a structural theory for varieties of tree languages potx

CRT CU A Structural Theory for Varieties of Tree Languages An Algebraic Study of the Theory of Formal Tree Languages and Tree Automata... Saeed Salehi A Structural Theory for Varietie

CRT CU

A Structural Theory

for Varieties of Tree Languages

An Algebraic Study of the Theory of Formal

Tree Languages and Tree Automata

Saeed Salehi

A Structural Theory for Varieties

of Tree Languages

An Algebraic Study of the Theory of Formal

Tree Languages and Tree Automata

VDM Verlag Dr Miller

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Preface

‘This book is au extended (ancl improved) version of my Ph.D, dissertation, Varieties of Tree Languages, University of Turkv, Finland 2005 (TUCS Dissertation Series 64) That rhosis was published by TUCS (Turk Centre for Computer Science wu tues.£4) before the defense in August 2005, aul his book appears by the kind invitation of the VM Verlag publishing house, almost five years after During those few years, some advance

Ihave hoon made on the subject For example, the papers [51 and 58] are published after the original thesis [BD and are now included in this hook Also the proolsof the main results of [5] and 55] which were exchided in fl, due to the space shortage, are ncluded hore ‘These ate essentially the main differences between [Gl] aud the present book:

J would like to thenk Mr Bulesu Body from the WDM for taking the initiative in publishing the book ind for his pationce in the teehnieal matters As moutioned in fl] also, I would lke to express

iy sincerest gratitude to Professor Maguus Steinby for teaching me, guiding me and supervising

me through those years that I studied in Turku, Finland I have learnt from him not only as a mathematician but also as a human boiug Ilis groat persouality cased my stay in Finland and

provided comfort to me Two collowgues of mine, also under the suporvision of Profesor Magnus Steinbs, Dr Tatjana Petlovig and Mr, Ville Pirsinon have helped me alot aud supported ase through stimulating discussions, Chuptors $ and 5 of the book are joint works with Dr ‘Tatjana Pethovié, and Me, Vils Piirainen has contributed to the hook, as several lemmas aad remarks ar his ious, Chaptors L and 6 are joint works with Professor Magnus Steinby: T would als lk to thank TUCS for Ñaancialy supporting my studies and its stuff for creating a worm ancl seentilic atmosphere

My sperial thanks go to Professor Juhani Karhumilki who invitee me ¢o Einland and openel a new Aloor in my life His support during my early days iu Finland helped mo much to settle down T an

nh geateful to Dr, Bija Jurvauen for her kind help and guidance, [highly appreciate the support

of Professor Jarkko Kari and Profesor Patrick Sibelius (Abo Akademi, Fuslond) During some

enjoyesl fruitful discussions with Professors Wolfgang Thomas (RWTH Aachen, Esk and Zoltan Falap (Uuiversity of Seeged, Hungory)

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Chapter 1

Tntroduction and preliminaries

Computer seienen 18 no more about computers than astronoma is about telecupes

~ Badeger Dijesten

1.1 Introduction

“Trees and teemis are important structuned! objects that cau be found almost everywhere in computer science, not only in connection with their mathematical fonndations." Jantzen BT] Also, almost every working mathematician has hard of “trees” as this notion appears iv many seeing different teas of mathematies from graph theory to universal algebra to logic In computer science trees are often regarded as a natural generalization of strings ‘Though it is not possible to presant @ complete history of the subject here, we quote the following from the survey paper 9]:

‘The theory of tree automate and treo languages emerged in the middle of the 196)s quite naturally from the view of finite automata as unary algebtas adwiented by J R Biichi and J B Wright, From this perspective the generalization from strings to traes means simply that any’ Bite algebra of finite typo can bơ mgardel ae an automaton which as inputs accepts terms over the nfo alphabet formed by the operation symbols of the algebra, and these terms again eat be seen

1s (formal representations of} labeled troos with a left-to-right ordoving of the branches Strings over a finite alphabot can then be rogarded as torus over a unery ranked alphabet and hence finite

by Bichi’s posthumous book [I] in whiels many of the ideas are traced back to people like Thue, Skolem, Post, and even Leibniz,”

2 CHAPTER 1, INTRODUCTION AND PRELIMINARIES

i true that the thoory of nee automata and tree languages may have come into existence by enoralizing string automata anc languages, but “(olf course, no branch of mathematics contd stay alive very long as @ mete gonotalization

Apart from its intrinsic interest, the theory of tree automata and tree languages has found several applications and it offers new perspectives ta vasious parts of mathematical linguistics Tt fas also hen applied to some decision problems of logic, and it provides tools for syntactic pattern recognition

se [H má

and extend previously known results, and also to find now results For Instance recent works use “Actually using tree automata has proved to he a powerful approach to simplify

‘ss automsta for application,

abstract interpretation using st constraints, rewriting, automated theorem proving and program verifeation, databases aud XML schema langunges” HT

Mathematicians who have hourd of tes my reall one or two deiitions of thems, Considering

te as terms over «ranked alphabet and « lef alphabet has become a custom i same schools, csptially in Turku, Finland An advautage of this approach is that the concopts and results of suivorsl algebra econ immediatly wsuble

is worth noting thot the impact of universal algebra on the thoory of tree automata and treo Jnnguages hus not boon in one dineetion only; developments of teee automata and te languages Ihave suggested now pols and concepts of universal algebra, Alo in this hock we have develope algebraic wotions aud proved theorems in universal algebra when the nocssity hus emerge, However, the book of Deneeke and Wistuath [I] isthe fist universal olgetna text where toe atansata and

‘ace languages are explicitly studied (ee Chapters 5 and § of I)

‘The main topic ofthis book is the variety thoory of toe Iauguages "The history of vaiety theory

‘gins with Bilebor's celebrated varity theon FI, As Pin FO} puts it, “be tos important tool for classifying eevognizable languages is Elenberg’s varity theorem (2, whieh gives a ono

‘theorem is the existenee ofits many instances AS a matter of fact, most of the intresting classes

of algebraic structures nte varieties, andl similarly, most of the interesting families of tree or string languages studied in the literature turn out to be varieties of scan kind, ‘The aforementioned varity

‘theorem conncets these interesting families to each other

Bilenberg theorem has since then been extended in various directions Oue ofthese extensions is

“Thếnien [TT] notion of varieties of congruences on free monoide, Another extension is Pin’ positive

‘variety theorem

which establishes a bijetive correspondonce betwoon postive varitios of string languages and varieties of ordered semigroup

Any variety theorems comncets Eamilios of tree languages with class of some structures via the

“syntactic structures” (Pil) One of those syutactic structures is the syntactic semigroup monoid

of a tree language introduced by ‘Thomas [FQ] aud further studied by Saloanaa (Gl) A different formalism, hase on essontally the same concept was considorod by Nivat and Podelski fll [5]

Sovoral varity thioreis for tees are proved in this book

“The varioty thoorem for famsilics of tree languages and varitios of finite ulgobras, provided by Steinby and Alida, is gencralized to many-sorted algebras in Chapter 2, which isa joint work with Sbinhy

Chapter 3 based on a joint paper with Pethovi HB) is inspired by Pin’s theory of positive voriotos of string languages and vasieties of ordered monoids, We prow « variety thoorem for positive vorotios of tov languages and varieties of Bnite ordered algebras vhich correspond to each other via syntactic ordered algebras

‘Troe languages definable by syntactic monoids are studied iu Chapter 4, It was alendy known that

ny family of toe languages definable by syntactic monoids isa (generalized) variety of tee langsages,

‘hough not every varity of tree languages is definable by syntactic monoids fH] Characterizing the varieties of tree languages which are definable by syntactic monois was a relatively long-standing open peoblom (BTLIZH Here we give an answer to this question by providing & variety theorem for failies of tree languages and varietis of finite monoids which correspond to each other via syntactic onside

‘This characterization is goneralizol to # characterization of positive varieties of tree languages definable by syntactic ordered! monoids in Chapter 5, ‘This generalization was obtained together with Petkovié IB), Also instance ofthis positive varity theorem and the variety theorem

J Chapter 4 is elahorated,

‘The rest of the book is a study of Wilke’ tree algebra formalism [FI] for binary trees A com pletenes theorem for the axiomatizstion of tree algebras anc a variety theorem for families of binary toe languages and varietios of ite tre algebras is proved in Chapter 6 ‘The first two sections of this chapter are based on a joint paper with Steinby-

4 CHAPTER 1 INTRODUCTION AND PRELIMINARIES

‘The above mentioned results ance again demonstrate the richness of the theory of tree automata, and tree Languages and thes also suggest some new perspectives of the vaniety theory of string languages when words ate viewed as nnary tioes ‘This, from a variety theory viewpoint, confirms

our beliof that not only trees are mare than mere generalization of words, but alba words are particular cases of tress

We have made an effort to-make the book self-contained for its expected readership Por basic

are mote than enong, and [I8)[87] provide the fundamental tools of universal algebra and term rewriting used throughout the book,

notions of tree automata and tree languages

Although we have tied to use a uniform notation throughout the book, some exceptions scemed Inevitable In particular in Chaptor 6 we have preserved most of Wilko's fF] notation, and thus

‘many letters got meanings different from the ones they have ithe provions chapters, For the reade convenience att Index of Notation is provided,

1.2 Preliminaries

Strings over a finite alphabet X are often regarded us elements of the free monoid X* generated

by X, Similary, any tree considered here may be viewed as an clement of a tert algebra, Also nite tree automata ean be defined as finite algobras Therefore, universal algebra provides a natural

‘mathematical foundation for the theory of finite tre automata and mocognizable tre languages Here

we frst recall some basic notions of algebras andl then we fist formal definitions and concepts of trees

A ranked alphabet % is « inive set of function syinbols each of whieh has w nnique non-negative

Integer arity, For any 9 2 0, Nye denotes the elements of with arity m In particular, Sp is the

set of constant symbols of S.A algetnu isn structure A = (A,3) where Ais « nonempty set

in which every symbol of 3 is realize, i., any e € T5 is realized by a constant ot € A, and any [Yq for m > 0 is realized by an m-aty function f4: A" + A The algebra A = (4,2) is called finite ifthe set A is inte

Recall that a binary relation on a sot Ais 9 subsot OS Ax A The fact that (a,8) € 0, for some

= (6a) | (a,l) € A), and if

6 is another relation on the set A, the composition of @ and 6” isthe eelation

Bod = (Cae) | (0,5) € O& (Oye) € O" for some be A}

“The diggonal relation on A, {(@,a) | a € 4], and the universal elation A x Aare respectively

0 0 isan equivalence on 4Á, thơ guofienf set 4/8 Is the set (a/0 |a € Á} where a/® = {b¢ A|a 6) is the

4 A, is often written ng 0b, The inverse of the elation 0 is

denoted by Ay aud V4 The relation @ is called sn equivalence on Aif A407! 00

L2 PRELIMINARIES 5 O-class ofa IEA ix Sint then @ is sail to be of finite inder or simply a finite elation, Note that

Ay and V1, are the lost and the grontest equivalence relations on A, respectively For sets ane B,

fs mapping gi 1 > B oan he view nho us a special relation, a subset of Ax B The image ofa)

ofan a € A is often written as mg For subsets CC A end DC B the image of C and the inverse imoge of D ave Cig ~ {a2 € Ba € C] and De — {a € Al ag € DỊ, mspnoiely

Fix a raked alphabet B and tet ,4 = (4, Đ), B — (E,Y) bơ algcbrns

The algebra B is said to be « sulalgebra of A, if BC A and every fuuetion J, for J € 3 is the

every ayy seate € As The fet that @ is a homomorphism: is expressed by weiting +A B,

‘The kernel of y is the relation kere = {ee} € Ax A] ag = ceh A homomorphism is called

a monomorphism if it is injective, and is an epimorphism if itis surjective, AN isomorphism is a jective homomorphism, Sometimes « hoxomoxphi u is called situply a morphism, We say that Bisa homomorphic image of A and write 8 — A when there exists an epimorphism from A onto

BB, aud we write AC B when there exists « mone

phim from A to B We say that A divides

B anil write A+ B if for some algebra C = (C.E) there exist « monomorphism + C+ B and

an epimorphiswn ¡ Ở —+ 4- TẾ thene exists an fsomorphisw) Between A and 6, then and B are iomorphie, and we write A B

An equivalence lation # on A is called a congruence on A, if for all fe Y

AML yee bye sesPae © AL HE Doo Oy eM FAA, oa) OFA Lasoo

to note that the kerael of any homomorphism is a congrucnee If # is a congruence on A then

© being a congrucnce om A ensures us that the above operations are well Isinsd on AYO The natural mapping 0: A+ A/O, a ++-0/0, is seen to be an epimorphisin Note hat ker® = 0, The Hemotnorplism Theonem in universal ulgebra states that if ¢ A+ B is an cpimonphismn, then A/leerig * 8 This theorems ean be generalized as follow For any congruence

em B and any howomorphism = A> B, the relation > 2g"? is w congruence on A, Now, if ix

kore

tn epimorphism, then Aza og"? © BY, Note that by definition œ sợ

A mapping 9: A+ Ais called an elementory translation of A, if

for some m > 0, f € "` © A, where € is 8 new variable

ranging Gver A, The set Tr{.A) of translutions of A is the smallest sot of unary operations on 4L that contains the identity mop Ly: A+ A, eer a, sad all thế clemenlary translations of A aa is

6 'CHAPTER l._INTRODUCTION AND PRELIMINARIES

closed under the composition It is easy to verify that an eqtdalehee Ø on Ä Í€ & congrlenoe nh Á ine € podopr! for every stanton p € Tr)

The divet product Ax B of A and B is the algebra (4 xB.) defined hy Ae! = (eA

and /49((6ị bị, (e,B„)) = (6y ) /(hu h

atl jy € Ay Bassa € B Tie flowing faets can be easily verified for any algetnas A and

Simply varieties from now on, there exists sm an

hooreus gives 9 logical earaeteriantion for those classes For the ps

ogue characterization with ultimately definabilty

by exquations (See e.g; ĐH), thongh we will uot tonch this subject ju this book, It is easy t0 soo that the intersection aF any class of varieties is a vaviety, So, for a collection of E-algebras C the Inersoction ofall vatieties containing C is a variety, called the variety generated by C

[Now we teview the theory of trees as terms, Ronghly speaking, 0 tree fsa strnetrod objet that

is branched rot a root which stands i the Wighest loved and every node in Uke auidlle is cither branched to other novles oF stands as a lea For « formal definition ler E he a ranked alphabet aud X be any fite set, called fenf alphabet, The set T(%,X) of EX-toes is defined to be the smallest se4 containing Vị U X ích thất € TY, X) whonevet / €

(On > 0) and

fiseesstn € TAS, XT In this formalism the leaves of EX-trees axe labelled by symbols from Bo UX

‘aud the inner nodes are label by the syinbols iE with non-zero avities Any subset of T,X)

i called a tree language

The algebra T(E.X) = (TUNIS) i defined by 78) = ¢ for any constant asmubol ¢ € By and J7 lu, ut„) = /ecce đa) Bát dU Ƒ € Sự {m > 0) an fecal PEAY This Ts callel thự ĐX-lemm alpcben, or simply a term algebra We note that TÍY, X) le the

generdiel bự X Le, for any algebra ¿1 = (Ä,Ÿ), nhy maphing ø + Ý -x 1 can mưiqhehy be estendl}

to a homomorphism a4: 7(2,X) + A A tase language TC T(E, X) is said to be recognized by

bra A = (A, when thote exist a homomorphism ý ý TÍY, X) + A and a subset FCA

such that Fz! = 7 A tree laugunge is called reeognizeble fa finite algebra recognizes it Ls

EX-recognizer as a triple (Aa F) whore A= (A,

steusion {© a homomorphism, The subset FC A is called the set of fal states, and thy tree

» (Ava F) is by definition (F€ T,X) | too F)

languagt recoguized

Let { be a now sytubol which does not appear in any 1

hore, The set of EX-conterts, denoted by C(Đ, X], consists of the L(X U {g}}teees in which & appears exactly ouce, For contexts P,Q € C(U.X) and tree ¢ € T(,X), the context Qs P, the composition of P aud Q, results from P by replacing the special leaf € with Q, and the term PCO), also denoted as ¢- P, results from P by replacing € with t For w tree language T © (2S, X) and context P€ C(S,X), the ianerse translation of F under P is PT) = {2 € T(,X) |t- P €T)

We shall now ontline the basic theory of varieties of recognizable tree languages that is the everal starting point of this work For a tree language TC T(E, X), the syntactic eamgruence =P

of T (ll is defined by t < + => YP ECE, NML-P eT + e-PET) (th € TSX)

Ie cam be easily soon that the relation 7 is really a congruence on T(E,X)- A tree language is

{ke alphabet or leaf alphabet considered

recognizable iff ts syntactic congvience is of finite index ‘The aynlactic algebra SA(T) of T is the

‘quotient alyobra T(S, X}/ =P Itcan be shown that a finite algebra A rooognizes a tree language T iff SA(T) = A Thus, SA(T) isthe smallest algebra recognizing 7 The folowing relations hold for any leaf alphabets X,Y, tre languages 7.7" © T(2, X), homomorphism = T(S.¥) + T(8.X), tnd context P € C(, X) (see Propositions 3:4 and 3.4 i [i

'CHAPTER l._INTRODUCTION AND PRELIMINARIES 670° € F(X) and yo oy € TCV}, and morever, Íf # is ahother congrlenoe on T(E, X) such that C9 then BE TX)

Lat VEA(S), VII), and VE

class ofall varieties of tree laugmages, ae the class of all vatietios of congruences, nespectivels, ‘Thove

encnated by (SACP) |Te ¥(8, X)}s ed

et = (¥°(2.X)} be the fanily of Guite congruences defined by setting for each leaf alphabet

KX, YUN) = (8 | 8 DSP Moi wt forsome Ty Ty © PDX) Finally, for a class P= (P(X) of finite congruences, let I be the variety of finite

{T(2.X)j/O | @ € PEN)}, and lot C= (P(E, X}} be the family of recognizable tove languages defined by E(B, X) = (7 © T(E, X) le F(E_X)} for eny leaf alphabet X

ave compatible with each ather, Len Ke = KE, IM = KS 8 = 9 7 = 5, P= EY ond P= P! for any K € VEA(E), ¥ € VTLS), aud Pe VEC(Z),

Chapter 2

Many-sorted variety theorem

‘Many-sorted algebras have found their way into computer seienoe through abstract data type specific

br other axiouus ane the

cations, Many-sortod algebras and their specifications in terms of equatios

athematical fundlament of rigorons approactos to abstract data types in programming aad specific

cation languages, It Is widely believed that many-sotted algebras ane the eight mathematical tools

co explain what abstenct data types are (soe [) It appears that Maibanm [IT] was the fest one

to conskler mauy-sorted tree languages, while the ides of recoguignble subsets of arbitrary algebras oes hack to Mezei and Weight [Il], Many-sorted trees are uso also by Engelfrier and Schmit

in their stndy of the equational semanties of context-ree tree languages Recognizable subsets of encral many-sorted algebras wore studied by Comreelle [IIIB In this chapter, we consider varieties

of recognizable subsets of mang-sorted finitely generated free algebras over a given variety, varieties

of congmences of ch algebras, and varieties of finite many-sorta algebras A variety theorem that,

“establishes bijections beewoen the clases of thew three types of varieties proved For this, appro- priate notions of many-sorted syntactic congrucnos and algebras are newed, Indeed, by developing

1 theory of varstce of recognizable subsets of fee many-sortedalgchras we generalize the theories

of (5) i] and fl] to the manysorted cas In Section EE] we present some basie definitions and our notation for many-sorted algebras, Also some more specialize notions relevant to our work are inteodced, ‘The references [20 5 2) aay be consulted for

of mauy-sorted ugebras Ia Section Zrecognizable subsets of many-sorted algebras are consered eral treatments of the theory

‘There are actually two types ofthese, recognizable sorted subsets and the “pure” recognizable sub- sets considered in (16221 1] in which ll elements are af some given sort (soe Section A) Abo

syntactic congrucnoes and syntactic laobras of subsets of many-sorted algebras ar introduco, and

it is shown that thoy onjoy all the same general properties as their counterparts for monoids 2109)

or torm algobras, or one-sorto algebras in general [I] §5)(66) In Soction we define our varieties

of recognizable sets and varities of eongrucneus For this a finite sut of sorts $ and variety V of somo finite Ssortod type ® are fixed A varioty of ragaizable V-sets consists then of rooogizable

10 CHAPTER 2._ MAN

SORTED VARIETY THEOREM, subsets of finitely generated fioe algebras over V Similarly, s variety of V-congrucnoss consists of congruences af finite index on these algebras Finally, a V-variety of finite algebras is defined asa

‘variety of finite algebras contained in V We define six mappings that transforin varieties of tog nizable V-sets, varioties of V-congrucnces and V-varietes of finite algebras to each other Then we prow our Variety Theorem that essentially says that these six mappings fort three pains of mutually inverse isomorphisms between the complete lattices of these three kinds of varictias The proaf is

but there are naturally some teehnienl very similar to the corresponding proof presented i

Jifforences and for the reader's convenionce a rather detailed proof ix presented In Section FT} we define varieties of pure recognizable V-sets in which each recognizable set is a subset of tho set of elements of some given sort of a Bnitely generated free algebra over V By establishing a natural cor=

respondence etmeen the two types of varities of recoguzable V-sts, a Variety Theorem ls derived albo for rales of pure recognizable V-ats

2.1 Many-sorted algebras

In what flows, Sis always a non-empty set of sorts, We consider various families of objocts indexed

by 8 Such families are said to be S-sorted, or just sorted The sort of an object is usually shown

fas subscript oF in parentheses (Lo avoid multiple subscripts), An S-sorted set A= (Ages is an S-indewod fnmily of sets; far each s € S, A, is the sot of elements of sorts iu Ay and we write i also

fs Als), The basic st-theoretie wotions are defined for S-sorted sots in the uatural sortwise manner

sorta sets A= (A,}yes and B= (Bayes, AC B means that Ay © By (ALU Byjacs and AN B= (AyM Bases and general sorted unions ancl Intersections are defined similaely The notation Mis used also for the S-sorted empty so¢ (Os:

In particular, for any

We shall also consider subsets of one given srt of sorted sets, With any subset TS Ay of some sort € Sof mn Sorted set A = (4,)es We associate the sort suet (7) C Asuch that (F}q = and (Ps Ai A sorted reation (BOs) om a § (Anacs Is

an S:sorta family of relations such that foreach 6 € $,&y isa Telation on as A sented equivalence

A sorted mopping g: A+ B from an Sorted sot A= (Aes to an S-sorted set LF = (B,)oes