryTCF+,aconstructivetheory
ofhigher-typepartialcomputablefunctionalsandapproximations. Weproveadefin-
abilitytheoremfortypesystemswithatmostunaryconstructorsviaatomic-coherent
information systems, and give a simple proof for the density property for arbitrary
finitary type systems using coherent information systems. We introduce the notions
of token matrices and eigen-neighborhoods, and use them to locate normal forms of
neighborhoods, as well as to demonstrate that even non-atomic information systems
feature implicit atomicity. We then establish connections between coherent informa-
tion systems and various point-free structures. Finally, we introduce a fragment of
TCF+andshowthatextensionalitycanbeeliminated.
Auf Deutsch. Diese Dissertation befasst sich mit nicht-flachen Scott-Informations-
systemen als geeignete denotationelle Semantik fu¨r die vorgestellte Theorie TCF+,
eine konstruktive Theorie von partiellen berechenbaren Funktionalen und Approxi-
mationen in ho¨heren Typen. Auf Basis von atomisch-koha¨renten Informationssyste-
menwirdeinDefinierbarkeitssatzfu¨rTypsystememitho¨chstenseinstelligenKonstruk-
torengegebenundeineinfacherBeweisdesDichtheitssatzesvonbeliebigenfinita¨ren
Typsystemen auf koha¨renten Informationssystemen erbracht. Token-Matrizen und
Eigenumgebungen werden eingefu¨hrt und verwendet, um Normalformen von Umge-
bungen aufzufinden und um aufzuzeigen, dass auch nicht-atomische Information-
ssysteme u¨ber implizite Atomizita¨t verfu¨gen. Im Anschluss werden Verbindungen
zwischenkoha¨rentenInformationssystemenundverschiedenenpunktfreienStrukturen
geknu¨pft. Schlussendlich wird ein Fragment von TCF+ vorgestellt und gezeigt, dass
Extensionalita¨tumgangenwerdenkann.

Dues
Mathematicalresearchtexts,accordingtothemodernnorm,beartomymindacertain
ironic resemblance to the sort of narratives you typically get when reading stories by
Raymond Carver or watching films by Clint Eastwood: you’re shown the waves, but
it’stheundercurrentthatmatters. It’sanironicresemblance, sincethemodernmath-
ematician takes frantic care in leaving out any possible hint at the undercurrent; it’s
perceivedasataboo. Mathematicsafterall,asweallsodutifullyagree,isnoart;it’s
nothistoric,norpolitical,stormyloveaffairs,familytragedies,internationalfinancial
trendsorcrisesareallirrelevant,andsoonandsoforth.
Yet the undercurrent is strong, and usually manages to sneak in the text anyway.
Likeintheacknowledgments. Soletmefollowthenorm.
It’snotwisetoletanendeavorlikethislastforsolong,nor,ratherequivalently,isit
wisetoallowyourselfatrulysociallifeinthemeantime: peopleyoubecomethankful
toaccumulateatanunwieldyrate. I’lltrynevertheless.
HelmutSchwichtenbergistheonewhoreallygavemethechancetoliveandwork
inMunich. He’sbeenpatientandtrustfulalltheseyears, hegavemealotofleeway,
andyethewasalwaysalertandreadytospringtoaidwhenIwouldgetlostinallthis
freedom. IguessIwillhavetoletitsinkinforawhilebeforeIcantelljusthowmuch
Ireallyowehim.
Iamdeeplythankfultomyfamily,mymom,mysister,andmylittlenieceChristina
whoalwaystriedtohelpmewithmymathematicalproblemsoverthephone—ifIgath-
eredhersuggestionsinalittlebook,I’dgetacollectionofmathyfairytalesreminiscent
ofbothEdwinAbbottandE.E.Cummings. I’mespeciallythankfultomyfather,who
neverthelesswon’tbereadingthis(asfarasIknow;mymetaphysicsisabitrusty).
AndIhavetothankallpeoplewhoIholddear, likeGiorgos, Thanos, Apostolos,
Kostas,Brent(whoalsodidagreatjobshapingupmyenglisch),Pavel,Ian’n’Leah,
Sarah,Spyros,Dirk,andcertainlyRhea,who’sbeentherewhenI’dotherwisebetruly
alone—let alone she taught me a couple of cool programming tricks. Most of these
peoplehaveevenhadatraceableimpactontheworkitself, buttofollowthesetraces
wouldbetodivedeepintotheundercurrent. TherearemoreIcouldmentionhere,but
IkeptittotheguysandgirlsthatIburdenedwiththethesisthemost,Iguess.
Thenthecolleagues:DianaRatiu,LucchinoChiarabini,BogomilKovachev,Trifon
Trifonov, Stefan Schimanski, Freiric Barral, Klaus Thiel; Florian Ranzi and Simon
Huber; Parmenides Garcia Cornejo; Josef Berger; Daniel Bembe´, Kathrin Bild, Sifis
Petrakis, Kenji Miyamoto, Davide Rinaldi, and Fredrik Nordvall Forsberg; Andreas
Abel,too. Allofthem,atonetimeoranother,havehelped,bothasfriendsaswellas
scientifically.
Andtheteachers: GiovanniSambin,PeterSchuster,DouglasBridges,DetlefDu¨rr,
SofiaKalpazidou,SymeonBozapalidis,GiorgosRachonis,ThanasisTzouvaras.Imust
x Dues
also mention Thierry Coquand, whose few and strictly scientific remarks guided this
worktoanunexpecteddegree.
The indirect teachers too: Thomas Kuhn [26], Imre Lakatos [27], and especially
Ludwig Wittgenstein’s [58], have been invaluable, instructive, and highly soothing
companions whose influence ran deep throughout my research experience. It’s sad
howdownplayed,attimesevenridiculed(seeGeorgKreisel’sunfortunate[23]),such
worksarebythemodernworkingmathematician,butthisisIbelieveatopicnotbest
suited here. These people are now dead, and aren’t going to read this text either, but
hopefullysomebodyelsewhoreadsthiswillreadthem.
Finally, a thanks to the places: Alter Simpl and Zeitgeist of Tu¨rkenstraße; Land-
hausatTal,a.k.a.“thetreebar”;CabaneofTheresienstraße;Flascheno¨ffnerofFraun-
hoferstraße;AltesKreuzofFalkenstraße;GartensalonofAmalienpassage;allofthem,
places fit to ponder over calculations, when the office or home felt too small and the
streetstooopen. Atleast,ofcourse,uptothetimewhenMunichwasstillfarthereast
than any U.S. city and you could still have a decent smoke indoors without feeling
guiltylikeateenagerdoesaftersex;it’sashamehowthispicturesquecity’sgottentoo
steriletoallowforanhonestwayoflife,butagain,Iguessthisisatopicforsomeother
kindoftext.
Acknowledgments
I was supported by a Marie Curie Early Stage Training fellowship (MEST-CT-2004-
504029)forthreeyears. Unfortunately,Idon’tknowwhotothankforthat,otherthan
thelocalMATHLOGAPScommitteeinMunichwhotrustedmewiththeallottedfunds,
andourdutifulsecretary,GerlindeBach,whodealtwithmuchofthedrearypaperwork;
fordespitetheoutrageousbureaucraticdisguise, MarieCuriefundingisablessingin
mostways.
I also want to thank the people who’ve trusted me to help as a TA with their
students, providing me with the means to pay my rent in my post-fellowship years:
SchwichtenbergandSchusteragain,andalsoGu¨ntherKrauss,ErwinScho¨rner,Rudolf
Fritsch,MichaelPra¨hofer,MaxvonRenesse,MartinHofmann,andFranzMerkl.
Atthispoint,Iwouldliketogiveabigthankstothestudentstoo,whosetrusthas
oftenbeenmorepreciousandmotivatingthanthatofpeersorsupervisors. Cliche´sare
oftenbasedontruths.
That’saboutit.
BasilK.,Summer2013