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Algorithmic Term Rewriting Systems AriyaIsihara Copyright(cid:176)c 2010 AriyaIsihara PrintedandboundbyWo¨hrmannPrintService 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 supervisor 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 science. 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 supervisor 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. Joërg,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 properties. 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