Table Of ContentKleene Coalgebra
Alexandra Silva
Copyright(cid:13)c 2010AlexandraSilva
ISBN:978-90-6464-433-7
IPADissertationSeries2010-08
TypesetwithLATEX2ǫ
CoverdesignbyAmadeuSantos
PrintedbyPonsen&Looijen
TheworkinthisthesishasbeencarriedoutattheCentrumWiskunde&Informatica,
undertheauspicesoftheresearchschoolIPA(InstituteforProgrammingresearchand
Algorithmics). The author was funded by the Fundação para a Ciência e Tecnologia
(Portugal), under grant number SFRH/ BD/ 27482/ 2006, and by the CWI group
FoundationsofSoftwareEngineering(SEN3).
Kleene Coalgebra
eenwetenschappelijkeproeveophetgebied
vandeNatuurwetenschappen,WiskundeenInformatica
Proefschrift
terverkrijgingvandegraadvandoctor
aandeRadboudUniversiteitNijmegen
opgezagvanderectormagnificusprof.mr.S.C.J.J.Kortmann,
volgensbesluitvanhetcollegevandecanen
inhetopenbaarteverdedigenopdinsdag21december2010
om10.30uurprecies
door
AlexandraSilva
geborenop10februari1984
teChaves,Portugal
Promotor:
Prof. dr.JanRutten
Copromotor:
Dr.MarcelloBonsangue (UL)
Manuscriptcommissie:
Dr.LuísBarbosa (UniversidadedoMinho,Braga,Portugal)
Prof. dr.HermanGeuvers
Prof. dr.BartJacobs
Prof. dr.DexterKozen (CornellUniversity,Ithaca,NewYork)
Dr.ErikdeVink (TUE)
CONTENTS
Contents v
Acknowledgments ix
1 Introduction 1
1.1 Coalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Kleene... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 ...andfollowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Ouraim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Kleenecoalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Thesisoutlineandsummaryofthecontributions . . . . . . . . . . . . . . . 5
1.7 Relatedwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Preliminaries 9
2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Automataascoalgebras 15
3.1 Deterministicautomataandregularexpressions . . . . . . . . . . . . . . . 15
3.1.1 Fromdeterministicautomatatoregularexpressions . . . . . . . . 21
3.1.2 Fromregularexpressionstodeterministicautomata . . . . . . . . 23
3.1.3 Non-deterministicautomataandthesubsetconstruction . . . . . 27
3.1.4 Kleenealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 AutomataonguardedstringsandKATexpressions . . . . . . . . . . . . . . 40
4 KleenemeetsMealy 49
4.1 Mealymachines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 RegularexpressionsforMealymachines . . . . . . . . . . . . . . . . . . . . 53
4.2.1 ExpressionsformaMealycoalgebra . . . . . . . . . . . . . . . . . . 55
4.2.2 AKleenetheoremforMealycoalgebras . . . . . . . . . . . . . . . . 56
4.3 AnalgebraforMealymachines . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Non-deterministic Kleenecoalgebras 73
5.1 Non-deterministiccoalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Alanguage ofexpressionsfornon-deterministiccoalgebras . . . . . . . . 75
5.2.1 Brzozowskiderivativesfornon-deterministicexpressions . . . . . 79
v
5.2.2 Fromcoalgebrastoexpressions . . . . . . . . . . . . . . . . . . . . . 82
5.2.3 Fromexpressionstocoalgebras . . . . . . . . . . . . . . . . . . . . . 87
5.3 Asoundandcompleteaxiomatization. . . . . . . . . . . . . . . . . . . . . . 94
5.4 Twomoreexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
5.5 Polynomialandfinitarycoalgebras . . . . . . . . . . . . . . . . . . . . . . . .107
5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
6 QuantitativeKleenecoalgebras 113
6.1 Themonoidalexponentiationfunctor . . . . . . . . . . . . . . . . . . . . . .116
6.2 Anon-idempotentalgebraforquantitativeregularbehaviors . . . . . . .120
6.3 Extendingtheclassoffunctors . . . . . . . . . . . . . . . . . . . . . . . . . .134
6.4 Probabilisticsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137
6.5 Aslightvariationonthefunctor . . . . . . . . . . . . . . . . . . . . . . . . .140
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142
7 Furtherdirections 145
Bibliography 147
Index 159
Summary 162
Samenvatting 165
Curriculumvitae 167
Acknowledgments
IthasnowbeenroughlyfouryearssinceIarrivedintheNetherlandstostartmyPhD.
Alongtheyears,anddespitethelackofsun,Igrewfondofthecountryanditspeople.
Thesefouryearswereveryactiveforme,withmany,manytrips...Iwasalsoluckyto
meetwonderfulpeople,whomIwouldliketothanknext.
This thesis would never exist if it was not for the help of my promotor, Jan Rutten,
and my co-promotor, Marcello Bonsangue. A famous cook once said: So much in
cooking,asintherestoflifeIsuppose,isaboutcontrast,aboutbalance. Iheretakeher
words and adapt them to the last four years: the contrast and balance between Jan
and Marcello made my journey richer and more interesting. I know how lucky I am
tohaveyouasmysupervisors!
Jan, I am infinitely grateful for all you taught me: among many many things, you
taught me how to construct a final coalgebra, you showed me how to handle every
situationdiplomatically and youevenexplained me howtoproperlycookpasta. My
contact with you has transformed me into a better researcher, but more importantly
into a more balanced and happier person (I hope ;-)). You always had your door
open for me and you patiently listened when I needed it the most. You were the
best company I could wish for in the many trips we had together, among which the
memorabletriptoBrazil! Iamalsogratefulfortheeffortyoumadeinfindingfunding
toallowmetocontinueattheCWI.Itriedmanytimestounderstandwhatmakesyou
suchagreatsupervisor. Maybeitisthebalancebetweenrespectandconfidencethat
youfindwithyourstudents. Maybeitisthefactthatyoualwaysshowusthatthere’s
somuchmoretolifethanresearch. UptothisdayIhavenotquitefigureditout,but
IsuspectIcouldwriteawholechapteraboutthisandaboutallthethingsinyouthat
inspire me. For me, you are and will always be super cool :-) Hartelijk bedankt Jan,
vooralleswatjevoormegedaanhebt!
MarcelloisthemostenthusiasticsupervisorIknow: healwaysmadesureIfeltthathe
wasinterestedinmyworkandhewasalwaysavailableforlongphoneconversations,
togooverproofs,draftsorjusttomakesureIwasdoingfine. Iamgratefulforyour
ix
supportalongtheyearsandforyourinfinitepatiencetoteachme. Youbecameagood
friend, always there to help me in my adaptation to the Dutch culture, to teach me
important lessons about Italian cuisine (creamy, but not frozen)or to comfort me in
the manymomentsI doubted about mydecision ofstaying in theNetherlands. I am
grateful to have you in my life and I know that I would have never made it without
you! Tisonoinfinitamentegratapertuttol’aiutoel’affettochemihaidato!
A special thanks to the members of my reading committee – Bart Jacobs, Dexter
Kozen,ErikdeVink,HermanGeuversandLuísBarbosa–fortakingthetimetocare-
fully read my thesis and for all the helpful comments. The title of this thesis as well
as the titles of some of the chapters/sections were inspired by titles used by Dexter
Kozeninhispapers.
Parts of my thesis were published in conferences/journals. I would like to thank my
co-authors,alltheanonymousrefereesandthefollowingpeopleforalltheimportant
remarks and comments: Bas Luttik, Bartek Klin, Clemens Kupke, Dave Clarke, Erik
deVink,FilippoBonchi,HelleHansen,PaulvanTilburg,PrakashPanangaden,Stefan
MiliusandYdeVenema. AspecialthankstoBartek,whose(verydetailed)comments
greatly improved the presentation of Chapter 5; to Erik, for the time and energy he
always found to carefully read my drafts and discuss them with me and for all the
LATEX trickshe taught me; and to Dave, forhis patience in correctingmyEnglish and
showing so much interest in my research! I would also like to thank Sam Staton
for having shared with me the nice fonts he used for his thesis, Ichiro and Helle for
sharingtheirstylefilesandJoostWinterforproofreadingpartsofthismanuscript!
MytimeattheCWIwasenjoyableandmanypeoplecontributedtothis: themembers
(frequent visitors and ex-members) of the SEN3 group, with a special mention to
Behnaz, Christian, Clemens, Dave, David, Einar, Erik, Farhad, Filippo, Frank, Helle,
Immo, Lara, Marjan, Milad, Jacopo, Jan, Joost, José, Marcello, Michiel, Stephanie,
Stijn, Sun Meng and Young-Joo; the support staff at the CWI, in particular Bikkie
Aldeias,KarinvanGemert,IrmavanLunemburg,HuibvandeBerg,JanSchipper,Jos
vanderWerf,MaartenDijkema,MinnieMiddelbergandSusannevanDam;myoffice
mates: Sun Meng & Young-Joo, Helle & Clemens & Marcello (with whom I had the
gezelligsteofficeever),Milad&Marcello,Filippo&Marcello.
To our friendly neighboring group, in particular Bas Baeten, Floor Sietsma, Jurgen
Vinju, Paul Klint, Tijs van der Storm and Yanjing Wang. Some special words for
Yanjing. We wrote our thesis almost in parallel and along the months you became a
valuable close friend. It was comforting, in the long days at the CWI, to know that
you were a few offices away always ready to have a chat about the thesis or about
life! I also thank youfor explaining me all kind of intriguing factsabout the chinese
cultureandforcookingdeliciousChinesefoodforme! Iwishyouallthebestinyour
newlifeinChina. Ihopethatyoufindhappinessandthatwekeepintouch!
TotheUvAcolleagues,Alessandra,Gaelle,Jacob,RaulandYde,whowelcomedmein
theirAlgebra-Coalgebra seminarsand alwaysshowedinterestinmywork,providing
mewithgreatfeedback!
Duringtheselast fouryears, Ihad thepleasure of collaboratingwith manypeople. I
wouldliketothankDorelLucanu,GeorgianaCaltaisandEugenGoriacfortheirefforts
