On the Continuity Set of an omega Rational Function

Theoretical Informatics and Applications Will be set by the publisher Informatique Th ´ eorique et Applications ON THE CO NTINUITY SET OF AN OMEGA RA TIONAL F UNC TION DEDICA TED TO SERG E GRIGORIEFF ON THE OCCASIO N OF HIS 60TH BIR THD A Y Olivier Car ton 1 , Olivier Finkel 2 and Pierre Simonnet 3 Abstract . In this paper, we study the contin uit y of rational functions realized by B ¨ uchi finite state transducers. It has b een sho wn by Prieur that it can b e decided whether such a function is con tinuous. W e pro ve here that surprisingly , it cannot b e decided whether such a function f has at least one p oint of con tinuit y and that its contin u ity se t C ( f ) cannot b e computed. In the case of a synchronous rational funct ion, w e show that its conti nuit y set is rational and that it can b e comput ed. F u rthermore w e prov e that any rational Π 0 2 -subset of Σ ω for some alphab et Σ is th e conti nuit y set C ( f ) of an ω -rational synchronous function f defined on Σ ω . 1991 Mathematics Sub ject Classification. 68Q05;68Q 45; 03D05. LIP Researc h Rep ort RR 2008-04 Keywor ds and phr ases: Infinitary rational relations; omega rational functions; topology; points of con tinuit y; decision problems; omega rational languages; omega con text-free languages. 1 LIAF A, Universit ´ e Paris 7 et CNRS 2 Place Jussieu 75251 P aris cedex 05, F rance. e-mail: Olivier.C arton@lia fa.jussieu.fr 2 Equipe M od` eles de Calcul et Complexit´ e L ab or atoir e de l’Informatique du Par al l ´ elisme (UMR 5668 - CNRS - ENS L y on - UCB Lyon - INRIA) CNRS et Ec ole Normale Sup ´ erieur e de Lyon 46, Al l´ ee d’Italie 69364 Lyon Ce dex 07, F r anc e. e-mail: Olivier.Finkel@e ns-lyon.fr 3 UMR 6134-Syst ` emes Physique s de l’ En vironnemen t F acult´ e des Sciences, Unive rsit´ e de Corse Quartier Gr ossetti BP52 20250, C or te, F r ance e-mail: simonnet@ univ-cors e.fr. c  EDP Scienc es 1999 2 TITLE WILL BE SE T BY THE PUBLISHER 1. Intr oduction Acceptance o f infinite words by finite automata was firstly c o nsidered in the sixties by B ¨ uchi in or der to study decida bility o f the monadic second o rder theor y of o ne s uccessor ov er the integers [B ¨ uc62]. Then the so called ω - regular la nguages hav e been intensiv ely studied a nd many applications ha ve been found. W e refer the r eader to [Tho90, Sta97, PP04] for many r esults a nd refer ences. Gire a nd Niv a t studied infinitar y r ational relatio ns accepted by B ¨ uc hi trans- ducers in [Gir81 , GN84]. Infinitary rational r e lations are subsets of Σ ω × Γ ω , where Σ and Γ are finite alphab ets, which ar e accepted by 2 -tap e finite B ¨ uchi automata with tw o asynchronous re a ding heads. They hav e been mu ch s tudied, in par ticular in connection with the rationa l functions they may define, see for example [CG9 9 , BCPS00, Sim92, Sta97, Pri00]. Gire pr ov ed in [Gir83] that one can decide whether an infinitary rational relatio n R ⊆ Σ ω × Γ ω recognized b y a given B ¨ uchi tr a nsducer T is the gr aph of a function f : Σ ω → Γ ω , (resp ectively , is the graph of a function f : Σ ω → Γ ω recognized by a sy nchronous B ¨ uc hi tra ns ducer). Such a function is called an ω -r ational function (resp ectively , a synchronous ω -rational function). The co ntin uity of ω -rational functions is a n imp ortant issue since it is rela ted to many asp ects. Let us mention tw o of them. First, sequential functions that may be rea lized by input deterministic a utomata a re contin uous but the co nv e rse is not true. Sec o nd, co nt inuous functions define a reduction b etw e en subsets of a topo - logical space that yields a hier arch y called the W adge hierar ch y . The r estriction of this hierar chy to r ational sets gives the W agner hierar chy . This pa p e r is fo cused on the contin uit y sets of rational functions. P rieur pr ov ed in [P ri00, Pri01] that it can be decided whether a given ω -r ational function is contin uo us. This means that it can b e decided whether the co ntin uity set is equal to the domain of the function. W e show how ever that it cannot b e decided whether a ratio nal function has a t leas t one p oint of co nt inuit y . W e show tha t in general the co ntin uity set of a ratio nal function is no t ratio na l and even not co ntext-free. F urthermo r e, we prov e that it cannot be decided whether this contin uity set is rational. W e pur sue this study with synchronous ratio nal functions. These functions are accepted by B ¨ uchi tr ansducers in which the tw o hea ds mov e synchronously . Contrary to the genera l case, the contin uit y set of synchronous rational function is alwa ys rational and it ca n be effectively computed. W e also g ive a characteriza tio n of contin uit y sets of synchronous functions. It is well known that a contin uit y set is a Π 0 2 -set. W e prov e co nv er sely that any rational Π 0 2 -set is the contin uit y se t of some sy nchronous ratio nal function. The pap er is org anized as follows. In section 2 we recall the notions of in- finitary rational relation, of ω -r a tional function, of synchronous or asynchronous ω -ra tional function, o f top o logy and contin uity; we r ecall a lso some recent results on the top o lo gical complexity of infinitary ra tional relatio ns. In sectio n 3 we study TITLE WILL BE SE T BY THE PUBLISHER 3 the contin uit y sets of ω -rationa l functions in the g eneral cas e, stating s ome unde- cidability results. Finally we study the ca se of sy nchronous ω -r ational functions in section 4. 2. Recall on ω -ra tional functions and topology 2.1. Infinit ar y ra tional rela tions and ω -ra tion al functions Let Σ b e a finite alphab et w ho se element s are ca lled le tter s. A non-empty finite word ov er Σ is a finite seque nc e of letter s: x = a 1 a 2 . . . a n where ∀ i ∈ [1; n ] a i ∈ Σ. W e shall denote x ( i ) = a i the i th letter of x and x [ i ] = x (1) . . . x ( i ) for i ≤ n . The length of x is | x | = n . The empty word will b e denoted by λ and has 0 letter. Its length is 0. The set of finite words over Σ is denoted Σ ⋆ . Σ + = Σ ⋆ − { λ } is the set of no n empty words over Σ. A (finitary) language L over Σ is a subs e t o f Σ ⋆ . The usua l concatenatio n pr o duct of u a nd v will b e denoted by u .v or just uv . F or V ⊆ Σ ⋆ , we denote by V ⋆ the se t R = { v 1 . . . v n | n ∈ N a nd ∀ i ∈ [1; n ] v i ∈ V } . The first infinite ordina l is ω . An ω -word ov er Σ is an ω -sequenc e a 1 a 2 . . . a n . . . , where for all integers i ≥ 1 a i ∈ Σ. When σ is an ω -w ord ov er Σ, w e write σ = σ (1) σ (2 ) . . . σ ( n ) . . . and σ [ n ] = σ (1) σ (2 ) . . . σ ( n ) the finite w ord of length n , prefix of σ . The set o f ω -words ov er the alpha be t Σ is denoted by Σ ω . An ω -lang uage over an alphab et Σ is a subset of Σ ω . F or V ⊆ Σ ⋆ , V ω = { σ = u 1 . . . u n . . . ∈ Σ ω | ∀ i ≥ 1 u i ∈ V } is the ω -p ow er of V . The conc a tenation pro duct is extended to the pr o duct o f a finite word u a nd an ω -word v : the infinite word u.v is then the ω -word such that: ( u .v )( k ) = u ( k ) if k ≤ | u | , and ( u.v )( k ) = v ( k − | u | ) if k > | u | . W e a ssume the r eader to be fa miliar with the theory of formal languages and of ω - regular languag es, see [B ¨ uc62, Tho90, EH93, Sta97, PP04] for ma ny res ults and r e ferences. W e reca ll that ω -r egular langua ges for m the class o f ω -la ng uages accepted by finite a uto mata with a B ¨ uchi acceptance condition and this class, denoted by RAT , is the omega Kleene closure o f the class of regular finitar y languages . W e are going now to recall the notion of infinitary ratio nal rela tion which ex- tends the notion of ω -re gular languag e, via definition by B ¨ uchi transducers: Definition 2.1. A B ¨ uchi transducer is a sextuple T = ( K , Σ , Γ , ∆ , q 0 , F ), wher e K is a finite set o f s ta tes, Σ a nd Γ ar e finite sets ca lle d the input and the output alphab ets, ∆ is a finite subset of K × Σ ⋆ × Γ ⋆ × K called the set of transitio ns, q 0 is the initial state, and F ⊆ K is the set of accepting states. A computation C of the tr ansducer T is an infinite sequence of consecutive tra n- sitions ( 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 co mputation is said to b e successful iff there exis ts a final state q f ∈ F and infinitely many in tegers i ≥ 0 such that q i = q f . The input word a nd output 4 TITLE WILL BE SE T BY THE PUBLISHER word of the computation a r e resp ectively u = u 1 .u 2 .u 3 . . . and v = v 1 .v 2 .v 3 . . . The input a nd the output words may be finite or infinite. The infinitar y rational relation R ( T ) ⊆ Σ ω × Γ ω accepted by the B ¨ uchi tr a nsducer T is the set of couples ( u, v ) ∈ Σ ω × Γ ω such that u and v are the input and the output words of some successful c omputation C of T . The set of infinitary ra tional relations will b e denoted R AT 2 . If R ( T ) ⊆ Σ ω × Γ ω is an infinitary ra tional re lation re cognized by the B ¨ uchi transducer T then we de no te Do m( R ( T )) = { u ∈ Σ ω | ∃ v ∈ Γ ω ( u, v ) ∈ R ( T ) } and Im( R ( T )) = { v ∈ Γ ω | ∃ u ∈ Σ ω ( u, v ) ∈ R ( T ) } . It is w ell known that, for each infinitary rationa l rela tion R ( T ) ⊆ Σ ω × Γ ω , the sets Dom( R ( T )) a nd Im( R ( T )) are regular ω -lang uages. The B ¨ uchi tr ansducer T = ( K, Σ , Γ , ∆ , q 0 , F ) is sa id to b e synchronous if the set of transitions ∆ is a finite subset o f K × Σ × Γ × K , i.e. if each transition is lab elled with a pair ( a, b ) ∈ Σ × Γ. An infinitary r ational re la tion r ecognized by a synchronous B ¨ uchi transducer is in fact a n ω -la nguage ov er the pro duct alphab et Σ × Γ which is accepted by a B ¨ uchi a uto maton. It is called a synchronous infinitary rational relation. An infinitary r ational relation is s aid to b e a synchronous if it can not b e recogniz e d by any s ynchronous B ¨ uchi transducer . Recall now the following undecidability r esult of C. F ro ugny and J . Sak arovitch. Theorem 2.2 ( [FS93 ]) . One c annot de cide whether a given infin itary r ational r elation is synchr onous. A B¨ uchi tra nsducer T = ( K , Σ , Γ , ∆ , q 0 , F ) is said to b e functional if for each u ∈ Dom ( R ( T )) there is a unique v ∈ Im( R ( T )) such that ( u , v ) ∈ R ( T ). The infinitary ra tio nal r elation rec ognized by T is then a functiona l re la tion a nd it defines an ω -r ational (partia l) function f T : Dom ( R ( T )) ⊆ Σ ω → Γ ω by: for ea ch u ∈ Dom ( R ( T )), f T ( u ) is the unique v ∈ Γ ω such that ( u, v ) ∈ R ( T ). An ω -rational (partial) function f : Σ ω → Γ ω is said to be sy nchronous if there is a synchronous B ¨ uchi transducer T suc h that f = f T . An ω -rational (partial) function f : Σ ω → Γ ω is sa id to be a s ynchronous if there is no synchronous B ¨ uc hi trans ducer T suc h that f = f T . Theorem 2.3 ( [Gir83 ]) . One c an de cide whether an infinitary r ational r elation r e c o gnize d by a given B ¨ uchi tr ansduc er T is a functional infinitary r ational r elation (r esp e ctively, a synchr onous functional infin itary r ational r elation). 2.2. Topology W e a ssume the reader to b e familiar w ith basic notions of top olo gy which may be found in [Mos8 0, Kec9 5, L T94, Sta97 , PP04]. There is a natural metric on the set Σ ω of infinite words ov er a finite alphab et Σ which is c a lled the prefix metric and defined as follows. F or u, v ∈ Σ ω and u 6 = v let d ( u, v ) = 2 − l pref ( u,v ) where l pref ( u,v ) is the lea s t integer n such that the ( n + 1) th letter of u is different fro m the ( n + 1 ) th letter o f v . This metric induce s on Σ ω the usual Ca nt or top olo gy for which o pe n subsets o f Σ ω are in the form W. Σ ω , where W ⊆ Σ ⋆ . Recall that a se t TITLE WILL BE SE T BY THE PUBLISHER 5 L ⊆ Σ ω is a close d set iff its co mplement Σ ω − L is an op en set. W e define now the next classes of the Bore l Hierarchy: Definition 2.4. The cla sses Σ 0 n and Π 0 n of the Bor el Hierar ch y on the top olo gical space Σ ω are defined as follows: - Σ 0 1 is the cla s s of op en sets of Σ ω . - Π 0 1 is the class of closed sets of Σ ω . And for any integer n ≥ 1: - Σ 0 n +1 is the cla s s of countable unio ns of Π 0 n -subsets o f Σ ω . - Π 0 n +1 is the class of countable intersections of Σ 0 n -subsets o f Σ ω . The Borel Hierarch y is also defined fo r transfinite levels: The classes Σ 0 α and Π 0 α , for a non-null countable o rdinal α , ar e defined in the following wa y . - Σ 0 α is the cla s s of countable unio ns of subsets of Σ ω in ∪ γ < α Π 0 γ . - Π 0 α is the class of countable intersections of subsets of Σ ω in ∪ γ < α Σ 0 γ . Let us re c a ll the characteriza tion o f ra tio nal Π 0 2 -subsets of Σ ω , due to Landwe- ber [Lan69]. This character iz ation will b e used in the pro of that any rational Π 0 2 -subset is the contin uit y set of so me ra tio nal synchronous function. Theorem 2.5 (Landweber ) . A r ational subset of Σ ω is Π 0 2 if and only if it c an b e r e c o gnize d by a deterministic B¨ uchi automaton. There ar e some s ubs ets of the Cantor set, (hence also o f the top o lo gical space Σ ω , for a finite alpha b et Σ having at lea st tw o elements) which a r e not Borel sets. There exists another hier arch y beyond the Borel hier arch y , c alled the pro jective hierarch y . The first clas s of the pr o jective hierar ch y is the cla ss Σ 1 1 of analytic sets. A set A ⊆ Σ ω is analytic iff there exists a Borel set B ⊆ (Σ × Y ) ω , with Y a finite alpha b e t, such that x ∈ A ↔ ∃ y ∈ Y ω such that ( x, y ) ∈ B , where ( x, y ) ∈ (Σ × Y ) ω is defined by: ( x, y )( i ) = ( x ( i ) , y ( i )) fo r all integers i ≥ 1. Remark 2. 6. An infinitary rationa l re lation is a subset of Σ ω × Γ ω for tw o finite alphab ets Σ a nd Γ. One can als o consider that it is a n ω - language over the finite alphab et Σ × Γ. If ( u, v ) ∈ Σ ω × Γ ω , one can consider this co uple o f infinite words as a single infinite word ( u (1) , v (1)) . ( u (2) , v (2)) . ( u (3) , v (3 )) . . . ov er the a lphab et Σ × Γ. Since the set (Σ × Γ) ω of infinite words over the finite alpha be t Σ × Γ is naturally eq uipped with the Can tor top o logy , it is natural to inv es tig ate the top olog ic al c o mplexity of infinitary r a tional relatio ns a s ω -la nguages, and to lo cate them with r egard to the B orel and pro jectiv e hie r archies. Every infinitary rational relation is a n analytic set and there exist some Σ 1 1 -complete, hence no n- Borel, infinitary rationa l rela tions [Fin0 3a]. The second author has recent ly prov ed the following very s urprising r esult: infinitary rationa l r elations have the same top ologica l complexity as ω -lang ua ges accepted b y B ¨ uchi T uring mac hines [Fin06b, Fin06a]. In particular , for every recursive non-null o rdinal α there exist so me Π 0 α - complete a nd so me Σ 0 α -complete infinitary r ational r elations, and the supr emum of the se t o f Bo r el r anks o f infinitary rational relations is the ordinal γ 1 2 . This 6 TITLE WILL BE SE T BY THE PUBLISHER ordinal is defined by A.S. Kechris, D. Marker, and R.L. Sami in [K MS8 9] and it is prov ed to b e stric tly g reater than the ordina l δ 1 2 which is the firs t non ∆ 1 2 ordinal. Thu s the o rdinal γ 1 2 is als o strictly gr eater than the first non-recurs ive ordinal ω CK 1 , usua lly called the C hurch-kleene ordina l. Notice that a mazingly the exact v alue of the o rdinal γ 1 2 may dep end on a xioms of set theory , see [KMS89 , F in06b]. Remark 2.7. Infinitary rationa l relations recognized b y sync hronous B¨ uchi trans- ducers a re regula r ω - la nguages thus they are bo olean combinations of Π 0 2 -sets hence ∆ 0 3 -sets [PP 04]. So we ca n see that ther e is a great difference b etw een the cases of synchronous and of a synchronous infinitar y rational rela tions. W e shall see in the s equel that these t wo ca ses ar e also very differ e n t when we inv es tigate the co ntin uity sets o f ω -r ational functions. 2.3. Continuity W e have alre ady seen that the Cantor top ology of a space Σ ω can b e defined by a distance d . W e r ecall that a function f : Dom( f ) ⊆ Σ ω → Γ ω , whose domain is Dom( f ) , is s aid to be co nt inuous a t p oint x ∈ Dom( f ) if : ∀ n ≥ 1 ∃ k ≥ 1 ∀ y ∈ Dom( f ) [ d ( x, y ) < 2 − k ⇒ d ( f ( x ) , f ( y )) < 2 − n ] The function f is said to b e contin uo us if it is contin uous at every p oint x ∈ Σ ω . The contin uit y s et C ( f ) of the function f is the set of p o int s o f contin uit y o f f . Recall that if X is a subset o f Σ ω , it is a lso a top ologica l spa c e whos e topo logy is induced by the top ology o f Σ ω . Op en sets of X ar e traces on X of op e n se ts of Σ ω and the sa me result holds for closed s ets. Then one can ea sily show by induction tha t for every in teger n ≥ 1, Π 0 n -subsets (r e sp. Σ 0 n -subsets) of X are traces on X o f Π 0 n -subsets (resp. Σ 0 n -subsets) of Σ ω , i.e. ar e intersections with X of Π 0 n -subsets (r esp. Σ 0 n -subsets) o f Σ ω . W e recall now the following well known result. Theorem 2. 8 (see [Kec9 5 ]) . Le t f b e a fun ction fr om Dom( f ) ⊆ Σ ω into Γ ω . Then the c ontinuity set C ( f ) of f is always a Π 0 2 -subset of Dom( f ) . Pr o of. Let f b e a function from D om ( f ) ⊆ Σ ω int o Γ ω . F or some integers n , k ≥ 1, we cons ider the set X k,n = { x ∈ D om ( f ) | ∀ y ∈ D om ( f ) [ d ( x, y ) < 2 − k ⇒ d ( f ( x ) , f ( y )) < 2 − n ] } W e know, from the definition of the distance d , tha t for tw o ω -words x and y over Σ, the inequality d ( x, y ) < 2 − k simply means that x and y have the same ( k + 1) first letters . Then it is eas y to see that the se t X k,n is an op en subset of Dom ( f ), be cause for each x ∈ X k,n , the set X k,n contains the op en ball (in D om ( f )) of all y ∈ D om ( f ) such that d ( x, y ) < 2 − k . TITLE WILL BE SE T BY THE PUBLISHER 7 By union we ca n infer that X n = S k ≥ 1 X k,n is an op en subset of D om ( f ) a nd then the countable intersection C ( f ) = T n ≥ 1 X n is a Π 0 2 -subset o f D om ( f ).  In the sequel we are going to inv estigate the contin uity sets of ω -r ational func- tions, firstly in the general cas e and next in the case o f synchronous ω -rationa l functions. 3. Continuity set of ω -ra tional functions Recall that C. Prieur prov ed the following result. Theorem 3.1 ( [Pri0 0, P ri01]) . One c an de cide whether a given ω -r ational fu n ction is c ontinuous. Prieur show ed that the clo sure (in the topo logical sense) of the gra ph of a ratio- nal r elation is still a ra tional rela tion that can b e effectively computed. F rom this closure, it is quite easy to decide whether a given ω -rational function is contin uous. So one can decide whether the co nt inuit y set of an ω - rational function f is equa l to its doma in Dom( f ). W e shall prov e b elow some undecidability results, using the undecida bilit y of the Post Corr esp ondenc e Pr oblem which we now reca ll. Theorem 3.2 (Post) . L et Γ b e an alphab et having at le ast two elements . Then it is u n de cidable to determine, for arbitr ary n-t uples ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ) of non-empty wor ds in Γ ⋆ , whether ther e ex ists a non-empty se quenc e of indic es i 1 , . . . , i k such that u i 1 . . . u i k = v i 1 . . . v i k . W e now state our first undecidability result. Theorem 3 .3. One c annot de cide whether the c ontinuity set C ( f ) of a given ω -r ational function f is empty. Pr o of. Let Γ b e a n alphab et having at least tw o elements a nd ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ) b e tw o sequences of n non-empty words in Γ ⋆ . Let A = { a, b } and C = { c 1 , . . . , c n } such that A ∩ C = ∅ and A ∩ Γ = ∅ . W e define the ω -rational function f o f doma in Dom( f ) = C + .A ω by: • f ( x ) = u i 1 . . . u i k .z if x = c i 1 . . . c i k .z and z ∈ ( A ⋆ .a ) ω . • f ( x ) = v i 1 . . . v i k .z if x = c i 1 . . . c i k .z and z ∈ A ⋆ .b ω . Notice that ( A ⋆ .a ) ω is simply the set of ω -words ov er the a lpha be t A having an infinite n umber o f o ccurrences of the letter a . And A ⋆ .b ω is the complement o f ( A ⋆ .a ) ω in A ω , i.e. it is the set o f ω -words over the a lpha b e t A containing only a finite num ber of letters a . The t wo ω -la nguages ( A ⋆ .a ) ω and A ⋆ .b ω are ω -r e gular, so they are accepted by B ¨ uc hi automata. It is then easy to see that the function f is ω -rationa l and we can co ns truct a B ¨ uc hi trans duce r T that acce pts the graph of f . W e are go ing to prov e firs tly that if x = c i 1 . . . c i k .z ∈ C + .A ω is a po int of contin uity of the function f then the Post C o rresp ondenc e Problem of instances 8 TITLE WILL BE SE T BY THE PUBLISHER ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ) w ould hav e a solution i 1 , . . . , i k such that u i 1 . . . u i k = v i 1 . . . v i k . W e distinguish tw o cases. First Case. Assume firstly that z ∈ ( A ⋆ .a ) ω . Then b y definition of f it holds that f ( x ) = u i 1 . . . u i k .z . Notice that there is a se quence of elements z n ∈ A ⋆ .b ω , n ≥ 1 , such tha t the sequence ( z n ) n ≥ 1 is conv ergent and l im ( z n ) = z . This is due to the fact that A ⋆ .b ω is dense in A ω . W e set x n = c i 1 . . . c i k .z n . So we hav e also l im ( x n ) = x . By definition of f , it holds that f ( x n ) = f ( c i 1 . . . c i k .z n ) = v i 1 . . . v i k .z n . If x = c i 1 . . . c i k .z is a p oint of contin uit y of f then w e must have li m ( f ( x n )) = f ( x ) = u i 1 . . . u i k .z . But f ( x n ) = v i 1 . . . v i k .z n conv erges to v i 1 . . . v i k .z . Thus this would imply that u i 1 . . . u i k = v i 1 . . . v i k and the Post Corres p o ndence Pr oblem o f instances ( u 1 , . . . , u n ) a nd ( v 1 , . . . , v n ) would hav e a solution. Second Case. Assume now that z ∈ A ⋆ .b ω . Notice that ( A ⋆ .a ) ω is also dense in A ω . Then reasoning as in the ca s e of z ∈ ( A ⋆ .a ) ω , we can prove that if x = c i 1 . . . c i k .z is a p oint of con tin uity of f then u i 1 . . . u i k = v i 1 . . . v i k so the P ost Co rresp ondence Problem of instances ( u 1 , . . . , u n ) a nd ( v 1 , . . . , v n ) would hav e a solution. Conv ersely ass ume that the Post Cor resp ondence Problem of ins tances ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ) has a solution, i.e. a non-empty sequence o f indices i 1 , . . . , i k such that u i 1 . . . u i k = v i 1 . . . v i k . Consider now the function f defined ab ov e. W e hav e: f ( c i 1 . . . c i k .z ) = u i 1 . . . u i k .z = v i 1 . . . v i k .z for every z ∈ A ω So it is ea sy to s e e tha t the function f is contin uous a t p o int c i 1 . . . c i k .z for every z ∈ A ω . Finally we hav e proved that the function f is contin uous at p oint c i 1 . . . c i k .z , for z ∈ A ω , if and only if the non-empty sequence of indices i 1 , . . . , i k is a solutio n of the Post Cor resp ondence Problem o f insta nc e s ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ). Thus one ca nno t decide whether the function f has (at least) one p oint of contin uity .  Theorem 3. 4 . One c ann ot de cide whether t he c ontinuity set C ( f ) of a given ω - r ational function f is a r e gular ω -language (r esp e ctively, a c ontext-fr e e ω - language). Pr o of. W e shall use a par ticular instance of Post Corr e sp ondence Pr oblem. F or t wo letters c, d , le t PCP 1 be the Post Corr esp ondence Pro blem of instances ( t 1 , t 2 , t 3 ) and ( w 1 , w 2 , w 3 ), wher e t 1 = c 2 , t 2 = t 3 = d and w 1 = w 2 = c , w 3 = d 2 . It is easy to see that its solutions are the se q uences of indices in { 1 i . 2 i . 3 i | i ≥ 1 } ∪ { 3 i . 2 i . 1 i | i ≥ 1 } . In pa rticular this la nguage over the alphab et { 1 , 2 , 3 } is no t context-free and this will b e useful in the sequel. Let now Γ b e a n a lpha b e t having a t leas t tw o elements such that Γ ∩ { c, d } = ∅ , a nd ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ) be tw o sequence s o f n non-empty words in Γ ⋆ . Let PCP b e the Post Corres po ndence Pro blem of instances ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ). TITLE WILL BE SE T BY THE PUBLISHER 9 Let A = { a, b } and C = { c 1 , . . . , c n } and D = { d 1 , d 2 , d 3 } b e three alphab ets t wo by tw o disjoints. W e a ssume a lso that A ∩ { c, d } = ∅ . W e define the ω -rational function f o f doma in Dom( f ) = C + .D + .A ω by : • f ( x ) = u i 1 . . . u i k .t j 1 . . . t j p .z if x = c i 1 . . . c i k .d j 1 . . . d j p .z a nd z ∈ ( A ⋆ .a ) ω . • f ( x ) = v i 1 . . . v i k .w j 1 . . . w j p .z if x = c i 1 . . . c i k .d j 1 . . . d j p .z and z ∈ A ⋆ .b ω . Reasoning as in the preceding pr o of a nd using the fact that ( A ⋆ .a ) ω and A ⋆ .b ω are b oth dense in A ω , we ca n pr ove that the function f is contin uous a t p oint x = c i 1 . . . c i k .d j 1 . . . d j p .z , where z ∈ A ω , if and only if the se quence i 1 , . . . , i k is a solution of the Post Corr esp ondence Problem PCP of instances ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ) and the seque nc e j 1 , . . . , j p is a solution of the Post Corresp ondence Problem PCP 1 . There are now tw o ca ses. First Case. The Post Corresp ondence Pro blem P C P has not any solution. Th us the function f ha s no points o f contin uit y , i.e. C ( f ) = ∅ . Second Case. The Post Corre sp ondence Problem P CP has at least one s olution i 1 , . . . , i k . W e now prove that in that c ase the contin uit y set C ( f ) is not a context-free ω - la nguage, i.e. is not accepted by any B ¨ uchi pushdown automa- ton. T ow a rds a contradiction, a ssume o n the contrary that C ( f ) is a context-free ω -lang uage. Consider now the intersection C ( f ) ∩ R where R is the reg ular ω - language c i 1 . . . c i k . ( d + 1 .d + 2 .d + 3 ) .A ω . The cla ss C F ω of context-free ω -language s is closed under intersection with regula r ω -lang uages, [Sta97], thus the languag e C ( f ) ∩ R would b e a lso context-free. But C ( f ) ∩ R = c i 1 . . . c i k . { d i 1 .d i 2 .d i 3 | i ≥ 1 } .A ω and this ω -language is not context-free b ecause the finitary la nguage { d i 1 .d i 2 .d i 3 | i ≥ 1 } is not context-free. So we hav e proved that C ( f ) is not a context-free ω -language. In the first ca se C ( f ) = ∅ so C ( f ) is a regular hence also context-free ω -language. In the sec ond case C ( f ) is not a co ntext-free ω - language so it is not ω -regular. But one cannot decide which case holds b eca use one canno t decide whether the Post Cor resp ondence P r oblem PCP has a t least one solution i 1 , . . . , i k .  4. Continuity s et of synchronous ω -ra tional functions W e hav e s hown that for non-synchronous rational functions, the contin uit y set can be very co mplex. In this s ection, we show that the la ndscap e is quite differ ent for synchronous rational functions. Their co nt inuit y set is alwa ys ra tional. F ur- thermore we show that any Π 0 2 rational set is the contin uity set of so me rationa l function. Theorem 4. 1 . L et f : A ω → B ω b e a r ational synchr onous function. The c onti- nuity set C ( f ) of f is r ational. Pr o of. W e actually prov e that the complement A ω \ C ( f ) is r ational. Since the inclusion C ( f ) ⊆ Dom( f ) holds and the set Dom( f ) is r ational, it suffices to prove that Dom( f ) \ C ( f ) is rational. Suppo se that f is r ealized by the synchronous tra nsducer T . Without loss of generality , it may b e as sumed that T is trim, that is, any state q app ears in an 10 TITLE WILL BE SET BY THE P UBLISHER accepting path. Let x be an element of the do main of f . W e claim that f is not contin uo us at x if there are tw o infinite pa ths γ a nd γ ′ in T s uch that the following prop erties hold, i) the path γ is accepting and y = f ( x ). ii) the lab els of γ and γ ′ are ( x, y ) and ( x, y ′ ) with y 6 = y ′ . The path γ exists since x b elongs to the domain of f . Remark that the path γ ′ cannot b e ac c epting since T realizes a function. It is clear that if such a path γ ′ exists, the function f cannot be co nt inuous at x . Suppo se that f is not contin uous at x . There is a seque nce ( x n ) n ≥ 0 of elements from the domain of f co nv er ging to x such that d ( f ( x ) , f ( x n )) > 2 − k for some int eger k . Since ea ch x n belo ngs to the domain of f , there is a path γ n whose lab el is the pair ( x n , f ( x n )). Since the set of infinite paths is a compact s pace, it c a n b e extracted fr om the sequence ( x n ) n ≥ 0 another sequence ( x s ( n ) ) n ≥ 0 such that the sequence ( γ s ( n ) ) n ≥ 0 conv erges to a path γ ′ . Let ( x ′ , y ′ ) b e the lab el of this path γ ′ . Since ( x n ) n ≥ 0 conv erges to x , x ′ is equal to x and since d ( f ( x ) , f ( x n )) > 2 − k , y ′ is different from y . This proves the claim. F rom the cla im, it is easy to build an a utomaton that accepts infinite words x such that f is not contin uous at x . Roughly sp eaking , the automaton checks whether there ar e tw o paths γ and γ ′ as ab ov e. Let T b e the transducer ( Q, A, B , E , q 0 , F ). W e build a non deterministic B ¨ uchi a utomaton A . The sta te set of A is Q × Q × { 0 , 1 } . The initial state is ( q 0 , q 0 , 0) a nd the set o f final states is F × Q × { 1 } . The set of transitions of this automa ton is G = { ( p, p ′ , 0) a → ( q , q ′ , 0) | ∃ b ∈ B p a | b → q ∈ E and p ′ a | b → q ′ ∈ E } ∪ { ( p, p ′ , 0) a → ( q , q ′ , 1) | ∃ b, b ′ ∈ B p a | b → q ∈ E , p ′ a | b ′ → q ′ ∈ E and b 6 = b ′ } ∪ { ( p, p ′ , 1) a → ( q , q ′ , 1) | ∃ b, b ′ ∈ B p a | b → q ∈ E and p ′ a | b ′ → q ′ ∈ E }  Theorem 4 .2. L et X b e a r ational Π 0 2 subset of A ω . Then X is the c ontinuity set C ( f ) of some r ational synchr onous function f of domain A ω . Pr o of. If A only contains one sy m b ol a , the result is trivial since A ω only contains the infinite word a ω . W e now as s ume that A co nt ains at least tw o symbols. Let b be a distinguished s ymbol in A let c a new symbol no t b elonging to A . W e define a synchronous function f that is of the following form : f ( x ) =          x if x ∈ X wc ω for so me prefix w of x if x ∈ X \ X wb ω for so me prefix w of x if x ∈ ( A ∗ b ) ω \ X wc ω for so me prefix w of x otherwise, where w is a word precised below. TITLE WILL BE SE T BY THE PUBLISHER 11 By Theorem 2 .5, ther e is a deterministic B¨ uchi automaton A = ( Q, A, E , { q 0 } , F ) accepting X . W e a ssume that A is trim. The function f is now defined as follows. If x b elongs to X , then f ( x ) is equal to x . If x b elongs to the clos ur e X of X but not to X , let w be the longest prefix of x which is the lab el of a path in A from q 0 to a final state. Then f ( x ) is equal to w c ω . If x do es not b elo ng to the closure of X , le t w be the longest pr efix which is the la b el of a path in A fro m q 0 . Then f ( x ) is equal to w b ω if b o ccurs infinitely many times in x and it is equa l to wc ω otherwise. It is ea sy to verify that the contin uity set o f f is exac tly X . W e now give a synchronous transducer T r ealizing the function f . Let R be the set of pairs ( q , a ) such that ther e is no tra nsition q a → p in A . The state set of T is Q × { 0 } ∪ ( Q \ F ) × { 1 } ∪ { q 1 , q 2 , q 3 , q 4 } . The initial state is q 0 and the set of final states is F × { 0 } ∪ ( Q \ F ) × { 1 } ∪ { q 2 , q 4 } . The set of transitio ns is defined as follows. G = { ( p, 0) a | a → ( q , 0) | p a → q ∈ E } ∪ { ( p, 0) a | c → ( q , 1) | p a → q ∈ E and q / ∈ F } ∪ { ( p, 1) a | c → ( q , 1) | p a → q ∈ E and p , q / ∈ F } ∪ { ( p, 0) a | b → q 1 | ( p, a ) ∈ R } ∪ { ( p, 0) a | c → q 3 | ( p, a ) ∈ R } ∪ { p a | b → q 1 | p ∈ { q 1 , q 2 } and a 6 = b } ∪ { p b | b → q 2 | p ∈ { q 1 , q 2 }} ∪ { q 3 a | c → q 3 | a ∈ A } ∪ { p a | c → q 4 | p ∈ { q 3 , q 4 } and a 6 = b }  Recall that a p oint x of a subset D o f a top o lo gical space X is isolate d if there is a neig hborho o d of x whose intersection with D is equal to { x } . Isolated points have the following pro pe rty with rega rd to contin uity . An y function from a domain D is co nt inuous at any isolated p oint of D . Therefore, if X is the contin uit y set of some function of domain D , X must con tain all isolated p oints of D . The following theorem sta tes that for rational Π 0 2 sets, this condition is also sufficient. Theorem 4.3. L et D and X b e t wo r ational subsets of A ω such that X ⊆ D . If ther e exists a ra tional Π 0 2 -subset X ′ of A ω such that X = X ′ ∩ D , and if X c ontains al l isolate d p oints of D , then it is the c ontinuity set C ( f ) of some synchr onous r ational fun ction f of domain D . 12 TITLE WILL BE SET BY THE P UBLISHER In the pro o f of Theor em 4.2, the complement of the set X has b een pa rtitioned int o t w o dense sets. The following lemma ex tends this result to any ra tional set of infinite words. Lemma 4.4. L et X b e a ra tional set c ontaining no isolate d p oints . Then, the set X c an b e p artitione d into two r ational sets X ′ and X ′′ such that b oth X ′ and X ′′ ar e dense in X . Pr o of. Let X be a rationa l set of infinite words with no isolated p o ints and let A be a deter ministic and trim Muller a utomaton accepting X . W e refer the rea der for instance to [Tho90 , P P04] for the definition and prop er ties of Muller automata. F rom a ny state q , either there is no a ccepting path sta rting in q or there ar e a t least tw o acc epting paths with different la bels starting from q . W e first consider the ca se where the table T of A o nly contains one a ccepting set F . Since the a utomaton is trim, all states o f F b elong to the same s tr ongly connected compo ne nt of A . W e consider tw o cases depending o n whether the s et F contains all states of its connected comp onent. Suppo se firs t that the state q do es not belo ng to F but is in the same s trongly connected comp onent as F . Let X ′ and X ′′ the sets of words which r esp ectively lab el an accepting path which g o es an o dd o r an even num b er of times through the sta te q . It is clear that X ′ and X ′′ hav e the requir ed prop er ty . Suppo se now that F contains a ll the states in its strongly connected comp o nent. Since X has no isolated p oint, there must b e a state q of F with tw o outgoing edges e and e ′ . Let X ′ be the set of words which lab el a path of the following form. The tr a ce of this path ov er the tw o edg es e a nd e ′ is an infinite sequence of the for m ( e + e ′ ) ∗ ( ee ′ ) ω . W e now co me back to the genera l ca s e where the ta ble T of A may contain several acce pting sets { F 1 , . . . , F n } . Let X i be the set of words accepted b y the table { F i } . Note fir st that if the b oth sets X i and X j can b e partitioned into dense ra tional sets in to X ′ i , X ′′ i , X ′ j and X ′′ j , the b oth s ets X ′ i ∪ X ′ j and X ′′ i ∪ X ′′ j are dense in X i ∪ X j . Note als o that if F i is ac c essible from F j , then any set dense in X i is als o dense in X j and therefor e in X i ∪ X j . It follows that if the set X i can b e pa rtitioned int o t w o dense r a tional sets X ′ i and X ′′ i , the set X i ∪ X j can b e partitioned in to X ′ i ∪ X j and X ′′ i . F rom the previous t wo remarks, it suffices to partition indep endently each X i such that the cor resp onding F i is maximal for access ibility . By maximal, w e mean that F i is maxima l whenever if F j is accessible from F i , then F i is also acces s ible from F j and bo th sets F i and F j are in the same strongly connected compo nent. This can b e done using the method descr ib ed ab ove.  W e now come to the pro of of the previous theorem. Pr o of. The pro o f is s imilar to the pro o f of The o rem 4.2 but the domain D has to be taken into account. By hyp o thesis there exists a r ational Π 0 2 -subset X ′ of A ω such that X = X ′ ∩ D . Le t A a trim a nd deterministic B ¨ uchi a utomaton accepting X ′ . TITLE WILL BE SE T BY THE PUBLISHER 13 W e define the function f of domain D as follows. F or any x in X , f ( x ) is still equal to x . If x b e longs to X ∩ D \ X , f ( x ) is equal to w c ω where w is the longest prefix o f x whic h is the lab el in A of a pa th form the initial state to a final state. By Lemma 4 .4, the set Z = D \ X c an b e par titio ned into tw o ra tional subsets Z 1 and Z 2 such that b oth Z 1 and Z 2 are dens e into Z . If x b elongs to D \ X , then f ( x ) is defined as follows. Let w b e the longest prefix o f x which is the lab el in A o f a path from the initial state. Then f ( x ) is equal to wb ω if x ∈ Z 1 and f ( x ) = w c ω if x ∈ Z 2 .  The following co rollar y provides a complete characteriza tion of s ets of contin uit y of synchronous rationa l functions of doma in D ⊆ A ω when D is the intersection of a ra tional Σ 0 2 -subset a nd o f a Π 0 2 -subset of A ω . This is in particula r the cas e if D is simply a Σ 0 2 -subset o r a Π 0 2 -subset o f A ω . Corollary 4.5. L et D and X b e two r ational subset s of A ω such t hat X is a Π 0 2 - subset of D and D = Y 1 ∩ Y 2 , wher e Y 1 is a r ational Σ 0 2 -subset of A ω and Y 2 is a Π 0 2 -subset of A ω . If X c ontains al l isolate d p oints of D , then it is the c ontinuity set C ( f ) of some synchr onous r ational function f of domain D . Pr o of. Let D and X satisfying the hypo theses of the corolla ry . Assume firstly that D = Y 1 is a Σ 0 2 -subset of A ω . Then it is eas y to s ee that X ′ = X ∪ ( A ω − D ) is a ratio na l Π 0 2 -subset o f A ω such tha t X = X ′ ∩ D . Thus in this case Corolla ry 4.5 follows fro m Theor em 4.3. Assume now that D = Y 1 ∩ Y 2 , where Y 1 is a ra tional Σ 0 2 -subset of A ω and Y 2 is a Π 0 2 -subset of A ω . By hypothesis X is a Π 0 2 -subset of D th us ther e is a Π 0 2 -subset X 1 of A ω such that X = X 1 ∩ D = X 1 ∩ ( Y 1 ∩ Y 2 ) = ( X 1 ∩ Y 2 ) ∩ Y 1 . 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