Semigroups and automata on infinite words

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1 Semigroups nd utomt on infinite words Dominique Perrin nd Jen-Éric Pin Pulished in Introduction This pper is n introduction to the lgeric theory of infinite words. Infinite words re widely used in computer science, in prticulr to model the ehviour of progrms or circuits. From mthemticl point of view, they hve rich structure, t the cross-rods of logic, topology nd lger. This pper emphsizes the comintoril nd lgeric spects of this theory ut theinterested reder is referred to thesurvey rticles [34, 44] or to thereport [30] for more informtion on the other spects. In prticulr, the importnt topic of the complexity of the lgorithms on infinite words is not treted in this pper. The pper is written with the perspective of generlizing the results on recognizle sets of finite words to infinite words. This does not exctly follow the historicl development of the theory, ut it gives good ide of the typeof prolems tht occur in this field. Some of these prolems re still open, or hve een solved quite recently so tht the definitions nd results presented elow my not e s yet finlized. The first result to e generlized is the equivlence etween finite utomt, finite deterministic utomt nd rtionl expressions. If one dds infinite itertion ( omeg opertion) to the stndrd rtionl opertions, union, product nd str, one gets nturl definition of the ω-rtionl sets of infinite words tht extends the definition of rtionl sets of finite words. Büchi [5] ws the first to propose definition of finite utomt cting on infinite words. This definition suffices to extend Kleene s theorem to infinite words: the sets of infinite words recognized y finite Büchi utomt re exctly the ω-rtionl sets. This result is now known s Büchi s theorem. However, Büchi s definition is not totlly stisfying since deterministic Büchi utomt re not equivlent to non deterministic ones. The con- Institut Gsprd Monge, Université de Mrne l Vllée, Noisy-le-Grnd, FRANCE. From 1st Sept 1993, LITP, Université Pris VI, Tour 55-65, 4 Plce Jussieu, Pris Cedex 05, FRANCE. E-mil: pin@litp.ip.fr 1

2 nection etween deterministic nd non deterministic Büchi utomt ws enlightened y deep theorem of McNughton: set of infinite words is recognized y non deterministic Büchi utomton if nd only if it is finite oolen comintion of sets recognized y deterministic Büchi utomt. This result ws prepred y suitle definition for utomt on infinite words given y Muller [22]. These utomt hve the sme power s the Büchi utomt, ut this time, non deterministic utomt re equivlent to deterministic ones. It is well known fct tht finite semigroups cn e viewed s twosided lgeric counterprt of finite utomt tht recognize finite words. Severl ttempts hve een mde to find n lgeric counterprt of finite utomt tht recognize infinite words. Since the notion of infinite word is symmetricl, finite semigroups re not suitle ny more. They cn e replced y ω-semigroups, which re, roughly speking, semigroups equipped with n infinite product. The sic definitions re quite promising, ut technicl difficulty rises lmost immeditely. Indeed, in order to design lgorithms on finite ω-semigroups one needs finite representtion for them, ut the definition of the infinite product pprently requires n infinite tle, even for 2-element ω-semigroup. However, Rmsey-type rgument shows tht the structure of finite ω-semigroup is totlly determined y three opertions of finite signture. Tht is, finite ω-semigroupsre equivlent to certin finite lgers of finite signture, the Wilke lgers. Now, the definitions of recognizle set, of syntctic congruence, etc., ecome nturl nd most results vlid for finite words cn e dpted to infinite words. Crrying on the work of Arnold [1], Pécuchet [24, 23] nd the first uthor [25, 26, 27], Wilke [45, 46] hs pushed the nlogy with the theory for finite words sufficiently fr to otin counterprt of Eilenerg s vriety theorem for finite or infinite words. This is pproximtively the point reched y the lgeric pproch tody, lthough current reserch my lredy hve pssed eyond. In ny cse, this is the plce where our rticle finishes. The pper is orgnized s follows: the sic definitions on words nd ω-rtionl sets re given in sections 2 nd 3. Büchi utomt re defined in section 4, deterministic Büchi utomt in section 5 nd Muller utomt in section 6. The equivlence etween finite ω-semigroups nd finite Wilke lgers is estlished in section 7 nd their connections with utomt re presented in sections 8, 9 nd 10. Syntctic ω-semigroups re introduced in section 11 nd the vriety theorem nd its consequences re discussed in section 12. Section 13 presents the conclusion of the rticle. 2

3 2 Words Let A e finite set clled n lphet, whose elements re letters. A finite word is finite sequence of letters, tht is, function u from finite set of the form {0,1,2,...,n} into A. If one puts u(i) = i for 0 i n, the word u is usully denoted y 0 1 n, nd the integer u = n+1 is the length of u. The unique word of length 0 is the empty word, denoted y 1. An infinite word is function u from N into A, usully denoted y 0 1 2, where u(i) = i for ll i N. A word is either finite word or n infinite word. Intuitively, the conctention or product of two words u nd v is the word uv otined y writing u followed y v. More precisely, if u is finite nd v is finite or infinite, then uv is the word defined y { u(i) if i < u (uv)(i) = v(i u ) if i u We denote respectively y A, A +, A N the set of ll finite words, finite nonempty words, nd infinite words, respectively. We lso denote y A = A + A N the set of ll non empty words. A word x is fctor of word w if there exist two words u nd v (possily empty) such tht w = uxv. 3 Rtionl sets The rtionl opertions re the four opertions union, product, plus nd str, defined on the set of susets of A s follows (1) Union: L 1 L 2 = {u u L 1 or u L 2 } (2) Product: L 1 L 2 = {u 1 u 2 u 1 L 1 nd u 2 L 2 } (3) Plus: L + = {u 1 u n n > 0 nd u 1,...,u n L} (4) Str: L = {u 1 u n n 0 nd u 1,...,u n L} Thus we hve the reltions L + = LL = L L nd L = L + {1} ThesetofrtionlsusetsofA isthesmllestsetofsusetsofa contining the finite sets nd closed under finite union, product nd str. For instnce, ( ) ( ) denotes rtionl set. Similrly, the set of rtionl susets of A + is the smllest set of susets of A + contining the finite sets nd closed under finite union, product nd plus. It is not difficult to verify tht the rtionl susets of A + re exctly the rtionl susets of A tht do not contin the empty word. It is possile to generlize the concept of rtionl sets to infinite words s follows. First, the product cn e extended, y setting, for X A nd 3

4 Y A N, XY = {xy x X nd y Y}. Next, we define n infinite itertion ω y setting, for every suset X of A + X ω = {x 0 x 1 for ll i 0,x i X} Equivlently, X ω is the set of infinite words otined y conctenting n infinite sequence of words of X. In prticulr, if u = 0 1 n, we set u ω = 0 1 n 0 1 n 0 1 n 0 1 Theset Rt(A ) of ω-rtionl susetsofa isthesmllest set R ofsusets of A such tht () R nd for ll A, {} R, () R is closed under finite union, (c) For every suset X of A + nd for every suset Y of A, X R nd Y R imply XY R, (d) For every suset X of A +, X R implies X + R nd X ω R. In other words, the set of ω-rtionl susets of A is the smllest set of susets of A contining the finite sets of A + nd closed under finite union, finite product, plus nd omeg. The ω-rtionl sets which re contined in A N re clled, y use of lnguge, the ω-rtionl susets of A N. There is very simple chrcteriztion of these sets, which is often used s definition. Proposition 3.1 A suset of A N is ω-rtionl if nd only if it is finite union of susets of the form XY ω where X nd Y re non-empty rtionl susets of A +. Exmple 3.1 The set of infinite words on the lphet {,} hving only finite numer of s is given y the expression {,} ω. 4 Automt A finite (non deterministic) utomton is triple A = (Q,A,E) where Q is finite set (the set of sttes), A is n lphet, nd E is suset of Q A Q, clled the set of trnsitions. Two trnsitions (p,,q) nd (p,,q ) re consecutive if q = p. A pth in A is finite sequence of consecutive trnsitions e 0 = (q 0, 0,q 1 ), e 1 = (q 1, 1,q 2 ),..., e n 1 = (q n 1, n 1,q n ) lso denoted q n 1 q1 q2 q n 1 q n 4

5 The stte q 0 is the origin of the pth, the stte q n+1 is its end, nd the word x = 0 1 n is its lel. An infinite pth in A is sequence p of consecutive trnsitions indexed y N. e 0 = (q 0, 0,q 1 ), e 1 = (q 1, 1,q 2 ),... lso denoted q q1 q2 The stte q 0 is the origin of the infinite pth nd the infinite word 0 1 is its lel. A stte q occurs infinitely often in p if q n = q for infinitely mny n. Exmple 4.1 Let A = (Q,A,E) e the utomton represented in Figure 4.1. Then Q = {1,2}, A = {,}, E = {(1,,1),(2,,1),(1,,2),(2,,2)} nd (1,,2)(2,,2)(2,,1)(1,,2)(2,,2)(2,,1)(1,,2)(2,,2)(2,,1)(1,,2) is n infinite pth of A. 1 2 Figure 4.1: A finite utomton. A finite Büchi utomton is quintuple A = (Q,A,E,I,F) where (1) (Q,A,E) is finite utomton, (2) I nd F re susets of Q, clled the set of initil nd finl sttes, respectively. A finite pth in A is successful if its origin is in I nd its end is in F. An infinite pth p is successful if its origin is in I nd if some stte of F occurs infinitely often in p. The set of finite (respectively infinite) words recognized y A is the set, denoted L + (A) (respectively L ω (A)), of the lels of ll successful finite (respectively infinite) pths of A. A set of finite (respectively infinite) words X is recognizle if there exists finite Büchi utomton A such tht X = L + (A) (respectively X = L ω (A)). Exmple 4.2 Let A e the Büchi utomton otined from exmple 4.1 y tking I = {1} nd F = {2}. Initil sttes re represented y n incoming rrow nd finl sttes y n rrow going out. 5

6 1 2 Figure 4.2: A Büchi utomton. Then L + (A) = {,} is the set of ll finite words whose first letter is n, L ω (A) = ( ) ω is the set of infinite words whose first letter is n nd contining n infinite numer of s. Exmple 4.3 Let A e the Büchi utomton represented elow. Then L ω (A) = A ω is the set of ll infinite words contining finite numer of s., 1 2 Figure 4.3: The utomton A recognizing A ω. Let A e the utomton otined from A y tking only 1 s initil stte. Then L ω (A ) = A ω. Exmple 4.4 Let A e the Büchi utomton represented elow,c, 1 2,c Then L ω (A) = ({,c} {}) ω. Figure 4.4: The reltionship etween rtionl nd recognizle sets of finite words is given y the fmous theorem of Kleene. Theorem 4.1 A suset of A is rtionl if nd only if it is recognizle. The counterprt of Kleene s theorem for infinite words is due to Büchi [4]. Theorem 4.2 A suset of A N is ω-rtionl if nd only if it is recognizle. 6

7 Sketch of the proof. (1) Let A = (Q,A,E,I,F) e finite Büchi utomton. For p,q Q, denote y L + (E,p,q) the set of non empty finite words recognized y the utomton (Q,A,E,{p},{q}). By Theorem 4.1, ll these sets re rtionl. Furthermore, it is not difficult to see tht X = L ω (A) = L + (E,i,f) ( L + (E,f,f) ) ω i I f F It follows tht L ω (A) is ω-rtionl. (2) It is esy to see tht recognizle sets re closed under finite union. Indeed, the disjoint union of two utomt A 1 nd A 2 recognizes exctly the union of the sets recognized respectively y A 1 nd A 2. In order to prove tht every ω-rtionl set is recognizle, it remins to show tht every set of the form XX ω, where X nd X re non-empty rtionl susets of A +, is recognizle. A smll comintoril rgument shows tht every rtionl set cn e recognized y norml utomton, tht is, n utomton with exctly one initil stte i, exctly one finl stte f nd no trnsition strting from f or ending in i. Now, if A nd A re norml utomt recognizing X ndx, thentheutomton otinedyidentifyingf, i ndf, sindicted in the figure elow, recognizes XX ω : i A f = i = f A Figure 4.5: An utomton recognizing X(X ) ω. 5 Deterministic Büchi utomt A Büchi utomton A = (Q,A,E,I,F) is sid to e deterministic if I is singletonndife continsnopiroftrnsitionsoftheform(q,,q 1 ),(q,,q 2 ) with q 1 q 2. In other words, given stte q nd letter, there is t most one stte q such tht (q,,q ) E. In prticulr, ech word u is the lel of t most one pth strting from the initil stte. A finite word u is ccepted if the end of this unique pth is finl stte. And n infinite word is ccepted if the unique pth visits infinitely often finl stte. It follows tht n infinite word is ccepted y A if nd only if it hs infinitely mny prefixes ccepted y A. This leds to the following definition: for ny suset L of A, put L = {u A N u hs infinitely mny prefixes in L}. Then one cn stte 7

8 Proposition 5.1 If A is finite deterministic Büchi utomton, then L ω (A) = L + (A). Exmple 5.1 If A = {,} nd L =, then L =. 1 2 Figure 5.1: L + (A) = nd L ω (A) =. Consider now the set L of ll finitewords of the form n. Then L = ( ) ω, the set of infinite words contining infinitely mny s. 1 2 Figure 5.2: L + (A) = ( ) + = {,} nd L ω (A) = ( ) ω. A set of infinite words is deterministic if it is recognized y finite Büchi utomton. Lndweer [14] hs shown the existence of non deterministic recognizle sets. Proposition 5.2 The set {,} ω is not deterministic. Proof. Let X = {,} ω nd suppose tht X = L for some suset L of A. Then ω hs prefix u 1 = n 1 in L nd similrly, n 1 ω hs prefix u 2 = n 1 n 2 in L, etc. so tht the infinite word u = n 1 n 2 n3 hs infinitely mny prefixes in L. Thus u L, contrdiction, since u hs n infinite numer of s. 6 Muller utomt We hve seen tht non deterministic Büchi utomt re not equivlent to deterministic ones. This led Muller [22] to introduce more stisfying definition. A finite Muller utomton is quintuple A = (Q,A,E,I,T ) where (1) (Q,A,E) is finite utomton, (2) I is suset of Q, clled the set of initil sttes. (3) T = {T 1,...,T n } (the stte tle) is set of susets T 1,...,T n of Q. 8

9 A pth p is successful in A if Infinite(p) T, where Infinite(p) is the set of sttes tht re visited infinitely often y p. The set L ω (A) (lso denoted L ω (E,i,T )) of infinite words ccepted y A is the set of lels of ll successful pths. Exmple 6.1 Let A = {,}. The utomton elow, with the stte tle T = {{2}}, recognizes the set A ω of ll infinite words contining finite numer of s. 1 2 Figure 6.1: A Muller utomton. The utomton A otined y tking s stte tle T = {{1},{2}}, recognizes the set A ω A ω. A finite Muller utomton A = (Q,A,E,I,T ) is deterministic if I is singletonndife continsnopiroftrnsitionsoftheform(q,,q 1 ),(q,,q 2 ) with q 1 q 2. This time, deterministic utomt re equivlent to non deterministic ones. The following result, minly due to McNughton [19], summrizes severl equivlences. Theorem 6.1 Let X e suset of A N. The following conditions re equivlent: (1) X is ω-rtionl, (2) X is recognized y finite Muller utomton, (3) X is recognized y finite deterministic Muller utomton, (4) X is recognized y finite Büchi utomton, (5) X is oolen comintion of sets recognized y finite deterministic Büchi utomton. Sketch of the proof. (3) implies (2) is cler. (2) implies (1). Let A = (Q,A,E,I,T ) e deterministic Muller utomton with T = {T 1,...,T n }. Then, it is esy to see tht L ω (A) = L ω (E,i,{T j }) 1 j n In other words, we my ssume tht the tle T contins only one set of sttes T = {t 0,...,t k 1 }. Let X = L + (E,i,t 0 ) nd, for 0 i k 1, let X i e the set of lels of ll finite pths from t i to t i+1 visiting only sttes of T (where the indices i re clculted modulo k). Now pth p is 9

10 successful if nd only if it strts in i nd Infinite(p) = T. In this cse, p cn e fctorized s follows: (finite) pth from i to t 0, one from t 0 to t 1, one from t 1 to t 2,..., one from t k 2 to t k 1, one from t k 1 to t 0, one from t 0 to t 1, etc. Therefore L ω (E,i,{T}) = X(X 0 X 1 X k 1 ) ω nd thus L ω (E,i,{T}) is ω-rtionl. (1) implies (4) follows from Theorem 4.2. (4) implies (3) is the more difficult prt of the proof. We shll present in section 10 semigroup theoretic proof of this result. The est known lgorithm to convert Büchi utomton into deterministic Muller utomton is Sfr s construction, which, given n-stte Büchi utomton, produces n equivlent deterministic Muller utomton with t most n n sttes. The reder is referred to the originl rticle of Sfr [37] or to [30] for presenttion of this difficult lgorithm. Thus conditions (1) to (4) re equivlent. We conclude y showing tht (2) implies (5) nd (5) implies (1). (2) implies (5). Let A = (Q,A,E,I,T ) e deterministic Muller utomton. As ove, we my ssume tht T = {T}. Then, for n infinite pth p, the condition Infinite(p) = T is equivlent to the conjunction of the two conditions (1) p visits every t T infinitely often, (2) p visits every t / T finitely often. This cn e converted into the following formul, which gives (5) L ω (E,i,{T}) = t T L ω (E,i,t)\ t/ T L ω (E,i,t) (5) implies (1). By Theorem 4.2, every set recognized y finite deterministic Büchi utomton is ω-rtionl. Furthermore, since (1) nd (3) re equivlent, ω-rtionl sets re closed under oolen opertions. Corollry 6.2 The recognizle sets of A N re closed under finite oolen opertions, morphisms nd inverse morphisms etween free semigroups. 7 ω-semigroups As it is shown elsewhere in this volume [33], one cn use finite semigroups in plce of utomt to define the recognizle sets of finite words. It is possile to extend this ide to infinite words y replcing semigroups y ω-semigroups, which re, siclly, lgers in which infinite products re defined. Although these lgers do not hve finitry signture, stndrd results on lgers still hold. In prticulr, A ppers to e the free lger on the 10

11 set A nd recognizle sets cn e defined, s efore, s the sets recognized y finite lgers. However, prolem rises since these finite lgers hve n infinitry signture nd thus re not finite ojects. This prolem cn e solved y Rmsey type rgument showing tht the structure of these finite lgers is totlly determined y only three opertions of finite signture. This defines new type of lgers of finite signture, the Wilke lgers, tht suffice to del with infinite products. 1 We now come to the precise definitions. An ω-semigroup is two-sorted lger S = (S f,s ω ) equipped with the following opertions: () A inry opertion defined on S f nd denoted multiplictively, () A mpping S f S ω S ω, clled mixed product, tht ssocites to ech pir (s,t) S f S ω n element of S ω denoted st, (c) A mpping π : S N f S ω, clled infinite product These three opertions stisfy the following properties : (1) S f, equipped with the inry opertion, is semigroup, (2) for every s,t S f nd for every u S ω, s(tu) = (st)u, (3) for every incresing sequence (k n ) n>0 nd for ll (s n ) n N S N f, π(s 0 s 1 s k1 1,s k1 s k1 +1 s k2 1,...) = π(s 0,s 1,s 2,...) (4) for every s S f nd for every (s n ) n N S N f sπ(s 0,s 1,s 2,...) = π(s,s 0,s 1,s 2,...) Conditions (1) nd (2) cn e thought of s n extension of ssocitivity. Conditions(3)nd(4)showthtonecnreplcethenottionπ(s 0,s 1,s 2,...) y the nottion s 0 s 1 s 2 without miguity. We shll use this simplified nottion in the sequel. Intuitively, n ω-semigroup is sort of semigroup in which infinite products re defined. A ω-semigroup S = (S f,s ω ) is complete if every element of S ω cn e written s n infinite product of elements of S f. It is hrmless hypothesis to ssume tht ll the ω-semigroups considered in this pper re complete. Exmple 7.1 We denote y A the ω-semigroup (A +,A N ) equipped with the usul conctention product. Given two ω-semigroups S = (S f,s ω ) nd T = (T f,t ω ), morphism of ω-semigroups S is pir ϕ = (ϕ f,ϕ ω ) consisting of semigroup morphism ϕ f : S f T f nd of mpping ϕ ω : S ω T ω preserving the mixed product nd the infinite product: for every sequence (s n ) n N of elements of S f, ϕ ω (s 0 s 1 s 2 ) = ϕ f (s 0 )ϕ f (s 1 )ϕ f (s 2 ) 1 Actully, the chronology is little it different. Rmsey type rguments hve een used for long time in semigroup theory [18, 40, 9], the Wilke lgers were introduced y Wilke in [45] under the nme of inoids to clrify the pproch of Arnold [1], Pécuchet [23] nd Perrin [25] nd the ide of using infinite products on semigroups cme lst [30]. 11

12 nd for every s S f, t S ω, ϕ f (s)ϕ ω (t) = ϕ ω (st) In the sequel, we shll often omit the suscripts, nd use the simplified nottion ϕ insted of ϕ f nd ϕ ω. Algeric concepts like ω-susemigroup, quotient nd division re esily dpted to ω-semigroups. The semigroup A + is clled the free semigroup on the set A ecuse it stisfies the following property (which defines free ojects in the generl setting of ctegory theory): every mp from A into semigroup S cn e extended in unique wy into semigroup morphism from A + into S. Similrly, it is not difficult to see tht the free ω-semigroup on (A, ) is the ω-semigroup A. A key result is tht when S is finite, the infinite product is totlly determined y the elements of the form s ω = sss, ccording to the following result of Wilke [46]. Theorem 7.1 Let S f e finite semigroup nd let S ω e finite set. Suppose tht there exists mixed product S f S ω S ω nd mp from S f into S ω, denoted s s ω, stisfying, for every s,t S f, the equtions s(ts) ω = (st) ω (s n ) ω = s ω for every n > 0 Then the pir S = (S f,s ω ) cn e equipped, in unique wy, with structure of ω-semigroup such tht for every s S, the product sss is equl to s ω. This is non trivil result, sed on consequence of Rmsey s theorem which is worth mentioning: Theorem 7.2 Let ϕ : A + S e morphism from A + into finite semigroup S. For every infinite word u A N, there exist pir (s,e) of elements of S such tht se = s, e 2 = e, nd fctoriztion u = u 0 u 1 of u s product of words of A + such tht ϕ(u 0 ) = s nd ϕ(u n ) = e for every n > 0. This motivtes the following definition: linked pir in finite semigroup S is pir (s,e) S S such tht e is idempotent nd se = s. Two linked pirs (s,e) nd (s,e ) re conjugte (nottion (s,e) (s,e )) if there exist x,y S 1 such tht e = xy, e = yx nd s = sx. Note tht these conditions lso imply s = s y, since s y = sxy = se = s. A Wilke lger is two-sorted lger (S f,s ω ) equipped with three opertions: n ssocitive product on S f, mixed product S f S ω S ω nd mp from S f into S ω, denoted s s ω, stisfying, for every s,t S f, the equtions s(ts) ω = (st) ω (s n ) ω = s ω for every n > 0 12

13 A Wilke lger is complete if every element of S ω cn e written s st ω for some s,t S f. Theorem 7.1 sttes tht every finite ω-semigroup is equivlent with finite Wilke lger. One cn ttch to every finite semigroup S finite complete ω-semigroup S = (S,S ω ), constructed s follows. Denote y π the exponent of S, tht is, the smllest positive integer n such tht s n is idempotent for ll s S. Let [s,e] e the conjugcy clss of linked pir (s,e). One defines S ω s the set of conjugcy clsses of linked pirs of S. The pir S is equipped with structure of Wilke lger y setting, for ll [s,e] S ω nd t S, t[s,e] = [ts,e] nd t ω = [t π,t π ] The definition is coherent, since if (s,e ) nd (s,e) re conjugte linked pirs, then (ts,e ) nd (ts,e) re conjugte linked pirs. One cn prove tht S is complete Wilke lger such tht every semigroup morphism ϕ : A + S cn e extended into morphism of ω-semigroups ϕ : A S defined y ϕ f = ϕ nd ϕ ω (u) = [s,e], where (s,e) is linked pir ssocited with u. The ω-semigroup S hs the following universl property. Proposition 7.3 Let T = (T f,t ω ) e finite ω-semigroup nd let S e finite semigroup. For every semigroup morphism ϕ from S into T f, there exists unique morphism of ω-semigroups ϕ : S T tht extends ϕ. Furthermore, if ϕ(s) genertes T s n ω-semigroup, then ϕ is onto. In prticulr, every finite ω-semigroup is quotient of n ω-semigroup of the form S. Corollry 7.4 Every finite complete ω-semigroup S = (S f,s ω ) is quotient of S f. Exmple 7.2 Let S = {,} e the finite semigroup given y the multipliction tle =, =, = nd =. There re four linked pirs nd two conjugcy clsses: (,) (,) nd (,) (,). Thus, S ω = {[,],[,]} nd the ω-power nd the mixed product re given in the following tles ω = [,] [,] = [,] = [,] ω = [,] [,] = [,] = [,] Exmple 7.3 Let S e the five element Brndt semigroup: S is the semigroup with zero presented on the set {,} y the reltions 2 = 0, 2 = 0, = nd =. Thus S = {,,,,0} nd the D-clss structure of S is represented in the figure elow: 13

14 0 There re seven linked pirs nd four conjugcy clsses: (0,) (0,), (,) (,), (,) (,) nd (0,0). Thus, S ω = {[,],[,],[0,],[0,0]} nd the ω-power nd the mixed product re given in the following tles () ω = [,] () ω = [,] ω = ω = [0,0] [,] = [0,] [,] = [,] [0,] = [0,] [0,0] = [0,0] [,] = [,] [,] = [0,] [0,] = [0,] [0,0] = [0,0] [,] = [,] [,] = [0,] [0,] = [0,] [0,0] = [0,0] [,] = [0,] [,] = [,] [0,] = [0,] [0,0] = [0,0] 0[,] = [0,] 0[,] = [0,] 0[0,] = [0,] 0[0,0] = [0,0] A surjective morphism of ω-semigroups ϕ : A S recognizes suset X of A N if there exists suset P of S ω such tht X = ϕ 1 (P). By extension, n ω-semigroup S recognizes X if there exists surjective morphism of ω- semigroups ϕ : A S tht recognizes X. This definition cn e extended to susets of A s follows. A morphism of ω-semigroups ϕ : A S recognizes suset X of A if nd only if ϕ 1 ϕ(x) = X, tht is, if there exists suset P = (P f,p ω ) of (S f,s ω ) such tht X A + = ϕ 1 f (P f) nd X A = ϕ 1 ω (P ω). A consequence of Corollry 7.4 is tht every suset of A recognized y finite ω-semigroup is recognized y n ω-semigroup of the form S. Proposition 7.5 Let S = (S f,s ω ) e finite ω-semigroup recognizing suset X of A. Then S f lso recognizes X. Exmple 7.4 Let S = ({,},{ ω, ω }), where the finite product is given y the equlities =, =, =, = nd the infinite product is given y the rules x 1 x 2 = ω nd x 1 x 2 = ω. One recognizes here the ω-semigroup of Exmple 7.2. Let ϕ : A S e the morphism of ω-semigroups defined y ϕ() = nd ϕ() =. Then ϕ 1 ( ω ) = A ω. As for finite words, the following theorem holds. Theorem 7.6 A suset of A N is recognizle if nd only if it is recognized y finite ω-semigroup. 14

15 In other words, finite ω-semigroups cn e converted into utomt nd vice vers. It is not difficult to see tht suset of A N recognized y finite ω-semigroup is ω-rtionl nd thus, y Theorem 4.2, recognizle. Indeed, if ϕ : A S recognizes X, then X is finite union of sets of the form ϕ 1 (s)ϕ 1 (e) ω. The lgorithm to pss from finite ω-semigroup to Büchi utomton presented in section 9 gives direct proof of this result, without using Theorem 4.2. In the opposite direction, n lgorithm to pss from Büchi utomt to ω-semigroups is presented in section 8. Finlly, the lgorithm to pss from finite ω-semigroup to Muller utomton given in section 10 gives proof of McNughton s theorem. 8 From Büchi utomt to ω-semigroups We give in this section the construction to pss from finite (Büchi) utomton to finite ω-semigroup. This construction is much more involved thn the corresponding construction for finite words. Given finite Büchi utomton A = (Q,A,E,I,F) recognizing suset of X of A N, we would like to otin finite ω-semigroup recognizing X. Our construction mkes use of the semiring k = {,0,1} in which ddition is the mximum for the ordering < 0 < 1 nd multipliction, which extends the oolen ddition, is given in the following tle Tle 1: The multipliction tle. To ech letter A is ssocited mtrix µ() with entries in k defined y if (p,,q) / E µ() p,q = 0 if (p,,q) E nd p / F nd q / F 1 if (p,,q) E nd (p F or q F) We hve lredy used similr technique to encode utomt, ut now we wnt to keep trck of the visits of finl sttes. We would like to extend µ into morphism of ω-semigroups. It is esy to extend µ to semigroup morphism from A + to the multiplictive semigroup of Q Q-mtrices over 15

16 k. If u is finite word, one gets if there exists no pth of lel u from p to q, 1 if there exists pth from p to q with lel u µ(u) p,q = going through finl stte, 0 if there exists pth from p to q with lel u ut no such pth goes through finl stte However, troule rises when one tries to equip k Q Q with structure of ω-semigroup. The solution consists in coding infinite pths not y squre mtrices, ut y column mtrices, in such wy tht ech coefficient µ(u) p codes the existence of n infinite pth of lel u strting t p. Let S = (S f,s ω ) where S f = k Q Q is the set of squre mtrices of size CrdQ with entries in k nd S ω = k Q is the set of column mtrices with entries in {, 1}. In order to define the opertion ω on squre mtrices, we need the following definition. If s is mtrix of S f, we cll infinite s-pth strting t p sequence p = p 0,p 1,... of elements of Q such tht, for 0 i n 1, s pi,p i+1. The s-pth p is successful if s pi,p i+1 = 1 for n infinite numer of coefficients. Then s ω is the element of S ω defined, for every p Q, y { s ω 1 if there exists successful s-pth of origin p, p = otherwise Note tht the coefficients of this mtrix cn e effectively computed. Indeed, computing s ω p mounts to checking the existence of circuits contining given edgein finitegrph. Thenonecn verify tht S, equippedwiththese opertions, is n ω-semigroup. The morphism µ cn now e extended in unique wy s morphism of ω-semigroups from A into S. Furthermore, we hve the following result. Proposition 8.1 The morphism of ω-semigroups from A into S induced y µ recognizes the set L ω (A). The ω-semigroup µ(a ) is clled the ω-semigroup ssocited with A. Exmple 8.1 Let X = ({,c} {}) ω. This set is recognized y the Büchi utomton represented elow:, 1 2,c,c Figure 8.1: 16

17 The ω-semigroup ssocited with this utomton contins 9 elements ( ) ( ) ( ) = = c = ( ) ( ) ( ) 1 1 = c = ω 1 = ( ) ( ) ( ) ω 1 = c ω = (c) ω = 1 1 It is defined y the following reltions 2 = = c = 2 = c = c c = c c 2 = c () ω = ω ω = ω ω = ω c ω = c ω (c) ω = ω ω = ω ω = ω c ω = c ω (c) ω = (c) ω c ω = (c) ω c ω = (c) ω cc ω = c ω c(c) ω = (c) ω 9 From Wilke lgers to Büchi utomt Given morphism of ω-semigroups recognizing suset X of A N, we construct Büchi utomton tht recognizes X. By Proposition 7.5, we my suppose tht the ω-semigroup is of the form S = (S,S ω ). Thus, let ϕ : A S e morphism of ω-semigroups recognizing X nd let P = ϕ(x). The construction of Büchi utomton tht recognizes X relies on the following result of Pécuchet [24] which gives, for ech idempotent e of S, representtion of S y reltions in S 1. Lemm 9.1 Let S e finite semigroup nd e n idempotent of S. The mp ϕ e : S B S1 S 1 defined, for ll s S nd for ll p,q S 1 y { 1 if sq = p or if sq = pe ϕ e (s) p,q = 0 otherwise is semigroup morphism. Let us consider, for every idempotent e of S, the Büchi utomton A e = (S 1,A,E e,i e,{1}) with I e = {s S (s,e) P} nd E e = {(p,,q) S 1 A S 1 ϕ()q = p or ϕ()q = pe} Then the Büchi utomton A tht recognizes X is the disjoint union of the utomt A e, where e rnges over the set of idempotents of S. Exmple 9.1 Let S = ({,},{ ω, ω }) e the ω-semigroup considered in Exmple 7.2 nd let ϕ : A S e the morphism of ω-semigroups defined 17

18 y ϕ() = nd ϕ() =. The utomton A is represented in the figure elow. 1 1 Figure 9.1: The utomt A (on the left) nd A (on the right). 10 From Wilke lgers to Muller utomt Given semigroup morphism ϕ : A + S recognizing suset X of A +, it is esy to uild deterministic utomton recognizing X. Denote y S 1 the monoid equl to S if S hs n identity nd to S {1} otherwise. Tke the right representtion of A + on S 1 defined y s. = sϕ(). This defines deterministic utomton A S = (S 1,A,E,{1},P}, where E = {(s,,s.) s S 1, A}, which recognizes L. It is tempting to do the sme construction to otin deterministic Muller utomton, given finite ω-semigroup S = (S,S ω ) recognizing suset X of A N. B. Le Sec [15] hs oserved tht such construction is possile if the right stilizers of S re nds. Given finite semigroup S nd n element s of S recll tht the right stilizer of s is the susemigroup St(s) of S defined y St(s) = {t st = s}. Theorem 10.1 Let S = (S,S ω ) e finite ω-semigroup such tht the right stilizers of S re idempotent semigroups. Let ϕ : A S e morphism of ω-semigroups recognizing suset X of A N. Then the utomton A S = (S 1,A,E,{1},T ), where E = {(s,,s.) s S 1, A} nd T = {Infinite(u) u X} is Muller utomton recognizing X. Exmple 10.1 Let S = ({,},{ ω, ω }) e the ω-semigroup considered in Exmples 7.2 nd 9.1 nd let ϕ : A S e the morphism of ω-semigroups defined y ϕ() = nd ϕ() =. Let P = { ω }, so tht ϕ 1 ( ω ) = A ω. Then the trnsitions of the utomton A S re represented elow nd its tle is T = {{}}. 18

19 1,, Figure 10.1: A deterministic Muller utomton recognizing A ω. Unfortuntely, the right stilizer of n element of finite semigroup is not lwys n idempotent semigroup. However, it is shown in [16, 17] tht every finite semigroup is quotient of finite semigroup in which the right stilizer of every element is n idempotent semigroup. In fct, slightly stronger result holds. Theorem 10.2 Every finite semigroup is quotient of finite semigroup in which the right stilizer of every element is semigroup which stisfies the identities x = x 2 nd xy = xyx. Therefore, one cn ssocite with every morphism from A onto finite ω-semigroup S Muller utomton s follows: first compute finite semigroup T whose right stilizers re idempotent nd commuttive such tht S f is quotient of T nd then pply the result of Le Sec. This gives the proof of (4) implies (3) in Theorem 6.1. To sum up, we hve otined the following result. Theorem 10.3 Let X e set of infinite words. The following conditions re equivlent: (1) X is recognizle, (2) X is recognized y finite ω-semigroup, (3) X is ω-rtionl. 11 Syntctic ω-semigroup Let S = (S f,s ω ) e n ω-semigroup. A congruence of ω-semigroup on S is pir = ( f, ω ) where f is semigroup congruence on S f nd ω is n equivlence on S ω such tht (1) for ll s,s S f nd for ll t,t S ω, s f s nd t ω t imply st ω s t (2) for ll infinite sequences s 0,s 1,s 2,... nd s 0,s 1,s 2,... of elements of S f such tht s i f s i for ll i, s 0s 1 ω s 0s 1 19

20 IfS isfiniteω-semigroup, thenonemy replce(2) ytheweker condition (2 ) (2 ) for ll s,s S f, s f s implies s ω ω s ω. In the sequel, we shll often omit the suscripts, nd use the simplified nottion insted of f nd ω. Let X e suset of A. A congruence of ω-semigroup on A recognizes X if the morphism of ω-semigroup ϕ : A A / recognizes X. Contrry to the cse of finite words, the lower ound of the congruences tht recognize X does not lwys recognize X. If this lower ound still recognizes X, it is clled the syntctic congruence of X nd is denoted X. Arnold hs shown tht the syntctic congruence lwys exists if X is recognized y finite ω-semigroup [1]. The reson is tht finite ω-semigroup cn e considered s finite Wilke lger nd tht syntctic congruences cn e defined on Wilke lgers s on ny lger of finite signture. More precisely, congruence on Wilke lger S is pir = ( f, ω ) where f is semigroup congruence on S f nd ω is n equivlence reltion on S ω tht is comptile with the ω-power nd the mixed product. In prctice, it is convenient to omit the suscripts f nd ω nd write for oth f nd ω. A congruence on S sturtes suset P = (P f,p ω ) of S if for ll x,y S f (resp. S ω ), x y nd x P f (resp. P ω ) implies y P f (resp. P ω ). Now the lower ound of the congruences tht sturte P lso sturtes P nd is clled the syntctic congruence of P. It is the congruence P defined on S f y u P v if nd only if, for every x,y S 1 f nd for every z S f, xuyz ω P ω xvyz ω P ω x(uy) ω P ω x(vy) ω P ω nd on S ω y u P v if nd only if, for every x S 1 f, xu P ω xv P ω (11.1) The syntctic Wilke lger of P is the quotient of S y the syntctic congruence of P. Now, to compute the syntctic ω-semigroup of set X recognized y finite Büchi utomton A, one first computes the finite ω-semigroup S ssocited with A nd the imge P of X in S nd then one computes the syntctic Wilke lger of P in S. This provides n lgorithm to compute the syntctic ω-semigroup of recognizle set. The syntctic congruence X of recognizle set X of A cn lso e defined directly s follows. On A +, u X v if nd only if, for every x,y A nd for every z A +, xuyz ω X xvyz ω X x(uy) ω X x(vy) ω X (11.2) 20

21 nd on A N, u X v if nd only if, for every x A, xu X xv X (11.3) The syntctic ω-semigroup of X is the quotient of A under the congruence of ω-semigroup X. This is lso the miniml (with respect to the quotient ordering) complete finite ω-semigroup recognizing X. Exmple 11.1 We come ck to the exmple 8.1. The set X = ({,c} {}) ω is recognized y the ω-semigroup S = {,,c,,c, ω, ω,(c) ω,0}, defined y the following reltions: 2 = = c = 2 = c = c c = c c 2 = c c ω = 0 () ω = ω ω = ω ω = ω (c) ω = ω ω = ω ω = ω (c) ω = (c) ω c ω = (c) ω c ω = (c) ω c(c) ω = (c) ω Since P = { ω, ω }, the congruence P is defined y P et ω P ω Therefore, the syntctic ω-semigroup is S(X) = {,,c,c, ω,(c) ω,0}, with the following reltions 2 = = c = = 2 = c = c c = c c 2 = c ω = ω c ω = 0 ω = ω (c) ω = ω ω = ω (c) ω = (c) ω c ω = (c) ω c(c) ω = (c) ω 12 Vrieties Sofr, wehveextended toinfinitewords thenotions of finiteutomt, of semigroup recognizing set of words nd finlly of syntctic semigroup. The next nturl step in this process is to extend the vriety theorem. Pécuchet [24, 23] mde first ttempt in this direction, ut the result of Wilke [45] is more stisfctory. A vriety of finite Wilke lgers is clss of finite Wilke lgers closed under the tking of finite direct products, quotients nd complete sulgers. Let X A nd let u A +. We set u 1 X = {v A uv X} Xu ω = {v A + (vu) ω X} Xu 1 = {v A + vu X} 21

22 An -vriety V ssocites with every lphet A set V(A ) of recognizle sets of A such tht (1) for every lphet A, V(A ) contins, A +, A N nd A nd is closed under finite oolen opertions (finite union nd complement), (2) for every semigroup morphism ϕ : A + B +, X V(B ) implies ϕ 1 (X) V(A ), (3) If X V(A ) nd u A, then u 1 X V(A ), Xu 1 V(A ) nd Xu ω V(A ). It is importnt to notice tht one works with A (nd not with A N ) in this definition. In other words, the elements of -vriety re sets of finite or infinite words. If V is vriety of finite Wilke lgers, we denote y V(A ) the set of recognizle sets of A whose syntctic ω-semigroup elongs to V. This is lso the set of sets of A recognized y n ω-semigroup of V. An -clss of recognizle sets is correspondence which ssocites with every finite lphet A, set C(A ) of recognizle sets of A. In prticulr, the correspondence V V ssocites with every vriety of finite Wilke lgers -clss of recognizle sets. The theorem of Wilke, which extends the corresponding theorem of Eilenerg, cn now e stted s follows. Theorem 12.1 The correspondence V V defines ijection etween the vrieties of finite Wilke lgers nd the -vrieties. We conclude y giving four exmples of correspondence etween vrietiesof finite Wilke lgers nd -vrieties. The first nd the second one extend to infinite words well known results on finite words (the chrcteriztion of str-free nd loclly testle lnguges), ut the lst two exmples re less fmilir since they concern two topologicl clsses. We refer to our rticle [33] in this volume for the definitions of the strfree nd loclly testle lnguges. The set of str-free susets of A is the smllest set S of susets of A contining the str-free lnguges of A + which is closed under finite oolen opertions nd such tht if X is strfree suset of A + nd Y S, then XY S. Equivlent chrcteriztions of the str-free susets of A N were proposed y Thoms [41, 42]. Theorem 12.2 Let X e suset of A N. The following conditions re equivlent: (1) X is str-free, (2) X is finite union of sets of the form UV ω where U nd V re strfree susets of A + nd V 2 V, (3) X is oolen comintion of sets of the form L where L is str-free lnguge. 22

23 The syntctic chrcteriztion is given in [26]. A finite ω-semigroup (S f,s ω ) is periodic if the semigroup S f is periodic. Theorem 12.3 A recognizle set of A is str-free if nd only if its syntctic ω-semigroup is periodic. It follows in prticulr tht the str-free sets form -vriety. Similr results hold for the loclly testle sets. A suset of A is loclly testle if nd only if it is finite oolen comintion of sets of the form ua, A u, A ua, ua ω, A ua ω nd A (ua ) ω, for ll u A +. The following results re due to Pécuchet [24, 23] Theorem 12.4 Let X e suset of A N. The following conditions re equivlent: (1) X is loclly testle, (2) X is finite union of sets of the form UV ω where U nd V re loclly testle susets of A + nd V 2 V, (3) X is oolen comintion of sets of the form L where L is loclly testle lnguge. A finite ω-semigroup (S f,s ω ) is loclly idempotent nd commuttive if the semigroup S f is loclly idempotent nd commuttive. Theorem 12.5 A recognizle set of A is loclly testle if nd only if its syntctic ω-semigroup is loclly idempotent nd commuttive. Onecn definetopology ona N yconsideringasdiscretespcend y tking the producttopology. Theopen sets of A N re the sets of the form XA N for some X A +. A set is closed if its complement is open. The sets of 1 re the clopen sets, sets which re t the sme time closed nd open. The sets of 2 re t the sme time countle unions of closed sets nd countle intersection of open sets. One cn show tht the recognizle sets of 2 re the sets which re ccepted y deterministic Büchi utomton s well s their complement. The chrcteriztion of these two clsses in terms of ω-semigroups is due to Wilke [46]. In these sttements, x π denotes the unique idempotent of the semigroup generted y x. Theorem 12.6 The clss 1 form -vriety. The corresponding vriety of finite Wilke lgers is the clss of lgers defined y the eqution x π yz ω = x π y z ω. Theorem 12.7 The clss 2 form -vriety. The corresponding vriety of finite Wilke lgers is the clss of lgers defined y the eqution (x π y π ) π x ω = (x π y π ) π y ω. 23

24 13 Conclusion Most known results on utomt on finite words hve found their counterprt on infinite words. However, the complexity of the solutions is pprently incresed y n order of mgnitude. This is true for instnce for the equivlence etween deterministic nd non deterministic utomt nd for the vriety theorem. Furthermore, severl questions which re solved for finite words re still open for infinite words, ut one cn e resonly optimistic out them. Acknowledgements The uthors would like to thnk Olivier Crton nd Pscl Weil for their comments on preliminry version of this rticle. References [1] A. Arnold, A syntctic congruence for rtionl ω-lnguges, Theoret. Comput. Sci. 39,2-3 (1985), [2] D. Beuquier nd J.-E. Pin, Fctors of words, in Automt, lnguges nd progrmming (Stres, 1989), pp , Springer, Berlin, [3] D. Beuquier nd J.-E. Pin, Lnguges nd scnners, Theoret. Comput. Sci. 84 (1991), [4] J. R. Büchi, Wek second-order rithmetic nd finite utomt, Z. Mth. Logik und Grundl. Mth. 6 (1960), [5] J. R. Büchi, On decision method in restricted second order rithmetic, in Logic, Methodology nd Philosophy of Science (Proc Internt. Congr.), pp. 1 11, Stnford Univ. Press, Stnford, Clif., [6] J. R. Büchi nd L. H. Lndweer, Definility in the mondic second-order theory of successor, J. Symolic Logic 34 (1969), [7] S. Eilenerg, Automt, lnguges, nd mchines. Vol. A, Acdemic Press [A susidiry of Hrcourt Brce Jovnovich, Pulishers], New York, Pure nd Applied Mthemtics, Vol. 58. [8] S. Eilenerg, Automt, lnguges, nd mchines. Vol. B, Acdemic Press [Hrcourt Brce Jovnovich Pulishers], New York, With two chpters ( Depth decomposition theorem nd Complexity of 24

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