Topological Complexity of Context-Free omega-Languages: A Survey

T opological Complexity of Context-Free ω -Languages : A Sur vey Oli vier Finkel Equipe de Logique Math´ ematique Institut de Math ´ ematiques de Jussieu - Paris Riv e Gauche UMR7586 CNRS et Univ ersit ´ e Paris Diderot Paris 7 B ˆ atiment Sophie Germain Case 7012 75205 Paris Cede x 13, France. finkel@math.u niv-paris-did erot.fr Abstract. W e surve y recent results on the top ological complexity of co ntext-free ω -languages which form the second lev el of the Chomsk y h ierarchy of languag es of inﬁnite words. In particular , we c onsider the Borel hierarchy and the W adg e h i- erarchy of n on-deterministic or deterministic conte xt-free ω -languages. W e study also decision problems, the links with the notions of ambiguity and of d egrees of ambiguity , and the special case of ω -po wers. K eywords: Inﬁnite words; pushdo wn automata; context-free ( ω )- languages; ω -powers; Cantor topology; topological comple xity; Borel hierarchy; W adge hierarchy; complete sets; decision problems. 1 Intr oduction The Chomsky hierarchy of form al languages of ﬁnite words over a ﬁnite alphab et is now well known, [49]. The class of re gular languages accepted by ﬁnite autom ata forms the ﬁrst le vel of this hierarchy an d th e class o f c ontext-free lang uages accep ted by push- down autom ata or generated by co ntext-free gramm ars form s its second le vel [3]. The third and the fo urth lev els a re formed by the class of context-sensitive languages ac- cepted by lin ear-bounded auto mata or gen erated b y T ype-1 grammars and the class of recursively enu merable lang uages accep ted by T u ring ma chines or gen erated b y T ype-0 grammar s [15]. In par ticular, context-free language s, ﬁrstl y in troduce d by Chomsky to analyse the syntax of natural languages, ha ve been very useful in Computer Science, in particular in the domain of program ming langu ages, for the construction of co mpilers used to verify correctness of programs, [48]. There is a h ierarchy of languages of inﬁnite words which is ana logous to the Chom sky hierarchy b ut where th e languages are f ormed by inﬁnite word s over a ﬁnite alp habet. The ﬁrst le vel of this hierarchy is formed by the class of regular ω - languag es accepted by ﬁnite automata. They were ﬁrst studied by B ¨ uchi in orde r to study decision prob- lems for logical theor ies. In particular, B ¨ u chi proved that the monadic second order theory of one successor over the integers is decida ble, using ﬁnite automata equ ipped with a certain accep tance condition fo r inﬁnite w ords, now called the B ¨ u chi acceptance condition . W e ll k nown pioneers in th is research are a are named Muller , Mc Naugh ton, Rabin, Landweber, Cho ueka, [61,6 2,68,52,16]. Th e the ory o f regular ω -lang uages is now well e stablished and has found m any applications for speciﬁcation an d veriﬁca- tion of non-term inating systems; see [81,7 8,67] for many resu lts a nd references. The second level of the hier archy is f ormed by the class of con text-free ω -langu ages. As in the case of lang uages of ﬁn ite words it turne d o ut th at an ω -langua ge is accep ted by a (n on-d eterministic) pushdown a utomaton ( with B ¨ uch i acc eptance con dition) if and only if it is generated by a context-free gram mar where in ﬁnite de riv ations are consid- ered. Context-free langu ages of inﬁnite words were ﬁrst studied by Cohen and Gold, [19,20], Linna, [56,57,58], Boasson, Ni vat, [ 64,63,7,8], Beauquier , [4], see the survey [78]. Notice that in the case of inﬁnite w ords T ype- 1 g rammars and T ype-0 grammars accept the same ω -langu ages which are also the ω -langu ages accepted by T uring ma- chines with a B ¨ uchi ac ceptance condition [21,7 8], see also the funda mental study of Engelfriet and Hoogeboom on X -automata, i.e. ﬁnite automata equippe d with a storage type X , accepting inﬁnite words,[29]. Context-free ω -lang uages h av e occurr ed recen tly in the works on games played on inﬁnite pu shdown gr aphs, following the fun damental study o f W aluk iewicz, [85,82] [74,40]. Since the set X ω of inﬁnite words over a ﬁn ite alphabet X is naturally equipped with the Cantor topo logy , a way to study th e complexity o f ω -lang uages is to stu dy their topolog ical complexity . The ﬁrst task is to locate ω -lan guages with regar d to the Borel and the pro jectiv e hierarchies, and next to the W adg e hierarch y which is a great reﬁne- ment of the Borel hierarch y . It is then natural to ask for decidab ility properties and to study decision problems like : is there an effective proce dure to determine the Bore l rank or th e W adge degree of any context-free ω -langu age ? Such question s were asked by Lescow an d Th omas in [55]. In this paper we survey some recent r esults o n th e topo- logical com plexity of co ntext-free ω -lang uages. Some o f them were very surprising as the two following o nes: 1. there is a 1 -coun ter ﬁnitar y languag e L such that L ω is analytic but no t Borel, [35]. 2. The W a dge hierarchy , hence also the Borel hier archy , o f ω - languag es accep ted by real time 1 -cou nter B ¨ uchi automata is th e same as the W adge hierarch y of ω - languag es accepted by B ¨ uchi T uring machines, [41]. The B orel and W ad ge hierarchies of non deterministic context-fr ee ω -lan guage s a re not effecti ve. One can neither decide wheth er a g i ven co ntext-free ω - languag e is a Borel set n or whe ther it is in a given Borel cla ss Σ 0 α or Π 0 α . On the oth er hand deterministic context-free ω -la nguage s are located at a lo w le vel of the Borel hierarchy: they are all ∆ 0 3 -sets. They en joy some decid ability pro perties althoug h som e impor tant q uestions in this area are still o pen. W e consider also the links with the no tions of ambig uity and of degrees of ambiguity , and the special case of ω -powers, i.e. of ω - languag es in the form V ω , where V is a (c ontext-free) ﬁnitary language. Finally we state some perspectives and give a list o f some questions which remain open for further study . The paper is organized as follows. In Section 2 we recall the n otions of con text-free ω -lang uages accepted b y B ¨ uchi or Muller pushd own automata. T opo logical notion s and Borel and W adge hiera rchies are recalled in Section 3. In Section 4 is studied the case of no n-deter ministic co ntext-free ω -langu ages while deterministic context-free ω - languag es are consid ered in Sec tion 5. L inks with notions of ambiguity in co ntext free languag es are studied in Section 6 . Section 7 is dev oted to the spe cial case o f ω -powers. Perspectives and som e open questions are presented in last Section 8. 2 Context-fr ee ω -languages W e a ssume the r eader to be familiar with the theor y of forma l ( ω )-lang uages [81,78]. W e shall use usual notatio ns of formal language theory . When X is a ﬁnite alphabe t, a non-em pty ﬁ nite wor d over X is any sequence x = a 1 . . . a k , where a i ∈ X for i = 1 , . . . , k , and k is an integer ≥ 1 . The length of x is k , denoted by | x | . The emp ty word has no letter s and is deno ted by λ ; its length is 0 . For x = a 1 . . . a k , we write x ( i ) = a i and x [ i ] = x (1) . . . x ( i ) for i ≤ k and x [0] = λ . X ⋆ is the set of ﬁnite wor ds (including the empty word) o ver X . For V ⊆ X ⋆ , the complem ent of V (in X ⋆ ) is X ⋆ − V den oted V − . The ﬁrst inﬁnite ordinal is ω . An ω - wor d over X is an ω -sequence a 1 . . . a n . . . , where for all integers i ≥ 1 , a i ∈ X . When σ is an ω -word over X , we write σ = σ (1) σ (2 ) . . . σ ( n ) . . . , wh ere f or all i , σ ( i ) ∈ X , an d σ [ n ] = σ (1) σ (2) . . . σ ( n ) for all n ≥ 1 and σ [0 ] = λ . The usual concatenation product of two ﬁnite w ords u and v is den oted u.v (and some- times just uv ). This p rodu ct is extended to the product of a ﬁn ite w ord u an d an ω - word v : the inﬁnite word u.v is th en the ω -word such that: ( u.v )( k ) = u ( k ) if k ≤ | u | , and ( u.v )( k ) = v ( k − | u | ) if k > | u | . The pr eﬁx relation is den oted ⊑ : a ﬁnite word u is a preﬁx of a ﬁnite word v (resp ec- ti vely , an inﬁn ite word v ), den oted u ⊑ v , if and only if there exists a ﬁnite word w (respectively , an inﬁnite word w ), such that v = u.w . The set of ω - words over th e alphabet X is deno ted by X ω . An ω - language over an alphabet X is a subset of X ω . The complem ent (in X ω ) of an ω - language V ⊆ X ω is X ω − V , deno ted V − . For V ⊆ X ⋆ , the ω -p ower of V is : V ω = { σ = u 1 . . . u n . . . ∈ X ω | ∀ i ≥ 1 u i ∈ V } . W e now deﬁne pushdown machines and the class of ω -context-f ree lan guages. Deﬁnition 1. A pushdo wn machine (PDM) is a 6 -tuple M = ( K, X , Γ , δ, q 0 , Z 0 ) , wher e K is a ﬁnite set of states, X is a ﬁ nite in put alphabet, Γ is a ﬁnite pushdo wn alphab et, q 0 ∈ K is the initial state, Z 0 ∈ Γ is the start symbol, an d δ is a ma pping fr om K × ( X ∪ { λ } ) × Γ to ﬁn ite subsets of K × Γ ⋆ . If γ ∈ Γ + describes th e pushdown store co ntent, the leftmost symbol will b e a ssumed to be on “top” of the stor e. A conﬁguration of a PDM is a pair ( q , γ ) wher e q ∈ K an d γ ∈ Γ ⋆ . F o r a ∈ X ∪ { λ } , β , γ ∈ Γ ⋆ and Z ∈ Γ , if ( p, β ) is in δ ( q , a, Z ) , then we write a : ( q , Z γ ) 7→ M ( p, β γ ) . 7→ ⋆ M is th e transitive and r e ﬂexive closur e o f 7→ M . (Th e sub script M will be omitted whenever the meaning r ema ins clear). Let σ = a 1 a 2 . . . a n . . . b e an ω -wor d over X . An inﬁn ite sequence of conﬁgurations r = ( q i , γ i ) i ≥ 1 is called a complete run of M on σ , starting in conﬁguration ( p, γ ) , iff: 1. ( q 1 , γ 1 ) = ( p, γ ) 2. for ea ch i ≥ 1 , th er e e xists b i ∈ X ∪ { λ } satisfying b i : ( q i , γ i ) 7→ M ( q i +1 , γ i +1 ) such that a 1 a 2 . . . a n . . . = b 1 b 2 . . . b n . . . F o r every such run, I n ( r ) is the set o f all states enter ed inﬁ nitely often during run r . A complete run r of M on σ , starting in con ﬁguration ( q 0 , Z 0 ) , will be simply called “a run of M on σ ”. Deﬁnition 2. A B ¨ uchi pu shdown automato n is a 7 -tuple M = ( K, X , Γ , δ, q 0 , Z 0 , F ) wher e M ′ = ( K , X , Γ , δ, q 0 , Z 0 ) is a PDM a nd F ⊆ K is the set of ﬁna l states. The ω -lan guage accepted by M is L ( M ) = { σ ∈ X ω | ther e exists a complete run r of M on σ such that I n ( r ) ∩ F 6 = ∅} Deﬁnition 3. A Muller p ushdown automa ton is a 7-tuple M = ( K, X , Γ , δ, q 0 , Z 0 , F ) wher e M ′ = ( K , X , Γ , δ, q 0 , Z 0 ) is a PDM and F ⊆ 2 K is the collection of designated state sets. The ω -la nguage accepted by M is L ( M ) = { σ ∈ X ω | ther e exists a complete run r of M on σ such that I n ( r ) ∈ F } Remark 4. W e con sider here two acc eptance c ondition s for ω -words, the B ¨ uchi an d the Muller acc eptance c ondition s, r espectively den oted 2- acceptan ce and 3- acceptan ce in [52] and in [20] and ( inf , ⊓ ) a nd ( inf , =) in [78]. W e r efer the r eader to [19,20,7 8,29] for consideration o f weak er acceptance condition s, an d t o [46,67] for the d eﬁnitions of other usua l ones like Ra bin, Street, o r pa rity accepta nce conditions. Notice however that it s eems that the latter ones have not been much conside r ed in the study of co ntext- fr ee ω -lang uages b ut they ar e often involved in construction s con cerning ﬁnite auto mata r eading inﬁnite wor ds. Notatio n. In the sequel we shall often abbreviate “Muller pushdown automato n” by MPD A and “B ¨ uc hi pushdown automaton ” by BPD A. Cohen and Go ld an d indepen dently Lin na e stablished a cha racterization theorem for ω - languag es accep ted b y B ¨ u chi or Mu ller pu shdown automata. W e shall n eed the no tion of “ ω -Kleen e closure” which we now ﬁrstly d eﬁne: Deﬁnition 5. F or any family L o f ﬁnitary languages, t he ω -Kleene closure of L is : ω − K C ( L ) = {∪ n i =1 U i .V ω i | ∀ i ∈ [1 , n ] U i , V i ∈ L} Theorem 6 (Linna [56], Cohen and G old [19]). Let C F L be the class of context-fr ee (ﬁnitary) langu ages. Then for any ω - language L the following thr ee condition s ar e equivalen t: 1. L ∈ ω − K C ( C F L ) . 2. Ther e exists a B P D A that accepts L . 3. Ther e exists a M P DA that accepts L . In [19] are also studie d ω -langua ges generated by ω -context-fre e gr ammars an d it is shown tha t each of th e cond itions 1 ), 2), and 3) of the above Theorem is also eq uiv a- lent to: 4) L is generated by a co ntext-free g rammar G by lef tmost deriv ations. Th ese grammar s are also studied by Nivat in [ 63,64]. T hen we c an let the following d eﬁnition: Deﬁnition 7. An ω -la nguage is a context-fr ee ω -lan guage iff it satisﬁes one of the condition s of the ab ove Theo r em. The class of context-fr ee ω - languages will be de noted by C F L ω . If we om it the pu shdown s tore in th e ab ove Theorem we obtain th e ch aracterization of languag es accepted by classical Muller automa ta (MA) or B ¨ uchi automata (BA) : Theorem 8. F or any ω -langu age L , the following condition s ar e equivalent: 1. L belongs to ω − K C ( RE G ) , wher e RE G is the class of ﬁnitary r e g ular languages. 2. Ther e exists a MA that accepts L . 3. Ther e exists a BA that accepts L . An ω -lang uage L satisfying one of the conditions of the above Theo r em is called a r e g ular ω - language. The class of re gu lar ω -la nguages will be denoted by RE G ω . It follows f rom Mc Naughton’ s The orem that the expressiv e power of deterministic MA (DMA) is equal to the expressiv e power of n on determin istic M A, i.e. that every r eg- ular ω -langu age is accep ted b y a d eterministic Mu ller automaton , [62,67]. Notice that Choueka gave a simpliﬁed pr oof of Mc Naughton ’ s Theorem in [16]. Another variant was given by Rabin in [68]. Un like th e case of ﬁnite auto mata, d eterministic M P D A do not deﬁne the same class of ω -la nguage s as n on deterministic M P D A . Let us now deﬁne determin istic pushdown machines. Deﬁnition 9. A P D M M = ( K, X , Γ , δ, q 0 , Z 0 ) is said to be deterministic iff for each q ∈ K , Z ∈ Γ , and a ∈ X : 1. δ ( q , a, Z ) contains at most one element, 2. δ ( q , λ, Z ) contains at most one element, and 3. if δ ( q , λ, Z ) is non empty , then δ ( q , a, Z ) is empty for all a ∈ X . It turned out that the class of ω -lang uages accepted by deterministic B P D A is strictly included into the class of ω -langu ages accepted by d eterministic M P D A . This lat- est class is the class D C F L ω of determin istic con text-free ω -langu ages. W e denote D C F L th e class of deterministic context-free (ﬁnitary) languages. Proposition 10 ([2 0]). 1. D C F L ω is c losed u nder co mplementation , but is n either clo sed u nder un ion, n or under intersection. 2. D C F L ω ( ω − K C ( D C F L ) ( C F L ω (these inclusions ar e strict). 3 T opology 3.1 B orel hierarch y and analytic sets W e assume the reader to b e familiar with basic notions of top ology which may be found in [60,55,50,78,67]. There is a natural metric on the set X ω of inﬁnite word s over a ﬁnite alphabet X contain ing at least two letters wh ich is called the pr eﬁ x metric a nd deﬁn ed as follows. For u, v ∈ X ω and u 6 = v let δ ( u , v ) = 2 − l pref ( u,v ) where l pref ( u,v ) is the ﬁrst integer n such that u ( n + 1) is different from v ( n + 1 ) . This metric induces o n X ω the usual Cantor topo logy for which open subsets of X ω are in the form W.X ω , where W ⊆ X ⋆ . A set L ⊆ X ω is a clo sed set iff its complem ent X ω − L is an open set. Deﬁne now the Bor el Hierar chy of subsets of X ω : Deﬁnition 11. F or a non-null co untable or dinal α , the classes Σ 0 α and Π 0 α of the Bor el Hierar chy on the topological space X ω ar e deﬁned as follows: Σ 0 1 is the class of open subsets of X ω , Π 0 1 is the class of closed subsets of X ω , and for any counta ble or dina l α ≥ 2 : Σ 0 α is the class of coun table unions of subsets of X ω in S γ < α Π 0 γ . Π 0 α is the class of countable intersections of subsets of X ω in S γ < α Σ 0 γ . Recall some basic results about these classes : Proposition 12. (a) Σ 0 α ∪ Π 0 α ( Σ 0 α +1 ∩ Π 0 α +1 , for each countable or d inal α ≥ 1 . (b) ∪ γ < α Σ 0 γ = ∪ γ < α Π 0 γ ( Σ 0 α ∩ Π 0 α , for each countable limit or d inal α . (c) A set W ⊆ X ω is in the class Σ 0 α iff it s comp lement 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 countable or d inal α ≥ 1 . For a cou ntable ordin al α , a subset of X ω is a Borel set of rank α iff it is in Σ 0 α ∪ Π 0 α but not in S γ < α ( Σ 0 γ ∪ Π 0 γ ) . There are also some subsets of X ω which are not Borel. Indeed there exists a nother hi- erarchy beyond the Borel hierarchy , wh ich is called the projec ti ve hierarchy and which is obtained from th e Borel hierarchy b y successi ve application s of oper ations of p ro- jection and complem entation. The ﬁrst le vel of the pro jectiv e hier archy is fo rmed by the class of a nalytic sets and the class of co-ana lytic sets wh ich are comp lements o f analytic sets. In particula r the class of Borel sub sets of X ω is strictly included into th e class Σ 1 1 of analytic sets which are obtained by projectio n of Borel sets. Deﬁnition 13. A sub set A of X ω is in the class Σ 1 1 of analy tic sets if f the r e exist a ﬁnite alphabet Y an d a Bor el subset B of ( X × Y ) ω such th at x ∈ A ↔ ∃ y ∈ Y ω such that ( x, y ) ∈ B , wher e ( x, y ) is the inﬁnite wor d over the alphabe t X × Y such that ( x, y )( i ) = ( x ( i ) , y ( i )) for each inte ger i ≥ 1 . Remark 14. In the above de ﬁnition we co uld take B in the class Π 0 2 . Mor eover analy tic subsets of X ω ar e the pr o jections of Π 0 1 -subsets o f X ω × ω ω , where ω ω is th e B air e space, [60]. W e now deﬁne completeness with regard to reduction b y contin uous func tions. For a countab le or dinal α ≥ 1 , a set F ⊆ X ω is said to be a Σ 0 α (respectively , Π 0 α , Σ 1 1 )- complete set iff for a ny set E ⊆ Y ω (with Y a ﬁnite alp habet): E ∈ Σ 0 α (respectively , E ∈ Π 0 α , E ∈ Σ 1 1 ) iff there exists a continu ous function f : Y ω → X ω such that E = f − 1 ( F ) . Σ 0 n (respectively Π 0 n )-comp lete sets, with n an in teger ≥ 1 , ar e thoroug hly characterized in [76]. In particular R = (0 ⋆ . 1) ω is a well kn own example of Π 0 2 -complete subset of { 0 , 1 } ω . It is the set of ω -words over { 0 , 1 } ha ving inﬁnitely many occurrenc es o f the letter 1 . Its complem ent { 0 , 1 } ω − (0 ⋆ . 1) ω is a Σ 0 2 -complete subset of { 0 , 1 } ω . W e recall n ow the d eﬁnition o f the arithmetical hierarchy of ω -lang uages wh ich fo rm the effecti ve an alogue to the hierarchy of Borel sets of ﬁnite rank. Let X be a ﬁnite alph abet. An ω - languag e L ⊆ X ω belongs to the class Σ n if an d only if there exists a recursiv e relation R L ⊆ ( N ) n − 1 × X ⋆ such that L = { σ ∈ X ω | Q 1 a 1 Q 2 a 2 . . . Q n a n ( a 1 , . . . , a n − 1 , σ [ a n + 1]) ∈ R L } where Q 1 is the existential quan tiﬁer ∃ , and e very other Q i , for 2 ≤ i ≤ n , is one of the quantiﬁers ∀ or ∃ (not necessarily in an alternating o rder). An ω -lang uage L ⊆ X ω belongs to the class Π n if and only if its compleme nt X ω − L belon gs to the class Σ n . The inclusion relatio ns that hold b etween the classes Σ n and Π n are th e same as for the c orrespon ding classes of the Bor el hie rarchy . The classes Σ n and Π n are in cluded in th e resp ectiv e classes Σ 0 n and Σ 0 n of the Borel h ierarchy , and cardinality arguments sufﬁce to show that these inclusions are strict. As in the case of the Borel hierarch y , projection s of arithm etical sets (of the second Π -class) lead beyond the arithmetical hierarch y , to th e analytical hierarchy of ω - languag es. The ﬁrst class of this h ierarchy is the class Σ 1 1 of effective an alytic sets which are obtained by p rojection of arithmetical sets. An ω -langu age L ⊆ X ω belongs to the class Σ 1 1 if an d only if th ere exists a re cursive relation R L ⊆ N × { 0 , 1 } ⋆ × X ⋆ such that: L = { σ ∈ X ω | ∃ τ ( τ ∈ { 0 , 1 } ω ∧ ∀ n ∃ m (( n, τ [ m ] , σ [ m ]) ∈ R L )) } Then an ω -lang uage L ⊆ X ω is in the class Σ 1 1 iff it is the projectio n of an ω - languag e ov er th e alphab et X × { 0 , 1 } which is in the class Π 2 . Th e class Π 1 1 of effective co -analytic sets is simply the class of complemen ts of effecti ve an alytic sets. W e deno te as usual ∆ 1 1 = Σ 1 1 ∩ Π 1 1 . Recall that an ω -langu age L ⊆ X ω is in the class Σ 1 1 iff it is accepted by a non deterministic T uring machin e ( reading ω -word s) with a B ¨ uch i or Muller acceptanc e condition [78]. The B orel ranks of ∆ 1 1 sets are the (recursive) ord inals γ < ω CK 1 , where ω CK 1 is the ﬁrst non-r ecursive o rdinal, usually called the Chur ch-Kleene or dinal. Moreover , for ev ery non null ord inal α < ω CK 1 , there exist some Σ 0 α -complete and some Π 0 α -complete sets in the class ∆ 1 1 . On th e oth er hand , Kechris, Marker and Sami proved in [51] tha t the supremum of th e set of Borel ranks of (effecti ve) Σ 1 1 -sets is the ordina l γ 1 2 . This or dinal is proved to be strictly greater th an the ord inal δ 1 2 which is the ﬁrst non ∆ 1 2 ordinal. In p articular, the ordinal γ 1 2 is strictly g reater than the ord inal ω CK 1 . Remark that the exact value of the ordinal γ 1 2 may depen d on axioms of set theory , see [51,41] for mo re details. Notice also that it seems still unknown whether every non null o rdinal γ < γ 1 2 is th e Bor el rank of a Σ 1 1 -set. 3.2 W adge hierarchy W e now intro duce th e W adge hierarchy , which is a great reﬁnement o f th e Borel hier- archy deﬁned via reductions by continuo us function s, [23,83]. Deﬁnition 15 (W adge [83]). Let X , Y be two ﬁnite alpha bets. F o r L ⊆ X ω and L ′ ⊆ Y ω , L is sa id to be W a dge r educible to L ′ ( L ≤ W L ′ ) iff th er e exists a continuo us function f : X ω → Y ω , such that L = f − 1 ( L ′ ) . L and L ′ ar e W adge eq uivalent iff L ≤ W L ′ and L ′ ≤ W L . This will be denoted by L ≡ W L ′ . And we shall say that L < W L ′ iff L ≤ W L ′ but not L ′ ≤ W L . A set L ⊆ X ω is sa id to b e self dua l iff L ≡ W L − , and oth erwise it is said to b e n on self dual. The relation ≤ W is reﬂexi ve and transitive, an d ≡ W is an equivalence relatio n. The equivalen ce classes of ≡ W are called W adge de gr ee s . The W adge hierarchy W H is the class of Borel sub sets of a set X ω , where X is a ﬁ nite set, equipp ed wit h ≤ W and with ≡ W . For L ⊆ X ω and L ′ ⊆ Y ω , if L ≤ W L ′ and L = f − 1 ( L ′ ) wher e f is a con tinuous function from X ω into Y ω , then f is called a continuous reduction of L to L ′ . Intuitively it mean s that L is less complicated than L ′ because to check whethe r x ∈ L it suf ﬁces to check whether f ( x ) ∈ L ′ where f is a continuo us functio n. Hence the W adge degree of an ω -lan guage is a m easure of its topolog ical complexity . Notice that in the above d eﬁnition, we consider that a subset L ⊆ X ω is giv en together with the alphab et X . This is importan t as it is shown by the following simple example. Let L 1 = { 0 , 1 } ω ⊆ { 0 , 1 } ω and L 2 = { 0 , 1 } ω ⊆ { 0 , 1 , 2 } ω . So th e languages L 1 and L 2 are e qual but co nsidered over the different alp habets X 1 = { 0 , 1 } and X 2 = { 0 , 1 , 2 } . It turns out that L 1 < W L 2 . In fact L 1 is open and closed in X ω 1 while L 2 is closed but non open in X ω 2 . W e can now deﬁne the W a dge class of a set L : Deﬁnition 16. Let L be a subset of X ω . The W adge class of L is : [ L ] = { L ′ | L ′ ⊆ Y ω for a ﬁnite alpha bet Y and L ′ ≤ W L } . Recall that each Bor el class Σ 0 α and Π 0 α is a W adge class . A set L ⊆ X ω is a Σ 0 α (respectively Π 0 α )- complete set if f for any set L ′ ⊆ Y ω , L ′ is in Σ 0 α (respectively Π 0 α ) iff L ′ ≤ W L . It follows from the study o f the W adge hier archy that a set L ⊆ X ω is a Σ 0 α (respectively , Π 0 α )- complete set if f it is in Σ 0 α but not in Π 0 α (respectively , in Π 0 α but not in Σ 0 α ). There is a clo se relationship between W a dge reduc ibility and games which we now introdu ce. Deﬁnition 17. Let L ⊆ X ω and L ′ ⊆ Y ω . The W adge g ame W ( L, L ′ ) is a g ame with perfect information between two p layers, p layer 1 who is in char ge of L and player 2 who is in char ge of L ′ . Player 1 ﬁrst writes a le tter a 1 ∈ X , th en p layer 2 writes a letter b 1 ∈ Y , then player 1 writes a letter a 2 ∈ X , an d so on. The two p layers alternatively write letters a n of X for player 1 a nd b n of Y for pla yer 2. After ω steps, playe r 1 ha s written an ω -wor d a ∈ X ω and p layer 2 ha s written a n ω -word b ∈ Y ω . Player 2 is allowed to skip, even inﬁnitely often, p r ovided he really writes an ω -wo r d in ω steps. Player 2 wins the play iff [ a ∈ L ↔ b ∈ L ′ ], i.e. if f : [( a ∈ L a nd b ∈ L ′ ) or ( a / ∈ L and b / ∈ L ′ and b is inﬁnite )]. Recall that a strate gy for player 1 is a fu nction σ : ( Y ∪ { s } ) ⋆ → X . And a strate gy for player 2 is a function f : X + → Y ∪ { s } . σ is a winning stategy f or playe r 1 iff he always wins a play whe n he u ses the strategy σ , i.e. when the n th letter h e writes is giv en by a n = σ ( b 1 . . . b n − 1 ) , where b i is the letter written by player 2 at step i and b i = s if playe r 2 skips at step i . A winning strategy for player 2 is deﬁned in a similar manner . Martin’ s The orem states th at every Gale- Ste wart Game G ( B ) , with B a Bore l set, is determined , i.e . tha t one o f th e two play ers h as a winning strategy in the game G ( B ) , see [50]. This implies the following determinacy r esult : Theorem 18 ( W adge). Let L ⊆ X ω and L ′ ⊆ Y ω be two Bo r el sets, wher e X and Y ar e ﬁnite a lphabets. Then the W adge g ame W ( L, L ′ ) is determined : o ne o f the two players has a winning str ate gy . And L ≤ W L ′ iff player 2 h as a winning s trate gy in the game W ( L, L ′ ) . Theorem 19 ( W adge). Up to the comple ment and ≡ W , the class of Bo r el subsets of X ω , for a ﬁ nite alphab et X , is a well or der ed hierar chy . Ther e is an ordinal | W H | , called the length of the hierar chy , and a map d 0 W fr om W H onto | W H | − { 0 } , such that for all L, L ′ ⊆ X ω : d 0 W L < d 0 W L ′ ↔ L < W L ′ and d 0 W L = d 0 W L ′ ↔ [ L ≡ W L ′ or L ≡ W L ′− ] . The W ad ge h ierarchy o f Bore l sets o f ﬁ nite rank has len gth 1 ε 0 where 1 ε 0 is th e limit of the ordina ls α n deﬁned b y α 1 = ω 1 and α n +1 = ω α n 1 for n a n on negative inte ger, ω 1 being th e ﬁrst non cou ntable or dinal. Then 1 ε 0 is th e ﬁrst ﬁxed point o f the ordinal exponentiation o f base ω 1 . The length of the W ad ge hierar chy o f Borel sets in ∆ 0 ω = Σ 0 ω ∩ Π 0 ω is the ω th 1 ﬁxed po int o f th e ordinal exponentiation of base ω 1 , which is a much larger or dinal. The length o f the whole W adge hierarc hy of Bore l sets is a huge ordinal, with re gard to the ω th 1 ﬁxed point o f the ordinal exponentiation of base ω 1 . It is described in [83,23] by the use of the V eblen functions. 4 T opologi cal complexity of context-fr ee ω -languages W e r ecall ﬁrst results ab out the topolo gical c omplexity o f regular ω -lang uages. T opo - logical properties of regular ω -lan guages wer e ﬁrst studied by L. H. Landwebe r in [52] where he charac terized r egular ω -lan guages in a given Borel class. It tur ned out th at a regular ω -lang uage is a Π 0 2 -set iff it is accepted by a deter ministic B ¨ uchi automaton. On the other han d M c Naughton’ s Th eorem implies that regular ω -lang uages, accep ted by dete rministic Muller au tomata, are b oolean combin ations of regular ω -langua ges accepted by deterministic B ¨ uch i automata. Thus they are boolean combination s of Π 0 2 - sets hence ∆ 0 3 -sets. Moreover Landweb er proved that one can effecti vely de termine the exact le vel of a giv en regular ω -lan guage with regard to the Borel h ierarchy . A great impr ovement of these results was ob tained by W agner who determined in an effecti ve way , using the notion s of chains and superchain s, the W adge hier archy of the class RE G ω , [8 4]. This hierar chy has length ω ω and is n ow called the W ag ner hierarchy , [6 9,71,72,70,78]. W ilke and Y oo proved in [86] that o ne can compu te in polyno mial tim e the W adg e degree of a regular ω -langu age. Later Carton and Perrin gave a p resentation of th e W ag ner hierarchy u sing a lgebraic n otions o f ω -semigr oups, [14,13,67]. This work was completed by Duparc and Riss in [27]. Context-free ω -lang uages beyond the class ∆ 0 3 have been constructed for the ﬁrst time in [32]. Th e co nstruction used a n o peration o f exponentiation of sets of ﬁn ite or in - ﬁnite word s introdu ced by Duparc in his stud y of the W adg e hierar chy [23]. W e are going now to recall these co nstructions althoug h so me stronger resu lts on the top o- logical com plexity of context-f ree ω -lang uages were ob tained later in [38,41] by oth er methods. Howe ver the methods of [32] using Dup arc’ s oper ation of exponentiation are also interesting and it g av e other results on am biguity and on ω -powers of context-free languag es we can not (yet ?) get by other method s, see Sections 6 and 7 below . W adge gave a description of the W adge hierarchy of Borel sets in [83 ]. Duparc recently got a new pro of of W adge’ s resu lts an d g av e in [22,23] a n ormal f orm o f Bor el sets in the class ∆ 0 ω , i.e. an in ductive constructio n o f a Borel set of e very gi ven degree smaller than the ω th 1 ﬁxed point of the ordinal expon entiation of base ω 1 . The construction relies on set theo retic op erations wh ich are the counter part of arithmetical o perations over ordinals needed to compute the W adge degrees. Actually Duparc studied the W adge hierarch y via the study o f the conciliating hierarch y . Conciliating sets ar e sets of ﬁnite or inﬁnite word s over an alphabe t X , i.e. subsets of X ⋆ ∪ X ω = X ≤ ω . It turned out that the conciliating hierarch y is isomorph ic to the W adge hierarchy of non-self-dua l Borel sets, via the correspon dence A → A d we recall now: For a word x ∈ ( X ∪ { d } ) ≤ ω we d enote by x ( /d ) th e seq uence o btained fro m x by removing ev ery occu rrence of the le tter d . T hen fo r A ⊆ X ≤ ω and d a letter n ot in X , A d is the ω -lang uage ov er X ∪ { d } which is deﬁned by : A d = { x ∈ ( X ∪ { d } ) ω | x ( /d ) ∈ A } . W e are g oing no w to introduce the operation of exponentiation of conciliating sets. Deﬁnition 20 (Duparc [2 3]). Let X b e a ﬁnite a lphab et, և / ∈ X , a nd let x b e a ﬁnite or inﬁnite wor d over the alphabet Y = X ∪ { և } . Then x և is indu ctively deﬁned by: λ և = λ , and for a ﬁnite wor d u ∈ ( X ∪ { և } ) ⋆ : ( u.a ) և = u և .a , if a ∈ X , ( u. և ) և = u և (1) .u և (2) . . . u և ( | u և | − 1) if | u և | > 0 , ( u. և ) և = λ if | u և | = 0 , and for u in ﬁnite: ( u ) և = lim n ∈ ω ( u [ n ]) և , wher e, g iven β n and v in X ⋆ , v ⊑ lim n ∈ ω β n ↔ ∃ n ∀ p ≥ n β p [ | v | ] = v . (The ﬁnite or inﬁnite wor d lim n ∈ ω β n is determined by the set of its (ﬁnite) pr eﬁxes). Remark 21. F or x ∈ Y ≤ ω , x և denotes the string x , once every և o ccuring in x has been “e valuated” to the ba ck space operation, pr oceeding fr o m left to right inside x . In other words x և = x fr om which every interval of the form “ a և ” ( a ∈ X ) is remo ved. For example if u = ( a և ) n , fo r n an integer ≥ 1 , or u = ( a և ) ω , or u = ( a ևև ) ω , then ( u ) և = λ . If u = ( ab և ) ω then ( u ) և = a ω and if u = b b ( և a ) ω then ( u ) և = b . Let us notice that in Deﬁnition 20 the limit is not deﬁned in the usual way: for example if u = bb ( և a ) ω the ﬁnite word u [ n ] և is alternatively equal to b or to ba : mor e precisely u [2 n + 1] և = b a nd u [2 n + 2] և = ba for every integer n ≥ 1 (it holds a lso tha t u [1] և = b and u [2] և = bb ). Thus De ﬁnition 20 implies th at lim n ∈ ω ( u [ n ]) և = b so u և = b . W e can now deﬁne the operation A → A ∼ of exponentiation of concilia ting s ets : Deﬁnition 22 (Duparc [23]). F or A ⊆ X ≤ ω and և / ∈ X , let A ∼ = d f { x ∈ ( X ∪ { և } ) ≤ ω | x և ∈ A } . The opera tion ∼ is mo notone with regard to the W ad ge order ing and pro duces some sets of higher complexity . Theorem 23 ( Duparc [23] ). Let A ⊆ X ≤ ω and n ≥ 1 . if A d ⊆ ( X ∪ { d } ) ω is a Σ 0 n -complete ( r espectively , Π 0 n -complete) set, then ( A ∼ ) d is a Σ 0 n +1 -complete ( r e- spectively , Π 0 n +1 -complete) set. It was p roved in [32] that the class of co ntext-free inﬁnitary languag es (which are u nions of a context-fre e ﬁnitary language and of a context-free ω -langua ge) is closed u nder th e operation A → A ∼ . On the other hand A → A d is an operatio n fro m th e class of context-free inﬁnitary lan guages into the class of context-free ω -lang uages. Th is im- plies that, fo r each integer n ≥ 1 , ther e exist some context-free ω -lang uages which are Σ 0 n -complete and some others which are Π 0 n -complete. Theorem 24 ( [32]). F o r each non ne gative integer n ≥ 1 , there exist Σ 0 n -complete context-fr ee ω -lang uages A n and Π 0 n -complete context-fr ee ω -la nguages B n . Proof. For n = 1 conside r the Σ 0 1 -complete regular ω -languag e A 1 = { α ∈ { 0 , 1 } ω | ∃ i α ( i ) = 1 } and the Π 0 1 -complete regular ω -languag e B 1 = { α ∈ { 0 , 1 } ω | ∀ i α ( i ) = 0 } . These languag es are context-free ω -lang uages because RE G ω ⊆ C F L ω . Now consider the Σ 0 2 -complete regular ω -languag e A 2 = { α ∈ { 0 , 1 } ω | ∃ <ω i α ( i ) = 1 } and the Π 0 2 -complete regular ω -languag e B 2 = { α ∈ { 0 , 1 } ω | ∃ ω i α ( i ) = 0 } , where ∃ <ω i m eans: ” there exist only ﬁnitely many i such that . . . ” , an d ∃ ω i means: ” there exist i nﬁnitely many i such that . . . ”. A 2 and B 2 are context-free ω -langu ages because they are r egular ω -lang uages. T o ob tain co ntext-free ω -langua ges of greater Borel ranks, consider now O 1 (respec- ti vely , C 1 ) subsets of { 0 , 1 } ≤ ω such that ( O 1 ) d (respectively , ( C 1 ) d ) are Σ 0 1 -complete ( respectiv ely Π 0 1 -complete ) . For example O 1 = { x ∈ { 0 , 1 } ≤ ω | ∃ i x ( i ) = 1 } and C 1 = { λ } . W e can apply n ≥ 1 times the operation of expone ntiation of sets. More precisely , we d eﬁne, for a set A ⊆ X ≤ ω : A ∼ . 0 = A A ∼ . 1 = A ∼ and A ∼ . ( n +1) = ( A ∼ .n ) ∼ . Now a pply n times (for an in teger n ≥ 1 ) the operation ∼ (with different new letter s և 1 , և 2 , և 3 , . . . , և n ) to O 1 and C 1 . By Theor em 23, it holds that for an integer n ≥ 1 : ( O ∼ .n 1 ) d is a Σ 0 n +1 -complete subset of { 0 , 1 , և 1 , . . . , և n , d } ω . ( C ∼ .n 1 ) d is a Π 0 n +1 -complete subset of { 0 , 1 , և 1 , . . . , և n , d } ω . And it is easy to see tha t O 1 and C 1 are in the form E ∪ F where E is a ﬁnitary context- free lang uage an d F is a context-free ω -languag e. Then the ω -lang uages ( O ∼ .n 1 ) d and ( C ∼ .n 1 ) d are c ontext-free. Hence the class C F L ω exhausts the ﬁnite ranks of the Borel hierarchy : we obtain the context-free ω -lang uages A n = ( O ∼ . ( n − 1) 1 ) d and B n = ( C ∼ . ( n − 1) 1 ) d , for n ≥ 3 .  This ga ve a partial answer to q uestions of Tho mas and Lesco w [5 5] about the hierarchy of context-free ω -langu ages. A natural question now arose: Do the decidability results of [52] extend to context-free ω -lang uages? Unfo rtunately the a nswer is no. Cohe n an d Gold proved that one cannot decide whethe r a given context-free ω -lang uage is in the class Π 0 1 , Σ 0 1 , or Π 0 2 , [19]. This result was ﬁrst extended to all classes Σ 0 n and Π 0 n , for n an integer ≥ 1 , using the undecid ability of the Post Correspon dence Problem, [32]. Later , the co ding of an inﬁnite number of er asers և n , n ≥ 1 , and an iteration of th e operation of expon entiation were used to prove that there exist some context-free ω - languag es which are Borel of inﬁnite rank, [36]. Using th e corresp onden ces b etween the operation of expon entiation of sets an d the o r- dinal expo nentiation of b ase ω 1 , an d betwe en the W adge’ s operation o f sum of sets, [83,23], and th e ordin al sum, it was proved in [33] that the leng th of the W adge hie rar- chy of the class C F L ω is at least ε 0 , the ﬁrst ﬁxed p oint of the o rdinal expo nentiation of base ω . Next were con structed some ∆ 0 ω context-free ω -lan guages in ε ω W adge d e- grees, where ε ω is the ω th ﬁxed point of the ordinal exponentiation of base ω , and also some Σ 0 ω -complete context-free ω -lan guages, [31,39]. Notice that the W a dge hierarchy of non-d eterministic context-free ω -languag es is not effecti ve, [ 33]. The qu estion of the existence of non- Borel c ontext-free ω -langu ages was solved by Finkel and Ressayre. Using a co ding of inﬁnite b inary trees labele d in a ﬁnite alphabet X , it was proved that ther e exist so me n on-Borel, an d even Σ 1 1 -complete, co ntext-free ω -lang uages, and that one cannot d ecide whe ther a giv en context-fre e ω -langu age is a Borel set, [35]. Amazingly there is a simple ﬁnitary lang uage V accep ted by a 1 - counter automaton such that V ω is Σ 1 1 -complete; we sha ll recall it in Section 7 below on ω -p owers. But a comp lete and very su rprising result was obtained in [ 38,41], which extend ed previous results. A simu lation of m ulticoun ter au tomata b y 1 -cou nter automata was used in [ 38,41]. W e ﬁrstly recall now the deﬁnition o f th ese automata, in order to sketch the construction s in volved in these simulations. Deﬁnition 25. Let k be an integer ≥ 1 . A k -counte r machine ( k -CM) is a 4-tuple M = ( K, X , ∆, q 0 ) , wher e K is a ﬁ nite set of states, X is a ﬁnite input a lphab et, q 0 ∈ K is th e initial state, and ∆ ⊆ K × ( X ∪ { λ } ) × { 0 , 1 } k × K × { 0 , 1 , − 1 } k is the transition r elation. The k -co unter machine M is s aid to b e r eal time iff: ∆ ⊆ K × X × { 0 , 1 } k × K × { 0 , 1 , − 1 } k , i.e. if f ther e ar e not any λ -transitions. If the machine M is in state q a nd c i ∈ N is the co ntent of the i th counter C i then the conﬁg uration (o r global state) of M is the ( k + 1) -tu ple ( q, c 1 , . . . , c k ) . F o r a ∈ X ∪ { λ } , q , q ′ ∈ K a nd ( c 1 , . . . , c k ) ∈ N k such that c j = 0 for j ∈ E ⊆ { 1 , . . . , k } and c j > 0 for j / ∈ E , if ( q , a, i 1 , . . . , i k , q ′ , j 1 , . . . , j k ) ∈ ∆ wher e i j = 0 for j ∈ E and i j = 1 for j / ∈ E , then we write: a : ( q , c 1 , . . . , c k ) 7→ M ( q ′ , c 1 + j 1 , . . . , c k + j k ) Thus we see that the transition r elatio n must satisfy: if ( q , a, i 1 , . . . , i k , q ′ , j 1 , . . . , j k ) ∈ ∆ and i m = 0 for some m ∈ { 1 , . . . , k } , then j m = 0 or j m = 1 (but j m canno t be equal to − 1 ). Let σ = a 1 a 2 . . . a n . . . be an ω - wor d over X . An ω - sequence of con ﬁgurations r = ( q i , c i 1 , . . . c i k ) i ≥ 1 is called a run of M on σ , starting in conﬁg uration ( p, c 1 , . . . , c k ) , iff: (1) ( q 1 , c 1 1 , . . . c 1 k ) = ( p, c 1 , . . . , c k ) (2) for ea ch i ≥ 1 , ther e exists b i ∈ X ∪ { λ } such tha t b i : ( q i , c i 1 , . . . c i k ) 7→ M ( q i +1 , c i +1 1 , . . . c i +1 k ) such that either a 1 a 2 . . . a n . . . = b 1 b 2 . . . b n . . . or b 1 b 2 . . . b n . . . is a ﬁnite pr eﬁx of a 1 a 2 . . . a n . . . The run r is said to b e complete when a 1 a 2 . . . a n . . . = b 1 b 2 . . . b n . . . F o r every such run, In( r ) is the set of all states en ter ed inﬁnitely often during run r . A co mplete run r of M on σ , starting in con ﬁguration ( q 0 , 0 , . . . , 0) , will be simply called “a run of M on σ ”. Deﬁnition 26. A B ¨ uchi k - counter automato n is a 5-tup le M = ( K, X , ∆, q 0 , F ) , where M ′ = ( K, X , ∆, q 0 ) is a k - counter machine a nd F ⊆ K is th e set of acceptin g states. The ω -la nguage accepted by M is L ( M ) = { σ ∈ X ω | ther e exists a run r of M on σ such that In( r ) ∩ F 6 = ∅} The notion of Muller k -coun ter automaton is deﬁned in a similar way . One can see that an ω -lang uage is accepted by a (real time) B ¨ uchi k -co unter automaton if f it is accepted by a (real time) Muller k -coun ter autom aton [29]. Notice that this result is no longer true in the deterministic case. W e d enote BC ( k ) (r espectiv ely , r - BC ( k ) ) the class of B ¨ uchi k - counter au tomata (re- spectiv ely , of real time B ¨ uch i k -coun ter automata. W e den ote B C L ( k ) ω (respectively , r - BCL ( k ) ω ) the class of ω - languag es accepted by B ¨ uchi k -co unter automata (respecti vely , by real time B ¨ uchi k -co unter automata). Remark that 1 -coun ter au tomata introdu ced above are equ iv a lent to pushdown au tomata whose stack alp habet is in the form { Z 0 , A } where Z 0 is th e bo ttom symbol wh ich al- ways re mains at the bo ttom of the stack a nd app ears o nly ther e an d A is an other stack symbol. The pu shdown stack may be seen like a counter whose content is the integer N if the stack content is the word A N .Z 0 . In the m odel introduced here the coun ter value can not b e increased by more than 1 dur - ing a single transition. Ho we ver th is does n ot change the class of ω -lang uages accepted by such automata. So the class BCL (1) ω is equal to the class 1 - ICL ω , introduce d in [33], and it is a strict subclass of the class CFL ω of context-free ω -langu ages acc epted by B ¨ uchi pushd own auto mata. W e state now the surprising result proved in [41], using mu lticounter-automa ta. Theorem 27 ( [41]). The W adge hierar chy of the class r - BCL (1) ω , hence also of the class CFL ω , or of every class C such that r - BCL (1) ω ⊆ C ⊆ Σ 1 1 , is the W a dge hierar chy of the class Σ 1 1 of ω -lan guages accepted b y T uring machines with a B ¨ uchi acceptan ce condition . W e n ow sketch the pro of of this result. It is well known that ev ery T ur ing machine can be simulated by a (non real time) 2 -coun ter au tomaton , see [49]. Thu s the W ad ge hierarchy of the class BCL (2) ω is also the W ad ge hierar chy of the class of ω -lan guages accepted by B ¨ uchi T uring machines. One can then ﬁnd, from an ω - languag e L ⊆ X ω in BCL (2) ω , an other ω -langu age θ S ( L ) which will be of the same topological complexity but accepted by a real-time 8-coun ter B ¨ uchi au tomaton. The idea is to add ﬁrstly a storag e type called a qu eue to a 2-coun ter B ¨ uch i automaton in ord er to r ead ω -words in r eal-time. Th en the qu eue can be simu lated b y two p ushdown stacks o r by four cou nters. This simulation is not d one in real-time but a crucial fact is that one can bound the number of transitions needed to simulate the queue. This allows to pad the strings in L with eno ugh extra letters so that the n ew words will be read in rea l-time b y a 8-cou nter B ¨ uchi automaton . The p adding is obtained via the function θ S which we deﬁne now . Let X be an alph abet having at least two letters, E be a new letter not in X , S be an integer ≥ 1 , and θ S : X ω → ( X ∪ { E } ) ω be the func tion deﬁned, for all x ∈ X ω , by: θ S ( x ) = x (1) .E S .x (2) .E S 2 .x (3) .E S 3 .x (4) . . . x ( n ) .E S n .x ( n + 1) .E S n +1 . . . It turns out that if L ⊆ X ω is in BCL (2) ω then there exists an integer S ≥ 1 such that θ S ( L ) is in the class r - BCL (8) ω , and, except fo r som e spec ial few cases, θ S ( L ) ≡ W L . The next step is to simulate a real-time 8- counter B ¨ u chi au tomaton, using only a r ea l- time 1-co unter B ¨ uchi automato n. Consider the produ ct of the eight ﬁrst prime numbers: K = 2 × 3 × 5 × 7 × 11 × 1 3 × 17 × 1 9 = 9699 6 90 Then an ω -word x ∈ X ω can be code d by the ω -word h ( x ) = A. 0 K .x (1) .B . 0 K 2 .A. 0 K 2 .x (2) .B . 0 K 3 .A. 0 K 3 .x (3) .B . . . B . 0 K n .A. 0 K n .x ( n ) .B . . . where A , B and 0 are new letters not in X . The mapping h : X ω → ( X ∪ { A, B , 0 } ) ω is co ntinuo us. It is easy to see that the ω -langu age h ( X ω ) − is an open sub set of ( X ∪ { A, B , 0 } ) ω and that it is in the class r - BCL (1) ω . If L ( A ) ⊆ X ω is ac cepted b y a real time 8 -coun ter B ¨ uchi automaton A , then one can construct e ffecti vely from A a 1 -counte r B ¨ u chi automaton B , rea ding word s over the alphabet X ∪ { A, B , 0 } , such that L ( A )= h − 1 ( L ( B )) , i.e. ∀ x ∈ X ω h ( x ) ∈ L ( B ) ← → x ∈ L ( A ) In fact, the simulation, d uring the read ing o f h ( x ) by the 1 -counter B ¨ u chi automaton B , of the b ehaviour of the rea l time 8 -cou nter B ¨ uchi automaton A reading x , can be achieved, using the coding of the content ( c 1 , c 2 , . . . , c 8 ) of eight counters by the prod- uct 2 c 1 × 3 c 2 × . . . × (1 7 ) c 7 × (19) c 8 , and the special shape of ω -word s in h ( X ω ) which allows the propag ation of th e value of the counte rs of A . A crucial fact her e is that h ( X ω ) − is in the class r - BCL (1) ω . Thus the ω -langu age h ( L ( A )) ∪ h ( X ω ) − = L ( B ) ∪ h ( X ω ) − is in the class BCL (1) ω and it has th e same top ological complexity as the ω -langu age L ( A ) , (except the special fe w cases where d W ( L ( A )) ≤ ω ). One can see, from the construction o f B , that at most ( K − 1 ) consecutive λ -transitions can occur du ring th e read ing of an ω -word x b y B . It is then easy to see tha t the ω - languag e φ ( h ( L ( A )) ∪ h ( X ω ) − ) is an ω -langu age in the class r - BCL (1) ω which has the same topolog ical com plexity as the ω -lang uage L ( A ) , where φ is the mapp ing from ( X ∪ { A, B , 0 } ) ω into ( X ∪ { A, B , F, 0 } ) ω , with F a new letter , which is deﬁned by: φ ( x ) = F K − 1 .x (1) .F K − 1 .x (2) .F K − 1 .x (3) . . . F K − 1 .x ( n ) .F K − 1 .x ( n +1) .F K − 1 . . . Altogether these construc tions are used in [41] to prove T heorem 2 7. As the W adge hierarchy is a reﬁn ement of the Bore l hiera rchy and, for any cou ntable o rdinal α , Σ 0 α - complete sets (respectively , Π 0 α -complete sets) form a single W a dge degree, this im plies also the following result. Theorem 28. Let C be a class of ω -lan guages su ch that: r - BCL (1) ω ⊆ C ⊆ Σ 1 1 . (a) The Bor el hierar chy of the class C is equal to the Bor el hierar chy of the class Σ 1 1 . (b) γ 1 2 = S up { α | ∃ L ∈ C such that L is a Bor el s et of rank α } . (c) F or every non null o r dinal α < ω CK 1 , th er e exists some Σ 0 α -complete and some Π 0 α -complete ω -lan guages in the class C . Notice that similar metho ds have next be used to ge t ano ther surprising resu lt: the W adge h ierarchy , he nce a lso the Bo rel hierarc hy , of in ﬁnitary ra tional relations accep ted by 2 -ta pe B ¨ uch i auto mata is eq ual to th e W adge hierarchy of the class r - BCL (1) ω or of the class Σ 1 1 , [42,43]. 5 T opologi cal complexity of deterministic context-fr ee ω -languages W e have seen in the previous sectio n that all non-d eterministic ﬁnite machines accept ω -lang uages of the same topological com plexity , as soon as they ca n simulate a r eal time 1 -counte r automaton . This r esult is still tru e in the deterministic case if we co nsider only the Borel hier- archy . Recall that regular ω -langu ages accep ted by B ¨ u chi automata are Π 0 2 -sets and ω -lang uages accepted by M uller a utomata ar e bo olean comb inations of Π 0 2 -sets hen ce ∆ 0 3 -sets. Engelfr iet an d Hoogeb oom proved th at this re sult ho lds also f or all ω -la nguage s accepted by dete rministic X -autom ata, i. e. autom ata equipped with a storage type X , including the cases o f k -counte r automata, pushd own automata, Petri n ets, T uring m a- chines. In par ticular, ω - languag es a ccepted by deterministic B ¨ uch i Turing mach ines are Π 0 2 -sets and ω -lang uages accepted by deterministic Muller Turing machines are ∆ 0 3 - sets. It turned out that this is no longer true if we consider the much ﬁner W adg e hierarchy to measure the complexity of ω -lang uages. The W adg e hierarchy is suitable to distinguish the accepting power of deterministic ﬁnite machines reading inﬁnite w ords. Recall that the W adge hierarchy of regular ω -lang uages, now called the W agner h ierarchy , has been effecti vely dete rmined by W agner; it has length ω ω [84,69,70]. Its extension to deterministic co ntext-free ω - languag es has be en d etermined by Du parc, its leng th is ω ( ω 2 ) [26,24]. T o determine the W adge h ierarchy of the class D C F L ω , Duparc ﬁr st d eﬁned o perations on DMPD A which co rrespon d to or dinal op erations of sum, m ultiplication by ω , and mu ltiplication by ω 1 , over W ad ge d egrees. In this way ar e constructed some DMPD A accepting ω -lan guage s of every W adge degree in the form : d 0 W ( A ) = ω n j 1 .δ j + ω n j − 1 1 .δ j − 1 + . . . + ω n 1 1 .δ 1 where j > 0 is an integer , n j > n j − 1 > . . . > n 1 are in tegers ≥ 0 , an d δ j , δ j − 1 , . . . , δ 1 are non null ordinals < ω ω . On the other hand it is kn own that the W adge degree α of a boo lean com bination of Π 0 2 -sets is smaller than the ordinal ω ω 1 thus it has a Cantor normal form : α = ω n j 1 .δ j + ω n j − 1 1 .δ j − 1 + . . . + ω n 1 1 .δ 1 where j > 0 is an integer , n j > n j − 1 > . . . > n 1 are in tegers ≥ 0 , an d δ j , δ j − 1 , . . . , δ 1 are non null ordinals < ω 1 , i.e. non null countable ordinals. In a second step it is prov ed in [24], using inﬁnite multi-play er gam es, that if such an ordin al α is the W adge de- gree of a determ inistic context-free ω -langua ge, then all the o rdinals δ j , δ j − 1 , . . . , δ 1 appearin g in its Cantor normal form are smaller than the o rdinal < ω ω . Thus the W adge hierarchy of the class DC F L ω is completely determine d. Theorem 29 ( Duparc [24]). The W adge degr ee s of deterministic context-fr e e ω -la nguages ar e e xactly the or dinals in the form : α = ω n j 1 .δ j + ω n j − 1 1 .δ j − 1 + . . . + ω n 1 1 .δ 1 wher e j > 0 is an integ er , n j > n j − 1 > . . . > n 1 ar e integ ers ≥ 0 , and δ j , δ j − 1 , . . . , δ 1 ar e non null or dinals < ω ω . The length of the W adge hierar chy of the class DC F L ω is the or dinal ( ω ω ) ω = ω ( ω 2 ) . Notice that theW adge hierarch y of D C F L ω is n ot determin ed in an effecti ve way in [24]. Th e qu estion o f th e de cidability of pro blems like: “g iv en two DMPDA A an d B , does L ( A ) ≤ W L ( B ) h old ?” or “given a DMPD A A can we com pute d 0 W ( L ( A )) ?” naturally arises. Cohen and Go ld proved that one can decide whether an effecti vely gi ven ω -langu age in D C F L ω is an o pen or a closed set [ 19]. Linna characterized the ω -lang uages accepted by DBPD A as the Π 0 2 -sets in D C F L ω and proved in [58] that one can d ecide wheth er an effecti vely giv en ω -lang uage accepted by a DMPD A is a Π 0 2 -set or a Σ 0 2 -set. Using a recent result of W aluk iewicz on inﬁnite gam es played o n push down grap hs, [85], these decidability results we re extend ed in [ 32] whe re it was proved that one c an decide whether a deterministic context-free ω -lang uage accepted by a given DMPDA is in a g iv en Bor el class Σ 0 1 , Π 0 1 , Σ 0 2 , or Π 0 2 or e ven in th e wadge c lass [ L ] given by any regular ω -langu age L . An effecti ve extension of the W agner hierarchy to ω -lang uages accepted by Muller deterministic r eal time blind (i. e. witho ut zero- test) 1 -cou nter automata has been de- termined in [30]. Recall th at blind 1 -cou nter au tomata form a subclass of 1 - counter automata hence also of pushdown auto mata. A blind 1 -coun ter Muller automa ton is just a Muller pushdown autom aton M = ( K, X , Γ , δ, q 0 , Z 0 , F ) such that Γ = { Z 0 , I } where Z 0 is the b ottom symbo l an d always remains at the b ottom of the sto re. More- over every tran sition which is enabled at zero le vel is also enabled at no n zero le vel, i.e. if δ ( q , a, Z 0 ) = ( p, I n Z 0 ) , fo r some p, q ∈ K , a ∈ X and n ≥ 0 , then δ ( q , a, I ) = ( p, I n +1 ) . But the con verse may not be true, i.e. some transition may be enabled at non zero le vel but not at zero level. Notice that blind 1 -cou nter au tomata ar e sometimes called partially blind 1 -coun ter automata as in [47]. The W ad ge hierarchy of blind co unter ω - languag es, accepted by deterministic Mu ller real tim e blind 1 -coun ter automata (MBCA), is studied in [30] in a similar way as W agner studied the W adge hierarch y of regular ω -langu ages in [84]. Chains an d su- perchain s for MBCA ar e deﬁned as W agner did fo r Muller auto mata. The essential difference b etween the two hierarch ies relies on the existence of superch ains o f trans- ﬁnite length α < ω 2 for MBCA whe n in the case of Muller au tomata th e superchains have only ﬁnite length s. The hier archy of ω -lang uages accepted by MBCA is effecti ve and leads to effecti ve winning strategies in W adge games between two play ers in ch arge of ω -langu ages accepted b y M BCA. Co ncerning th e len gth of the W a dge hierarchy of MBCA the following result is proved : Theorem 30 ( Finkel [30]). (a) The length of the W adge hierar chy of blind counter ω -lang uages in ∆ 0 2 is ω 2 . (b) The length of the W adge hierar chy of blind counter ω -la nguages is th e or dinal ω ω (hence it is equal to the length of the W agner hierar chy). Notice that the length of the W adge h ierarchy of blind counter ω -lang uages is equal to the length of the W agner hierarc hy although it is actually a strict extension of the W ag- ner hierarch y , as sh own alread y in item (a) of the above theo rem. The W adge degrees of blind counter ω -lang uages are the ordinals in the form : α = ω n j 1 .δ j + ω n j − 1 1 .δ j − 1 + . . . + ω n 1 1 .δ 1 where j > 0 is an integer , n j > n j − 1 > . . . > n 1 are in tegers ≥ 0 , an d δ j , δ j − 1 , . . . , δ 1 are non null ordinals < ω 2 . Recall that in the case of Muller autom ata, the ordinals δ j , δ j − 1 , . . . , δ 1 are non-n egati ve in tegers, i.e. non null ordinals < ω . Notice that Seliv anov h as recently determine d the W ad ge hierarch y of ω -lan guages ac- cepted b y d eterministic T u ring m achines; its length is ( ω CK 1 ) ω [72,71]. The ω -lan guages accepted by deterministic Mu ller Turing mach ines or equivalently which ar e b oolean combinatio ns of arithmetical Π 0 2 -sets have W adge degrees in the form : α = ω n j 1 .δ j + ω n j − 1 1 .δ j − 1 + . . . + ω n 1 1 .δ 1 where j > 0 is an integer , n j > n j − 1 > . . . > n 1 are in tegers ≥ 0 , an d δ j , δ j − 1 , . . . , δ 1 are non null ordinals < ω CK 1 . 6 T opology and ambiguity in context-fr ee ω -languages The notions of ambiguity and of degrees of ambiguity are well known an d important in the stud y of co ntext-free lang uages. T hese notions h av e bee n extend ed to context-f ree ω -lang uages accepted by B ¨ u chi or Muller pushdown automata in [34]. Notice that it is p roved in [ 34] th at these notions ar e in depend ent of the B ¨ uchi or Muller a cceptance condition . So in the sequel we shall only consider the B ¨ uchi acceptan ce condition. W e now ﬁrstly intro duce a slight mod iﬁcation in the deﬁnition of a run o f a B ¨ uchi pushdown automaton, which will be used in this section. Deﬁnition 31. Let A = ( K , X , Γ , δ, q 0 , Z 0 , F ) be a B ¨ uchi pushdown automaton. Let σ = a 1 a 2 . . . a n . . . be a n ω -wor d over X . A run of A on σ is a n inﬁn ite seque nce r = ( q i , γ i , ε i ) i ≥ 1 wher e ( q i , γ i ) i ≥ 1 is an inﬁn ite sequence of conﬁg urations of A and, for all i ≥ 1 , ε i ∈ { 0 , 1 } a nd: 1. ( q 1 , γ 1 ) = ( q 0 , Z 0 ) 2. for each i ≥ 1 , ther e e xists b i ∈ X ∪ { λ } satisfying b i : ( q i , γ i ) 7→ A ( q i +1 , γ i +1 ) and ( ε i = 0 iff b i = λ ) and such that a 1 a 2 . . . a n . . . = b 1 b 2 . . . b n . . . As before the ω -lan guage accepted by A is L ( A ) = { σ ∈ X ω | th er e e xists a run r of A on σ such that I n ( r ) ∩ F 6 = ∅} Notice that the num bers ε i ∈ { 0 , 1 } are introd uced in th e above d eﬁnition in order to distinguish run s of a BPDA which go th rough the same inﬁnite sequence of co nﬁgura- tions but for which λ -transition s do not occur at the same steps of the computations. As usual the card inal o f ω is den oted ℵ 0 and the car dinal of the continu um is denote d 2 ℵ 0 . Th e latter is also the card inal of the set of real number s or of the set X ω for every ﬁnite alphabet X having at l east two letters. W e are now ready to deﬁne degrees of a mbiguity fo r BPD A an d for context-free ω - languag es. Deﬁnition 32. Let A be a BPDA r eading inﬁn ite wor ds over the a lphabet X . F or x ∈ X ω let α A ( x ) b e the car dinal of the set of accepting runs of A on x . Lemma 33 ([34]). Let A be a BPD A r eading inﬁnite wor ds over the alphabet X . Then for all x ∈ X ω it holds that α A ( x ) ∈ N ∪ {ℵ 0 , 2 ℵ 0 } . Deﬁnition 34. Let A be a BPD A r ead ing inﬁnite wor ds over the alph abet X . (a) If sup { α A ( x ) | x ∈ X ω } ∈ N ∪ { 2 ℵ 0 } , then α A = sup { α A ( x ) | x ∈ X ω } . (b) If sup { α A ( x ) | x ∈ X ω } = ℵ 0 and th er e is no wor d x ∈ X ω such that α A ( x ) = ℵ 0 , then α A = ℵ − 0 . ( ℵ − 0 does not repr esent a car dinal but is a new symb ol th at is intr oduced her e to conveniently spea k of this situation). (c) If sup { α A ( x ) | x ∈ X ω } = ℵ 0 and th er e exists (at least) one word x ∈ X ω such that α A ( x ) = ℵ 0 , then α A = ℵ 0 Notice that for a BPD A A , α A = 0 iff A does not accep t an y ω -word. W e shall co nsider below that N ∪ {ℵ − 0 , ℵ 0 , 2 ℵ 0 } is linearly orde red by the relation < , which is deﬁned by : ∀ k ∈ N , k < k + 1 < ℵ − 0 < ℵ 0 < 2 ℵ 0 . Deﬁnition 35. F or k ∈ N ∪ { ℵ − 0 , ℵ 0 , 2 ℵ 0 } let C F L ω ( α ≤ k ) = { L ( A ) | A is a B P D A with α A ≤ k } C F L ω ( α < k ) = { L ( A ) | A is a B P D A with α A < k } N A − C F L ω = C F L ω ( α ≤ 1) is the class of n on ambigu ous context-fr ee ω -la nguages. F o r every inte ger k such that k ≥ 2 , or k ∈ {ℵ − 0 , ℵ 0 , 2 ℵ 0 } , A ( k ) − C F L ω = C F L ω ( α ≤ k ) − C F L ω ( α < k ) If L ∈ A ( k ) − C F L ω with k ∈ N , k ≥ 2 , or k ∈ { ℵ − 0 , ℵ 0 , 2 ℵ 0 } , then L is said to be inher ently ambiguous of de gr ee k . Notice th at on e can deﬁne in a similar w ay the degree of ambiguity of a ﬁnitar y co ntext- free langua ge. If M is a pushd own automaton accepting ﬁnite words by ﬁnal states (or by ﬁnal states and topmost stack letter) then α M ∈ N or α M = ℵ − 0 or α M = ℵ 0 . Howe ver e very context-free lan guage is accepted by a pu shdown au tomaton M with α M ≤ ℵ − 0 , [3]. W e denote the class of non amb iguou s context-free lang uages by N A − C F L and th e class o f inher ently ambigu ous co ntext-free lan guages by A − C F L . Then one can state the following result. Theorem 36 ( [34]). N A − C F L ω ( ω − K C ( N A − C F L ) A − C F L ω * ω − K C ( A − C F L ) W e now come to the study o f links betwee n topo logy and ambiguity in context-fr ee ω -lang uages [34,45]. Using a The orem of Lusin and Novikov , and another theorem o f descriptive set theo ry , see [50, page 123], Simonnet proved the following stro ng result which shows that non- Borel context-free ω -langua ges hav e a maximum degree of ambiguity . Theorem 37 ( Simonnet [45]). Let L ( A ) be a context-fr ee ω -lang uage accep ted by a BPD A A such that L ( A ) is an analytic but n on Bor el set. The set o f ω -wor ds, which have 2 ℵ 0 accepting runs by A , has car d inality 2 ℵ 0 . On the other hand , it turn ed out that, infor mally speaking , the oper ation A → A ∼ conserves g lobally the degrees of ambiguity of inﬁnitar y co ntext-free lan guages (wh ich are u nions of a ﬁnitary context-fr ee language and of a context-free ω -lang uage). Then, starting from known examples o f ﬁnitary context-fr ee lan guages o f a given d egree of ambiguity , are con structed in [34] some context-free ω -lang uages of any ﬁnite Borel rank and which are non-amb iguou s o r o f any ﬁnite degree of ambiguity or o f degree ℵ − 0 . Theorem 38. 1. F or each n on n e gative inte ger n ≥ 1 , th er e exis t Σ 0 n -complete non a mbiguo us context-fr ee ω -lang uages A n and Π 0 n -complete non ambigu ous con text-fr ee ω - languages B n . 2. Let k be an integer ≥ 2 or k = ℵ − 0 . Then for each inte ger n ≥ 1 , ther e e xist Σ 0 n -complete con text-fr ee ω -lang uages E n ( k ) and Π 0 n -complete con text-fr ee ω - languages F n ( k ) which ar e in A ( k ) − C F L ω , i.e. which ar e inher ently ambiguous of degr ee k . Notice that the ω -lan guages A n and B n are simply those which were constructed in the proof of T heorem 24. On th e other ha nd it is easy to see that the BPD A accep ting the context-free ω -langu age which is Bor el of inﬁnite rank, constructed in [36] using an iteration of the op eration A → A ∼ , has a n inﬁnite d egree of ambiguity . And 1 -co unter B ¨ uchi automata ac cepting c ontext-free ω -langu ages of any Borel rank o f an effectiv e analytic set, co nstructed v ia simulation of m ulticounter automata, m ay also hav e a great degree of a mbiguity . So this left open some questions we shall detail in the last section. W e in dicate now a n ew result wh ich follows easily from the proo f of Theorem 27 sketched in Section 4 above, see [41]. Consider an ω -lang uage L acc epted by a deter- ministic M uller T ur ing machine or eq uiv alently by a deterministic 2 -coun ter Muller automaton . W e g et ﬁrst an ω -langu age θ S ( L ) ⊆ X ω which ha s the sam e top ological complexity (except fo r ﬁnite W adge d egrees), and which is accepted by a deterministic real time 8 -cou nter Muller automaton A . Then o ne can construct from A a 1 -coun ter Mu ller autom aton B , reading words over the alphabet X ∪ { A, B , 0 } , such that h ( L ( A )) ∪ h ( X ω ) − = L ( B ) ∪ h ( X ω ) − , where h : X ω → ( X ∪ { A, B , 0 } ) ω is the m apping deﬁned in Section 4. Notice that the 1 - counter Muller automa ton B which is constructed is now also deterministic . On th e other h and it is easy to see, fro m the decom position given in [41, Pr oof o f Lem ma 5.3], that the ω -lan guage h ( X ω ) − is accepted by a 1 -coun ter B ¨ u chi automato n wh ich has degree of ambigu ity 2 and the ω -langu age L ( B ) is in N A − C F L ω = C F L ω ( α ≤ 1) because it is accepted by a deterministic 1 -counter Muller automaton. Then we can easily infer, using [34, Theo rem 5. 16 (c)] th at the ω -lang uage h ( L ( A )) ∪ h ( X ω ) − = L ( B ) ∪ h ( X ω ) − is in C F L ω ( α ≤ 3) . And this ω -langu age h as the same complexity as L ( A ) Thus we can state the following result. Theorem 39. F or ea ch ω -lan guage L accep ted by a deter ministic Muller T urin g ma - chine ther e is an ω - language L ′ ∈ C F L ω ( α ≤ 3) , accepted b y a 1 -c ounter Muller automato n D with α D ≤ 3 , such that L ≡ W L ′ . 7 ω -Po wers of context-fr ee languages The ω -powers of ﬁnitary languages are ω -langua ges in the form V ω , where V is a ﬁnitary lang uage over a ﬁnite alphabe t X . Th ey appear very naturally in the character- ization of the class RE G ω of regular ω - languag es (respectively , of the class C F L ω of context-fre e ω - languag es) as the ω -Klee ne c losure of the family RE G of regular ﬁnitary languag es (respectively , of the family C F o f context-free ﬁ nitary langu ages) . The question of the topolog ical complexity of ω -powers naturally arises and w as raised by Niwinski [66], Simonnet [75], and Staiger [79]. An ω -power of a ﬁnitary langu age is always an ana lytic set because it is either the continuo us image of a compact set { 0 , 1 , . . . , n } ω for n ≥ 0 or of the Baire space ω ω . The ﬁrst example of ﬁnitar y langu age L such that L ω is analytic but not Borel, and ev en Σ 1 1 -complete, was o btained in [35]. Amazing ly the langu age L was very simple and e ven accepted by a 1 -coun ter automaton . It was obtained via a cod ing of inﬁnite labelled binary trees. W e now give a simple co nstruction o f this langu age L using th e n otion o f sub stitution which we now reca ll. A substitution is deﬁned by a mapp ing f : X → P ( Γ ⋆ ) , whe re X = { a 1 , . . . , a n } an d Γ are two ﬁnite alphabets, f : a i → L i where for a ll integers i ∈ [1; n ] , f ( a i ) = L i is a ﬁnitary langu age o ver t he alphabet Γ . Now th is mapping is extended in the usual m anner to ﬁnite word s: f ( a i 1 . . . a i n ) = L i 1 . . . L i n , an d to ﬁnitary languages L ⊆ X ⋆ : f ( L ) = ∪ x ∈ L f ( x ) . If fo r ea ch in teger i ∈ [1; n ] the language L i does not con tain th e empty w ord, then the mapping f may be extended to ω -words: f ( x (1) . . . x ( n ) . . . ) = { u 1 . . . u n . . . | ∀ i ≥ 1 u i ∈ f ( x ( i )) } and to ω -lan guage s L ⊆ X ω by setting f ( L ) = ∪ x ∈ L f ( x ) . Let now X = { 0 , 1 } and d be a new letter not i n X and D = { u.d.v | u, v ∈ X ⋆ and ( | v | = 2 | u | ) or ( | v | = 2 | u | + 1) } D ⊆ ( X ∪ { d } ) ⋆ is a co ntext-free langu age accep ted by a 1 -cou nter autom aton. Let g : X → P (( X ∪ { d } ) ⋆ ) be the substitution deﬁned b y g ( a ) = a.D . As W = 0 ⋆ 1 is regular , L = g ( W ) is a c ontext-free languag e and it is accepted by a 1 -co unter automaton . Moreover one c an p rove that ( g ( W )) ω is Σ 1 1 -complete, he nce a non Borel set. This is done by reducing to this ω -lang uage a well-known example of Σ 1 1 -complete set : the set of inﬁnite binary trees labe lled in the alphabet { 0 , 1 } wh ich ha ve an inﬁnite branch in the Π 0 2 -complete set (0 ⋆ . 1) ω , see [35] for more details. Remark 40. The ω -lang uage ( g ( W )) ω is con text-fr ee. By Theorem 3 7 every BPDA accepting ( g ( W )) ω has the maximum ambiguity a nd ( g ( W )) ω ∈ A (2 ℵ 0 ) − C F L ω . On th e other ha nd we ca n pr ove that g ( W ) is a non amb iguou s conte xt-fr e e langu age. This is u sed in [45] to pr ove that n either unambiguity no r am biguity o f co ntext-fr ee languages ar e preserved under the operation V → V ω . Concernin g Bor el ω - powers, it has been proved in [32] that for each integer n ≥ 1 , there exist som e ω -powers of co ntext-free langua ges which are Π 0 n -complete Borel sets. These results were obtained by the use of a new operation V → V ≈ over ω -lang uages, which is a sligh t modiﬁcation of the o peration V → V ∼ . The new operation V → V ≈ preserves ω - powers an d context-fr eeness. M ore pre cisely if V = W ω for some context- free lang uage W , then V ≈ = T ω for som e con text-free languag e T wh ich is obtained from W by ap plication of a gi ven context-fre e substitution. And it follows easily fr om [23] that if V ⊆ X ω is a Π 0 n -complete set, for some integer n ≥ 2 , then V ≈ is a Π 0 n +1 -complete set. Then , starting from the Π 0 2 -complete set (0 ⋆ . 1) ω , we get some Π 0 n -complete ω -p owers of con text-free languages for each integer n ≥ 3 . An iteration o f the op eration V → V ≈ was used in [ 37] to prove that there exists a ﬁnitary lang uage V such that V ω is a Borel set of inﬁnite ran k. The lan guage V was a simple r ecursive language but it was n ot co ntext-free. Later , with a mo diﬁcation of th e construction , using a coding of an inﬁnity o f era sers previously deﬁned in [36], Finkel and Duparc got a co ntext-free lan guage V such that V ω is a Borel set above the class ∆ 0 ω , [25]. The qu estion of the Borel h ierarchy of ω -powers of ﬁnitar y langu ages has bee n solved very re cently by Finkel and Lec omte in [44], where a very surprising r esult is proved, showing that actu ally ω -powers exhibit a great top ological com plexity . For every non- null countab le o rdinal α there exist some Σ 0 α -complete ω -p owers and also some Π 0 α - complete ω -powers. B ut the ω - powers constructed in [44] are not ω -powers o f context- free languag es, excep t fo r the case of a Σ 0 2 -complete set. Notice also that an examp le of a regular languag e L such that L ω is Σ 0 1 -complete was giv en by Simonnet in [75], see also [54] . 8 Pe rspectiv es and open questions W e give b elow a list of some open questions which arise naturally . Th e problems listed here seem impo rtant f or a better comprehension of context-free ω -lan guages but the list is not exhaustiv e. 8.1 Effect ive results In th e non- deterministic c ase, the B orel a nd W ad ge h ierarchies of context-free ω -lang ua ges are no t ef fecti ve, [32,35,33]. This is not surprising sin ce most decision problems on context-free lang uages are undec idable. On the oth er h and we can expect some decid- ability results in th e case of deterministic c ontext-free ω -langu ages. W e ha ve alread y cited some o f them : we can d ecide whether a de terministic context-free ω -lan guage is in a given Borel class or even in the W ad ge c lass [ L ] of a given regular ω -langu age L . The most challenging question in this area would b e to ﬁnd an effeci ve procedu re to determine the W adge degree of an ω -lang uage in the class DC F L ω . Recall that the W adge h ierarchy of the class DC F L ω is d etermined in a n on-effectiv e way in [24]. On the o ther hand the W adge hier archy o f the class of blind counter ω - languag es is determin ed in an effecti ve way , usin g notions of chain s and superchains, in [30]. Ther e is a gap between the two hierar chies because (blind) 1 -counte r automata are much less expressi ve tha n p ushdown auto mata. One cou ld tr y to extend the methods of [30] to the study of deterministic pushdown automata. Another question c oncern s the co mplexity of decidab le prob lems. A ﬁr st q uestion would be the following one. C ould we extend th e results o f W ilke and Y oo t o the class of blind counter ω -lan guages, i. e. is the W ad ge degree o f a blind co unter ω -lang uage co mputable in polyn omial time ? Otherwise what is the complexity of this problem ? Of course the question m ay b e fu rther asked for classes of ω -la nguage s which are located between the classes of blind counter ω -lang uages and of deterministic context-free ω -lang uages. Another interesting q uestion would be to determ ine the W adge hierarchy of ω -lan guage s accepted by deterministic high er order pushdown au tomata (even ﬁrstly in a non effec- ti ve way), [28 ,11]. 8.2 T opolo gy and ambiguity Simonnet’ s Th eorem 37 states that n on-Borel context-fr ee ω - languag es have a m axi- mum degree of am biguity , i.e. ar e in the c lass A (2 ℵ 0 ) − C F L ω . On th e other h and, there exist some non -ambigu ous context-free ω -lang uages of every ﬁn ite Borel r ank. The qu estion naturally arises wh ether ther e exist som e non- ambigu ous context-free ω - languag es which are W ad ge equ i valent to any gi ven Borel context-free ω -lan guage (or equiv alently to any Borel Σ 1 1 -set, by Theorem 28). This may be con nected to a result of Arnold who proved in [2] th at every Borel subset o f X ω , fo r a ﬁnite alphabet X , is accepted by a non-amb iguous ﬁnitely br anching transition system with B ¨ u chi accep- tance condition. By Theorem 38, if k is a n integer ≥ 2 or k = ℵ − 0 , then fo r each in teger n ≥ 1 , there exist Σ 0 n -complete context-fr ee ω - languag es E n ( k ) an d Π 0 n -complete context-free ω -lan guages F n ( k ) which are in A ( k ) − C F L ω , i.e. which are in her- ently amb iguou s of degree k . More g enerally the q uestion arises : d etermine the Bo rel ranks and th e W adge degrees of context-free ω -langu ages in classes C F L ω ( α ≤ k ) o r A ( k ) − C F L ω where k ∈ N ∪ {ℵ − 0 , ℵ 0 , 2 ℵ 0 } ( k ≥ 2 in the case of A ( k ) − C F L ω ). A ﬁrst result in this direction is Theorem 39 stated in Section 6. 8.3 ω -Powers The results o f [32,35,37,44] show that ω - powers o f ﬁnitary lang uages have a ctually a great topological co mplexity . Concernin g ω -p owers of context-fr ee languag es we do not know y et what are all their inﬁnite Borel ranks. Howe ver the results of [41] suggest that ω -powers of co ntext-free languag es or even of languages accepted by 1 - counter automata exhibit also a great topological complexity . Indeed Theorem 2 8 states that th ere are ω -lang uages acce pted by B ¨ u chi 1 -counter au - tomata of every Bore l rank (an d even of ev ery W adge d egree) o f an effectiv e ana lytic set. On the other hand e ach ω -langu age accepte d b y a B ¨ uc hi 1 -counte r automa ton can be written as a ﬁnite union L = S 1 ≤ i ≤ n U i .V ω i , where for each integer i , U i and V i are ﬁnitary langu ages acc epted by 1 -coun ter au tomata. Then we can co njecture th at there exist some ω -powers of languages ac cepted b y 1 -coun ter automata which have Borel ranks up to the ordinal γ 1 2 , although these languag es are lo cated at the very low le vel in the complexity hierarchy of ﬁnitary languages. Recall that a ﬁnitary lan guage L is a cod e ( respectively , an ω - code) if every word of L + (respectively , every ω -word of L ω ) has a u nique decom position in words of L , [6] . It is proved in [45] that if V is a context-free langu age such that V ω is a non Borel set then there are 2 ℵ 0 ω -words of V ω which hav e 2 ℵ 0 decomp ositions in words of V ; in p articular, V is really not an ω -co de althou gh it is proved in [ 45] th at V may be a code (see the example V=g(W) g i ven in S ection 7). The following question about Borel ω -powers now arises : are there som e con text-free codes (respectively , ω -codes) V suc h that V ω is Σ 0 α -complete or Π 0 α -complete for a given co untable ordinal α < γ 1 2 ? Refer ences 1. A. Andretta an d R. Camerlo. The u se of co mplexity hierarch ies in des criptiv e set theory and automata theory . T ask Quarterly , 9(3):337–356, 2005. 2. A. Arno ld. T opological chara cterizations of inﬁnite beh aviou rs of transition systems. 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