An omega-Power of a Finitary Language Which is a Borel Set of Infinite Rank

An ω -P o w er of a Finitary Lang u age Whic h is a Borel Set o f Infinite Rank Olivi er Fink el Equip e de L o gique Math ´ ematique U.F.R. de Math ´ emati ques, Universit ´ e Paris 7 2 Plac e Jussieu 75251 Paris c e dex 05 , F r anc e. E Mail: fink el@log i que.jussieu. fr Abstract ω -p ow ers of finitary languag es are ω -languages in the form V ω , where V is a finitary language o ver a finite alphab et Σ. Since the set Σ ω of infinite w ord s o v er Σ can b e equipp ed with the usual Can tor top ology , the question of the top ological complexit y of ω -p o wers naturally arises and has b een raised by Niwinski [Niw90], by Simonnet [Sim92], and b y Staiger [Sta97b]. It has b een pro ved in [Fin01] that for eac h integ er n ≥ 1, there exist some ω -p ow ers of con text free languages which are Π 0 n -complete Borel sets, and in [Fin03] that there exists a conte xt free language L suc h that L ω is analytic but not Borel. But the qu estion w as still op en whether there exists a finitary language V suc h that V ω is a Borel set of in fi nite rank . W e answer this question in this pap er, giving an example of a fin itary language whose ω -p o wer is Borel of infi nite rank. Keyw ords: Infinite wo rds; ω -languages; ω -pow ers; Can tor top ology; top olo gical complexit y; Borel sets; infi nite rank. 1 In tro duction ω - p o w ers are ω - languages in the form V ω , where V is a finitary la ng uage. The operatio n V → V ω is a fundamen tal op eration o ve r finitary la nguages leading to ω - languages. This operation app ears in the c ha r a cterization of the class RE G ω of ω - regular lang uages (r esp ectiv ely , of the class C F ω of 1 con text free ω - languages) as the ω -Kleene closure of the family R E G of reg- ular finitary languages (respective ly , of the family C F of context free finitary languages) [Sta97a]. The set Σ ω of infinite w o rds ov er a finite alphab et Σ is usually equipped with the Can tor t op ology whic h may b e defined b y a distance, see [Sta97a] [PP04]. One can then study the complexit y of ω -languages, i.e. languages of infinite w ords, by considering their t o p ological complexity , with regard to the Borel hierarc hy ( a nd b ey ond to the pro jectiv e hierarc h y) [Sta97a] [PP04 ]. The question of t he top ological complexit y of ω -p o wers of finitary languages naturally arises. It ha s b een p osed by Niwinski [Niw90], by Simonnet [Sim92] and by Staiger [Sta97b]. The ω -p ow er of a finita r y language is alw ays an an- alytic set b ecause it is the con tinuous image of a compact set { 0 , 1 , . . . , n } ω for n ≥ 0 or of the Baire sp ace ω ω , [Sim92] [F in01]. It has b een prov ed in [Fin01] that for eac h in teger n ≥ 1, there exist some ω - p o w ers of con text free languages whic h are Π 0 n -complete Borel sets, and in [Fin03] that there exists a con text f r ee language L such that L ω is analytic but not Bo rel. But the que stion w as still op en whether there exists a finitary language V suc h that V ω is a Borel set of infinite rank. W e answ er this question in this pa p er, giving an example of a finitary la n- guage whose ω -p ow er is Borel o f infinite rank. The pap er is organized as follows. In Section 2 we recall definitions of Borel sets and previous results and w e prov ed our main result in Section 3. 2 Recall on B o rel s ets and previ o us resu l t s W e assume the reader t o b e familiar with the theory of formal ω -languages [Tho90], [Sta97a]. W e shall use usual notatio ns of formal la nguage theory . When Σ is a finite alphab et, a non-em pty finite wor d o ve r Σ is any sequence x = a 1 . . . a k , where a i ∈ Σ for i = 1 , . . . , k , and k is an in teger ≥ 1. The length of x is k , denoted b y | x | . The empty wor d has no letter and is denoted b y λ ; its len gth is 0. F or 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 . Σ ⋆ is the set of fini te wor ds (including the empty w ord) o v er Σ. The first infinite or dinal is ω . An ω - wor d o ve r Σ is an ω -sequence a 1 . . . a n . . . , where for all in tegers i ≥ 1, a i ∈ Σ. When σ is a n ω -word o ver Σ, w e write σ = σ (1) σ (2) . . . σ ( n ) . . . , where for all i , σ ( i ) ∈ Σ, and σ [ n ] = σ (1) σ (2) . . . σ ( n ) for all n ≥ 1 and σ [0] = λ . 2 The pr efix r elation is denoted ⊑ : a finite word u is a pr efix of a finite w o rd v (resp ectiv ely , an infinite w o rd v ), denoted u ⊑ v , if and only if there exists a finite w ord w ( r espective ly , an infinite w o r d w ), suc h that v = u.w . The set of ω - wo r ds ov er the alphab et Σ is denoted by Σ ω . An ω - language ov er a n alphab et Σ is a subset of Σ ω . F or V ⊆ Σ ⋆ , the ω - p ower of V is the ω -language: V ω = { σ = u 1 . . . u n . . . ∈ Σ ω | ∀ i ≥ 1 u i ∈ V − { λ }} W e assume the reader to b e familiar with basic notions of top ology whic h ma y b e found in [Mos80] [L T94] [Kec95 ] [Sta9 7a] [PP04]. F or a finite alphab et X , w e cons ider X ω as a top ological space with the Can tor top o logy . The o p en sets o f X ω are t he sets of the form W .X ω , where W ⊆ X ⋆ . A set L ⊆ X ω is a close d set iff its complemen t X ω − L is an op en set. Define no w the B or el Hier ar chy of subsets of X ω : Definition 2.1 F or a non-nul l c ountable or d inal α , the classes Σ 0 α and Π 0 α of the Bor el Hier ar chy on the top o l o g ic al sp ac e X ω ar e de fine d as fol lows: Σ 0 1 is the class of op en subsets o f X ω . Π 0 1 is the class of close d subsets of X ω . and for any c ountable or dinal α ≥ 2 : Σ 0 α is the class of c ountable unions of subsets of X ω in ∪ γ <α Π 0 γ . Π 0 α is the class of c ountable interse ctions of s ubse ts of X ω in ∪ γ <α Σ 0 γ . F or a countable ordinal α , a subset o f X ω is a Borel set of r ank 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 a lso some subsets of X ω whic h are not Bor el. In particular t he class o f Borel subsets o f X ω is strictly included into the class Σ 1 1 of analytic sets whic h are obtained by pro jection of Borel sets, see for example [Sta97a] [L T94] [PP04] [Kec95] for more details. W e now define completeness with regard to reduction b y contin uous func- tions. F or a coun table ordinal α ≥ 1, a set F ⊆ X ω is said to b e a Σ 0 α (resp ectiv ely , Π 0 α , Σ 1 1 )- c omplete set iff for any set E ⊆ Y ω (with Y a finite alphab et): E ∈ Σ 0 α (resp ectiv ely , E ∈ Π 0 α , E ∈ Σ 1 1 ) iff there exists a con- tin uous function f : Y ω → X ω suc h that E = f − 1 ( F ). Σ 0 n (resp ectiv ely Π 0 n )-complete sets, with n an in teger ≥ 1, are thoroughly c haracterized in [Sta86]. 3 In part icular R = (0 ⋆ . 1) ω is a we ll kno wn example of Π 0 2 -complete subset of { 0 , 1 } ω . It is the set of ω -w o rds o v er { 0 , 1 } ha ving infinitely man y o ccurrenc es of the letter 1. Its complemen t { 0 , 1 } ω − (0 ⋆ . 1) ω is a Σ 0 2 -complete subset of { 0 , 1 } ω . W e shall recall the definition of the op eration A → A ≈ o v er sets of infinite w ords we introduced in [Fin01] and whic h is a simple v ariant of Dupar c’s op eration of exp onen tiatio n A → A ∼ [Dup01]. F or a finite alphab et Σ w e denote Σ ≤ ω = Σ ω ∪ Σ ⋆ . Let no w և a letter not in Σ a nd X = Σ ∪ { և } . F or x ∈ X ≤ ω , x և denotes the string x , o nce eve ry և o ccuring in x has b een “ev a luated” to the bac k space o p eration ( the one familiar to your computer!), pro ceed ing from left to right inside x . In other w ords x և = x from whic h ev ery in terv al of the form “ a և ” ( a ∈ Σ) is remo ved . W e add the conv ention that ( u. և ) և is undefined if | u և | = 0, i.e. when the last letter և can not b e used a s an eraser (b ecaus e eve ry letter of Σ in u has already b een erased by some erasers և placed in u ) . Remark that the resulting word x և ma y b e finite or infinite. F or example if u = ( a և ) n , for n ≥ 1, or u = ( a և ) ω then ( u ) և = λ , if u = ( ab և ) ω then ( u ) և = a ω , if u = bb ( և a ) ω then ( u ) և = b , if u = և ( a և ) ω or u = a ևև a ω or u = ( a ևև ) ω then ( u ) և is undefined. Definition 2.2 F or A ⊆ Σ ω , A ≈ = { x ∈ (Σ ∪ { և } ) ω | x և ∈ A } . The follow ing result follo ws easily f r o m [Dup01] and w a s applied in [Fin01] to study the ω -p o wers of finitary con text free la nguages. Theorem 2.3 L et n b e an inte ger ≥ 2 and A ⊆ Σ ω b e a Π 0 n -c omplete set. Then A ≈ is a Π 0 n + 1 -c omplete subset of (Σ ∪ { և } ) ω . F or each ω -language A ⊆ Σ ω , the ω - language A ≈ can b e easily described from A b y the use of the not ion o f substitution whic h w e r ecall now. A substitu tion is define d b y a mapping f : Σ → P (Γ ⋆ ), where Σ = { a 1 , . . . , a n } and Γ are tw o finite alphabets, f : a i → L i where for all in tegers i ∈ [1; n ], f ( a i ) = L i is a finitary language o v er the alphab et Γ. No w this mapping is e xtended in the usual manner to finite w ords: f ( a i 1 . . . a i n ) = L i 1 . . . L i n , and to finitary languages L ⊆ Σ ⋆ : f ( L ) = ∪ x ∈ L f ( x ). If fo r each in teger i ∈ [1; n ] the language L i do es not con tain the empt y word, then the mapping f ma y b e extended to ω -w ords: f ( x (1) . . . x ( n ) . . . ) = { u 1 . . . u n . . . | ∀ i ≥ 1 u i ∈ f ( x ( i )) } 4 and t o ω -languages L ⊆ Σ ω b y setting f ( L ) = ∪ x ∈ L f ( x ). Let L 1 = { w ∈ (Σ ∪ { և } ) ⋆ | w և = λ } . L 1 is a contex t f ree (finitary) language generated b y the con text free gr a mmar with the followin g pro duction rules: S → aS և S with a ∈ Σ; and S → λ (where λ is the empt y w or d) . Then, for eac h ω -language A ⊆ Σ ω , the ω -language A ≈ ⊆ (Σ ∪ { և } ) ω is obtained by substituting in A the languag e L 1 .a for eac h letter a ∈ Σ. By definition the op eration A → A ≈ conserv es the ω -p ow ers of finitary lan- guages. Indeed if A = V ω for some language V ⊆ Σ ⋆ then A ≈ = g ( V ω ) = ( g ( V )) ω where g : Σ → P ((Σ ∪ { և } ) ⋆ ) is the substitution defined b y g ( a ) = L 1 .a fo r ev ery letter a ∈ Σ. 3 An ω -p o wer whic h is Borel of infi n ite rank W e can now iterate k times t his op eration A → A ≈ . More prec isely , we define, f o r a set A ⊆ Σ ω : A ≈ . 0 k = A , A ≈ . 1 k = A ≈ , A ≈ . 2 k = ( A ≈ . 1 k ) ≈ , and A ≈ .k k = ( A ≈ . ( k − 1) k ) ≈ , where w e apply k time s the op eration A → A ≈ with different new letters և k , և k − 1 , . . . , և 3 , և 2 , և 1 , in suc h a wa y t ha t we ha ve successiv ely: A ≈ . 0 k = A ⊆ Σ ω , A ≈ . 1 k ⊆ (Σ ∪ { և k } ) ω , A ≈ . 2 k ⊆ (Σ ∪ { և k , և k − 1 } ) ω , . . . . . . . . . . . . A ≈ .k k ⊆ (Σ ∪ { և k , և k − 1 , . . . , և 1 } ) ω . F or a reason whic h will b e clear later w e hav e c hosen to successiv ely call the erasers և k , և k − 1 , . . . , և 2 , և 1 , in this precis e order. W e set no w A ≈ .k = A ≈ .k k so it holds that A ≈ .k ⊆ (Σ ∪ { և k , և k − 1 , . . . , և 1 } ) ω Notice that definitions of A ≈ . 1 k , A ≈ . 2 k , . . . , A ≈ . ( k − 1) k w ere just some in termediate steps fo r the definition of A ≈ .k and will not b e used later. W e can also describ e the op eratio n A → A ≈ .k in a similar manner as in the case o f the op eration A → A ≈ , by the use of the notion of substitution. 5 Let L k ⊆ (Σ ∪ { և k , և k − 1 , . . . , և 1 } ) ⋆ b e the language con taining (finite) w ords u , suc h that a ll letters of u hav e b een erased whe n the op era t io ns of erasing using the erasers և 1 , և 2 , . . . , և k − 1 , և k , are succes siv ely applied t o u . Notice tha t the operatio ns of erasing hav e to b e do ne in a g o o d order: the first op eration of erasing uses the eraser և 1 , then the se cond one use s the eraser և 2 , and so on . . . So an eraser և j ma y only erase a letter o f Σ or an o ther “eraser” և i for some in teger i > j . It is easy t o see that L k is a con text fr ee language. In fact L k b elongs to the sub class of iterated counter languages whic h is the closure under substitution of the class of o ne counte r languag es, see [ABB96] [F in0 1] for more details. Let no w h k b e the substitution: Σ → P ((Σ ∪ { և k , և k − 1 , . . . , և 1 } ) ⋆ ) defined b y h k ( a ) = L k .a fo r ev ery letter a ∈ Σ. Then it holds that, for A ⊆ Σ ω , A ≈ .k = h k ( A ), i.e. A ≈ .k is obtained b y substituting in A the language L k .a fo r eac h letter a ∈ Σ. The ω -language R = (0 ⋆ . 1) ω = V ω , where V = (0 ⋆ . 1), is Π 0 2 -complete. Then, b y Theorem 2.3 , f o r eac h in teger p ≥ 1, h p ( V ω ) = ( h p ( V )) ω is a Π 0 p + 2 - complete set. W e can see that the languages L k , for k ≥ 1, form a sequen ce whic h is strictly increasing for the inclusion relation: L 1 ⊂ L 2 ⊂ L 3 ⊂ . . . ⊂ L i ⊂ L i +1 . . . In order to construct some ω -p o w er whic h is Borel of infinite rank, we could try to substitute the languag e ∪ k ≥ 1 L k .a to each letter a ∈ Σ. But the language ∪ k ≥ 1 L k .a is defined o v er the infinite a lpha b et Σ ∪ { և 1 , և 2 , և 3 , . . . } , so we shall first co de e v ery eraser և j b y a finite w ord ov er a fixed finite alphab et. The eraser և j will b e co ded b y the finite w ord α .β j .α o v er the alphab et { α, β } , where α and β are tw o new letters. The morphism ϕ p : (Σ ∪ { և 1 , . . . , և p } ) ⋆ → (Σ ∪{ α, β } ) ⋆ defined b y ϕ p ( c ) = c for eac h c ∈ Σ and ϕ p ( և j ) = α .β j .α for eac h in t eger j ∈ [1 , p ], can b e naturally extended to a contin uous function ψ p : (Σ ∪ { և 1 , . . . , և p } ) ω → (Σ ∪ { α, β } ) ω . 6 Let now L = [ n ≥ 1 ϕ n ( L n ) and h : Σ → P ((Σ ∪ { α, β } ) ⋆ ) b e the substitution defined by h ( a ) = L .a for each a ∈ Σ. W e can now state our main result: Theorem 3.1 L et V = (0 ⋆ . 1) . Then the ω -p ower ( h ( V )) ω ⊆ { 0 , 1 , α, β } ω is a B or el set of infinite r ank. T o prov e this theorem, w e shall pro ceed by succes siv e le mmas. Lemma 3.2 F or al l inte gers p ≥ 1 , the ω -language ψ p ( R ≈ .p ) is a Π 0 p + 2 - c omplete subset of (Σ ∪ { α, β } ) ω . Pro of. First w e prov e that ψ p ( R ≈ .p ) is in the class Π 0 p + 2 . F or Σ = { 0 , 1 } , ψ p ((Σ ∪ { և 1 , . . . , և p } ) ω ) is the contin uous image b y ψ p of the compact set (Σ ∪ { և 1 , . . . , և p } ) ω , hence it is also a compact set. The function ψ p is injectiv e and contin uous th us it induces an homeomor- phism ψ ′ p b et w een the t w o compact sets (Σ ∪ { և 1 , . . . , և p } ) ω and ψ p ((Σ ∪ { և 1 , . . . , և p } ) ω ). W e hav e a lready seen that, for each in teger p ≥ 1, the ω - language R ≈ .p is a Π 0 p + 2 -complete subset of (Σ ∪ { և 1 , . . . , և p } ) ω . Then ψ ′ p ( R ≈ .p ) is a Π 0 p + 2 - subset o f ψ p ((Σ ∪ { և 1 , . . . , և p } ) ω ), b ecause the function ψ ′ p is an ho meomor- phism. But one can pro v e, by induction ov er the in teger j ≥ 1, that each Π 0 j subset K of ψ p ((Σ ∪ { և 1 , . . . , և p } ) ω ) is also a Π 0 j subset o f (Σ ∪ { α, β } ) ω . Th us ψ ′ p ( R ≈ .p ) = ψ p ( R ≈ .p ) is a Π 0 p + 2 -subset of (Σ ∪ { α, β } ) ω . Remark now that the set R ≈ .p b eing Π 0 p + 2 -complete, eve ry Π 0 p + 2 -subset of X ω , for X a finite alphab et, is the in v erse image of R ≈ .p b y a contin uous function. But it holds tha t R ≈ .p = ψ − 1 p ( ψ p ( R ≈ .p )), where ψ p is a con tin uous function. Th us ev ery Π 0 p + 2 -subset of X ω , for X a finite alphab et, is the in vers e image of ψ p ( R ≈ .p ) b y a con tin uous function. Therefore ψ p ( R ≈ .p ) is also a Π 0 p + 2 -complete subset of (Σ ∪ { α, β } ) ω .  Lemma 3.3 The set ( h ( V )) ω is n ot a Bor el set of finite r ank. 7 Pro of. Cons ider, for eac h inte ger p ≥ 1, the regular ω -language R p = ψ p ( { 0 , 1 , և 1 , և 2 , . . . , և p } ω ) = { 0 , 1 , α .β .α, α.β 2 .α, . . . , α .β p .α } ω W e hav e seen that R p is compact hence it is a closed set. And b y construction it holds that ( h ( V )) ω ∩ R p = ψ p ( h p ( V ω )) = ψ p ( R ≈ .p ) where R = V ω , so by Lemma 3.2 this set is a Π 0 p + 2 -complete subset o f { 0 , 1 , α , β } ω . If ( h ( V )) ω w as a Borel set of finite rank it w ould b e in the class Π 0 J for some in teger J ≥ 1. But then ( h ( V )) ω ∩ R p w ould b e the in tersection of a Π 0 J -set and of a closed, i.e. Π 0 1 -set. Th us, fo r each inte ger p ≥ 1, the set ( h ( V )) ω ∩ R p w ould b e a Π 0 J -set. This would lead to a contradiction b ecause , f or J = p , a Π 0 J -set cannot b e a Π 0 p + 2 -complete set.  Lemma 3.4 Every ω -wor d x ∈ ( h ( V )) ω has a unique de c omp osition of the form x = u 1 .u 2 . . . u n . . . wher e for al l i ≥ 1 u i ∈ h ( V ) . Pro of. T ow ards a con tradiction a ssume o n the con trary that some ω -w ord x ∈ ( h ( V )) ω = ( h (0 ⋆ . 1)) ω has (at least) tw o distinct decomp ositions in words of h ( V ). So there are some w ords u j , u ′ j ∈ h ( V ), for j ≥ 1, suc h that x = u 1 .u 2 . . . u n . . . = u ′ 1 .u ′ 2 . . . u ′ n . . . and a n intege r J ≥ 1 suc h that u j = u ′ j for j < J and u J ⊏ u ′ J , i.e. u J is a strict prefix of u ′ J . Then for eac h in teger j ≥ 1, there are in tegers n j , n ′ j ≥ 0 suc h that u j ∈ h (0 n j . 1) and u ′ j ∈ h (0 n ′ j . 1). Th us there are some finite w or ds v j i ∈ L , where i is an integer in [1 , n j + 1], and w j i ∈ L , where i is an in teger in [1 , n ′ j + 1], suc h that u j = v j 1 . 0 .v j 2 . 0 . . . v j n j . 0 .v j n j +1 . 1 and u ′ j = w j 1 . 0 .w j 2 . 0 . . . w j n ′ j . 0 .w j n ′ j +1 . 1 W e consider now x given b y its first decomp osition x = u 1 .u 2 . . . u n . . . Let no w x (1) b e the ω -w or d obtained from x b y using the (co de of the) eraser և 1 as an eraser which may erase letters 0, 1, and ( co des of the) erasers և p for p > 1. Remark that b y construction these op era t ions of erasing o ccur inside the words v j i for j ≥ 1 and i ∈ [1 , n j + 1]. Next let x (2) b e the ω -w ord obt a ined fr o m x (1) b y using the (co de of t he) eraser և 2 as an eraser whic h ma y era se letters 0, 1, and (co des of the) erasers և p for p > 2. Aga in these erasing op eratio ns o ccur inside the w or ds v j i . W e can no w iterate this pro cess. Assume that, after ha ving successiv ely used the erasers և 1 , և 2 , . . . , և n , for some in teger n ≥ 1, w e hav e go t the ω - w ord 8 x ( n ) from the ω -w ord x . W e can now define x ( n +1) as the ω - w ord obtained from x ( n ) b y using the (co de of the) eraser և n +1 as an era ser which may erase letters 0, 1, and (co des of the) erasers և p for p > n + 1. W e shall denote K ( i,j ) = min { k ≥ 1 | v j i ∈ ϕ k ( L k ) } . Then v j i ∈ ϕ K ( i,j ) ( L K ( i,j ) ) for all in tegers j ≥ 1 and i ∈ [1 , n j + 1 ]. Th us after K j = max { K ( i,j ) | i ∈ [1 , n j + 1 ] } steps all words v j i , for i ∈ [1 , n j + 1], ha ve b een completely erased and, f rom the finite w o rd u j , it remains only the finite w ord 0 n j . 1. After K J = max { K j | j ∈ [1 , J ] } steps, from the word u 1 .u 2 . . . u J , it re- mains the word 0 n 1 . 1 . 0 n 2 . 1 . 0 n 3 . 1 . . . 0 n J . 1 whic h is a strict prefix of x ( K J ) . In particular the J -th letter 1 o f x ( K J ) is the last letter of u J , whic h has not b een erased, and it will not b e erased b y next erasing op erations. Notice that the success iv e erasing op erations are in fact applied to x inde- p enden tly of the decompo sition of x in words of h ( V ). So consider now the ab ov e erasing op erations applied to x giv en b y its second decomp osition. Let K ′ ( i,j ) = min { k ≥ 1 | w j i ∈ ϕ k ( L k ) } , and K ′ J = max { K ′ ( i,j ) | j ∈ [1 , J ] a nd i ∈ [1 , n ′ j + 1] } . K j = K ′ j holds for 1 ≤ j < J and K J ≤ K ′ J . W e see tha t , after K ′ J steps, the w ord 0 n 1 . 1 . 0 n 2 . 1 . 0 n 3 . 1 . . . 0 n J − 1 . 1 . 0 n ′ J . 1 is a strict prefix of x ( K ′ J ) . The J -t h letter 1 of x ( K ′ J ) is the last letter of u ′ J (whic h has not b een erased). But w e hav e seen ab ov e that it is also the la st letter of u J (whic h has not b een erased). Th us we w o uld ha v e u J = u ′ J and t his leads to a contradiction.  Remark that in terms of co de theory Lemma 3.4 states that the language h ( V ) is an ω -co de. Lemma 3.5 The set ( h ( V )) ω is a Bor el set. Pro of. Assume on the con t r ary that ( h ( V )) ω is an analytic but non Borel set. Recall that lemma 4.1 of [FS03] states that if X and Y are finite alphab ets ha ving at least t w o letters a nd B is a Borel subset of X ω × Y ω suc h that P RO J X ω ( B ) = { σ ∈ X ω | ∃ ν ( σ, ν ) ∈ B } is not Borel, then there are 2 ℵ 0 ω - w ords σ ∈ X ω suc h that B σ = { ν ∈ Y ω | ( σ, ν ) ∈ B } has cardinality 2 ℵ 0 (where 2 ℵ 0 is the cardinal of the contin uum). W e can now reason as in the pro of of F act 4.5 in [FS03]. Let θ b e a recursiv e enum eration of the set h ( V ). The function θ : N → h ( V ) is a bijection and w e denote u i = θ ( i ). Let now D b e t he set of pairs ( σ, ν ) ∈ { 0 , 1 } ω × { 0 , 1 , α, β } ω suc h that: 9 1. σ ∈ (0 ⋆ . 1) ω , so σ may b e written in the form σ = 0 n 1 . 1 . 0 n 2 . 1 . 0 n 3 . 1 . . . 0 n p . 1 . 0 n p +1 . 1 . . . where ∀ i ≥ 1 n i ≥ 0, and 2. ν = u n 1 .u n 2 .u n 3 . . . u n p .u n p +1 . . . D is a Borel subset o f { 0 , 1 } ω × { 0 , 1 , α, β } ω b ecause it is accepted by a deterministic T uring mach ine with a B ¨ uc hi acceptance condition [Sta97a]. But P RO J { 0 , 1 ,α,β } ω ( D ) = ( h ( V )) ω w ould b e a non Borel set th us there w ould b e 2 ℵ 0 ω - w ords ν in ( h ( V )) ω suc h that D ν has car dina lity 2 ℵ 0 . This means that there w ould exist 2 ℵ 0 ω - w ords ν ∈ ( h ( V )) ω ha ving 2 ℵ 0 decomp ositions in words in h ( V ). This would lead to a (strong) contradiction with Lemma 3 .4 .  Theorem 3 .1 follows now dir ectly fr o m Lemmas 3.3 a nd 3.5. 4 Conclud ing Remarks W e already knew that there are ω -p o w ers of ev ery finite Borel rank [Fin01]. W e ha ve prov ed that there exists some ω -p o w ers of infinite Borel rank. The language h ( V ) is v ery simple to de scrib e. It is obtained b y substituting in the regular language V = (0 ⋆ . 1) the lang uage L .a to eac h lette r a ∈ { 0 , 1 } , where L = S n ≥ 1 ϕ n ( L n ). Notice that the language L is not con text free but it is the union of the increasing sequence of con text free lang ua ges ϕ n ( L n ). Then L is a v ery simple recursiv e language and so is h ( V ). The question is left op en whether there is a con text free language W suc h that W ω is Borel of infinite rank. The question also naturally arises to kno w what are all the p ossible infinite Borel ranks of ω -p o wers of finitary languages or of finitary la ng ua ges b elong- ing to some natural class lik e t he class o f contex t free languages (resp ective ly , languages a ccepted b y stac k auto mata, recursiv e languag es, recursiv ely en u- merable languages, . . . ). Ac kno wledgemen ts. W e thank t he anon ymous referee fo r useful commen ts on a preliminary v ersion of this pap er. 10 References [ABB96] J.-M. Auteb ert, J. Berstel, a nd L. Boasson. Con text free languages and pushdo wn automata. In Handb o ok of formal languages, V ol. 1 . Springer-V erlag, 1 9 96. [Ber79] J. Berstel. T r ansductions and c ontext fr e e languages . T eubner Stu- dien b ¨ uc her Informatik, 1979. [Dup01] J. Duparc. W adge hierarc hy and v eblen hierarc h y: Part 1: Borel sets of finite rank. Journal of Symb olic L o gic , 66( 1 ):56–86, 2001. [Fin01] O. Fink el. T op olo g ical prop erties of omega conte xt f r ee languages. The or etic al Comp uter Scienc e , 2 6 2(1–2):669 – 697, 2001. [Fin03] O. Fink el. Borel hierarc hy and omega context free languages. The- or etic al C o mputer Scienc e , 290(3):13 8 5–1405, 2003. [FS03] O. Fink el and P . Simonnet. T op ology and ambiguit y in omega con- text free languag es. Bul letin of the B elgian Mathematic al S o c iety , 10(5):707– 722, 2003. [HU79] J. E. Hop croft and J. D . Ullman. I ntr o duction to a utomata the- ory, languages, and c o m putation . Addison-W esley Publishing Co., Reading, Mass., 1979 . Addison-W esley Series in Computer Science. [Kec95] A. S. Kechris . Classic al descriptive set the ory . Springer-V erlag, New Y ork, 1 995. [Kur66] K. Kurato wski. T op olo gy . Academic Press, New Y ork, 1966. [Lec02] D. Lecom te. Sur les ensem bles de phrases infinies constructibles a partir d’un dictionnaire sur un alphab et fini. In S´ eminair e d’Initiation a l’Analyse, V ol. 1 . Univ ersit´ e P aris 6, 20 01–2002. [L T94] H. Lesco w and W. Thomas. Logical sp ecifications o f infinite com- putations. In J. W. de Bakke r, Willem P . de Ro ev er, and Grzegorz Rozen b erg , editors, A De c a de of Concurr ency , v olume 803 of L e c- tur e Notes in Computer Sc ienc e , pages 5 83–621. Springer, 1 994. [Mos80] Y. N. Mosc hov akis. D escriptive set the ory . North- Ho lla nd Publish- ing Co., Amsterdam, 19 8 0. [Niw90] D. Niwinski. A problem on ω -p ow ers. In 1990 Worksho p o n L o gics and R e c o gnizabl e Sets , Unive rsit y of Kiel, 1990. 11 [PP04] D. P errin and J.- E. Pin. Infinite wor ds, automata, semigr oups, lo g ic and games , v o lume 141 of Pur e and Applie d Mathematics . Elsevier, 2004. [Sim92] P . Simonnet. Automates et th ´ eorie descriptive . PhD thesis, Uni- v ersit ´ e P a r is VI I, 1992 . [Sta86] L. Staiger. Hierarc hies of recursiv e ω -languages. Elektr onische In - formationsver arb eitung und Kyb ernetik , 22(5-6 ) :2 19–241, 1986. [Sta97a] L. Staiger. ω - languages. In Handb o ok of form al lan guages, V ol. 3 , pages 33 9 –387. Springer, Berlin, 1997 . [Sta97b] L. Staiger. On ω -p ow er lang uages. In New T r en d s in F ormal L anguages, Contr ol, Cop er ation, and Combinatorics , volume 1 218 of L e ctur e Notes in Computer Scienc e , pages 377–393. Springer- V erlag, Berlin, 1997. [Tho90] W. Thomas. Automata on infinite ob jects. In J. v an Leeuw en, edi- tor, Handb o ok of Th e or etic al C o mputer Scienc e , v olume B, F ormal mo dels and seman tics, pages 135–19 1. Elsevier, 1990. 12