Table Of ContentAlgorithmic Term Rewriting Systems
AriyaIsihara
Copyright(cid:176)c 2010 AriyaIsihara
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PublishedbyVUUniversityPress
ISBN:978-90-8659-442-9
CoverdesignbyAriyaIsihara
FrontCover: “TheCreationofInfinity”
MichelangeroBuonarroti–AriyaIsihara
BackCover: “TheCreationofNaturalNumbers”
MichelangeroBuonarroti–AriyaIsihara
InvitationCard: “AnnunciationofInfinity”
ElGreco–AriyaIsihara
VRIJEUNIVERSITEIT
Algorithmic Term Rewriting Systems
ACADEMISCHPROEFSCHRIFT
terverkrijgingvandegraadDoctoraan
deVrijeUniversiteitAmsterdam,
opgezagvanderectormagnificus
prof.dr.L.M.Bouter,
inhetopenbaarteverdedigen
tenoverstaanvandepromotiecommissie
vandefaculteitderExacteWetenschappen
opdonderdag25maart2010om15.45uur
indeaulavandeuniversiteit,
DeBoelelaan1105
door
AriyaIsihara
geborenteNagoya,Aichi,Japan
promotor: prof.dr.J.W.Klop
copromotor: dr.R.C.deVrijer
v
Acknowledgement
I would like to follow the classical custom to declare that this thesis is
organizedandwrittenallbymyself. But,thethesiswouldobviouslyhave
neverbeenfinishedwithouthelpfrommanypeopleIamacquaintedwith,
orthingsthathavemademewhatIam.
I learned analytic number theory in my bachelor years, supervised
by Yoshio Tanigawa. I owe him a lot; his words and his smile in my
memoryhavebeenalwaysencouragingtome.
Then,IswitchedmymajortocomputerscienceinmyMastercourseat
Kyoto University. I would like to thank Masahito Hasegawa, my supervi-
sor there. During the Master course, he has been an expert guide for me.
My acknowlegement includes the other professors, Jacques Garrigue and
Susumu Nishimura. I have learned from them the basis of computer sci-
ence. Inaddition,Iowedtheenjoyableatmospheretheretomycolleagues:
Hidehisa Arikawa, Kazuyuki Asada, Naoya Enomoto, Yuichiro Hoshi,
HirakuKawanoue,Sin-yaKatsumata,KeikoNakata,TakeshiNozawa,and
HisanoriOhashi.
Forobtainingmypresentposition, IamgratefultoJanWillemKlopand
Roel de Vrijer for having invited me to Amsterdam. Repeating the same
names, I would acknowledge my supervisor Roel de Vrijer and my super-
visor Jan Willem Klop. It has been the most pleasant stage of researches
to show an idea to Jan Willem. Roel has encouraged me very much when
I was in the most unpleasant stage of researches, putting my results on
paper. HereIwouldquotehisproverb: thebestdefinitionisnodefinition.∗
That considerably reduced my uncomfortability in the writing process.
UnfortunatelythethesiscontainsalotofDefinitions,however,thepaperI
dedicatedtohim[28]containsnodefinition,inappreciationofhisproverb.
My acknowledgement also includes the other colleagues for an excellent
atmosphere at the university: Rena Bakhshi, Taolue Chen, Jo¨rg Endrullis,
WanFokkink,ClemensGrabmayer,HelleHvidHansen,DimitriHendriks,
CynthiaKop,FemkevanRaamsdonk,JanRutten. Jo¨erg,Clemens,Dimitri,
and Jan Willem are the first coauthors I have ever had. The work we did
togetherunderliesmanyideasinthisthesis.
Special thanks go to Ichiro Hasuo; he studied at Radboud Universiteit
NijmegenwhenIcametoAmsterdam,andheshowedmehowaJapanese
manadaptsinHolland.
∗ParaphrasingPaulHalmos[24]:thebestnotationisnonotation.
vi Acknowledgement
As to my leisure life in Holland, I am grateful to the late Professional
Go Player Kaoru Iwamoto for the present prevalence of the European Go
society, andIthankthemembersoftheGoclubs‘TweeOgen’and‘Goog’.
Especially,veryspecialthanksgotoJudithvanDamandHarryWeerheijm
forhostingsomepokerparties,whichwerereallyenjoyable.
I am grateful to the members of the reading committee: Stefan Blom,
Jan Rutten, Andreas Weiermann, and Hans Zantema for their reviews.
Nevertheless,theremainingerrorsofthethesisareallmine.
Last,butnotleast,Iamgratefultomyfamily,especiallymywifeKeiko
and my son Haruka. Thanks to them, I have been able to keep my sanity;
tobeprecise,Iwasalwayseventuallyabletoreturntosanity.
AriyaIsihara
Amsterdam,January2010
vii
Contents
Acknowledgement v
Contents vii
0 Introduction 1
0.1 Theconceptofproductivity . . . . . . . . . . . . . . . . . . . 2
0.2 Theconceptofproperness . . . . . . . . . . . . . . . . . . . . 5
0.3 Theconceptofalgorithmicity . . . . . . . . . . . . . . . . . . 6
0.4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.4.1 Thesubjectofproductivity . . . . . . . . . . . . . . . 7
0.4.2 Thesubjectoftreeordinals . . . . . . . . . . . . . . . 8
0.5 Contributionandoverview . . . . . . . . . . . . . . . . . . . 9
0.6 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1 Algorithmictermrewritingsystems 13
1.1 Infinitarytermrewritingsystems . . . . . . . . . . . . . . . . 14
1.1.1 Sorts,symbols,andterms . . . . . . . . . . . . . . . . 14
1.1.2 Operationsonterms . . . . . . . . . . . . . . . . . . . 15
1.1.3 Rulesandreductions . . . . . . . . . . . . . . . . . . . 16
1.1.4 Constructors . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1.5 Sortedsystems . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Semanticallysortedsystems . . . . . . . . . . . . . . . . . . . 18
1.2.1 Inductiveandcoinductivesorts. . . . . . . . . . . . . 18
1.2.2 Semanticallysortedtermrewritingsystems . . . . . . 19
1.3 Algorithmicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Technicalpreliminaries 35
2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.1 Sortsandconstructors . . . . . . . . . . . . . . . . . . 35
2.1.2 Somealgorithmicsystems . . . . . . . . . . . . . . . . 37
2.2 Quasiorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Inductiveheight . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Algebraicinterpretation . . . . . . . . . . . . . . . . . . . . . 45
2.6 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
viii Contents
2.6.1 Formalization . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.2 Adequacy . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Ensuringproductivity 59
3.1 Domainnormalization . . . . . . . . . . . . . . . . . . . . . . 60
3.1.1 Characterizationviapathgeneration . . . . . . . . . . 60
3.1.2 Inductiveheightestimation . . . . . . . . . . . . . . . 62
3.2 Infinitarynormalization . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Characterizationbyquasiorder . . . . . . . . . . . . . 66
3.2.2 Rootactivityestimation . . . . . . . . . . . . . . . . . 67
3.3 Constructornormalization . . . . . . . . . . . . . . . . . . . . 70
3.3.1 Strongconstructornormalization . . . . . . . . . . . . 71
3.3.2 Characterizationbyobservation . . . . . . . . . . . . 72
3.3.3 Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3.4 Pre-requirementandκ-requirements . . . . . . . . . . 80
3.3.5 Dependency . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Treeordinals 93
4.1 Representationlimit . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 AsystemuptotheFeferman–Schu¨tteordinal . . . . . . . . . 95
4.2.1 ThebinaryVeblenfunction . . . . . . . . . . . . . . . 95
4.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . 99
4.2.3 Productivityandrepresentationlimit . . . . . . . . . 101
4.3 ThesmallVeblenordinal . . . . . . . . . . . . . . . . . . . . . 104
4.3.1 TheVeblenmeta-hierarchy . . . . . . . . . . . . . . . 104
4.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . 106
4.3.3 Productivityandrepresentationlimit . . . . . . . . . 108
4.4 Hydragames . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Bibliography 115
Listofnotations 119
Index 121
Samenvatting 125
CurriculumVitae 127
1
Chapter 0
Introduction
In theoretical computer science, in particular the area of algebraic specifi-
cationofabstractdatatypes,sortedspecificationsbasedontheprincipleof
induction are well-known. Here one is concerned with finite data such as
naturalnumbers,booleans,finitetrees,finitelists,andsoon. Morerecently,
a ‘dual’ specification method has become prominent: that of coalgebraic
specifications of infinite data, also called codata. Typical codata are lazy
natural numbers, infinite trees, infinite lists or streams. Whereas in the
world of finite data, induction is the salient principle of definition and
proof,itisreplacedbycoinductionintherealmofcodata.
In this thesis we develop a general framework that extends both the
inductiveandcoinductivespecifications: theframeworkofalgorithmicterm
rewriting systems. The class of algorithmic term rewriting systems pro-
vides a scheme for function specifications employing both inductive and
coinductive constructions. When a function specification is given, we are
concerned whether the specification is well-defined or not. This leads to
the primary desired property of algorithmic term rewriting systems: all
thespecificationsthatcanbegivenasexpressionsinthesystemshouldbe
well-defined. We shall call this property ‘productivity’. This description of
productivity is still very informal. A contribution of this thesis is giving it
a technically precise interpretation. The resulting notion of productivity
of an algorithmic system is fundamental, on a par with the notions of
terminationandconfluencefor(finitary)termrewritingsystems.
Productivity turns out to be the consequence of three secondary prop-
erties. First, infinitary normalization (WN) guarantees that an expression
has a possibly infinite normal form. Secondly, domain normalization (DN)
guarantees that the normal form is within the intended domain of results.
Thirdly, constructor normalization (CN) guarantees that the normal form
is built solely from constructors without defined function symbol. We
give conditions for each of the three properties WN, DN, CN, and in
someinstancesevencharacterizations. Together,theyformconditionsthat
ensureproductivityofthealgorithmictermrewritingsystem.
As an application of the theory developed here, we consider in the
finalchapterafairlycomplicatedinfinitedatatypeknownastreeordinals.
2 Chapter0. Introduction
These are important in the theory of ordinal notations in proof theory, a
branch of mathematical logic. They also embody a study of the expres-
sivity of the first order term rewriting framework, and we show that this
expressivityislarge: wecanexpressordinalsfarlargerthantheFeferman–
Schu¨tteordinalknownasΓ .
0
In order to set the stage, in this chapter we show how the framework
of algorithmic systems and the notion of productivity arise. This chapter
gives an informal introduction; later the notions will be formally intro-
duced.
0.1 The concept of productivity
Let us start by dealing with finitary objects. We think of representing
natural numbers as a paradigmatic example. The set of natural numbers
Ncanbeinductivelyspecifiedasfollows:
1. 0isanaturalnumber.
2. Ifnisanaturalnumber,thensoisthesuccessorn+1.
So,weemploytheconstructors
0 : () → NAT
s : NAT → NAT
to represent natural numbers, where NAT denotes the set of terms (repre-
sentations)ofnaturalnumbers. WecallsuchobjectsasNATconstructedby
inductivemeans‘inductiveobjects’.
We have the representing function (cid:112)−(cid:113) : N → NAT inductively defined
by
(cid:112)0(cid:113) ≡ 0
(cid:112)n+1(cid:113) ≡ s(cid:112)n(cid:113)
andthesemanticsfunction(cid:74)−(cid:75) : NAT → Ninductivelygivenby
(cid:74)0(cid:75) = 0
(cid:74)st(cid:75) = (cid:74)t(cid:75)+1
Weuse≡todenotesyntacticidentityofterms,sincethesymbol=usually
stands for the equivalence relation generated by the reduction relation.
Figure1illustratesthosefunctions.
Next,weconsiderdefiningfunctionsonnaturalnumbers;additionand
multiplication,forexample. Givennaturalnumbersnandm,boththesum
