On Infinite Real Trace Rational Languages of Maximum Topological Complexity

On infinite real trace rational languages of maxim um top ological complexit y ∗ 1 Olivier Fink el and 2 Jean-Pierre Ressa yre and 3 Pierre Simonnet 1 , 2 Equip e de L o gi q ue Math ´ ematique U.F.R. de Math ´ ematiques, Univ ersit ´ e P aris 7 2 Plac e Jussieu 75251 Paris c e dex 05, F r anc e. fink el@logique.jussieu.fr ressa yre@logique.jussieu.fr 3 UMR CNRS 6134, F acult ´ e des Sciences, Univ ersit ´ e de Corse Quartier Gr ossetti BP5 2 20250, Corte, F r anc e simonnet@univ-corse.fr Abstract W e consider the set R ω (Γ , D ) of infinite real traces, ov er a depen dence alphab et (Γ , D ) with no isolated letter, eq uipp ed with the topology in- duced by the prefi x metric. W e then prov e that all rational languages of infinite real traces are analytic sets. W e re prov e also that there exist some rational languages of infinite re al traces whic h are analytic but non Borel sets, and even Σ 1 1 -complete, hence of maxim um possible top ological com- plexity . F or that purp ose we give an example of Σ 1 1 -complete language whic h is fun d amental ly different from the known example of Σ 1 1 -complete infinitary rational relation given in [Fin03c]. Keywords: R eal traces; rational la nguages; t opological prop erties; analytic and Bo rel sets. 1 In tro duction T race monoids were firstly c o nsidered by Cartier and F o ata for studying co m- binatorial problems , [CF69]. Next Mazur kiewicz introduce d finite tra ces as a semantic mo del for concurrent systems, [Maz7 7]. Since then tr a ces ha ve b een m uch inv estiga ted by v ario us authors a nd they have b een extended to infinite traces to model sys tems which may no t terminate; see the handb o ok [DR95] and its chapter ab out infinite traces [GP95], for many results and r eferences. In par ticular, real traces hav e b een studied by Gastin, Petit and Zielonk a, who characterized in [GPZ94] the tw o imp or tant families of reco gnizable and ratio - nal langua ges of rea l traces, over a dependence alphab et (Γ , D ), in connection with rationa l lang uages of finite or infinite words. Several metrics ha ve b een defined on the set R (Γ , D ) of real traces over (Γ , D ). In ∗ The results of this paper hav e b een exp osed during the In ternational Conference JAF 22, 22nd Journ´ ees sur les Ar ithm´ etiques F aibles, June 11-14, 2003, Napoli , Italy . 1 particular, the prefix metric defined by Kwiatko wsk a [Kwi90] and the F oata nor- mal form metric defined by Bonnizzo ni, Mauri and Pighizzini [BMP 90]. Kum- metz and Kuske stated in [KK0 3] tha t f or finite dep endence alphabets these t wo metrics define the same top olo gy on R (Γ , D ). Moreo ver, if we consider only infinite rea l traces ov e r a dep endence alphab e t (Γ , D ) without isolated let- ter, the top ologica l subspace R ω (Γ , D ) = R (Γ , D ) − M (Γ , D ) of R (Γ , D ) (where M (Γ , D ) is the set o f finite traces over (Γ , D )), is ho meomorphic to the Cant or set 2 ω , or e q uiv alently to any set Σ ω of infinite words ov er a finite alphab et Σ , equipp e d with the pro duct o f the discrete top o lo gy on Σ, [KK03, Sta97a, PP 04]. W e can then define, from op en subsets of the top o lo gical s pace R ω (Γ , D ), the hierarch y of Borel sets by successive op erations of coun table intersections and countable unions. F ur thermore, it is w ell known that there exis t some subsets of the Cantor set, hence also some subsets of R ω (Γ , D ), which are not Borel. There is a nother hierarch y b e yond the Bor el one, called the pro jective hierarch y . It is then natural to try to lo ca te clas sical la nguages of infinite rea l traces with regar d to these hierar chies and this que s tion is p osed by Lescow and Tho mas in [L T94] (for the g eneral ca se of infinite lab elled par tial order s like traces). In the case of infinite words, Mc Naughton’s Theore m implies that every ω -re gular language is a bo olean combination o f Π 0 2 -sets hence a ∆ 0 3 = ( Π 0 3 ∩ Π 0 3 )-set. Landweber studied first the to po logical prop erties of ω -re gular languages and characterized the ω -regula r languag es in each o f the Bo rel c lasses Σ 0 1 , Π 0 1 , Σ 0 2 , Π 0 2 , [Lan6 9]. W e study in this pap er the topolo gical co mplexity of rational languages of i nfinite real traces . W e sho w b elow that all ra tional languages o f infinite real tr aces are analytic sets and that there exist s ome rational langua ges of infinite real traces which are analytic but non Bo rel sets, and ev en Σ 1 1 -complete, hence of maximum po ssible top ologica l complexity , giving a partial answer to the question of the compariso n b etw ee n the top olog ical complexity o f ratio na l languag e s of infinite words and o f infinite traces [L T94]. The first a uthor r ecently s how ed in [Fin0 3c] that ther e exis ts a Σ 1 1 -complete in- finitary rational rela tion R ⊆ Σ ω 1 × Σ ω 2 where Σ 1 and Σ 2 are tw o finite a lphab ets having at leas t tw o letters. W e could hav e used this result to prov e that there ex ists a Σ 1 1 -complete ratio nal language of infinite rea l tra c e s L ⊆ R ω (Γ , D ), whenever (Γ , D ) is a dep e ndence alphab et and Γ ⊇ Σ 1 ∪ Σ 2 , where Σ 1 and Σ 2 are t wo indep endent dep endence cliques having at leas t tw o letters. This ca n b e done by c o nsidering the natural embedding i : Σ ω 1 × Σ ω 2 → R ω (Γ , D ). The la nguage R ′ = i ( R ) is then Σ 1 1 - complete. But this way the langua ge R ′ would hav e in fact the structur e of an infinitary ratio na l relation. On the other side the Σ 1 1 -complete lang ua ge L giv e n in this pa p er is a new example whose str ucture is r adically different from that o f R ′ . In particular, L do es not contain any Σ 1 1 -complete la nguage of infinite tr a ces having the struc- ture of an infinitar y ra tional relation. This is imp ortant b ecaus e in T race Theor y the structure of dep endence alpha - bets is very imp or tant: so me results are known to b e true for some dep endence alphab ets and fa lse for other dep endence alphab ets (see for example [HM97 ]). Moreov er w e think that the pr esentation of this new example ha s a lso some 2 int erest for the following r easons. The pro of g iven in this pap er is self co n tained and is stated in the gener al context of traces. The problem is exp ose d in this general context and we use her e the g eneral prop erties of traces instead of the particular pro p er ties of infinitary ra tional relations. W e use the characteriza tio n of rationa l languag e s of infinite rea l traces given by Theorem 2.1 of Ga stin, Petit and Zie lonk a, whic h s tates a connection betw een r ational languag es of infinite words and rational languages of infinite real traces instea d of the notion o f B ¨ uchi transducer. W e prov e also that all rational lang uages of infinite rea l tra ces ar e ana lytic. Our pro of is not difficult, but it is original, using the Baire space ω ω and the characterization of rational languages o f infinite re a l tra ces given b y T he o rem 2.1: ev ery r ational lang uage of infinite tra ces is a finite union of sets of the form R.S ω where S and R are r ational monoalpha b etic languages o f finite tra c e s. This pro o f is rather different from usual ones in the theory of ω -lang uages. It uses a connection betw een the to po logical complexit y of ω -powers of languages of finite tr aces, i.e. o f la nguages o f the for m S ω , where S is a langua ge of finite traces, and the top olog ical complexity of rational langua ges of infinite trac e s. The clos ure under countable union o f the c la ss of analytic sets is als o imp ortant in the pr o of. W e e x pe c t that the inv erse way could also b e fruitful: it s eems to us that there should exist some context free ω - la nguages and some infinitary rational rela tions of high tr ansfinite Bore l rank. W e think that we could use this fact to show that there exist ω -p ow ers o f finitar y languag es o f hig h trans finite B orel rank. Notice that the ques tio n of the top ologica l complexity of ω -p ow ers (of language s of finite words) has b een r aised by se veral authors [Niw90, Sim92, Sta97 a, Sta97b] and some new r esults hav e b een r ecently proved [Fin01, Fin03 a, Fin04b, Lec01]. The pap er is orga nized as follows. In section 2 we r ecall the notion of w ords and traces. In section 3 we reca ll definitions of Bo rel a nd analytic s ets, and we prov e our main r esults in sec tion 4. 2 W ords and traces Let us now int ro duce notations for words. F or Σ a finite a lphab et, a non empty finite word o ver Σ is a finite sequence of le tter s: x = a 1 a 2 . . . a n where ∀ i ∈ [1; n ] a i ∈ Σ. W e shall denote x ( i ) = a i the i th letter o f x and x [ i ] = x (1) . . . x ( i ) for i ≤ n . The length of x is | x | = n . The empty word will be deno ted by ε and has no letters. Its length is 0 . The set of non empt y finite words ov er Σ is denoted Σ + . Σ ⋆ = Σ + ∪ { ε } is the se t of finite words ov er Σ. A (finitary) language L ov er Σ is a subset of Σ ⋆ . The usua l conca tenation pr o duct of u a nd v will b e denoted by u.v o r just uv . F or V ⊆ Σ ⋆ , we denote V ⋆ = { v 1 . . . v n | n ≥ 1 and ∀ i ∈ [1; n ] v i ∈ V } ∪ { ε } . The first infinite or dinal is ω . An ω -word o ver Σ is an ω -s equence a 1 a 2 . . . a n . . . , where ∀ i ≥ 1 a i ∈ Σ . When σ is an ω -word ov er Σ, we write σ = σ (1) σ (2) . . . σ ( n ) . . . and σ [ n ] = σ (1) σ (2) . . . σ ( n ) the finite w ord of leng th n , prefix of σ . The set of ω -words ov er the alphab et Σ is denoted by Σ ω . An ω -language ov er an alphab et Σ is a subset of Σ ω . F or V ⊆ Σ ⋆ , V ω = { σ = u 1 . . . u n . . . ∈ Σ ω | ∀ i ≥ 1 u i ∈ V } 3 is the ω -power of V . The conca tenation pr o duct is extended to the pro duct o f a finite word u and a n ω -word v : the infinite word u.v is then the ω -word such that: ( u.v )( k ) = u ( k ) if k ≤ | u | , and ( u.v )( k ) = v ( k − | u | ) if k > | u | . The prefix r elation is denoted ⊑ : the finite word u is a prefix of the finite word v (resp ectively , the infinite word v ), denoted u ⊑ v , if and only if there exis ts a finite word w (r esp ectively , an infinite word w ), s uch that v = u.w . W e shall denote Σ ∞ = Σ ⋆ ∪ Σ ω the set of finite or infinite w ords ov er Σ. W e introduce firstly traces a s dep endence gr aphs, [DR95, GP9 5, KK0 3]. A de- pendenc e rela tion over an alphab et Γ is a reflexive and symmetric relation on Γ. Its complement I D = (Γ × Γ) − D is the indep endence relation induced b y the relation D ; the relatio n I D is irreflexive and symmetric. A dep endence a lpha b e t (Γ , D ) is formed by a finite alphab et Γ and a dep endence relation D ⊆ Γ × Γ. A dep e ndence gr aph [ V , E , λ ] over the dep endence a lphab et (Γ , D ) is an iso- morphism cla ss of a no de lab elled gr aph ( V , E , λ ) s uch that ( V , E ) is a dir ected acyclic graph, V is at most count ably infinite, λ : V → Γ is a function whic h asso ciates a la b el λ ( a ) to e a ch no de a ∈ V , a nd such that: (1) ∀ v, w ∈ V ( λ ( v ) , λ ( w )) ∈ D ↔ ( v = w or ( v , w ) ∈ E or ( w, v ) ∈ E ) (2) The reflexive and transitive c losure E ⋆ of the edge relation E is w ell founded, i.e. there is no infinite strictly decr easing sequence of vertices. Let us r emark that, since in this definition ( V , E ) is acyclic, E ⋆ is a par tial o r der on V . The empty tra ce has no vertice and will be denoted by ε as in the ca se o f words. As usually the c o ncatenation o f tw o dep endence graphs g 1 = [ V 1 , E 1 , λ 1 ] and g 2 = [ V 2 , E 2 , λ 2 ], whe r e we can assume, without loss of generality , that V 1 and V 2 are disjoint, is the dep endence graph g 1 .g 2 = [ V , E , λ ] such that V = V 1 ∪ V 2 , E = E 1 ∪ E 2 ∪ { ( v 1 , v 2 ) ∈ V 1 × V 2 | ( λ 1 ( v 1 ) , λ 2 ( v 2 )) ∈ D } , and λ = λ 1 ∪ λ 2 . The alpha b e t al ph ( t ) o f a trace t = [ V , E , λ ] is the set λ ( V ) . The a lphab et at infinit y of t is the set al phinf ( t ) = { a ∈ Γ | λ − 1 [ a ] is infinite } of all a ∈ al ph ( t ) o ccurring infinitely o ften in t . The set M (Γ , D ) o f finite trace s ov er (Γ , D ) is the set of traces having o nly finitely many vertices. F or t ∈ M (Γ , D ) the length of t is the num b er of vertices of t denoted | t | . The star o per ation T → T ⋆ and the op eratio n T → T ω are natura lly extended to subsets T of M (Γ , D ): T ⋆ = { t 1 .t 2 . . . t n | n ≥ 1 a nd ∀ i ∈ [1; n ] t i ∈ T } ∪ { ε } T ω = { t 1 .t 2 . . . t n . . . | ∀ i t i ∈ T } . A real tr ace ov er (Γ , D ) is a dep endence graph [ V , E , λ ] s uch that fo r all v ∈ V the set { u ∈ V | ( u, v ) ∈ E ⋆ } is finite. The set of real traces ov er (Γ , D ) is denoted R (Γ , D ) and the set R (Γ , D ) − M (Γ , D ) of infinite real tra ces will b e denoted by R ω (Γ , D ). 4 The prefix order over words can b e extended to rea l traces in the following wa y . F o r a ll s, t ∈ R (Γ , D ) s ⊑ t iff there exists z ∈ R (Γ , D ) such that s.z = t iff s is a down wards closed subgraph o f t . The corresp onding suffix z is then unique. Real traces may also be viewed as equiv alence classes of (finite or infinite) words. Let (Γ , D ) b e a dep endence alpha be t and let ϕ : Γ ∞ → R (Γ , D ) b e the map- ping defined b y ϕ ( a ) = [ { x } , ∅ , x → a ] for each a ∈ Γ and ϕ ( a 1 .a 2 . . . ) = ϕ ( a 1 ) .ϕ ( a 2 ) . . . for eac h a 1 .a 2 . . . in Γ ∞ . Let us remark that if there is some ( a, b ) in I D then the mapping is not injective b ecause for instance ϕ ( ab ) = ϕ ( ba ). One can define an equiv alence relation ∼ I on Γ ∞ by: for all u, v ∈ Γ ∞ u ∼ I v iff ϕ ( u ) = ϕ ( v ). Then ϕ induces a surjective morphism from the free mono id Γ ⋆ onto the monoid of finite dep endence g raphs M (Γ , D ) = Γ ⋆ / ∼ . And the set ϕ (Γ ∞ ) = Γ ∞ / ∼ is the set of rea l traces R (Γ , D ). The empt y trace is the image ϕ ( ε ) of the empt y word a nd is still denoted by ε . W e a ssume the r eader to b e familiar with the theory of forma l lang uages and of ω -re gular la nguages, s ee [Tho90, Sta97a, PP0 4] fo r many results and references. W e recall that ω - regular languag es ar e accepted by B¨ uchi automata and tha t the clas s of ω -re gular languages is the omeg a Kleene closure o f the class of regular finitary la nguages. The family of r ational r eal tr ace languages over (Γ , D ) is the s mallest family which co ntains the e mpt yset, all the s ingletons { [ { x } , ∅ , x → a ] } , for a ∈ Γ, and which is close d under finite union, conca tenation pro duct, ⋆ -iteration and ω -itera tion on real traces. Let us now r ecall the following characteriza tio n of rationa l langua ges of infinite real traces (ther e exists also a version for finite or infinite traces , [GPZ94]). A real trace language R is said to be monoalphab etic if alp h ( s ) = al ph ( t ) for all s, t ∈ R . Theorem 2 .1 ([GPZ94]) L et T ⊆ R ω (Γ , D ) b e a language of infinite r e al tr ac es over the dep endenc e alpha b et (Γ , D ) . The fol lowing assertions ar e e quiv- alent: (1) T is r ational. (2) T is a finite union of sets of t he form R.S ω wher e S and R ar e r ational mono alphab etic languages of finite t r ac es over (Γ , D ) and ε / ∈ S . (3) T = ϕ ( L ) for some ω - r e gu lar language L ⊆ Γ ω . 3 T op ology W e assume the reader to b e familiar with basic notions o f top olog y which may be found in [Kec95, L T94, Sta 9 7a, PP04]. There is a na tural metric on the set Σ ω of infinite words over a finite alphab et Σ whic h is called the prefix metric and defined a s follows. F or u, v ∈ Σ ω and u 6 = v let d ( u, v ) = 2 − l pref ( u,v ) where l pref ( u,v ) is the first integer n such that the ( n + 1) th letter of u is different from the ( n + 1) th letter of v . This metric induces o n Σ ω the usual Can tor to po logy for which o p en subsets of Σ ω are in the form W. Σ ω , where W ⊆ Σ ⋆ . 5 (Notice that this prefix metric may b e extended to the set Σ ∞ of finite or infinite words ov er the alphab et Σ). The prefix metric has b een extended to rea l trac e s by Kw ia tko ws k a in [Kwi9 0] by defining firs tly for all s, t ∈ R (Γ , D ) with s 6 = t : l pref ( s, t ) = su p { n ∈ N | r ⊑ s ↔ r ⊑ t for all r ∈ M (Γ , D ) with | r | ≤ n } and next d pref ( s, t ) = 2 − l pref ( s,t ) Notice that we consider in this paper infinite tr aces and Kwiatko wsk a defined the pr efix metric over finite or infi nite traces a s one could also hav e do ne in the case of words. If D = Γ × Γ the prefix metric on infinite r e a l tra ces over (Γ , D ) coincide with the preceding definition in the cas e of infinite words ov er Γ. If (Γ , D ) is a dep endence a lphab et, a letter a ∈ Γ is s aid to b e a n isolated letter if a is independent from a ll other letters of Γ, i.e. ∀ b ∈ Γ − { a } , ( a, b ) ∈ I D . F r om now o n we supp ose that a dep endence a lphab et has no iso lated letter. Then the set R ω (Γ , D ) of infinite real tra ces over (Γ , D ), equipp ed with t he topol- ogy induced by the prefix metric, is homeomorphic to the Cantor set { 0 , 1 } ω , hence also to Σ ω for every finite alphab et Σ having at le a st tw o letters [KK 03]. Borel subsets of the Cantor set (hence a lso of top o logical spaces Σ ω or R ω (Γ , D )) form a s trict infinite hier arch y , the Borel hier a rch y , which is defined from op en sets by succes s ive op erations o f countable unions a nd of co untable intersections. W e give the definition in the cas e of a top ologica l space Σ ω , the definition b eing similar in the case of the top olog ical s pa ce R ω (Γ , D ). Then we recall some well known prop erties of B orel sets. Definition 3.1 The classes Σ 0 n and Π 0 n of the Bor el Hier ar chy on the top olo g- ic al sp ac e Σ ω ar e define d as fol lows: Σ 0 1 is the class of op en subsets of Σ ω . Π 0 1 is the class of close d subsets, i.e. c omplements of op en subsets, of Σ ω . And for any inte ger n ≥ 1 : Σ 0 n + 1 is the class of c oun table unions of Π 0 n -subsets of Σ ω . Π 0 n + 1 is the class of c oun table interse ctions of Σ 0 n -subsets of Σ ω . The Bor el Hier ar chy is also define d for tr ansfin ite levels. The classes Σ 0 α and Π 0 α , for a non-nu l l c oun table or dinal α , ar e define d in the fol lowing way: Σ 0 α is the class of c oun table unions of subsets of Σ ω in ∪ γ <α Π 0 γ . Π 0 α is the class of c oun table interse ctions of subset s of Σ ω in ∪ γ <α Σ 0 γ . Theorem 3 .2 (a) Σ 0 α ∪ Π 0 α ( Σ 0 α + 1 ∩ Π 0 α + 1 , for e ach c oun table or dinal α ≥ 1 . (b) ∪ γ <α Σ 0 γ = ∪ γ <α Π 0 γ ( Σ 0 α ∩ Π 0 α , for e ach c ountable limit or dinal α . (c) A set W ⊆ Σ ω is in the class Σ 0 α iff its c omplement is in the class Π 0 α . (d) Σ 0 α − Π 0 α 6 = ∅ and Π 0 α − Σ 0 α 6 = ∅ hold for every c ountable or dinal α ≥ 1 . 6 W e shall s ay that a subset of Σ ω is a Borel set o f ra nk α , for a c o untable ordina l α , iff it is in Σ 0 α ∪ Π 0 α but not in S γ <α ( Σ 0 γ ∪ Π 0 γ ). Let us recall the characterization of Π 0 2 -subsets of Σ ω , inv olving the δ -limit W δ of a finitary langua ge W . F or W ⊆ Σ ⋆ and σ ∈ Σ ω , σ ∈ W δ iff σ has infinitely many pr efixes in W , i.e. W δ = { σ ∈ Σ ω / ∃ ω i such that σ [ i ] ∈ W } , see [Sta 97a]. Prop ositi o n 3.3 A subset L of Σ ω is a Π 0 2 -subset of Σ ω iff ther e exists a set W ⊆ Σ ⋆ such that L = W δ . Example 3.4 L et Σ = { 0 , 1 } and A = (0 ⋆ . 1) ω ⊆ Σ ω b e the set of ω -wor ds over the alphab et Σ with infinitely many o c cu rr enc es of the letter 1 . It is wel l known that A is a Π 0 2 -subset of Σ ω b e c ause A = ((0 ⋆ . 1) + ) δ holds. There are s ome subsets of the Cantor set, (hence also of the to p o logical spaces Σ ω or R ω (Γ , D )) whic h are not Borel sets. There exis ts ano ther h ierar ch y beyond the Bo rel hier arch y , called the pro jective hier arch y . Pro jective s e ts a r e defined from B orel sets by success ive op era tions of pro jection and co mplement ation. W e sha ll only need in this pap er the first class of the pro jective hier arch y: the class Σ 1 1 of analytic sets. A set A ⊆ Σ ω is ana lytic iff there exists a Borel se t B ⊆ (Σ × Y ) ω , with Y a finite alphab et, such that x ∈ A ↔ ∃ y ∈ Y ω such that ( x, y ) ∈ B , where ( x, y ) ∈ (Σ × Y ) ω is defined by: ( x, y )( i ) = ( x ( i ) , y ( i )) for a ll int egers i ≥ 1. Analytic sets ar e also characterize d as contin uous images of the Ba ire space ω ω , which is the set o f infinite sequence s of non nega tive integers. It ma y be seen as the set of infinite words ov e r the infinite alphab et ω = { 0 , 1 , 2 , . . . } . The top ology of the Baire space is then defined b y a prefix metric whic h is just an extension of the previous one to the case of a n infinite a lphab et. A set A ⊆ Σ ω (resp ectively A ⊆ R ω (Γ , D )) is then analytic iff there exists a contin uous function f : ω ω → Σ ω (resp ectively f : ω ω → R ω (Γ , D )) such that f ( ω ω ) = A . A Σ 0 α (resp ectively Π 0 α , Σ 1 1 )-complete set is a Σ 0 α (resp ectively Π 0 α , Σ 1 1 )- set which is in some s ense a set of the highest top olog ical complexity amo ng the Σ 0 α (resp ectively Π 0 α , Σ 1 1 )- sets. This no tio n is defined via r eductions by contin uo us functions. More precisely a set F ⊆ Σ ω is said to b e a Σ 0 α (resp ectively Π 0 α , Σ 1 1 )-complete set iff for any set E ⊆ Y ω (with Y a finite alphab et): E ∈ Σ 0 α (resp ectively E ∈ Π 0 α , Σ 1 1 ) iff there exists a contin uous function f such that E = f − 1 ( F ). Σ 0 n (resp ectively Π 0 n )-complete sets, with n an integer ≥ 1, are thoroughly characterized in [Sta86]. The ω -re gular langua g e A = (0 ⋆ . 1) ω given in Example 3.4 is a well known example of Π 0 2 -complete set. 4 Rational languages of infinite traces W e wan t now to inv estig ate the top olog ical complexity of rational lang uages of infinite rea l traces. In a first step we s ha ll give a n upper bound of this complexity , s howing that all rationa l lang uages T ⊆ R ω (Γ , D ) a re analytic sets. 7 W e would like to us e the characteriz ation of r ational languag es T ⊆ R ω (Γ , D ) given in item 3 of Theorem 2.1: T = ϕ ( L ) for s ome ω -regula r langua ge L ⊆ Γ ω . Indeed every ω -regular language is a Bo rel set (of rank at most 3) and the contin uous ima ge o f a Bor el set is an analytic set. Unfortunately , the ma pping ϕ is not contin uous as the following example shows. Let ( a, b ) ∈ I D and x n ∈ Γ ω defined by x n = a n ba ω for each integer n ≥ 1. Then in Γ ω the sequence ( x n ) n ≥ 1 is conv ergent and its limit is a ω . But the sequence ( ϕ ( x n )) n ≥ 1 is constant in R ω (Γ , D ) b ecause for all n ≥ 1 ϕ ( x n ) = ϕ ( ba ω ). Thus the sequenc e ( ϕ ( x n )) n ≥ 1 is conv ergent but its limit is ϕ ( ba ω ) which is different fr o m ϕ ( a ω ). W e shall use the c harac terization of rational languages T ⊆ R ω (Γ , D ) given in item 2 of Theo rem 2.1: T is a finite union o f s e ts of the form R.S ω where S and R a re rational mo no alphab etic langua ges of finite tr aces ov e r (Γ , D ) and ε / ∈ S . W e consider firstly such rational lang ua ges in the simple form S ω where S is a monoa lphab etic languag e of finite traces ov er (Γ , D ) whic h do es not contain the e mpt y tra ce. The set S is at most countable so it c a n b e finite or countably infinite. In the first case card( S )= p and w e ca n fix an enumeration of S b y a bijectiv e function ψ : { 0 , 1 , 2 , . . . , p − 1 } → S a nd in the seco nd c ase we can fix an enumeration o f S b y a bijective function ψ : ω = { 0 , 1 , 2 , . . . } → S . Let now H b e the function defined from { 0 , 1 , 2 , . . . , p − 1 } ω (in the first case) or from ω ω (in the sec o nd case) into R ω (Γ , D ) by: H ( n 1 n 2 . . . n i . . . ) = ψ ( n 1 ) .ψ ( n 2 ) . . . ψ ( n i ) . . . for a ll sequences n 1 n 2 . . . n i . . . in { 0 , 1 , 2 , . . . , p − 1 } ω (in the fir st case) or in ω ω (in the second case). It holds that H ( { 0 , 1 , 2 , . . . , p − 1 } ω ) = S ω (in the first case) or that H ( ω ω ) = S ω (in the sec o nd case). It is easy to see that H is a contin uo us function. A crucial p oint is that S is a monoal phab etic language, i.e. ther e exists Γ ′ ⊆ Γ s uch that for all s ∈ S , al ph ( s ) = Γ ′ . Let N = ( n i ) i ≥ 1 and M = ( m i ) i ≥ 1 be tw o infinite sequences of int egers in { 0 , 1 , 2 , . . . , p − 1 } ω or in ω ω such that for all i ≤ k n i = m i . Then r ⊑ H ( N ) ↔ r ⊑ H ( M ) holds (at lea s t) for all r ∈ M (Γ , D ) with | r | ≤ k . Th us l pref ( H ( N ) ,H ( M )) ≥ k a nd d pref ( H ( N ) , H ( M )) = 2 − l pref ( H ( N ) ,H ( M )) ≤ 2 − k . This implies that the function H is contin uous (and even uniformly contin uous). If S is finite, then the s e t S ω is the contin uous image of the compact set { 0 , 1 , 2 , . . . , p − 1 } ω th us it is a clo sed hence also a n analy tic subset o f R ω (Γ , D ). If S is infinite, then the set S ω is the contin uous image of the Baire spa ce ω ω th us it is an analytic se t. Let now R ⊆ M (Γ , D ) b e a language o f finite traces. F o r r ∈ R let θ r : R ω (Γ , D ) → R ω (Γ , D ) b e the function defined by θ r ( t ) = r.t for a ll t ∈ R ω (Γ , D ). It is easy to see that this function is contin uous. Then r .S ω = θ r ( S ω ) is an analytic set because the image of a n ana lytic set by a contin uo us function is still an analytic set. The lang uage R .S ω = S r ∈ R r .S ω is a countable union of analytic s ets (b ecause R is countable) but the cla ss of a nalytic subsets of R ω (Γ , D ) is closed under c ountable unions thus R.S ω is an a nalytic set. A rational lang uage T ⊆ R ω (Γ , D ) is a finite union of sets of the fo rm R .S ω where S and R ar e rationa l monoalpha betic lang uages of finite traces ov er (Γ , D ) and ε / ∈ S . Then by finite union this langua ge is an analy tic s et. 8 Notice that we have not use d the fact that S and R ar e ratio na l so the ab ov e pro of can be applied to finite unions of s e ts of the form R.S ω where S is a monoalphab etic lang uage of finite tr aces and we have g ot the following result. Prop ositi o n 4.1 L et (Γ , D ) b e a dep endenc e alphab et without isolate d letter, and let S i , R i , 1 ≤ i ≤ n , b e languages of finite tr ac es over (Γ , D ) , wher e, for al l i , S i do es not c ontain the empty t r ac e and is mono alphab etic. Then the language of infinite tr ac es T = [ 1 ≤ i ≤ n R i .S ω i is an analytic set. In p articular every r ational language T ⊆ R ω (Γ , D ) is an analytic set. In order to prov e the exis tence of Σ 1 1 -complete rational languag e of infinite traces, we s ha ll use results a bo ut languages of infinite binary tree s whose no des are lab elled in a finite alphab et Σ having at least tw o letters. A no de o f an infinite bina ry tree is r epresented by a finite word ov er the alphab et { l , r } where r means “right” and l means “left”. Then an infinite binary tr ee whose no des are la be lle d in Σ may b e viewed as a function t : { l , r } ⋆ → Σ. The set of infinite binary trees lab elled in Σ will b e denoted T ω Σ . There is a natural top ology on this set T ω Σ which is defined by the following distance, [L T94]. Let t and s b e tw o dis tinct infinite trees in T ω Σ . Then the distance b etw een t and s is 1 2 n where n is the s mallest integer such that t ( x ) 6 = s ( x ) for s ome word x ∈ { l , r } ⋆ of length n . The op en sets are then in the for m T 0 .T ω Σ where T 0 is a set of finite lab elled trees. T 0 .T ω Σ is the set of infinite binary tr ees which ex tend s ome finite lab elled binary tree t 0 ∈ T 0 , t 0 is her e a so rt of pr e fix, a n “initial subtre e ” of a tree in t 0 .T ω Σ . It is well known that the top olo g ical s pace T ω Σ is homeo mo rphic to the Ca nt or set hence als o to the top olo gical spaces Σ ω or R ω (Γ , D ). The Borel hierar ch y a nd the pro jective hier arch y on T ω Σ are defined fro m op en sets as in the cases o f the top ologica l space s Σ ω or R ω (Γ , D ). Let t b e a tree. A branch B of t is a subset o f the set of no des of t which is linearly ordered b y the tree partial order ⊑ and which is closed under prefix relation, i.e. if x a nd y a re no des of t such that y ∈ B and x ⊑ y then x ∈ B . A br anch B of a tree is said to b e maximal iff ther e is no o ther br anch of t which strictly contains B . Let t b e an infinite bina ry tr e e in T ω Σ . If B is a maxima l br anch o f t , then this branch is infinite. Let ( u i ) i ≥ 0 be the enumeration of the nodes in B which is strictly increas ing for the pre fix order. The infinite seq uence of lab els of the no des of such a maximal branch B , i.e. t ( u 0 ) t ( u 1 ) ....t ( u n ) ..... is c alled a path. It is an ω -word over the alphab et Σ. F o r L ⊆ Σ ω we denote P ath ( L ) the set of infinite trees t in T ω Σ such that t has at least o ne path in L . 9 It is well known that if L ⊆ Σ ω is a Π 0 2 -complete subset of Σ ω (or a Bo rel s et of higher complexity in the Bor e l hiera rch y ) then the set P ath ( L ) is a Σ 1 1 -complete subset of T ω Σ , [Niw85, Sim93 ], [PP04, exer cise]. In or der to use this result we shall fir stly co de trees la b elled in Σ b y infinite words ov er the finite alphab et Γ = Σ ∪ Σ ′ ∪ { A, B } where Σ ′ = { a ′ | a ∈ Σ } is a disjoint copy of the alpha be t Σ and A, B ar e additio na l letters no t in Σ ∪ Σ ′ . Consider now the set { l , r } ⋆ of no des o f binary infinite tr ees. F or each integer n ≥ 0, ca ll C n the set o f words of length n of { l , r } ⋆ . Then C 0 = { ε } , C 1 = { l , r } , C 2 = { l l , l r, rl , rr } and so o n. C n is the set of no des which a pp ear at the ( n + 1) th level of an infinite binary tree. The num ber of nodes of C n is car d ( C n ) = 2 n . W e c o nsider now the lexicogr a phic order on C n (assuming that l is befo re r for this or der). Then, in the enumeration o f the no des with reg ard to this or de r , the no des of C 1 will b e: l , r ; the no des of C 3 will b e: l l l , l l r, l rl , lr r, rl l , rl r , r r l , r r r . Let u n 1 , . . . , u n j , . . . , u n 2 n be such a n enumeration of C n in the lexicogr a phic or der and let v n 1 , ..., v n j , ..., v n 2 n be the enumeration of the elements of C n in the reverse order. Then for all integers n ≥ 0 a nd i , 1 ≤ i ≤ 2 n , it holds that v n i = u n 2 n +1 − i . F o r t ∈ T ω Σ let U t n = t ( u n 1 ) t ( u n 2 ) . . . t ( u n 2 n ) b e the finite word enumerating the lab els of no des in C n in the lexicog r aphic order, and let V t n = t ( v n 1 ) t ( v n 2 ) . . . t ( v n 2 n ) be the reverse sequence. Let V ′ t n = ψ ( V t n ) where ψ is the mor phism from Σ ⋆ int o Σ ′ ⋆ defined by ψ ( a ) = a ′ for all a ∈ Σ. The co de g ( t ) of t is then g ( t ) = V ′ t 0 .A.U t 1 .B .V ′ t 2 .A.U t 3 .B .V ′ t 4 .A . . . A.U t 2 n +1 .B .V ′ t 2 n +2 .A . . . The ω -w ord g ( t ) enumerates the lab els o f the no des of the tree t which app ear at successive levels 1 , 2 , 3 , . . . The (images by ψ o f ) la b e ls of no des o ccuring a t o dd level 2 n + 1 are enum erated in the reverse lexico graphic o r der by the sequence V ′ t 2 n and the lab els of no des o ccuring at even level 2 n are en umerated in the lexicogra phic order by the sequence U t 2 n − 1 . La be ls o f no des of distinct levels are alter na tively s e parated by a le tter A or a le tter B . Let now (Γ , D ) be a dependence alphab et where Γ = Σ ∪ Σ ′ ∪ { A, B } and the independenc e relation I D = Γ × Γ − D is defined by I D = Σ × ( { A } ∪ Σ ′ ) [ ( { A } ∪ Σ ′ ) × Σ [ Σ ′ × { B } [ { B } × Σ ′ i.e. letters of Σ may o nly commut e with A and with letters in Σ ′ while letters of Σ ′ may only commute with B and with letters in Σ, letter A (resp ectively B ) may only commute with letters in Σ (res pec tively , with letters in Σ ′ ). Let now h : T ω Σ → R ω (Γ , D ) b e the function defined by: ∀ t ∈ T ω Σ h ( t ) = ϕ ( g ( t )) W e firstly state the following res ult. Lemma 4.2 The ab ove define d function h : T ω Σ → R ω (Γ , D ) is c ontinuous. 10 Pro of. Let us rema rk tha t in a se g ment B .V ′ t 2 n .A.U t 2 n +1 .B .V ′ t 2 n +2 .A of an ω -w ord g ( t ) wr itten as a b ove, le tters of U t 2 n +1 may only comm ute with the preceding letter A a nd letters of V ′ t 2 n . In a similar manner letters of V ′ t 2 n +2 may only commute with the pr eceding letter B and letters of U t 2 n +1 . So if tw o infinite binar y tr ees t, s ∈ T ω Σ hav e the same la bels o n their k first levels, ( k > 1), then for a ll r ∈ M (Γ , D ) s uch that | r | ≤ ( k − 1) + 1 + 2 + 2 2 + . . . + 2 k − 2 it holds that r ⊑ h ( t ) ↔ r ⊑ h ( s ) So l pref ( h ( t ) ,h ( s )) ≥ ( k − 1) + 1 + 2 + 2 2 + . . . + 2 k − 2 ≥ 2 k − 1 and d pref (( h ( t ) , h ( s )) = 2 − l pref ( h ( t ) ,h ( s )) ≤ 2 − 2 k − 1 so we hav e prov ed: ∀ t, s ∈ T ω Σ d ( t, s ) ≤ 2 − k → d pref (( h ( t ) , h ( s )) ≤ 2 − 2 k − 1 Thu s the function h is co nt inuous (and even uniformly contin uous).  Let no w R ⊆ Σ ω be a reg ular ω -language. W e are g oing to define fro m R a language of infinite real traces L ov er the dep endence alphab et (Γ , D ) defined ab ov e . Then we sha ll prov e that L is rationa l and that P ath ( R ) = h − 1 ( L ). Let us firstly define L as be ing the set o f infinite traces ϕ ( σ ) wher e σ ∈ Γ ω may be written in the following form: σ = x (1) .u 1 .A.v 1 .x (2) .u 2 .B .v 2 .x (3) .u 3 .A . . . A.v 2 n +1 .x (2 n +2) .u 2 n +2 .B .v 2 n +2 .x (2 n +3) .u 2 n +3 .A . . . where for all integers i ≥ 0, x (2 i + 2 ) ∈ Σ and x (2 i + 1 ) ∈ Σ ′ u 2 i +2 , v 2 i +1 ∈ Σ ⋆ and u 2 i +1 , v 2 i +2 ∈ Σ ′ ⋆ | v i | = 2 | u i | or | v i | = 2 | u i | + 1 and the ω -w ord x = ψ − 1 ( x (1)) x (2) ψ − 1 ( x (3)) . . . x (2 n ) ψ − 1 ( x (2 n + 1)) x (2 n + 2) . . . is in R . Lemma 4.3 The ab ove define d language L of infinite r e al tr ac es is r ational. 11 Pro of. E very ω -word σ = x (1) .u 1 .A.v 1 .x (2) .u 2 .B .v 2 .x (3) .u 3 .A . . . A.v 2 n +1 .x (2 n +2) .u 2 n +2 .B .v 2 n +2 .x (2 n +3) .u 2 n +3 .A . . . written as ab ove is equiv alent, mo dulo the equiv alence relation ∼ I D ov er infinite words in Γ ω , to the infinite word σ ′ = x (1) .u 1 .v 1 .A.x (2) .u 2 .v 2 .B .x (3) .u 3 .v 3 .A . . . A.x (2 n +2) .u 2 n +2 .v 2 n +2 .B .x (2 n + 3) .u 2 n +3 .v 2 n +3 .A . . . But letters of Σ may commute also with letters of Σ ′ and for a ll integers i , | v i | = 2 | u i | or | v i | = 2 | u i | + 1 by definition of L . Th us every ω -word σ w r itten as ab ov e is also equiv alent, mo dulo ∼ I D , to an infinite word in Γ ω in the for m σ ′′ = x (1) .W 1 .A.x (2) .W 2 .B .x (3) .W 3 .A . . . A.x (2 n +2) .W 2 n +2 .B .x (2 n + 3) .W 2 n +3 .A . . . where for all integers i ≥ 0, W 2 i +1 ∈ (Σ ′ Σ 2 ) ⋆ . (Σ ∪ { ε } ) W 2 i +2 ∈ (ΣΣ ′ 2 ) ⋆ . (Σ ′ ∪ { ε } ) Let L b e the ω - language over the alphab et Γ fo rmed by all suc h ω -words σ ′′ such that x = ψ − 1 ( x (1)) x (2) ψ − 1 ( x (3)) . . . x (2 n ) ψ − 1 ( x (2 n + 1)) x (2 n + 2) . . . is in R . It is ea sy to see that L is a n ω -re gular lang uage. Moreover L = ϕ ( L ) thus we can infer from Theor em 2.1 that L is a rational languag e of infinite rea l tra c e s.  W e are g o ing now to prov e the following res ult. Lemma 4.4 F or L define d as ab ove fr om the ω -language R , it holds that P ath ( R ) = h − 1 ( L ) , i.e. ∀ t ∈ T ω Σ h ( t ) ∈ L ← → t ∈ P ath ( R ) . Pro of. Supp ose that h ( t ) ∈ L for some t ∈ T ω Σ . Then h ( t ) = ϕ ( g ( t )) = ϕ ( σ ) where σ ∈ Γ ω may b e written in the following for m: σ = x (1) .u 1 .A.v 1 .x (2) .u 2 .B .v 2 .x (3) .u 3 .A . . . A.v 2 n +1 .x (2 n +2) .u 2 n +2 .B .v 2 n +2 .x (2 n +3) .u 2 n +3 .A . . . where for all integers i ≥ 0, x (2 i + 2 ) ∈ Σ and x (2 i + 1 ) ∈ Σ ′ u 2 i +2 , v 2 i +1 ∈ Σ ⋆ and u 2 i +1 , v 2 i +2 ∈ Σ ′ ⋆ | v i | = 2 | u i | or | v i | = 2 | u i | + 1 and the ω -w ord x = ψ − 1 ( x (1)) x (2) ψ − 1 ( x (3)) . . . x (2 n ) ψ − 1 ( x (2 n + 1)) x (2 n + 2) . . . is in R . 12 Then it is ea sy to see that σ = g ( t ), b ecause of the definition of σ , of g ( t ), and of the indep endence relation I D on Γ. Then ψ − 1 ( x (1)) = t ( v 0 1 ) and u 1 = ε , then | v 1 | = 2 | u 1 | = 0 o r | v 1 | = 2 | u 1 | + 1 = 1. If | v 1 | = 0 then x (2) = t ( u 1 1 ) and if | v 1 | = 1 then x (2) = t ( u 1 2 ). Then the choice of | v 1 | = 2 | u 1 | o r of | v 1 | = 2 | u 1 | + 1 implies tha t x (2) is the lab el of the left or the rigth succe ssor of the r o ot no de v 0 1 = ε . This phenomenon will happen for next levels. The choice of | v i | = 2 | u i | or o f | v i | = 2 | u i | + 1 determines one of the tw o success or no des of a no de at level i (whose label is x ( i ) if i is even or ψ − 1 ( x ( i )) if i is o dd) and then the lab el o f this success or is ψ − 1 ( x ( i + 1 )) if i is even, or x ( i + 1 ) if i is o dd. Thu s the suc c essive choices determine a br anch of t a nd the labels of no des of this branch (changing o nly x (2 n + 1) in ψ − 1 ( x (2 n + 1))) form a path x = ψ − 1 ( x (1)) x (2) ψ − 1 ( x (3)) . . . x (2 n ) ψ − 1 ( x (2 n + 1)) x (2 n + 2 ) . . . which is in R . Then t ∈ P ath ( R ). Conv er sely it is easy to see that if t ∈ P ath ( R ), the infinite word g ( t ) may b e written as a word σ in the a b ove form. Then h ( t ) = ϕ ( g ( t )) = ϕ ( σ ) is in L .  W e can now sta te the following Theorem 4 .5 Ther e exist some Σ 1 1 -c omplete, henc e non Bor el, r ational lan- guages of infinite r e al tr ac es. Pro of. Supp os e R ⊆ Σ ω is a Π 0 2 -complete ω -regular languag e. Let then L ⊆ R ω (Γ , D ) b e defined a s a b ove. L is a rational languag e of infinite rea l traces by Lemma 4.3. Then L is an analytic subset of R ω (Γ , D ) by Prop os ition 4.1. But P ath ( R ) is a Σ 1 1 -complete s et and P ath ( R ) = h − 1 ( L ) holds by Lemma 4.4 th us L is also Σ 1 1 -complete. In particula r L is not a B orel set.  5 Concluding r emarks The exis tence of a Σ 1 1 -complete infinitary rational rela tion has b een used to get many undecidability r e sults in [Fin03d] a nd the e x istence of a Σ 1 1 -complete context free ω -la nguage led to other undecida bility re s ults in [Fin0 3a, Fin03 b, FS03]. In particular, the to p o logical co mplexity and the degree of ambiguit y of an infinitary rational relation or o f a context free ω -langua ge a re highly unde- cidable. In a similar way , the existence of a Σ 1 1 -complete reco gnizable langua g e of in- finite pictures, prov ed in [A TW0 2] by Altenbernd, Thomas, and W¨ ohrle, has bee n used in [Fin04a] to prove many undecidability r esults, giv ing in particular the answer to some op en questions of [A TW02]. T opologic al a rguments following fro m the existence of Σ 1 1 -complete r ational languages of infinite rea l tra ces can also b e used to prove similar undec ida bilit y results for la ng uages of infinite traces. In [FS0 3] have b een established so me links b e t ween the ex istence of a Σ 1 1 - complete ω -lang uage in the form V ω and the num b er o f decomp ositions of ω - words of V ω in words of V . W e think that such fa c ts could b e useful in the domain of combinatorics of 13 traces. The co de problem for tr aces is impor ta nt in T race Theo ry and s everal questions are still ope ne d [HM97]. The a nalogue of the no tion of ω -co de and the study o f the num ber of decomp ositions of infinite traces o f V ω , where V is a se t of finite traces, in infinite pro duct of tr aces of V , is also an impo rtant sub ject related to practical applications and to the notion of a mbiguit y (se e [Aug01, AA01] for r elated r esults in the ca se of words). W e think that to p o logi- cal arg umen ts could b e useful in this resear ch ar ea, and the existence of several Σ 1 1 -complete language s of infinite traces, having differe nt structures, could b e useful in the cases of different dep endence a lphab ets. Ac knowledgemen ts. Thank s to the anonymous referees fo r useful comments on a pr eliminary version of this pap er. References [Aug01] X. Aug r os, Des Algo rithmes Autour des Co de s Rationnels, Ph.D. The- sis, universit ´ e de Nice-So pia Antipolis, December 2 001. [AA01] M. K . Ahmad and X. Augro s, Some Results on Co des for Gener alized F a ctorizatio ns , Journa l of Automata, Language s a nd Combinatorics, V o l- ume 6 (3), p. 2 39-25 1, 200 1. [A TW0 2 ] J- H. Altenbernd, W. Thomas and S. W¨ ohrle, Tiling Systems ov er Infinite Pictures and their Acceptance Conditions, in the Pro ceeding s of the 6th International Conference on Developements in Language Theory , DL T 2002 , L ecture Notes in Computer Science, V olume 2450, p. 2 97-30 6, Springer, 2003 . [BMP90] P . Bo nnizz o ni, G. Mauri and G. Pighizzini, Ab out Infinite T races, in V. Diek ert, editor, Pro ceeding s of the ASMICS W or kshop F re e Partially Commutativ e Monoids, Ko chel am See, Octob er 1989 , Rep or t TUM-I9002 , pages 1-10 , TU M ¨ unchen, 1990 . [CF69] P . Cartier and D. F oata, P robl` emes Combinatoires de C o mmut ation et de R´ earrange ments, V olume 8 5 of Lectur e Notes in Mathematics, Spr inger- V er lag, Berlin-Heidelb erg -New Y ork, 1969 . [DR95] V. Diekert a nd G. Rozenber g, E dito rs, The Bo ok of T races, W orld Sci- ent ific, Singap or e, 1995 . [Fin01] O . Finkel, T opo logical Pro p erties of Omega Context F r e e La ng uages, Theoretical Computer Science, V ol. 2 6 2 (1 -2), July 20 0 1, p. 669- 6 97. [Fin03a] O. Finkel, Bor e l Hierar ch y a nd O mega Context F ree La nguages, The- oretical Computer Science, V o l 29 0 (3), 2003, p. 138 5-140 5. [Fin03b] O . Finkel, Am biguity in Omega Co nt ext F ree Languag es, Theoretica l Computer Science, V olume 301 (1- 3), 2003 , p. 21 7 -270 . [Fin03c] O . Fink el, O n the T opologic a l Co mplexity of Infinitary Ratio nal Re- lations, RAIRO-Theoretical Infor matics and Applications, V olume 37 (2), 2003, p. 105- 113. 14 [Fin03d] O . Finkel, Undecidability of T op ologica l and Arithmetical Pr op erties of Infinitar y Rational Relations, RAIRO-Theoretical Informa tics a nd Ap- plications, V olume 37 (2), 20 03, p. 115 -126. [FS03] O . Finkel and P . Simonnet, T opo lo gy and Ambiguit y in O meg a Context F r ee La ng uages, Bulletin o f the Belg ia n Mathematica l So ciet y , V o lume 10 (5), 2003, p. 7 07-72 2. [Fin04a] O. Finkel, On Recog nizable Languag e s of Infinite Pic tur es, Interna- tional Journa l of F o undations of Computer Science, to a ppe ar. Av ailable from http://www.logique.jussieu.fr/www.finkel [Fin04b] O . Fink el, An ω -Pow er o f a Finitary Lang uage Which is a Borel Set of Infinite Rank, F undamen ta Info r maticae, V olume 6 2 (3- 4), 20 04, p. 333- 342. [Gas91] P . Gastin, Recogniza ble and Ra tio nal Languag es of Finite and Infinite T races, Actes du ST A CS’91, Le c ture Notes in Co mputer Science 480, p. 89-10 4, 199 1 . [GP95] P . Gas tin, A. Petit, Infinite T races, Chapter in The Bo ok of T races, edited by V. Diekert et G. Rozenberg, W orld Scientific, p. 393 -486 , 1995. [GPZ94] P . Ga stin, A. Petit, W. Zielonk a, An E xtension o f Klee ne ’s a nd Ochmanski’s Theorems to Infinite T races, F undamen ta l Study , Theoreti- cal Computer Science 125, p. 16 7-20 4 , 19 94. [HM97] H.J. Hoo geb o om and A. Muscholl, The Co de Problem for T races - Improving the Bo unda ries, Theor etical Co mputer Science 1 72, p. 30 9-321 , 1997. [Kec95] A.S. Kechris, Classica l Descriptive Set Theo ry , Springer-V erlag, 1995 . [KK03] R. K ummetz and D. Kuske, The T opolog y of Ma zurkiewicz T races, Theoretical Computer Science, V olume 30 5 (1 -3), p. 23 7-258 , 20 03. [Kwi90] M. K w ia tko ws k a, A Metr ic for T races, I nfo r mation P ro cessing Letters, 35, p. 129- 135, 1 9 90. [Lan69] L. H. L andweber, Decision Problems for ω -Automata, Math. Sys t. The- ory 3 (1969 ) 4,376- 384. [Lec01] D. Lecomte, Sur le s Ensembles de Phrases Infinies Co nstructibles a Partir d’un Dictionnaire sur un Alphab et Fini, S ´ eminair e d’Initiatio n a l’Analyse, V olume 1, ann´ ee 20 01-20 02. [L T94] H. Lescow and W. Thoma s, Logical Sp ecifications o f Infinite Computa- tions, In:” A Decade o f Concurrency ” (J . W. de Bakker et a l., eds), Spr inger LNCS 803 (199 4), 5 8 3-62 1. [Maz77] A. Mazurkiewicz , Concurrent P rogra ms Schemes and their Interpreta- tion, D AIMI Repo rt PB-78 , Aar hus University , Aarhus, 19 77. [Mos80] Y. N. Moschov akis, Descriptive Set Theo r y , North-Holland, Amster - dam 1980 . 15 [Niw85] D. Niwinsk i, An example of No n B o rel Set of Infinite T rees Reco gniz- able by a Rabin Automato n, in Polish, Manuscript, Universit y of W arsaw, 1985. [Niw90] D. Niwinski, Problem on ω -Pow er s po s ed in the P ro ceedings of the 1990 W orkshop “ Logics and Recogniza ble Sets” (Univ. Kiel). [PP04] D. Perrin and J.- E. Pin, Infinite W o rds, Automata, Semigroups, Log ic and Games , V o lume 141 o f P ure and Applied Mathematics , Elsevier, 200 4. [Sim92] P . Simonnet, Automa tes et Th´ eorie Descriptive, Ph. D. Thesis, Univer- sit´ e Paris 7, Ma rch 199 2. [Sim93] P . Simonnet, Automate d’ Arbres Infinis et Choix Bor´ elien, C.R.A.S. Paris, t.3 16, S´ erie 1, p. 97 -100, 1993. [Sta86] L. Staig e r, Hierar chies of Recur sive ω -La nguages, Jour. Inform. P r o cess. Cyb ernetics EIK 2 2 (1 9 86) 5/6 , 21 9-241 . [Sta97a] L. Staiger, ω -Lang ua ges, Chapter of the Handb o ok of F ormal la n- guages, V ol 3 , edited by G. Ro zenberg and A. Sa lo maa, Spring er-V erlag, Berlin. [Sta97b] L. Staige r, O n ω -Pow er Langua g es, in New T rends in F ormal Lan- guages, Control, Cop era tion, a nd Combinatorics, Lecture Notes in Com- puter Science 1 218, Springer- V er lag, Ber lin 1 997, 377 -393. [Tho90] W. Tho ma s, Automata on Infinite Ob jects, in: J. V an Leeuw en, ed., Handbo ok of Theoretical Computer Science, V ol. B ( Elsevier , Amsterdam, 1990 ), p. 13 3-191 . 16