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Mthemticl Foundtions of Automt Theory Jen-Éric Pin Version of November 30, Prefce These notes form the core of future book on the lgebric foundtions of utomt theory. This book is still incomplete,

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Mthemticl Foundtions of Automt Theory Jen-Éric Pin Version of November 30, 2016 2 Prefce These notes form the core of future book on the lgebric foundtions of utomt theory. This book is still incomplete, but the first eleven chpters now form reltively coherent mteril, covering roughly the topics described below. The erly yers of utomt theory Kleene s theorem [66] is usully considered s the strting point of utomt theory. It shows tht the clss of recognisble lnguges (tht is, recognised by finite utomt), coincides with the clss of rtionl lnguges, which re given by rtionl expressions. Rtionl expressions cn be thought of s generlistion of polynomils involving three opertions: union (which plys the role of ddition), product nd the str opertion. It ws quickly observed tht these essentilly combintoril definitions cn be interpreted in very rich wy in lgebric nd logicl terms. Automt over infinite words were introduced by Büchi in the erly 1960s to solve decidbility questions in first-order nd mondic second-order logic of one successor. Investigting two-successor logic, Rbin ws led to the concept of tree utomt, which soon becme stndrd tool for studying logicl definbility. The lgebric pproch The definition of the syntctic monoid, monoid cnoniclly ttched to ech lnguge, ws first given by Schützenberger in 1956 [133]. It lter ppered in pper of Rbin nd Scott [124], where the notion is credited to Myhill. It ws shown in prticulr tht lnguge is recognisble if nd only if its syntctic monoid is finite. However, the first clssifiction results on recognisble lnguges were rther stted in terms of utomt [84] nd the first nontrivil use of the syntctic monoid is due to Schützenberger [134]. Schützenberger s theorem (1965) sttes tht rtionl lnguge is str-free if nd only if its syntctic monoid is finite nd periodic. This elegnt result is considered, right fter Kleene s theorem, s the most importnt result of the lgebric theory of utomt. Schützenberger s theorem ws supplemented few yers lter by result of McNughton [80], which estblishes link between str-free lnguges nd first-order logic of the order reltion. Both results hd considerble influence on the theory. Two other importnt lgebric chrcteristions dte bck to the erly seventies: Simon [137] proved tht rtionl lnguge is piecewise testble if nd only if its syntctic monoid is J-trivil nd Brzozowski-Simon [21] nd independently, McNughton [79] 3 4 chrcterised the loclly testble lnguges. The logicl counterprt of the first result ws obtined by Thoms [158]. These successes settled the power of the lgebric pproch, which ws xiomtized by Eilenberg in 1976 [40]. Eilenberg s vriety theory A vriety of finite monoids is clss of monoids closed under tking submonoids, quotients nd finite direct products. Eilenberg s theorem sttes tht vrieties of finite monoids re in one-to-one correspondence with certin clsses of recognisble lnguges, the vrieties of lnguges. For instnce, the rtionl lnguges re ssocited with the vriety of ll finite monoids, the str-free lnguges with the vriety of finite periodic monoids, nd the piecewise testble lnguges with the vriety of finite J-trivil monoids. Numerous similr results hve been estblished over the pst thirty yers nd, for this reson, the theory of finite utomt is now intimtely relted to the theory of finite monoids. Severl ttempts were mde to extend Eilenberg s vriety theory to lrger scope. For instnce, prtil order on syntctic semigroups were introduced in [94], leding to the notion of ordered syntctic semigroups. The resulting extension of Eilenberg s vriety theory permits one to tret clsses of lnguges tht re not necessrily closed under complement, contrry to the originl theory. Other extensions were developed independently by Strubing [153] nd Ésik nd Ito [43]. The topologicl point of view Due llownce being mde, the introduction of topology in utomt theory cn be compred to the use of p-dic nlysis in number theory. The notion of vriety of finite monoids ws coined fter similr notion, introduced much erlier by Birkhoff for infinite monoids: Birkhoff vriety of monoids is clss of monoids closed under tking submonoids, quotient monoids nd direct products. Birkhoff proved in [13] tht his vrieties cn be defined by set of identities: for instnce the identity xy = yx chrcterises the vriety of commuttive monoids. Almost fifty yers lter, Reitermn [126] extended Birkhoff s theorem to vrieties of finite monoids: ny vriety of finite monoids cn be chrcterised by set of profinite identities. A profinite identity is n identity between two profinite words. Profinite words cn be viewed s limits of sequences of words for certin metric, the profinite metric. For instnce, one cn show tht the sequence x n! converges to profinite word denoted by x ω nd the vriety of finite periodic monoids cn be defined by the identity x ω = x ω+1. The profinite pproch is not only powerful tool for studying vrieties but it lso led to unexpected developments, which re t the hert of the current reserch in this domin. In prticulr, Gehrke, Grigorieff nd the uthor [45] proved tht ny lttice of recognisble lnguges cn be defined by set of profinite equtions, result tht subsumes Eilenberg s vriety theorem. The logicl pproch We lredy mentioned Büchi s, Rbin s nd McNughton s remrkble results on the connexion between logic nd finite utomt. Büchi s sequentil clculus 5 is logicl lnguge to express combintoril properties of words in nturl wy. For instnce, properties like word contins two consecutive occurrences of or word of even length cn be expressed in this logic. However, severl prmeters cn be djusted. Different frgments of logic cn be considered: first-order, mondic second-order, Σ n -formuls nd lrge vriety of logicl nd nonlogicl symbols cn be employed. There is remrkble connexion between first-order logic nd the conctention product. The polynomil closure of clss of lnguges L is the set of lnguges tht re sums of mrked products of lnguges of L. By lternting Boolen closure nd polynomil closure, one obtins nturl hierrchy of lnguges. The level 0 is the Boolen lgebr {,A }. Next, for ech n 0, the level 2n+1 is the polynomil closure of the level 2n nd the level 2n+2 is the Boolen closure of the level 2n+1. A very nice result of Thoms [158] shows tht recognisble lnguge is of level 2n+1 in this hierrchy if nd only if it is definble by Σ n+1 -sentence of first-order logic in the signture { ,() A }, where is predicte giving the positions of the letter. There re known lgebric chrcteristions for the three first levels of this hierrchy. In prticulr, the second level is the clss of piecewise testble lnguges chrcterised by Simon [136]. Contents of these notes The lgebric pproch to utomt theory relies mostly on semigroup theory, brnch of lgebr which is usully not prt of the stndrd bckground of student in mthemtics or in computer science. For this reson, n importnt prt of these notes is devoted to n introduction to semigroup theory. Chpter II gives the bsic definitions nd Chpter V presents the structure theory of finite semigroups. Chpters XIII nd XV introduce some more dvnced tools, the reltionl morphisms nd the semidirect nd wreth products. Chpter III gives brief overview on finite utomt nd recognisble lnguges. It contins in prticulr complete proof of Kleene s theorem which relies on Glushkov s lgorithm in one direction nd on liner equtions in the opposite direction. For comprehensive presenttion of this theory I recommend the book of my collegue Jcques Skrovitch [131]. The recent book of Olivier Crton [25] lso contins nice presenttion of the bsic properties of finite utomt. Recognisble nd rtionl subsets of monoid re presented in Chpter IV. The notion of syntctic monoid is the key notion of this chpter, where we lso discuss the ordered cse. The profinite topology is introduced in Chpter VI. We strt with short synopsis on generl topology nd metric spces nd then discuss the reltionship between profinite topology nd recognisble lnguges. Chpter VII is devoted to vrieties of finite monoids nd to Reitermn s theorem. It lso contins lrge collection of exmples. Chpter VIII presents the equtionl chrcteristion of lttices of lnguges nd Eilenberg s vriety theorem. Exmples of ppliction of these two results re gthered in Chpter IX. Chpters X nd XI present two mjor results, t the core of the lgebric pproch to utomt theory: Schützenberger s nd Simon s theorem. The lst five chpters re still under construction. Chpter XII is bout polynomil closure, Chpter XIV presents nother deep result of Schützenberger bout unmbiguous str-free lnguges nd its logicl counterprt. Chpter XVI gives brief introduction to sequentil functions nd the 6 wreth product principle. Chpter XVIII presents some logicl descriptions of lnguges nd their lgebric chrcteristions. Nottion nd terminology The term regulr set is frequently used in the literture but there is some confusion on its interprettion. In Ginzburg [49] nd in Hopcroft, Motwni nd Ullmn [60], regulr set is set of words ccepted by finite utomton. In Slom [132], it is set of words defined by regulr grmmr nd in Croll nd Long [24], it is set defined by regulr expression. This is no rel problem for lnguges, since, by Kleene s theorem, these three definitions re equivlent. This is more problemtic for monoids in which Kleene s theorem does not hold. Another source of confusion is tht the term regulr hs wellestblished mening in semigroup theory. For these resons, I prefer to use the terms recognisble nd rtionl. I tried to keep some homogeneity in nottion. Most of the time, I use Greek letters for functions, lower cse letters for elements, cpitl letters for sets nd clligrphic letters for sets of sets. Thus I write: let s be n element of semigroup S nd let P(S) be the set of subsets of S. I write functions on the left nd trnsformtions nd ctions on the right. In prticulr, I denote by q u the ction of word u on stte q. Why so mny computer scientists prefer the wful nottion δ(q,u) is still mystery. It leds to hevy formuls, like δ(δ(q,u),v) = δ(q,uv), to be compred to the simple nd intuitive (q u) v = q uv, for bsolutely no benefit. I followed Eilenberg s trdition to use boldfce letters, like V, to denote vrieties of semigroups, nd to use clligrphic letters, like V, for vrieties of lnguges. However, I hve dopted Almeid s suggestion to hve different nottion for opertors on vrieties, like EV, LV or PV. I use the term morphism for homomorphism. Semigroups re usully denoted by S or T, monoids by M or N, lphbets re A or B nd letters by, b, c,...but this nottion is not frozen: I my lso use A for semigroup nd S for lphbet if needed! Following trdition in combintorics, E denotes the number of elements of finite set. The nottion u is lso used for the length of word u, but in prctice, there is no risk of confusion between the two. To void repetitions, I frequently use brckets s n equivlent to respectively, like in the following sentence : semigroup [monoid, group] S is commuttive if, for ll x,y S, xy = yx. Lemms, propositions, theorems nd corollries shre the sme counter nd re numbered by section. Exmples hve seprte counter, but re lso numbered by section. References re given ccording to the following exmple: Theorem 1.6, Corollry 1.5 nd Section 1.2 refer to sttements or sections of the sme chpter. Proposition VI.3.12 refers to proposition which is externl to the current chpter. Acknowledgements Severl books on semigroups helped me in prepring these notes. Clifford nd Preston s tretise [28, 29] remins the clssicl reference. My fvourite source for the structure theory is Grillet s remrkble presenttion[53]. I lso borrowed lot from the books by Almeid [3], Eilenberg [40], Higgins [58], Lllement [73] 7 nd Lothire [75] nd lso of course from my own books [91, 86]. Another source of inspirtion (not yet fully explored!) re the reserch rticles by my collegues Jorge Almeid, Krl Auinger, Jen-Cmille Birget, Olivier Crton, Mi Gehrke, Victor Gub, Rostislv Horčík, John McCmmond, Sturt W. Mrgolis, Dominique Perrin, Mrk Spir, Imre Simon, Ben Steinberg, Howrd Strubing, Pscl Tesson, Denis Thérien, Mish Volkov, Pscl Weil nd Mrc Zeitoun. I would like to thnk my Ph.D. students Lure Dviud, Luc Drtois, Chrles Ppermn nd Ynn Pequignot, my collegues t LIAFA nd LBRI nd the students of the Mster Prisien de Recherches en Informtique (notbly Aiswry Cyric, Nthnël Fijlkow, Agnes Köhler, Arthur Milchior, Anc Nitulescu, Pierre Prdic, Léo Stefnesco, Boker Udi, Jill-Jênn Vie nd Furcy) for pointing out mny misprints nd corrections on erlier versions of this document. I would like to cknowledge the ssistnce nd the encourgements of my collegues of the Picsso project, Adolfo Bllester-Bolinches, Antonio Cno Gómez, Rmon Estebn-Romero, Xro Soler-Escrivà, Mri Belén Soler Monrel, Jorge Plnc nd of the Pesso project, Jorge Almeid, Mário J. J. Brnco, Vítor Hugo Fernndes, Grcind M. S. Gomes nd Pedro V. Silv. Other creful reders include Achim Blumensth nd Mrtin Beudry (with the help of his student Cédric Pinrd) who proceeded to very creful reding of the mnuscript. George Hnsoul, Sbstien Lbb, Anne Schilling nd Herbert Toth sent me some very useful remrks. Specil thnks re due to Jen Berstel nd to Pul Gstin for providing me with their providentil L A TEX pckges. Pris, November 2016 Jen-Éric Pin Contents I Algebric preliminries Subsets, reltions nd functions Sets Reltions Functions Injective nd surjective reltions Reltions nd set opertions Ordered sets Exercises II Semigroups nd beyond Semigroups, monoids nd groups Semigroups, monoids Specil elements Groups Ordered semigroups nd monoids Semirings Exmples Exmples of semigroups Exmples of monoids Exmples of groups Exmples of ordered monoids Exmples of semirings Bsic lgebric structures Morphisms Subsemigroups Quotients nd divisions Products Idels Simple nd 0-simple semigroups Semigroup congruences Trnsformtion semigroups Definitions Full trnsformtion semigroups nd symmetric groups Product nd division Genertors A-generted semigroups i ii CONTENTS 5.2 Cyley grphs Free semigroups Universl properties Presenttions nd rewriting systems Idempotents in finite semigroups Exercises III Lnguges nd utomt Words nd lnguges Words Orders on words Lnguges Rtionl lnguges Automt Finite utomt nd recognisble lnguges Deterministic utomt Complete, ccessible, coccessible nd trim utomt Stndrd utomt Opertions on recognisble lnguges Boolen opertions Product Str Quotients Inverses of morphisms Miniml utomt Rtionl versus recognisble Locl lnguges Glushkov s lgorithm Liner equtions Extended utomt Kleene s theorem Exercises Notes IV Recognisble nd rtionl sets Rtionl subsets of monoid Recognisble subsets of monoid Recognition by monoid morphisms Opertions on sets Recognisble sets Connexion with utomt Trnsition monoid of deterministic utomton Trnsition monoid of nondeterministic utomton Monoids versus utomt The syntctic monoid Definitions The syntctic monoid of lnguge Computtion of the syntctic monoid of lnguge Recognition by ordered structures CONTENTS iii 5.1 Ordered utomt Recognition by ordered monoids Syntctic order Computtion of the syntctic ordered monoid Exercises Notes V Green s reltions nd locl theory Green s reltions Inverses, wek inverses nd regulr elements Inverses nd wek inverses Regulr elements Rees mtrix semigroups Structure of regulr D-clsses Structure of the miniml idel Green s reltions in subsemigroups nd quotients Green s reltions in subsemigroups Green s reltions in quotient semigroups Green s reltions nd trnsformtions Summry: complete exmple Exercises Notes VI Profinite words Topology Generl topology Metric spces Compct spces Topologicl semigroups Profinite topology The free profinite monoid Universl property of the free profinite monoid ω-terms Recognisble lnguges nd clopen sets Exercises Notes VII Vrieties Vrieties Free pro-v monoids Identities Wht is n identity? Properties of identities Reitermn s theorem Exmples of vrieties Vrieties of semigroups Vrieties of monoids Vrieties of ordered monoids Summry iv CONTENTS 5 Exercises Notes VIII Equtions nd lnguges Equtions Equtionl chrcteristion of lttices Lttices of lnguges closed under quotients Strems of lnguges C-strems Vrieties of lnguges The vriety theorem Summry Exercises Notes IX Algebric chrcteristions Vrieties of lnguges Loclly finite vrieties of lnguges Commuttive lnguges R-trivil nd L-trivil lnguges Some exmples of +-vrieties Lttices of lnguges The role of the zero Lnguges defined by density Cyclic nd strongly cyclic lnguges Exercises Notes X Str-free lnguges Str-free lnguges Schützenberger s theorem Exercises Notes XI Piecewise testble lnguges Subword ordering Simple lnguges nd shuffle idels Piecewise testble lnguges nd Simon s theorem Some consequences of Simon s theorem Exercises Notes XII Polynomil closure Polynomil closure of lttice of lnguges A cse study CONTENTS v XIII Reltionl morphisms Reltionl morphisms Injective reltionl morphisms Reltionl V-morphisms Aperiodic reltionl morphisms Loclly trivil reltionl morphisms Reltionl ese e -morphisms Four exmples of V-morphisms Ml cev products Three exmples of reltionl morphisms Conctention product Pure lnguges Flower utomt XIV Unmbiguous str-free lnguges Unmbiguous str-free lnguges XV Wreth product Semidirect product Wreth product Bsic decomposition results Exercises XVI Sequentil functions Definitions Pure sequentil trnsducers Sequentil trnsducers Composition of sequentil functions Sequentil functions nd wreth product The wreth product principle nd its consequences The wreth product principle Applictions of the wreth product principle The opertions T U 1 T nd L LA The opertions T 2 T nd L L The opertion T U 2 T nd str-free expressions Exercises XVII Conctention hierrchies Conctention hierrchies XVIIIAn excursion into logic Introduction The formlism of logic Syntx Semntics Logic on words Mondic second-order logic on words First-order logic of the liner order vi CONTENTS 4.1 First order nd str-free sets Logicl hierrchy Annex 289 A A trnsformtion semigroup References Index Chpter I Algebric preliminries 1 Subsets, reltions nd functions 1.1 Sets The set of subsets of set E is denoted by P(E) (or sometimes 2 E ). The positive Boolen opertions on P(E) comprise union nd intersection. The Boolen opertions lso include complementtion. The complement of subset X of E is denoted by X c. Thus, for ll subsets X nd Y of E, the following reltions hold (X c ) c = X (X Y) c = X c Y c (X Y) c = X c Y c We let E denote the number of elements of finite set E, lso clled the size of E. A singleton is set of size 1. We shll frequently identify singleton {s} with its unique element s. Given two sets E nd F, the set of ordered pirs (x,y) such tht x E nd y F is written E F nd clled the product of E nd F. 1.2 Reltions Let E nd F be two sets. A reltion on E nd F is subset of E F. If E = F, it is simply clled reltion on E. A reltion τ cn lso be viewed s function 1 from E to P(F) by setting, for ech x E, τ(x) = {y F (x,y) τ} By buse of lnguge, we sy tht τ is reltion from E into F. The inverse of reltion τ E F is the reltion τ 1 F E defined by τ 1 = {(y,x) F E (x,y) τ} Note tht if τ is reltion from E in F, the reltion τ 1 cn be lso viewed s function from F into P(E) defined by τ 1 (y) = {x E y τ(x)} 1 Functions re formlly defined in the next section, but we ssume the reder is lredy fmilir with this notion. 1 2 CHAPTER I. ALGEBRAIC PRELIMINARIES A reltion from E into F cn be extended to function from P(E) into P(F) by setting, for ech subset X of E, τ(x) = τ(x) = {y F for some x X, (x,y) τ} x X If Y is subset of F, we then hve τ 1 (Y) = τ 1 (y) = {x E there exists y Y such tht y τ(x)} y Y = {x E τ(x) Y } Exmple 1.1 Let τ be the reltion from E = {1,2,3} into F = {1,2,3,4} defined by τ = {(1,1),(1,2),(2,1),(2,3),(2,4)} E F Figure 1.1. The reltion τ. Then τ(1) = {1,2}, τ(2) = {1,3,4}, τ(3) =, τ 1 (1) = {1,2}, τ 1 (2) = {1}, τ 1 (3) = {2}, τ 1 (4) = {2}. Given two reltions τ 1 : E F nd τ 2 : F G, we let τ 1 τ 2 or τ 2 τ 1 denote the com

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