Topology and Ambiguity in Omega Context Free Languages

T op ology and am biguit y in ω -con text free languages Olivi er Fink el Equip e de L o gique Math´ ematique U.F.R. de Math ´ emati ques, Universit ´ e P aris 7 2 Plac e Jussieu 75251 Paris c e dex 05, F r anc e. E Mail: fink el@log i que.jussieu . fr Pierr e Simo n n et UMR CNRS 6134 F acult ´ e des Sciences, Universit ´ e de Corse Quartier Gr ossetti BP52 20250, Corte, F r anc e E Mail: simonnet @univ-cor se.fr. Abstract W e study the links b et we en the top ological complexit y of an ω -con text free language and its degree of ambig uit y . In particular, using kno wn facts fr om classical descrip tive set theory , we pro v e that n on Borel ω -con text free lan- guages whic h are reco gnized b y B ¨ uc hi push do wn automata ha v e a maxim um degree of am biguit y . This resu lt implies th at degrees of am biguit y are really not preserv ed by the op eration W → W ω , defined o ver finitary con text free languages. W e pro v e also that taking the adherence or the δ -limit of a finitary language preserves neither am b iguit y nor inherent ambiguit y . On the other side we s ho w that metho ds u sed in the s tu dy of ω -con text fr ee languages can also b e a pplied to stu dy the notion of ambiguit y in infin itary rational relations accepted by B ¨ uchi 2-tap e automata and we get fi r st results in that dir ection. Keyw ords: con text free languag es; infin ite w ords; in finitary r ational relatio ns; am biguit y; degrees of am biguit y; top ological pr op erties; b orel hierarch y; analytic sets. AMS Sub ject C lassification: 68Q45; 03D05; 03D55; 03E15. 1 1 In t r o ducti on ω -con text free languages ( ω -CFL) form th e class C F L ω of ω -languages accepted by pu sh- do wn automata with a B ¨ uc hi or Muller acceptance condition. They w ere fi rstly s tu died b y Cohen and Gold, Linna, Boasson, Niv at, [CG77] [Lin76 ] [BN8 0] [Niv77], see Staiger’s pap er for a survey of th ese w orks [Sta97a]. A wa y to study the ric hness of the class C F L ω is to consider the top ological complexit y of ω -con text free languages when the set Σ ω of infinite words o ver th e alphab et Σ is equip p ed with the usual C an tor top ology . It is w ell kno wn that all ω -CFL as w ell as all ω -languages accepted b y T uring mac hines with a B ¨ uc hi or a Muller acceptance condition are analytic sets. ω -CFL accepted b y determinis- tic B ¨ uchi pushd o w n automata are Π 0 2 -sets, while ω -CFL accepted b y deterministic Muller pushd o wn automata are b o olean com b inations of Π 0 2 -sets. It w as recen tly p ro v ed that the class C F L ω exhausts the finite r anks of th e Borel hierarch y , [Fin01], that there exists some ω -CFL which are Borel sets of infi n ite rank, [Fin03b], or eve n analytic b u t n on Borel sets, [Fin03a]. Using kno wn facts fr om Descriptive Set Th eory , we pr o ve here that non Borel ω -CFL ha v e a maxim um degree of am b iguit y: if L ( A ) is a non Borel ω -CFL which is accepted b y a B ¨ uc hi pu shdown automaton (BPD A) A then there exist 2 ℵ 0 ω -wo rds α suc h that A has 2 ℵ 0 accepting runs reading α , where 2 ℵ 0 is the cardinal of the con tin uum. The ab o v e r esu lt of th e second author led the first author to the inv estigation of th e notion of am biguit y and of d egrees of ambiguit y in ω -cont ext free languages, [Fin03c ]. There exist some non ambiguous ω -CFL of ev ery fi n ite Borel rank, bu t all kno wn examples of ω -CFL whic h are Borel sets of infinite rank are acc epted by am b iguous BPD A. T hus one can mak e the h yp othesis that there are some links b et ween the top ologica l complexity and the degree of am biguit y for ω -C FL and su c h connections were fi r stly stu died in [Fin03c]. The op erations W → Adh ( W ) and W → W δ , where Adh ( W ) is the adherence of the finitary language W ⊆ Σ ⋆ and W δ is the δ -limit of W , app ear in the charact erization of Π 0 1 (i.e. closed)-subsets and Π 0 2 -subsets of Σ ω , for an alphab et Σ, [Sta97a]. Moreo ver it turned out that the fi rst one is usefu l in the stud y of top ological pr op erties of ω - con text fr ee languages of a giv en degree of am biguity [Fin03c]. W e show that eac h of these op erations preserv es neither unambiguit y n or inherent ambiguit y from fi nitary to ω - con text free languages. W e d educe also from the ab o ve resu lts that neither un am biguit y nor inheren t ambiguit y is preserve d b y the op eration W → W ω . This imp ortan t op eration is d efined o ver finitary languages and is inv olve d in the charac terization of the class of ω - regular languages (resp ectiv ely , of ω -con text fr ee languages) as the ω -Kleene closure of the class of regular (resp ecti v ely , cont ext free) languages [Tho90] [PP 02 ] [Sta97a] [Sta97b]. On the other side w e prov e that the same theorems of classical descriptiv e set theory can also b e applied in the case of infinitary rational relations accepted by 2-ta p e B ¨ uc hi automata. The top olog ical complexit y of infinitary rational relations has b een stu died by the fi rst author who sho w ed in [Fin03d] th at th ere exist some in finitary rational relations 2 whic h are not Borel. Moreo v er some undecidabilit y prop erties ha ve b een established in [Fin03e]. W e then p ro v e some first r esu lts ab out am biguit y in in finitary rational relations. The pap er is organized as follo w s. In sec tion 2, w e recall defin itions and results about ω -CFL and ambiguit y . In section 3, Borel and an alytic sets are defined. In section 4, w e study links b etw een top ology and am biguit y in ω -C FL. In section 5, we show some results ab out infinitary rational r elations. 2 ω -con te xt free language s W e assu me the r eader to b e familiar w ith the theory of formal languages and of ω - regular languages, [Ber79] [Th o90] [Sta97a] [PP02]. W e shall use usu al notations of formal language theory . When Σ is a finite alphab et, a non-empt y fi n ite wo rd ov er Σ is any sequence x = a 1 . . . a k , wh ere a i ∈ Σ for i = 1 , . . . , k , and k is an in teger ≥ 1. The length of x is k , denoted b y | x | . W e write x ( i ) = a i and x [ i ] = x (1) . . . x ( i ) for i ≤ k . W e w rite also x [0] = λ , where λ is the empty word, whic h has no letter; its length is | λ | = 0. Σ ⋆ is the set of finite wo rds o ver Σ, and Σ + is the set of finite non-empt y w ords o ve r Σ. The mirror image of a finite wo rd u will b e denoted by u R . The first in finite ordinal is ω . An ω -w ord o ver Σ is an ω -sequence a 1 . . . a n . . . , where ∀ i ≥ 1 a i ∈ Σ. The set of ω -w ord s o ver the alphab et Σ is denoted by Σ ω . An ω - language o ve r an alphab et Σ is a subset of Σ ω . F or V ⊆ Σ ⋆ , the ω -p o wer of V is the ω -language V ω = { σ = u 1 . . . u n . . . ∈ Σ ω | ∀ i ≥ 1 u i ∈ V − { λ }} . LF ( v ) is the set of finite prefixes (or left factors) of the word v , and LF ( V ) = ∪ v ∈ V LF ( v ) for ev ery language V of finite or infi nite words. W e introdu ce now ω -con text free languages via B ¨ uc hi pus hdo wn automata. Definition 2.1 A B¨ uchi pushdown automaton is a 7-tuple A = ( K , Σ , Γ , δ , q 0 , Z 0 , F ) , wher e K is a finite set of states, Σ is a finite input alphab et, Γ is a finite pushdown alphab et, q 0 ∈ K is the initial state, Z 0 ∈ Γ is the start symb ol, F ⊆ K is the set of final states, and δ is a mapping fr om K × (Σ ∪ { λ } ) × Γ to finite su b sets of K × Γ ⋆ . If γ ∈ Γ + describ es the pushdown stor e c ontent, the leftmost symb ol wil l b e assume d to b e on “top” of the stor e. A c onfigur ation of the BPDA A is a p air ( q , γ ) wher e q ∈ K and γ ∈ Γ ⋆ . F or a ∈ Σ ∪ { λ } , γ , β ∈ Γ ⋆ and Z ∈ Γ , if ( p, β ) is in δ ( q , a, Z ) , then we write a : ( q , Z γ ) 7→ A ( p, β γ ) . L et σ = a 1 a 2 . . . a n . . . b e an ω -wor d over Σ . A run of A on σ is an infinite se quenc e r = ( q i , γ i , ε i ) i ≥ 1 wher e ( q i , γ i ) i ≥ 1 is an infinite se quenc e of c onfigur ations of A and, for al l i ≥ 1 , ε i ∈ { 0 , 1 } and: 1. ( q 1 , γ 1 ) = ( q 0 , Z 0 ) 3 2. for e ach i ≥ 1 , ther e exists b i ∈ Σ ∪ { 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 . . . I n ( r ) is the set of al l states enter e d infinitely often during run r . The ω - language ac c epte d by A is L ( A ) = { σ ∈ Σ ω | ther e exists a run r of A on σ such that I n ( r ) ∩ F 6 = ∅} The class C F L ω of ω -con text f ree languages is the class of ω -languages accepted by B ¨ uc hi pu shdo wn automata. It is also the ω -Kleene clo sure of the class C F L of con text free finitary languages, where for an y family L of finitary languages, the 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} . If w e omit the p ushd own stac k and the λ -transitions, we get the classical n otion of B ¨ uc hi automaton. Recall that th e class RE G ω of ω -regular languages is the class of ω -languages accepted b y finite automata with a B ¨ uc hi acceptance condition. It is also the ω -Kleene closure of the class RE G of regular fin itary languages, [Th o90] [Sta97a] [PP02]. Notice that w e in trod uced in th e ab o v e definition the num b ers ε i ∈ { 0 , 1 } in order to distinguish runs of a BPD A wh ic h go th rough the same infinite sequ ence of configur ations but for whic h λ -trans itions do n ot o ccur at the same steps of the compu tations. W e can no w briefly recall some defin itions of [Fin03c] ab out am biguit y . W e shall denote ℵ 0 the cardinal of ω , and 2 ℵ 0 the card in al of the cont in uum. It is also the cardinal of the set of real num b ers and of the s et Σ ω for ev ery finite alphab et Σ havi ng at least t w o letters. Definition 2.2 L et A b e a BP DA ac c epting infinite wor ds over the alphab e t Σ . F or x ∈ Σ ω let α A ( x ) b e the c ar dinal of the set of ac c epting runs of A on x . Lemma 2.3 ([Fin03c]) L et A b e a BPDA ac c epting infinite wor ds over the alphab et Σ . Then for al l x ∈ Σ ω it holds that α A ( x ) ∈ N ∪ {ℵ 0 , 2 ℵ 0 } . Definition 2.4 L et A b e a BPDA ac c epting infinite wor ds over the alphab et Σ . (a) If sup { α A ( x ) | x ∈ Σ ω } ∈ N ∪ { 2 ℵ 0 } , then α A = sup { α A ( x ) | x ∈ Σ ω } . (b) If sup { α A ( x ) | x ∈ Σ ω } = ℵ 0 and ther e is no wor d x ∈ Σ ω such that α A ( x ) = ℵ 0 , then α A = ℵ − 0 . ( ℵ − 0 do es not r epr esent a c ar dinal but is a new symb ol that we intr o duc e to c onve- niently sp e ak of this situation). (c) If sup { α A ( x ) | x ∈ Σ ω } = ℵ 0 and ther e exists (at le ast) one wor d x ∈ Σ ω such that α A ( x ) = ℵ 0 , then α A = ℵ 0 4 Notice that for a BPD A A , α A = 0 iff A do es not accept an y ω -word. N ∪ {ℵ − 0 , ℵ 0 , 2 ℵ 0 } is linearly ordered by the relation < defined by ∀ k ∈ N , k < k + 1 < ℵ − 0 < ℵ 0 < 2 ℵ 0 . No w we can defin e a hierarc h y of ω -CFL: Definition 2.5 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 non ambiguous ω -c ontext fr e e languages. F or e v ery 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 b e inher ently ambiguous of de gr e e k . Recall that on e can defin e in a s imilar wa y the degree of am biguity of a finitary con text free language. If M is a push d o wn automaton accepting finite w ords b y final states (or b y final s tates and topmost stac k letter) then α M ∈ N or α M = ℵ − 0 or α M = ℵ 0 . Ho w ev er ev ery con text free language is accepted by a pu shdown automaton M with α M ≤ ℵ − 0 , [ABB96]. W e shall denote, with similar notations as ab o ve , th e class of non ambiguous con text free languages by N A − C F L and the class of inh er ently ambiguous con text free languages of degree k ≥ 2 by A ( k ) − C F L . Then A ( ℵ − 0 ) − C F L is u s ually called the class of con text free languages which are inherently ambiguous of infin ite degree, [Her97]. No w we can state some links b et ween cases of finite and in finite w ords. Prop osition 2.6 ([Fin03c]) L et V ⊆ Σ ⋆ b e a finitary c ontext fr e e language and d b e a new letter not in Σ , then the fol lowing e quivalenc e hold s for al l k ∈ N ∪ {ℵ − 0 } : V .d ω is in C F L ω ( α ≤ k ) iff V is in C F L ( α ≤ k ) 3 Borel and analytic sets W e assume the reader to b e familiar with basic notions of top ology which ma y b e found in [Mos80] [L T94] [Kec95] [Sta97a] [PP 02]. F or a finite alphab et X w e shall consider X ω as a topological space with the Can tor top ology . The op en sets of X ω are the sets in the form W .X ω , w here W ⊆ X ⋆ . A set L ⊆ X ω is a closed s et iff its complement X ω − L is an op en set. Define no w the hierarc h y of Borel sets of finite ranks: Definition 3.1 The classes Σ 0 n and Π 0 n of the Bor el hier ar chy on the top olo gic al sp ac e X ω ar e define d as fol lows: Σ 0 1 is the class of op en sets of X ω . Π 0 1 is the class of close d sets of X ω . And for any inte ger n ≥ 1 : Σ 0 n + 1 is the class of c ountable u nions of Π 0 n -subsets of X ω . Π 0 n + 1 is the class of c ountable i nterse ctions of Σ 0 n -subsets of X ω . 5 The Borel h ierarc h y is also d efined for transfi nite lev els, but w e shall not n eed th em in the presen t stud y . The class of Bo rel subsets of X ω is the closure of the class of op en subsets of X ω under complementa tion and counta ble u nions (hence also und er count able in tersections) There are also some s ubsets of X ω whic h are not Borel. In particular th e class of Borel s ubsets of X ω is strictly included in to the class Σ 1 1 of analytic sets w hic h are obtained by pro jection of Borel sets. Notice that if Σ and Γ are t w o fin ite alph ab ets then th e pro d uct Σ ω × Γ ω can b e id entified with the sp ace (Σ × Γ ) ω and w e alwa ys consid er in the sequel th at such a space Σ ω × Γ ω is equipp ed with the C an tor top ology . Definition 3.2 A set A ⊆ Σ ω is an analytic set if ther e i s a finite alphab et Γ and a Bor el set B ⊆ Σ ω × Γ ω such that A = { α ∈ Σ ω | ∃ β ∈ Γ ω ( α, β ) ∈ B } . A set C ⊆ Σ ω is c o analytic if its c omplement Σ ω − C is analytic. The class of analytic sets is denote d Σ 1 1 and the class of c o analy tic sets is denote d Π 1 1 . Recall also the notion of completeness with regard to reduction by con tin uous fun ctions. F or an int eger n ≥ 1, a set F ⊆ X ω is said to b e a Σ 0 n (resp ectiv ely , Π 0 n , Σ 1 1 , Π 1 1 )- complete set iff for an y set E ⊆ Y ω (with Y a fi nite alphab et): E ∈ Σ 0 n (resp ectiv ely , E ∈ Π 0 n , E ∈ Σ 1 1 , E ∈ Π 1 1 ) iff th er e exists a cont in 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 integer ≥ 1, are th oroughly characte rized in [Sta86]. 4 T op ology and am big uit y in ω -con te xt free lan- guages Let Σ and X b e tw o finite alphab ets. I f B ⊆ Σ ω × X ω and α ∈ Σ ω , the sectio n in α of B is B α = { β ∈ X ω | ( α, β ) ∈ B } and th e pro jection of B on Σ ω is th e set P RO J Σ ω ( B ) = { α ∈ Σ ω | B α 6 = ∅} = { α ∈ Σ ω | ∃ β ( α, β ) ∈ B } . W e are going to pr o ve the follo win g lemma wh ich will b e u seful in the sequel: Lemma 4.1 L et Σ and X b e two finite alphab ets having at le ast two letters and B b e a Bor el subset of Σ ω × X ω such that P RO J Σ ω ( B ) is not a Bor e l subset of Σ ω . Then ther e ar e 2 ℵ 0 ω -wor ds α ∈ Σ ω such that the se ction B α has c ar dinality 2 ℵ 0 . Pro of. Let Σ and X b e tw o finite alphab ets having at least t w o letters and B b e a Borel subset of Σ ω × X ω suc h that P RO J Σ ω ( B ) is not Borel. In a first step we shall pr o ve that there are un coun tably man y α ∈ Σ ω suc h that the section B α is u ncoun table. 6 Recall that b y a Theorem of Lusin and No vik ov, see [Kec95 , page 123], if for all α ∈ Σ ω , the section B α of the Borel set B w as counta ble, then P RO J Σ ω ( B ) w ould b e a Borel subset of Σ ω . Th us there exists at least one α ∈ Σ ω suc h that B α is un coun table. In fact w e ha v e not only one α such that B α is u ncoun table. F or α ∈ Σ ω w e hav e { α } × B α = B ∩ [ { α } × X ω ]. But { α } × X ω is a closed hence Borel subset of Σ ω × X ω th us { α } × B α is Borel as int ersection of t w o Borel sets. If there was only one α ∈ Σ ω suc h that B α is uncountable, th en C = { α } × B α w ould b e Borel so D = B − C wo uld b e b orel b ecause the class of Borel sets is closed under b o olean op erations. But all sections of D w ould b e countable thus P RO J Σ ω ( D ) would b e Borel by Lus in and No viko v’s T heorem. Then P R O J Σ ω ( B ) = { α } ∪ P R O J Σ ω ( D ) w ould b e also Borel as union of t w o Borel sets, an d this w ould lead to a con tradiction. In a similar manner we can pro ve that the set U = { α ∈ Σ ω | B α is uncountable } is uncounta ble, otherwise U = { α 0 , α 1 , . . . α n , . . . } wo uld b e Borel as the countable u nion of the closed sets { α i } , i ≥ 0. F or eac h n ≥ 0 the set { α n } × B α n w ould b e Borel, and C = ∪ n ∈ ω { α n } × B α n w ould b e Borel as a coun table union of Borel sets. S o D = B − C would b e b orel to o. But all sections of D w ould b e countable thus P RO J Σ ω ( D ) would b e Borel by Lus in and No viko v’s Theorem. Then P RO J Σ ω ( B ) = U ∪ P RO J Σ ω ( D ) wo uld b e also Borel as union of t w o Borel sets, and this wo uld lead to a con tradiction. So we h a v e prov ed that th e set { α ∈ Σ ω | B α is u ncoun table } is uncounta ble. On th e other h and we kno w from another Th eorem of Descriptiv e Set Th eory that the set { α ∈ Σ ω | B α is countable } is a Π 1 1 -subset of Σ ω , see [Kec95, page 123]. Th us its complemen t { α ∈ Σ ω | B α is u ncoun table } is analytic. But by Sus lin ’s Theorem an analytic su bset of Σ ω is either countable or h as cardin alit y 2 ℵ 0 , [Kec95 , p. 88]. T herefore the set { α ∈ Σ ω | B α is uncountable } has cardinalit y 2 ℵ 0 . Recall no w that w e ha v e already seen that, for eac h α ∈ Σ ω , the set { α } × B α is Borel. W e can then infer that B α itself is Borel by considering the function h : X ω → Σ ω × X ω defined b y h ( σ ) = ( α, σ ) f or all σ ∈ X ω . The fun ction h is contin uous and B α = h − 1 ( { α } × B α ). So B α is Borel b ecause the inv erse image of a Borel s et by a con tin u ous fu nction is a Borel set. Again by Suslin’s Theorem B α is either count able or h as cardinalit y 2 ℵ 0 . F rom this w e d educe that { α ∈ Σ ω | B α is uncountable } = { α ∈ Σ ω | B α has card inalit y 2 ℵ 0 } h as cardinalit y 2 ℵ 0 .  W e can n o w infer some r esults f or ω -con text free languages. 7 Theorem 4.2 L et L ( A ) b e an ω -CFL ac c epte d by a BP DA A such tha t L ( A ) is an an- alytic but non Bor el set. The set of ω -wor ds, which have 2 ℵ 0 ac c epting runs by A , has c ar dinality 2 ℵ 0 . Pro of. Let A = ( K, Σ , Γ , δ , q 0 , Z 0 , F ) b e a BPD A suc h that L ( A ) is an analytic bu t non Borel set. T o an infinite sequence r = ( q i , γ i , ε i ) i ≥ 1 , where for all i ≥ 1, q i ∈ K , γ i ∈ Γ + and ε i ∈ { 0 , 1 } , we asso ciate an ω -word ¯ r ov er the alphab et X = Γ ∪ K ∪ { 0 , 1 } d efined by ¯ r = q 1 .γ 1 .ε 1 .q 2 .γ 2 .ε 2 . . . q i .γ i .ε i . . . Then to an infin ite word σ ∈ Σ ω and an in finite sequence r = ( q i , γ i , ε i ) i ≥ 1 , w e asso ciate the couple ( σ, ¯ r ) ∈ Σ ω × (Γ ∪ K ∪ { 0 , 1 } ) ω . Recall no w that Π 0 2 -subsets of a Cantor set Σ ω are c h aracterized in the follo win g wa y . F or W ⊆ Σ ⋆ the δ -limit W δ of W is the set of ω -w ords o v er Σ having infinitely many p refixes in W : W δ = { σ ∈ Σ ω | ∃ ω i suc h that σ (1) . . . σ ( i ) ∈ W } . T hen a subset L of Σ ω is a Π 0 2 -subset of Σ ω iff there exists a set W ⊆ Σ ⋆ suc h that L = W δ , [Sta97a] [PP02]. It is then easy to see that the set R = { ( σ , ¯ r ) | ¯ r is th e co de of an accepting run of A o v er σ } is a Π 0 2 -subset of Σ ω × X ω = (Σ × X ) ω as in tersection of t w o Π 0 2 -sets. In fact we hav e R = ( R ′ ) δ ∩ ( R ′′ ) δ where R ′ ⊆ (Σ × X ) + is the set of couples of wo rds ( u, v ) in the f orm : u = a 1 .a 2 . . . . a p v = q 1 .γ 1 .ε 1 .q 2 .γ 2 .ε 2 . . . q n .γ n .ε n where for eac h i ∈ [1 , p ] a i ∈ Σ, for eac h i ∈ [1 , n ] q i ∈ K , γ i ∈ Γ + and ε i ∈ { 0 , 1 } . Moreo ver | u | = | v | , ε n = 1, and 1. ( q 1 , γ 1 ) = ( q 0 , Z 0 ) 2. for eac h i ∈ [1 , n − 1], there exists b i ∈ Σ ∪ { 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 suc h that b 1 b 2 . . . b n − 1 is a prefix of u = a 1 .a 2 . . . . a p . And R ′′ ⊆ (Σ × X ) + is the set of couples of words ( u, v ) ∈ Σ + × X + suc h th at | u | = | v | and the last letter of v is an elemen t q ∈ F . In particular R is a Borel subs et of Σ ω × X ω . But by definition of R it tur ns out that P RO J Σ ω ( R ) = L ( A ) so P R O J Σ ω ( R ) is not Borel. Thus Lemma 4.1 implies that there are 2 ℵ 0 ω -wo rds α ∈ Σ ω suc h that R α has cardinalit y 2 ℵ 0 . Th is means that these w ords ha v e 2 ℵ 0 accepting runs by the B ¨ uc hi pushd o w n automaton A .  8 Example 4.3 L et Σ = { 0 , 1 } and d b e a new letter not in Σ and D = { u.d.v | u, v ∈ Σ ⋆ and ( | v | = 2 | u | ) or ( | v | = 2 | u | + 1) } D ⊆ (Σ ∪ { d } ) ⋆ is a c ontext fr e e language. L et g : Σ → P ((Σ ∪ { d } ) ⋆ ) b e the substitution define d by g ( a ) = a.D . As W = 0 ⋆ 1 is r e gular, g ( W ) is a c ontext fr e e language, thus ( g ( W )) ω is an ω -CFL. It is pr ove d in [Fin03a] that ( g ( W )) ω is Σ 1 1 -c omplete. In p articular ( g ( W )) ω is an analytic non Bor el set. Thus every BPDA ac c epting ( g ( W )) ω has the maximum ambiguity and ( g ( W )) ω ∈ A (2 ℵ 0 ) − C F L ω . On the other hand we c an pr ove tha t g ( W ) is a non ambiguous c ontext fr e e language. F or that purp ose c onsider a (finite) wor d x ∈ g ( W ) ; then x ∈ g (0 n . 1) for some i nte ger n ≥ 0 . Ther efor e x may b e written in the form x = 0 .u 1 .d.v 1 . 0 .u 2 .d.v 2 . . . 0 .u n .d.v n . 1 .u n +1 .d.v n +1 wher e u i .d.v i ∈ D holds for al l i ∈ [1 , n + 1] . It is e asy to se e that the length | v n +1 | and the wor d v n +1 ar e determine d by the wor d x : v n +1 is the suffix o f x fol lowing th e last letter d of x , and | v n +1 | = 2 | u n +1 | (if | v n +1 | is even) or | v n +1 | = 2 | u n +1 | + 1 (if | v n +1 | is o dd) thus | u n +1 | is determine d by | v n +1 | henc e u n +1 is also determine d. Next one c an se e that v n also is fixe d b y x (the wor d v n . 1 .u n +1 is the se gment of x which is lo c ate d b etwe en the n th and the ( n + 1) th o c curr enc es of the letter d in x and knowing u n +1 gives us v n ). We c an similarly pr ove by induction on the inte ger k tha t the wor ds v n +1 − k and u n +1 − k , for k ∈ [0 , n ] , ar e uniquely determine d by x . Ther efor e the wor d x admits a unique de c omp osition in the ab ove form. We c an then e asily c onstruct a pushdow n automaton (and even a one c ounter automato n) which ac c e pts the language g ( W ) and which is non ambiguous. So the language g ( W ) is a non ambiguous c ontext fr e e language. The ab o v e example sho ws that the ω -p ow er of a non am b iguous con text free language ma y ha v e maxim um ambiguit y . Conv ersely consider the con text free language V = V 1 ∪ V 2 ⊆ { a, b, c } ⋆ where V 1 = { a n b n c p | n ≥ 1 , p ≥ 1 } and V 2 = { a n b p c p | n ≥ 1 , p ≥ 1 } . V 1 and V 2 are deterministic con text free, hence they are non am biguous con text fr ee languages. But their union V is an inh eren tly am biguous con text free language [Mau69]. V ⋆ is a con text free language wh ic h is inherently am biguous of infi n ite d egree (and it is prov ed in [Na j 98] that it is eve n exp onen tially am b iguous in the sense of Na ji and Wic h, see also [Wic99] ab out this notion). Let then L = V ⋆ ∪ { a, b, c } . The language L is still a cont ext free language which is inh eren tly am biguous of infin ite d egree and L ω = { a, b, c } ω is an ω -regular language hence it is a n on ambiguous ω -con text f ree language. W e h a v e then prov ed th at neither unambiguit y nor inherent ambiguit y is preserved by th e op eration L → L ω : 9 Prop osition 4.4 1. Ther e exists a non ambiguous c ontext fr e e finitary language L such th at L ω is in A (2 ℵ 0 ) − C F L ω . 2. Ther e exists a c ontext fr e e finitary language L , which is inher ently ambiguous of infinite de gr e e, such that L ω is a non ambiguous ω -c ontext fr e e language. W e can also consider the ab o ve men tioned language g ( W ) in the con text of co de th eory . W e ha v e pr ov ed that g ( W ) is a n on am biguous con text free language. By a similar reasoning w e can p ro v e that g ( W ) is a co d e, i.e. that ev ery (fin ite) w ord y ∈ g ( W ) + has a u nique decomp osition y = x 1 .x 2 . . . x n in w ords x i ∈ g ( W ). On the other sid e g ( W ) is not an ω -code, i.e. some w ords z ∈ g ( W ) ω ha v e sev eral decomp ositions in th e form z = x 1 .x 2 . . . x n . . . where for all i ≥ 1 x i ∈ g ( W ). In fact we can get a m uc h stronger result, usin g Lemma 4.1: F act 4.5 Ther e ar e 2 ℵ 0 ω -wor ds in g ( W ) ω which have 2 ℵ 0 de c omp ositions in wor ds i n g ( W ) . Pro of. W e can fix a recursive enumeratio n θ of the set g ( W ). So th e f unction θ : N → g ( W ) is a bijection and we d enote u i = θ ( i ). Let no w D b e the set of coup les ( σ, x ) ∈ { 0 , 1 } ω × (Σ ∪ { d } ) ω suc h that: 1. σ ∈ (0 ⋆ . 1) ω , so σ ma y 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. x = u n 1 .u n 2 .u n 3 . . . u n p .u n p +1 . . . D is a Borel subset of { 0 , 1 } ω × (Σ ∪ { d } ) ω b ecause it is accepted by a d etermin is- tic T uring m achine with a B ¨ uchi acceptance cond ition [S ta97a]. On the other hand P RO J (Σ ∪{ d } ) ω ( D ) = g ( W ) ω is not Bo rel and Lemma 4.1 im p lies that there are 2 ℵ 0 ω - w ords x in g ( W ) ω suc h that D x has cardinalit y 2 ℵ 0 . T his means that there are 2 ℵ 0 ω -wo rds x ∈ g ( W ) ω whic h ha v e 2 ℵ 0 decomp ositions in wo rds in g ( W ). W e can say that the co de g ( W ) is r eally not an ω -co de !  The r esu lt giv en by Theorem 4.2 ma y b e compared w ith a general study of top olog ical prop erties of transition systems due to Arnold [Arn83a]. If w e consider a BPDA as a tran- sition system with infin itely man y states, Arn old’s r esults imply that ev ery n on am biguous ω -CFL is a Borel set. O n the other side d eterministic ω -CFL hav e not a great top ologica l complexit y , b ecause they are b o olean com binations of Π 0 2 -sets. W e know some examples of non am biguous ω -CFL of ev ery finite Borel rank, but none of infi nite Borel rank. These results led the first auth or to the follo wing question: are there s ome more links b et ween 10 the top ological complexity of an ω -CFL and the ambiguit y of BPD A which accept it? In [Fin03c] the well kn o wn notions of degrees of am biguit y for CFL are extended to ω -CFL and suc h sup p osed connections are in vestig ated. In particular, using results of Duparc on the W adge h ierarc h y , whic h is a great refinement of the Borel hierarch y [Dup 01], it is pro v ed that f or eac h k su c h that k is an inte ger ≥ 2 or k = ℵ − 0 and for eac h inte ger n ≥ 1, there exist in A ( k ) − C F L ω some Σ 0 n -complete ω -CFL an d some Π 0 n -complete ω -CFL. In the pr o ofs of these results is used the operation W → Adh ( W ) where for a finitary language W ⊆ Σ ⋆ , Adh ( W ) = { σ ∈ Σ ω | LF ( σ ) ⊆ LF ( W ) } is the adher en ce of W . W e recall that a set L ⊆ Σ ω is a closed set of Σ ω iff there exists a finitary language W ⊆ Σ ⋆ suc h that L = Adh ( W ). It is well known that if W is a con text fr ee language, th en Adh ( W ) is in C F L ω . Moreo v er ev ery closed (deterministic ) ω -CFL is the adherence of a (deterministic ) con text free language, [Sta97a]. So th e qu estion of the p reserv ation of ambiguit y b y th e op eratio n W → Adh ( W ) naturally arises. Prop osition 4.6 Neither unambiguity nor inher ent ambiguity is pr eserve d by taking the adher e nc e of a finitary c ontext fr e e language. Pro of. (I) W e are fir stly lo oking for a non am b iguous finitary context fr ee language whic h ha v e an inheren tly am biguous a dherence. Let then the follo wing finitary language o ver the alphab et { a, b, c, d } : L 1 = { a n b n c p .d 2 i | n, p, i are in tegers ≥ 1 } ∪ { a n b p c p .d 2 i +1 | n, p, i are in tegers ≥ 1 } L 1 is the d isjoin t un ion of t w o deterministic (hence non ambiguous) finitary con text free languages thus it is a non ambiguous C FL b ecause the class N A − C F L is closed under finite disjoint union. It is easy to see that th e adherence of L 1 is Adh ( L 1 ) = { a ω } [ a + .b ω [ { a n b n | n ≥ 1 } .c ω [ ( V 1 ∪ V 2 ) .d ω where V 1 = { a n b n c p | n ≥ 1 , p ≥ 1 } and V 2 = { a n b p c p | n ≥ 1 , p ≥ 1 } . Then it h olds that Adh ( L 1 ) ∩ a + .b + .c + .d ω = ( V 1 ∪ V 2 ) .d ω = V .d ω , where V = V 1 ∪ V 2 . By prop osition 2.6, the ω -con text free language V .d ω is inherentl y am biguous b ecause V is inherent ly ambiguous [Mau69]. Thus Adh ( L 1 ) is inh eren tly am biguous b ecause otherw ise V .d ω w ould b e n on am biguous b ecause the class N A − C F L ω is closed und er in tersection with ω -regular languages [Fin03c], and a + .b + .c + .d ω is an ω -regular language. (I I) W e are no w lo oking for an inherent ly am biguous con text fr ee language whic h hav e a non am b iguous adheren ce. W e shall u se a result of Crestin, [Cre72]: the language C = { u.v | u, v ∈ { a, b } + and u R = u and v R = v } is a con text free language whic h is inherent ly am biguous (of infin ite degree). In fact C = L 2 p where L p = { v ∈ { a, b } + | v R = v } is th e language of palindromes ov er the alphab et { a, b } . Consider n ow the adherence of the language C . Adh ( C ) = { a, b } ω holds b ecause ev ery wo rd u ∈ { a, b } ⋆ is a prefix of a palindrome (for example of the palindrome u.u R ) hence it is also a p refix of a word of C . 11 Th us C is inherently am biguous and Adh ( C ) is a non am biguous ω -co n text free language b ecause it is an ω -r egular language.  W e hav e seen that closed sets are charact erized as adherences of fin itary languages. Sim- ilarly w e hav e already seen, in th e p ro of of Theorem 4.2, that Π 0 2 -subsets of Σ ω are c haracterize d as δ -limits W δ of finitary languages W ⊆ Σ ⋆ . Recall that W ∈ RE G implies th at W δ ∈ R E G ω . But there exist some context fr ee lan- guages L su c h that L δ is n ot in C F L ω ; see [Sta97a] for an example of such a language L . In the case W ∈ C F L and W δ ∈ C F L ω , the question naturally arises of the p reserv ation of am biguit y by the op eration W → W δ . The answe r is giv en by the follo w ing: Prop osition 4.7 Neither unambiguity nor inher ent ambiguity is pr eserve d by taking the δ -limit of a finitary c ontext f r e e language. Pro of. (I) Let again L 1 b e the follo wing fin itary language o v er the alph ab et { a, b, c, d } : L 1 = { a n b n c p .d 2 i | n, p, i are in tegers ≥ 1 } ∪ { a n b p c p .d 2 i +1 | n, p, i are in tegers ≥ 1 } L 1 is a non ambiguous CFL. And the δ -limit of the language L 1 is ( L 1 ) δ = ( V 1 ∪ V 2 ) .d ω = V .d ω . W e ha v e already seen that this ω -language is an inherently ambiguous ω -CFL. (I I) Consider no w the inherentl y am b iguous con text f r ee language V = { a n b n c p | n, p ≥ 1 } ∪ { a n b p c p | n, p ≥ 1 } . Its δ -limit is V δ = { a n .b n | n ≥ 1 } .c ω . It is easy to see that V δ is a deterministic ω -CFL hence it is a non am biguous ω -C FL.  5 T op ology and am biguity in infinitary rational relations Infinitary rational relations are su bsets of Σ ω × Γ ω , where Σ and Γ are fin ite alph ab ets, whic h are accepted by 2-tap e B ¨ uchi automata. W e are going to see in this section that some ab ov e metho ds can also b e us ed in the case of infinitary rational relations. Definition 5.1 A 2-tap e B ¨ uchi automaton (2-BA) is a sextuple T = ( K, Σ , Γ , ∆ , q 0 , F ) , wher e K is a finite set of states, Σ and Γ ar e finite alpha b ets, ∆ is a finite su b set of K × Σ ⋆ × Γ ⋆ × K c al le d the set of tr ansitions, q 0 is the i ni tial state, and F ⊆ K is the set of ac c epting states. A c omputation C of the 2-tap e B¨ uchi automaton T is an infinite se qu enc e of tr ansitions ( q 0 , u 1 , v 1 , q 1 ) , ( q 1 , u 2 , v 2 , q 2 ) , . . . ( q i − 1 , u i , v i , q i ) , ( q i , u i +1 , v i +1 , q i +1 ) , . . . The c omputation is said to b e suc c essful iff ther e exists a final state q f ∈ F and infinitely many inte gers i ≥ 0 su ch that q i = q f . 12 The input wor d of the c omputat ion is u = u 1 .u 2 .u 3 . . . The output wor d of the c omputa tion i s v = v 1 .v 2 .v 3 . . . Then the input and the output wor ds may b e finite or infinite. The infinitary r ational r elation R ( T ) ⊆ Σ ω × Γ ω ac c epte d by the 2-tap e B ¨ uchi automaton T is the set of c ouples ( u, v ) ∈ Σ ω × Γ ω such that u a nd v ar e the input and the output wor ds of some suc c essful c omputation C of T . The set of infinitary r ational r elations wil l b e denote d RAT ω . One can defin e degrees of am biguit y for 2-tap e B ¨ uc hi automata and for infinitary rational relations as in the case of BPD A and ω -CFL. Definition 5.2 L et T b e a 2-BA ac c epting c ouples of infinite wor ds of Σ ω × Γ ω . F or ( u, v ) ∈ Σ ω × Γ ω , let α T ( u, v ) b e the c ar dinal of the set of ac c epting c omputa tions of T on ( u, v ) . Lemma 5.3 L et T b e a 2-BA ac c epting c ouples of infinite wor ds ( u, v ) ∈ Σ ω × Γ ω . Then for al l ( u, v ) ∈ Σ ω × Γ ω it holds that α T ( u, v ) ∈ N ∪ {ℵ 0 , 2 ℵ 0 } . The pro of that a v alue b et ween ℵ 0 and 2 ℵ 0 is imp ossible follo ws from S uslin’s Theorem b ecause one can obtain the set of co des of accepting computations of T on ( u, v ) as a section of a Borel set (see p ro of of next theorem) hence as a Borel set. A similar reasoning w as used in the pro of of Lemma 2.3, [Fin03c]. Definition 5.4 L et T b e a 2-BA ac c epting c ouples of infinite wor ds ( u, v ) ∈ Σ ω × Γ ω . (a) If su p { α T ( u, v ) | ( u, v ) ∈ Σ ω × Γ ω } ∈ N ∪ { 2 ℵ 0 } , then α T = su p { α T ( u, v ) | ( u, v ) ∈ Σ ω × Γ ω } . (b) If sup { α T ( u, v ) | ( u, v ) ∈ Σ ω × Γ ω } = ℵ 0 and ther e i s no ( u, v ) ∈ Σ ω × Γ ω such that α T ( u, v ) = ℵ 0 , then α T = ℵ − 0 . (c) If sup { α T ( u, v ) | ( u, v ) ∈ Σ ω × Γ ω } = ℵ 0 and ther e exists (at le ast) one c ouple ( u, v ) ∈ Σ ω × Γ ω such that α T ( u, v ) = ℵ 0 , then α T = ℵ 0 The set N ∪ {ℵ − 0 , ℵ 0 , 2 ℵ 0 } is lin early ordered as ab ov e by the relation < . Definition 5.5 F or k ∈ N ∪ {ℵ − 0 , ℵ 0 , 2 ℵ 0 } , let RAT ω ( α ≤ k ) = { R ( T ) | T is a 2 − B A with α T ≤ k } RAT ω ( α < k ) = { R ( T ) | T is a 2 − B A with α T < k } N A − RAT ω = RAT ω ( α ≤ 1) is the class of non ambiguous infinitary r ational r elations. F or e v ery inte ger k ≥ 2 , or k ∈ {ℵ − 0 , ℵ 0 , 2 ℵ 0 } , A ( k ) − R AT ω = RAT ω ( α ≤ k ) − RAT ω ( α < k ) is the c lass of infinitary r ationa l r elations which ar e inher ently ambiguous of de gr e e k . As for ω -conte xt fr ee languages, one can use Lemma 4.1 to pro v e th e follo win g result. 13 Theorem 5.6 L et R ( T ) ⊆ Σ ω × Γ ω b e an infinitary r ational r e lation ac c epte d by a 2-tap e B¨ uchi automaton T suc h that R ( T ) is an analytic but non Bor el set. The set of c ouples of ω -wor ds, which have 2 ℵ 0 ac c epting c omputations by T , has c ar dinality 2 ℵ 0 . Pro of. It is very similar to pro of of Th eorem 4.2. Let R ( T ) ⊆ Σ ω × Γ ω b e an infin itary rational relation accepted by a 2-tap e B ¨ uchi automaton T = ( K, Σ , Γ , ∆ , q 0 , F ). W e assume also that R ( T ) is an analytic but n on Borel set. T o an in finite s equence C = ( q 0 , u 1 , v 1 , q 1 ) , ( q 1 , u 2 , v 2 , q 2 ) , . . . ( q i − 1 , u i , v i , q i ) , ( q i , u i +1 , v i +1 , q i +1 ) , . . . where for all i ≥ 0, q i ∈ K , for all i ≥ 1, u i ∈ Σ ⋆ and v i ∈ Γ ⋆ , we asso ciate an ω -w ord ¯ C o ver the alphab et X = K ∪ Σ ∪ Γ ∪ { e } , wh ere e is an additional letter. ¯ C is defined b y: ¯ C = q 0 .u 1 .e.v 1 .q 1 .u 2 .e.v 2 .q 2 . . . q i .u i +1 .e.v i +1 .q i +1 . . . Then the set { ( u, v , ¯ C ) ∈ Σ ω × Γ ω × X ω | ¯ C is the co de of an accepting computation of T o v er ( u, v ) } is accepted by a determin istic T u r ing mac hine with a B ¨ uchi acceptance condition thus it is a Π 0 2 -set. W e can conclude as in pro of of Th eorem 4.2.  The first author sho w ed that there exist some Σ 1 1 -complete, hen ce non Borel, infinitary rational relations [Fin03d]. So we can deduce the follo w ing r esu lt. Corollary 5.7 Ther e exist some infinitary r ational r elations which ar e inher ently am- biguous of de gr e e 2 ℵ 0 . Remark 5.8 L o oking c ar eful ly at the examp le of non Bor el infinitary r ational r elation given in [Fin03d], we c an find a r ational r elation S over finite wor ds such that S i s non ambiguous and S ω is non Bor el. So S is a finitary r ational r elation which is non ambiguous but S ω has maximum ambiguity b e c ause S ω ∈ A (2 ℵ 0 ) − R AT ω holds b y The or em 5.6. Moreo ver the question of the decidability of ambiguit y for infin itary r ational relations naturally arises. It can b e solv ed, u sing another recent result of the fi rst author. Prop osition 5.9 ([Fin03e]) L et X and Y b e finite alphab ets c ontaining at le ast two letters, then ther e exists a family F of infinitary r ational r elations which ar e subsets of X ω × Y ω , such that, for R ∈ F , either R = X ω × Y ω or R is a Σ 1 1 -c omplete subset of X ω × Y ω , bu t one c annot de ci de which c ase holds. Corollary 5.10 L et k b e an inte ger ≥ 2 or k ∈ {ℵ − 0 , ℵ 0 } . Then it is unde cidable to deter- mine whether a given infinitary r ational r elation is in the class R AT ω ( α ≤ k ) (r esp e ctively RAT ω ( α < k ) ). In p articular one c annot de c i de whether a given infinitary r ational r elation is non ambigu- ous or is inher ently ambiguous of de g r e e 2 ℵ 0 . 14 Pro of. Consider the family F giv en by Prop ositio n 5.9 and let R ∈ F . If R = X ω × Y ω then R is obvio usly non am biguous b ut if R is a Σ 1 1 -complete subset of X ω × Y ω then by Theorem 5.6 th e infi nitary r ational relation R is inherently ambiguous of degree 2 ℵ 0 . But one cannot decide w hic h case holds and th is end s the pro of.  Ac knowledgemen ts. W e thank Dominique Lecom te and Jean-Pierre Ressa yr e for useful discussions and the anon ymous referees for u seful commen ts on a preliminary v ersion of this pap er. References [Arn83b] A. Arnold, Ra tional O mega-Lang uages are Non-Ambiguous, Theor etical Computer Sci- ence 26 (198 3), 22 1-223 [Arn83a] A. 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