Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language

Three Applicatio n s to Rational Relatio n s of the Hi g h Undecid abi lit y of the In finite P ost Corres p o n dence Problem in a Regular ω -Language Olivier Fink el Equip e de L o gique Math´ ematique CNRS et Univ ersit ´ e P aris Diderot P aris 7 UFR de Math ´ ematiques case 7012, site Chev aleret, 75205 P aris Cedex 1 3, F rance. fink el@logique.jussieu.fr Abstract It was noticed b y Harel in [Har86] that “one can define Σ 1 1 -complete versions of the w ell-known Post Corresp ondence Pr oblem”. W e fir st give a complete proof of this result, showing that the infinite P ost Corresp ondenc e Problem in a regular ω - la nguage is Σ 1 1 -complete, hence lo cated beyond the arithmetical hierar ch y and highly undecidable. W e infer from this result that it is Π 1 1 -complete to determine whether t wo given infinitary rational rela tions are disjoint. Then we prov e that there is an ama z ing g ap b etw een t wo decision problems a bo ut ω -ra tional functions realized by finite s tate B ¨ uchi tra nsducers. Indeed Prieur prov ed in [Pri01, Pr i02] that it is dec idable whether a g iven ω -ratio nal function is con tinuous, while we show her e that it is Σ 1 1 -complete to determine whether a giv en ω -rationa l function has at leas t one p oint of contin uit y . Next w e prove that it is Π 1 1 -complete to determine whether the contin uit y set o f a g iven ω -ra tional function is ω -reg ular. This gives the ex act complexity of t w o problems which were shown to be undecidable in [CFS0 8]. Keyw ords. Decision problems; infin ite Po st C orresp onden ce Problem; an- alytical hierarc h y; high undecidabilit y; infinitary rational relatio ns; omega rational fun ctions; top ol ogy; p oin ts of con tin uity . 1 In tro duction Man y classical decision p roblems arise natur ally in the fields of F ormal Lan- guage Theory and of Au tomata Theory . It is we ll kno wn that most prob- lems ab out r egular languages accepted by fi nite automata are decidable. 1 On the other h and, at the second lev el of the Chomsky Hierarc hy , most problems ab out con text-free languages accepted by pushdown automata or generated by conte xt-free grammars are und ecidable. F or in stance it fol- lo ws from the und ecidability of the P ost Corresp ondence Problem that the unive rsalit y problem, the inclusion and the equiv alence p roblems for conte xt- free languages are also u ndecidable. Notic e that some few problems ab out con text-free languages remain decidable lik e the follo wing ones: “Is a giv en con text-free languag e L emp t y ?” “Is a given con text-free language L infi- nite ?” “Do es a giv en word x b elong to a giv en con text-free language L ?” S ´ enizergues pro v ed in [S´ en01] that th e d ifficult pr oblem of the equiv alence of t w o deterministic push do wn automata is d ecidable. Notice that almost pro ofs of undecidabilit y results ab out con text-free languages rely on the un- decidabilit y of the Post Corresp ondence Problem wh ic h is complete for the class of r ecursiv ely en umerable pr oblems, i.e. complete at the fi rst lev el of the arithmetical hierarc h y . Th us un decidabilit y pro ofs ab out con text-free languages p ro vided only hard ness resu lts for the fi rst lev el of the arithmeti- cal hierarc hy . On the other hand , s ome decision problems are kno wn to b e lo cated b e- y ond the arithm etical hierarc h y , in s ome classes of the analytical h ierarch y , and are then usu ally called “ highly undecidable”. Harel pro v ed in [Har86] that many domino or tiling p roblems are Σ 1 1 -complete or Π 1 1 -complete. F or instance the “recurring domino problem” is Σ 1 1 -complete. It is also Σ 1 1 - complete to determine whether a giv en T uring mac hine, when started on a blank tap e, admits an infinite computation that reen ters infi n itely often in the initial state. Alur and Dill us ed this latter r esult in [AD 94 ] to p ro v e that the universali t y problem fot timed B¨ uc hi automa ta is Π 1 1 -hard. In [CC89], Castro and Cuc k er studied man y decision problems for ω -language s of T ur- ing machines. In particular, they pr o v ed that th e n on-emptiness and the infiniteness p r oblems for ω -languages of T uring mac hines are Σ 1 1 -complete, and that the univ ersalit y problem, the inclusion pr ob lem, and the equiv a- lence problem are Π 1 2 -complete. Thus these problems are lo cated at the first or the second lev el of the analytical hierarc hy . Using Castro and Cuck er’s results, some reductions of [Fin06a, Fin06b], and top ologica l argument s, w e hav e p ro v ed in [Fin09] that many d ecision problems ab out 1-coun ter ω - languages, con text fr ee ω -languages, or infinitary rational relations, lik e the unive rsalit y pr oblem, th e inclusion problem, the equiv alence p roblem, the determinizabilit y problem, the complemen tabilit y problem, a nd the unam- biguit y problem are Π 1 2 -complete. Notice that the exac t complexit y of nu- merous problems remains still un kno wn. F or instance the exact complexities of the u niv ersalit y problem, the determinizabilit y , or the complementabil- it y p roblem for timed B ¨ uc hi automata whic h are kno wn to b e Π 1 1 -hard, see [AD94, Fin06c]. 2 W e intend to in tro duce here a n ew metho d for pro ving high un decidabilit y results w hic h seems to b e u nexplored. It w as actuall y noticed b y Ha rel in [Har86] that “one can define Σ 1 1 -complete v ersions of the w ell-kno wn P ost Corresp ond ence Pr oblem”, but it seems th at this p ossibilit y has not b een later inv estiga ted. W e first giv e a complete pro of of this result, sho wing that the infinite P ost Corresp o nd ence Problem in a r egular ω -language is Σ 1 1 -complete, hen ce lo cated b eyond the arithm etical hierarch y and highly undecidable. W e infer f r om this resu lt a n ew high u n decidabilit y result, pro ving that it is Π 1 1 -complete to determine wh ether t w o give n infi nitary rational relations are disjoin t. Then w e apply this Σ 1 1 -complete v ersion of the P ost Corresp ondence Problem to the study of con tinuit y problems for ω -rational fun ctions realized by finite state B ¨ uc hi transd ucers, considered b y Pr ieur in [Pri01, Pri02] and b y Carton, Fink el and Simonnet in [CFS08]. W e pro v e that there is an amazing gap b et wee n t w o decision problems ab ou t ω -rational fu nctions. Ind eed Prieur pro ve d in [Pri01, Pr i02] that it is decid- able wh ether a giv en ω -rational function is conti nuous, while we show here that it is Σ 1 1 -complete to determine whether a giv en ω -rati onal function h as at least one p oint of contin u it y . Next we prov e that it is Π 1 1 -complete to determine wh ether the cont inuit y set of a give n ω -ratio nal fun ction is ω - regular. This give s the exact complexit y of t w o problems which w ere sho wn to b e und ecidable in [CFS08]. The pap er is organized as follo ws. W e recall basic notions on automat a and on th e analytica l hierarch y in S ection 2. W e state in S ection 3 the Σ 1 1 -completeness of the infinite Post Corr esp ondence Problem in a regular ω -language. W e pr o v e our main n ew results in Section 4. Some concluding remarks are giv en in Section 5. 2 Recall of basic notions W e assume no w the reader to b e familiar with the theory of formal ( ω )- languages [Tho90, Sta97]. W e recall some usual notations of formal language theory . When Σ is a finite alphab et, a non-empty finite wor d o v er Σ is an y sequence x = a 1 . . . a k , where a i ∈ Σ for i = 1 , . . . , k , and k is an in teger ≥ 1. Σ ⋆ is the set of finite wor ds (including the empty w ord) o v er Σ . The first infinite or dinal is ω . An ω - wo r d o v er Σ is an ω -sequen ce a 1 . . . a n . . . , where for all in tegers i ≥ 1, a i ∈ Σ . When σ is an ω -word o ver Σ , w e w rite σ = σ (1) σ (2) . . . σ ( n ) . . . , w h ere for all i , σ ( i ) ∈ Σ , and σ [ n ] = σ (1) σ (2) . . . σ ( n ). The usual co ncatenation p ro duct of t w o finite w ords u and v is denoted u · v and sometimes ju st uv . This pro d uct is extended to the p ro duct of a finite 3 w ord u and an ω -w ord v : the infinite word u · v is then th e ω -w ord su c h that: ( u · v )( k ) = u ( k ) if k ≤ | u | , and ( u · v )( k ) = v ( k − | u | ) if k > | u | . The set of ω - wo r ds o v er the alphab et Σ is denoted by Σ ω . An ω - language o v er an alphab et Σ is a sub set of Σ ω . Definition 2.1 A B¨ uchi automaton is a 5-tuple A = ( K , Σ , δ, q 0 , F ) , wher e K is a finite set of states, Σ is a finite input alp hab et, q 0 ∈ K is the initial state and δ is a mapp ing fr om K × Σ into 2 K . The B¨ uchi automaton A is said to b e deterministic i ff: δ : K × Σ → K . L et σ = a 1 a 2 . . . a n . . . b e an ω -wor d over Σ . A se quenc e of states r = q 1 q 2 . . . q n . . . is c al le d an (infinite) run of A on σ , starting in state p , iff: 1) q 1 = p and 2) for e ach i ≥ 1 , q i +1 ∈ δ ( q i , a i ) . In c ase a run r of A on σ starts in state q 0 , we c al l it simply “a run of A on σ ”. F or every run r = q 1 q 2 . . . q n . . . of A , I n ( r ) is the set of states in K e nter e d by A infinitely many times 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 = ∅} . An ω -language L ⊆ Σ ω is said to b e r e gular iff it is ac cpte d by some B¨ uchi automato n A . W e recall that the class of regular ω -languages is the ω -Kleene closure of the class of regular finitary languages, see [Tho90, Sta97]: an ω -language L ⊆ Σ ω is r egular iff it is of the form L = S 1 ≤ i ≤ n U i · V ω i , for some r egular finitary languages U i , V i ⊆ Σ ⋆ . Acceptance of in finite w ords by other finite mac hines lik e pu shdown automata, T u ring mac h ines, P etri nets, . . . , with v arious a cceptance conditions, has also b een consider ed . In particular, the class of conte xt-free ω -la nguages is the class of ω -languages accepted by B ¨ uc hi pus hdo wn automata, see [Tho90, Sta97, EH93]. The set of natural num b ers is denoted by N , and the set of f unctions from N in to N is denoted by F . W e assu m e the reader to b e familia r with the arithmetical hierarch y on sub sets of N . W e no w recall the definition of classes of the analytical h ierarch y whic h may b e f ound in [Rog67, Odi89, Od i99]. Definition 2.2 L et k, l > 0 b e some inte gers. Φ is a p artial c omputable functional of k function var iables and l numb er variables if ther e exists z ∈ N such that for any ( f 1 , . . . , f k , x 1 , . . . , x l ) ∈ F k × N l , we have Φ( f 1 , . . . , f k , x 1 , . . . , x l ) = τ f 1 ,...,f k z ( x 1 , . . . , x l ) , wher e the right ha nd side is the output of the T uring machine with index z and or acles f 1 , . . . , f k over the input ( x 1 , . . . , x l ) . F or k > 0 and l = 0 , Φ is a p artial c omputable functional if, for some z , Φ( f 1 , . . . , f k ) = τ f 1 ,...,f k z (0) . 4 The value z is c al le d the G¨ odel numb er or index for Φ . Definition 2.3 L et k , l > 0 b e so me inte gers and R ⊆ F k × N l . The r elation R is said to b e a c omputable r elation of k function variables a nd l numb er variables if its char acteristic function is c omputable. W e n o w defin e analytical subsets of N l . Definition 2.4 A subset R of N l is analytic al if it is c omputable or i f ther e exists a c omputable se t S ⊆ F m × N n , with m ≥ 0 and n ≥ l , such that R = { ( x 1 , . . . , x l ) | ( Q 1 s 1 )( Q 2 s 2 ) . . . ( Q m + n − l s m + n − l ) S ( f 1 , . . . , f m , x 1 , . . . , x n ) } , wher e Q i is either ∀ or ∃ for 1 ≤ i ≤ m + n − l , and wher e s 1 , . . . , s m + n − l ar e f 1 , . . . , f m , x l +1 , . . . , x n in some or der. The expr ession ( Q 1 s 1 )( Q 2 s 2 ) . . . ( Q m + n − l s m + n − l ) S ( f 1 , . . . , f m , x 1 , . . . , x n ) is c al le d a pr e dic ate form for R . A qu antifier applying over a function variable is of typ e 1 , otherwise it is of typ e 0 . In a pr e dic ate form the (p ossibly empty) se que nc e of qu antifiers, indexe d by their typ e, is c al le d the pr efix of the form. The r e duc e d pr efix is the se quenc e of quantifiers obtaine d by su ppr essing the quantifiers of typ e 0 fr om the pr efix. W e can no w d istinguish the lev els of the analytical hierarch y b y considering the n umb er of alternations in the reduced pr efix. Definition 2.5 F or n > 0 , a Σ 1 n -pr efix is one whose r e duc e d pr efix b e gins with ∃ 1 and has n − 1 al ternations o f qu antifiers. A Σ 1 0 -pr efix is on e whose r e duc e d pr efix is empty. F or n > 0 , a Π 1 n -pr efix is one wh ose r e duc e d pr e fix b e gins with ∀ 1 and has n − 1 alternations of qu antifiers. A Π 1 0 -pr efix is one whose r e duc e d pr efix is empty. A pr e dic ate form is a Σ 1 n ( Π 1 n )-form if it has a Σ 1 n ( Π 1 n )-pr efix. The class of sets in some N l which c an b e expr esse d in Σ 1 n -form (r esp e ctively, Π 1 n -form) is denote d by Σ 1 n (r esp e ctively, Π 1 n ). The class Σ 1 0 = Π 1 0 is the c lass of arithmetic al sets. W e n o w recall s ome well kno wn r esults ab out the analytical hierarch y . Prop osition 2.6 L et R ⊆ N l for some inte ger l . Then R is an analytic al set iff ther e is so me inte ger n ≥ 0 such that R ∈ Σ 1 n or R ∈ Π 1 n . Theorem 2.7 F or e ach inte ge r n ≥ 1 , (a) Σ 1 n ∪ Π 1 n ( Σ 1 n +1 ∩ Π 1 n +1 . (b) A set R ⊆ N l is in the class Σ 1 n iff its c omplement is in the class Π 1 n . (c) Σ 1 n − Π 1 n 6 = ∅ and Π 1 n − Σ 1 n 6 = ∅ . 5 T r ansformations of prefixes are often used, follo wing the rules giv en b y th e next theorem. Theorem 2.8 F or any pr e dic ate form with the given pr efix, an e qui v alent pr e dic ate form with the new one c an b e obtaine d, fol lowing the al lowe d pr e fix tr ansformations g i ven b elow : (a) . . . ∃ 0 ∃ 0 . . . → . . . ∃ 0 . . . , . . . ∀ 0 ∀ 0 . . . → . . . ∀ 0 . . . ; (b) . . . ∃ 1 ∃ 1 . . . → . . . ∃ 1 . . . , . . . ∀ 1 ∀ 1 . . . → . . . ∀ 1 . . . ; (c) . . . ∃ 0 . . . → . . . ∃ 1 . . . , . . . ∀ 0 . . . → . . . ∀ 1 . . . ; (d) . . . ∃ 0 ∀ 1 . . . → . . . ∀ 1 ∃ 0 . . . , . . . ∀ 0 ∃ 1 . . . → . . . ∃ 1 ∀ 0 . . . ; W e now recall the notions of 1-reduction and of Σ 1 n -completeness (resp ec- tiv ely , Π 1 n -completeness). Giv en t wo sets A, B ⊆ N we say A is 1-reducible to B and wr ite A ≤ 1 B if there exists a total compu table in jectiv e fun ction f from N to N with A = f − 1 [ B ]. A set A ⊆ N is said to b e Σ 1 n -complete (resp ectiv ely , Π 1 n -complete) iff A is a Σ 1 n -set (resp ectiv ely , Π 1 n -set) and for eac h Σ 1 n -set (resp ectiv ely , Π 1 n -set) B ⊆ N it holds that B ≤ 1 A . W e n o w recall an example of a Σ 1 1 -complete decision pr ob lem which will b e useful in the sequel. Definition 2.9 A non deterministic T uring machine M is a 5 -tuple M = ( Q, Σ , Γ , δ , q 0 ) , wher e Q is a finite set of states, Σ is a finite i nput alphab et, Γ is a finite tap e alphab et satisfying Σ ⊆ Γ and c ontaining a sp e cial blank symb ol  ∈ Γ \ Σ , q 0 is the initial state, and δ is a ma pping fr om Q × Γ to subsets of Q × Γ × { L, R, S } . Harel pro ve d the follo wing resu lt in [Har86]. Theorem 2.10 The fol lowing pr oblem is Σ 1 1 -c omp lete: Given a T uring ma- chine M z , of index z ∈ N , do es M z , when starte d on a blank tap e , admit an infinite c omputation that r e enters infinitely often in the initial state q 0 ? 3 The infinite P ost Corresp ondence Problem Recall first the w ell kno wn resu lt ab out the un decidabilit y of the P ost Cor- resp onden ce Problem, d enoted PCP . 6 Theorem 3.1 (P ost, see [HMU01 ]) L et Γ b e an alphab et having at le ast two elements. Then it is unde ci dable to determine, for arbitr ary n-tu ples ( x 1 , x 2 . . . , x n ) and ( y 1 , y 2 . . . , y n ) o f non-empty wor ds in Γ ⋆ , wheth er ther e exists a non-empty se quenc e of indic es i 1 , i 2 . . . , i k such that x i 1 x i 2 . . . x i k = y i 1 y i 2 . . . y i k . On the other hand, the infinite P ost Corresp o nd ence Pr oblem, also called ω -PCP , has b ee n sh own to b e un decidable b y Ru ohonen in [Ruo85] and b y Gire in [Gir86]. Theorem 3.2 L et Γ b e an alphab et having at le ast two elements. Then it is unde cidable to determine, for arbitr ary n-tuples ( x 1 , . . . , x n ) an d ( y 1 , . . . , y n ) of non-empty wor ds in Γ ⋆ , whether ther e exists an infinite se quenc e of indic es i 1 , i 2 , . . . , i k . . . such that x i 1 x i 2 . . . x i k . . . = y i 1 y i 2 . . . y i k . . . Notice that an instance of the ω -PCP is giv en by t w o n-tuples ( x 1 , . . . , x n ) and ( y 1 , . . . , y n ) of n on-empt y w ords in Γ ⋆ , and if there exist some solutions, these ones are infin ite words o v er the alphab et { 1 , . . . , n } . W e are going to consider n o w a v ariant of the infin ite P ost Corresp on d ence Problem where we r estrict solutions to ω -w ords b elonging to a giv en ω - regular language L ( A ) accepted by a giv en B ¨ uc hi automaton A . An instance of the ω -PCP in a regular ω -language, also denoted ω -PCP(Reg) , is give n by t wo n-tuples ( x 1 , . . . , x n ) and ( y 1 , . . . , y n ) of non-empt y wo rds in Γ ⋆ along with a B ¨ uc hi automaton A accepting words o ve r { 1 , . . . , n } . A solution of this problem is an in fi nite sequence of indices i 1 , i 2 , . . . , i k . . . suc h that i 1 i 2 . . . i k . . . ∈ L ( A ) and x i 1 x i 2 . . . x i k . . . = y i 1 y i 2 . . . y i k . . . . Notice that one can asso ciate in a recursive and injectiv e wa y an un ique in teger z to eac h B ¨ uc hi automaton A , this in teger b ei ng called the index of the automato n A . W e denote also A z the B¨ uchi automat on of ind ex z . Then eac h instance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A z ) can b e also charact erized b y an index ¯ I ∈ N . W e can no w state pr ecisely the follo wing result. Theorem 3.3 It is Σ 1 1 -c omp lete to determine, for a given instanc e I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A z ) , given by its index ¯ I , whether ther e is an infi- nite se quenc e of indic es i 1 , i 2 , . . . , i k . . . such that i 1 i 2 . . . i k . . . ∈ L ( A z ) a nd x i 1 x i 2 . . . x i k . . . = y i 1 y i 2 . . . y i k . . . . Pro of. W e fi rst prov e that this problem is in the class Σ 1 1 . It is easy to see that there is an inj ective computable function Φ : N → N su c h th at for all ¯ I ∈ N the B ¨ uc hi T uring machine M Φ( ¯ I ) of ind ex Φ( ¯ I ), wh ere I = 7 (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A z ), acc epts the s et of in finite w ords i 1 i 2 . . . i k . . . ∈ L ( A z ) suc h that x i 1 x i 2 . . . x i k . . . = y i 1 y i 2 . . . y i k . . . . Then the ω -PCP(Reg) of instance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A z ) has a solution if and o nly if the ω -language of the B ¨ uc hi T urin g machine M Φ( ¯ I ) is non-empty . Th us the ω -PCP(Reg) is redu ced to the n on-emptiness problem of B ¨ uc hi T uring mac hines which is kno wn to b e in the class Σ 1 1 . Indeed for a giv en B ¨ uchi T u ring m ac hine M z reading infinite words o ve r an alphab et Σ we can ex- press L ( M z ) 6 = ∅ b y the formula “ ∃ x ∈ Σ ω ∃ r [ r is an accepting run of M z on x ]”; this is a Σ 1 1 -form ula b ec ause the existen tial second order quan tifica- tions are follo w ed b y an arithmetical formula, see [CC89, Fin09] for related results. Th erefore the ω -PCP(Reg) is also in the class Σ 1 1 . W e n ow pr o v e the completeness part of the theorem. Recall that the follo w- ing problem ( P ) is Σ 1 1 -complete by Theorem 2.10. ( P ): Give n a T uring mac hine M z , of in dex z ∈ N , d o es M z , when started on a blank tap e, admit an infin ite computatio n that reent ers infinitely ofte n in the initial state q 0 ? W e can r educe this problem to the ω -PCP in a regular ω -language in the follo wing wa y . Let M = ( Q, Σ , Γ , δ , q 0 ) b e a T uring mac hine, w h ere Q is a finite set of states, Σ is a fin ite input alphab et, Γ is a fi nite tap e alphab et s atisfying Σ ⊆ Γ and con taining a sp ecial blank symb ol  ∈ Γ \ Σ , q 0 is the in itial state, and δ is a mapp ing from Q × Γ to subsets of Q × Γ × { L, R , S } . W e are going to associate to this T u ring mac h ine an instance of the ω - PCP(Reg). First w e define the t w o follo wing lists x = ( x i ) 1 ≤ i ≤ n and y = ( y i ) 1 ≤ i ≤ n of finite w ords ov er the alphab et Σ ∪ Γ ∪ Q ∪ { # } , where # is a sym b ol not in Σ ∪ Γ ∪ Q . x y # = x 1 # q 0 # = y 1 # # a a for eac h a ∈ Γ qa q’b if ( q ′ , b, S ) ∈ δ ( q , a ) qa b q’ if ( q ′ , b, R ) ∈ δ ( q , a ) cqa q’cb if ( q ′ , b, L ) ∈ δ ( q , a ) q# b q’# if ( q ′ , b, R ) ∈ δ ( q ,  ) cq# q’cb# if ( q ′ , b, L ) ∈ δ ( q ,  ) q# q’b# if ( q ′ , b, S ) ∈ δ ( q ,  ) 8 The in teger n is the num b er of w ords in the list x and also in the list y . W e assu me that these t w o lists are indexed so that x = ( x i ) 1 ≤ i ≤ n and y = ( y i ) 1 ≤ i ≤ n . Let n o w E ⊆ { 1 , 2 , . . . , n } b e the set of int egers i suc h that the initial sta te q 0 of the T urin g mac hine M app ears in the w ord y i . The ω - language L ⊆ { 1 , 2 , . . . , n } ω of infinite words o v er the alphab et { 1 , 2 , . . . , n } whic h b egin b y th e letter 1 and ha ve in finitely man y let ters in E is a regular ω -language an d it is accepted by a (deterministic) B ¨ uchi automaton A . W e no w consider the instance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) of the ω - PCP(Reg). It is easy to c heck th at this instance of the ω -PC P(Reg) has a solution i 1 , i 2 , . . . , i k . . . if and only if the T u ring mac hine M , when started on a blank tap e, admits an infin ite compu tation that reen ters infinitely often in the initial state q 0 . Th us the Σ 1 1 -complete pr ob lem ( P ) is redu ced to the ω -PCP in a regular ω -language and th is latter problem is also Σ 1 1 -complete.  4 Applications to infinitary rational relations 4.1 Infinitary rational relations W e no w recall the definition of infinitary r ational r elations, via definition b y B ¨ u c hi transd ucers: Definition 4.1 A B¨ uchi tr ansduc er is a sextuple T = ( K , Σ , Γ , ∆ , q 0 , F ) , wher e K is a finite se t of states, Σ and Γ ar e finite sets c al le d the input and the output alpha b ets, ∆ is a finite subset of K × Σ ⋆ × Γ ⋆ × K c al le d the se t of tr ansitions, q 0 is the initial state, and F ⊆ K is the set of ac c epting states. A c omputation C of th e tr ansduc er T is an infinite se quenc e of c onse cutive 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 essfu l iff ther e exists a final state q f ∈ F and infinitely many inte gers i ≥ 0 su ch that q i = q f . The input wor d and output wor d of the c omputation ar e r esp e ctively u = u 1 .u 2 .u 3 . . . and v = v 1 .v 2 .v 3 . . . The input and the outp ut wor ds ma y b e finite or infinite. The infinitary r ational r elation R ( T ) ⊆ Σ ω × Γ ω ac c epte d by the B ¨ uchi tr ansduc er T is the set of c ouples ( u, v ) ∈ Σ ω × Γ ω such that u and 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 2 . If R ( T ) ⊆ Σ ω × Γ ω is an infinitary rational r elation recognized b y the B ¨ uc hi transducer T then we d enote D om ( R ( T )) = { u ∈ Σ ω | ∃ v ∈ Γ ω ( u, v ) ∈ R ( T ) } 9 and I m ( R ( T )) = { v ∈ Γ ω | ∃ u ∈ Σ ω ( u, v ) ∈ R ( T ) } . It is well kno wn that, f or eac h infinitary rational r elation R ( T ) ⊆ Σ ω × Γ ω , the sets D om ( R ( T )) and I m ( R ( T )) are regular ω -languages and that one can construct, from the B ¨ u c hi transducer T , some B ¨ u c hi automat a A and B accepting the ω -la nguages D om ( R ( T )) and I m ( R ( T )). T o eac h B ¨ uchi transducer T can b e asso ciated in an injectiv e and recursive w a y an index z ∈ N and we shall denote in the sequel T z the B ¨ uchi trans ducer of index z . W e prov ed in [Fin09 ] that many decision problems ab out infinitary rational relations are highly un decidable. In fact many of them, lik e the univ ersalit y problem, th e equ iv alence problem, the inclusion problem, the cofiniteness problem, the u n am biguit y problem, are Π 1 2 -complete, hence located at the second lev el of th e analytical hierarc hy . W e can no w u se the Σ 1 1 -completeness of the ω -PCP in a regular ω -l anguage to obtain a new result of high undecidabilit y . Theorem 4.2 It is Π 1 1 -c omp lete to determine whether two given infinitary r ational r elations ar e disjoint, i.e . the set { ( z , z ′ ) ∈ N 2 | R ( T z ) ∩ R ( T z ′ ) = ∅} is Π 1 1 -c omp lete. Pro of. W e are going to s h o w that the complement of this set is Σ 1 1 -complete, i.e. that the set { ( z , z ′ ) ∈ N 2 | R ( T z ) ∩ R ( T z ′ ) 6 = ∅} is Σ 1 1 -complete. Firstly , it is easy to see that, for t wo giv en B ¨ uchi transducers T z and T z ′ , one can define a B ¨ uc hi T uring mac hine M Φ( z ,z ′ ) of index Φ( z , z ′ ) accepting the ω -language R ( T z ) ∩ R ( T z ′ ). Moreo v er one can constru ct the function Φ : N 2 → N as an injectiv e computable function. This sho ws that the set { ( z , z ′ ) ∈ N 2 | R ( T z ) ∩ R ( T z ′ ) 6 = ∅} is r ed uced to the set { z ∈ N | L ( M z ) 6 = ∅} whic h is in the class Σ 1 1 , since the non-emptiness p r oblem for ω -la nguages of T uring mac hines is in the class Σ 1 1 . Thus th e set { ( z , z ′ ) ∈ N 2 | R ( T z ) ∩ R ( T z ′ ) 6 = ∅} is in the class Σ 1 1 . Secondly , we ha v e to sho w the completeness part of the theorem. W e are going to reduce the ω -PCP in a regular ω -language to the problem of the non-emptiness of the intersect ion of t wo infinitary rational relations. Let then I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) b e an instance of the ω -PCP(Reg), where the x i and y i are w ords o v er an alphab et Γ . W e can then con- struct B ¨ uc hi transducers T ψ 1 ( ¯ I ) and T ψ 2 ( ¯ I ) suc h that the infinitary ratio- nal relation R ( T ψ 1 ( ¯ I ) ) ⊆ { 1 , 2 , . . . , n } ω × Γ ω is the set of pairs of in finite 10 w ords in the form ( i 1 i 2 i 3 · · · ; x i 1 x i 2 x i 3 · · · ) with i 1 i 2 i 3 · · · ∈ L ( A ). And sim- ilarly R ( T ψ 2 ( ¯ I ) ) ⊆ { 1 , 2 , . . . , n } ω × Γ ω is the set of pairs of infinite words in the form ( i 1 i 2 i 3 · · · ; y i 1 y i 2 y i 3 · · · ) with i 1 i 2 i 3 · · · ∈ L ( A ). Thus it holds that R ( T ψ 1 ( ¯ I ) ) ∩ R ( T ψ 2 ( ¯ I ) ) is n on-empt y iff there is an infi nite sequence i 1 i 2 · · · i k · · · ∈ L ( A ) su c h that x i 1 x i 2 · · · x i k · · · = y i 1 y i 2 · · · y i k · · · . The re- duction is now giv en by the injectiv e computable function Ψ : N → N 2 giv en b y Ψ( z ) = (Ψ 1 ( z ) , Ψ 2 ( z )).  4.2 Con tin uit y of ω -rational functions Recall that an infinitary rational relation R ( T ) ⊆ Σ ω × Γ ω is said to b e functional iff it is th e graph of a function, i.e. iff [ ∀ x ∈ D om ( R ( T )) ∃ ! y ∈ I m ( R ( T )) ( x, y ) ∈ R ( T )] . Then the f unctional relation R ( T ) defines an ω -rational (partial ) fu nction F T : D om ( R ( T )) ⊆ Σ ω → Γ ω b y: for eac h u ∈ Dom ( R ( T )), F T ( u ) is the unique v ∈ Γ ω suc h that ( u, v ) ∈ R ( T ). Recall the follo wing p revious decidabilit y r esult. Theorem 4.3 ([Gir86]) One c an de cide whether an infinitary r ational r e- lation r e c o gnize d by a give n B¨ uchi tr ansduc er T is a functional infinitary r ational r elation. One can then asso ciate in a recursive and injectiv e wa y an index to eac h B ¨ u c hi transducer T accepting a functional infinitary rational relation R ( T ). In the sequel w e consider only th ese B ¨ uc hi transducers and w e shall denote T z the B ¨ uc hi trans ducer of index z (su c h that R ( T z ) is functional). It is v ery natural to consider the notion of cont inuit y for ω -rational fun ctions defined by B ¨ uc hi trans d ucers. W e assume the reader to b e familiar with basic notions of top ology whic h ma y b e found in [Kec95, Tho90, S ta97, PP04]. There is a natural metric on the set Σ ω of infinite w ords o v er a finite alphab et Σ whic h is cal led the prefix metric and d efined as follo ws. F or u, v ∈ Σ ω and u 6 = v let d ( u, v ) = 2 − l pr ef ( u,v ) where l pr ef ( u,v ) is the least in teger n suc h that the ( n + 1) th letter of u is differen t fr om the ( n + 1) th letter of v . This metric induces on Σ ω the u sual Canto r top ology for wh ic h op en subsets of Σ ω are in the form W · Σ ω , where W ⊆ Σ ⋆ . W e recall that a function f : D om ( f ) ⊆ Σ ω → Γ ω , wh ose domain is D om ( f ), is said to b e con tinuous at p oint x ∈ D om ( f ) if : 11 ∀ n ≥ 1 ∃ k ≥ 1 ∀ y ∈ D om ( f ) [ d ( x, y ) < 2 − k ⇒ d ( f ( x ) , f ( y )) < 2 − n ] The contin uit y set C ( f ) of the function f is the s et of p oints of con tin uit y of f . The function f is said to b e cont inuous if it is con tin uous at eve ry p oint x ∈ D om ( f ), i. e. if C ( f ) = D om ( f ). Prieur pro ve d the follo wing decidabilit y result. Theorem 4.4 (Prieur [Pri01, Pri02 ]) One c an de ci de whether a giv e n ω -r ational f u nction is c ontinuous. On the other hand the f ollo wing u ndecidabilit y result was p ro v ed in [CFS08]. Theorem 4.5 (see [CFS08]) One c annot de cide whether a given ω -r ational function f has at le ast one p oint of c ontinuity. W e can no w give the exact complexit y of this undecidable prob lem. Theorem 4.6 It is Σ 1 1 -c omp lete to determine whether a g iven ω -r ational function f has at le ast one p oint of c ontinuity, i.e. whether the c ontinuity set C ( f ) of f is non-empty. In other wor ds the set { z ∈ N | C ( F T z ) 6 = ∅} is Σ 1 1 -c omp lete. W e fi rst pro ve the follo wing lemma. Lemma 4.7 The set { z ∈ N | C ( F T z ) 6 = ∅} is in th e class Σ 1 1 . Pro of. Let F b e a f u nction from D om ( F ) ⊆ Σ ω in to Γ ω . F or some in tegers n, k ≥ 1, we consider the set X k ,n = { x ∈ D om ( F ) | ∀ y ∈ Dom ( F ) [ d ( x, y ) < 2 − k ⇒ d ( F ( x ) , F ( y )) < 2 − n ] } F or x ∈ D om ( F ) it holds that: x ∈ C ( F ) ⇐ ⇒ ∀ n ≥ 1 ∃ k ≥ 1 [ x ∈ X k ,n ] W e sh all denote X k ,n ( z ) the set X k ,n corresp onding to the fun ction F T z defined by the B ¨ uc hi transducer of index z . Then it h olds th at: x ∈ C ( F T z ) ⇐ ⇒ ∀ n ≥ 1 ∃ k ≥ 1 [ x ∈ X k ,n ( z ) ] And w e denote R k ,n ( x, z ) the relation giv en by: R k ,n ( x, z ) ⇐ ⇒ [ x ∈ X k ,n ( z ) ] W e n o w pr ov e that this relation is a Π 0 3 -relation. 12 F or x ∈ Σ ω and k ∈ N , w e denote B ( x, 2 − k ) the op en b all of cen ter x and of radius 2 − k , i.e. th e set of y ∈ Σ ω suc h that d ( x, y ) < 2 − k . W e kno w, from the definition of the distance d , that for t wo ω -w ords x and y o v er Σ , the inequalit y d ( x, y ) < 2 − k simply means that x and y hav e the same ( k + 1) first letters. T h us B ( x, 2 − k ) = x [ k + 1] · Σ ω . Bu t b y definition of X k ,n ( z ) it holds that: x ∈ X k ,n ( z ) ⇐ ⇒ ( x ∈ D om ( F T z ) and F T z [ B ( x, 2 − k ) ∩ Dom ( F T z )] ⊆ B ( F T z ( x ) , 2 − n )) W e claim that there is an algo rithm whic h, give n x ∈ Σ ω and z ∈ N , can decide whether F T z [ B ( x, 2 − k ) ∩ D om ( F T z )] ⊆ w · Γ ω , for some finite wo rd w ∈ Γ ⋆ suc h that | w | = n + 1. Indeed th e ω -language B ( x, 2 − k ) ∩ D om ( F T z ) = x [ k + 1] · Σ ω ∩ D om ( F T z ) is the intersect ion of tw o regula r ω -languages and one can construct a B¨ uc hi automaton accepting it. The grap h of th e restriction of the fun ction F T z to th e set x [ k + 1] · Σ ω ∩ D om ( F T z ) is also an infinitary rational relation and one can then also fin d a B ¨ uc hi automaton B accepting F T z [ x [ k + 1] · Σ ω ∩ D om ( F T z )]. On e can then find the set of prefixes of length n + 1 of infinite w ords in L ( B ). If there is only one such prefix w then F T z [ B ( x, 2 − k ) ∩ D om ( F T z )] ⊆ w · Γ ω and otherwise w e ha v e F T z [ B ( x, 2 − k ) ∩ D om ( F T z )] * w ′ · Γ ω for ev er y w ord w ′ ∈ Γ ⋆ suc h that | w ′ | = n + 1. W e no w write S ( x, k, n, z ) iff F T z [ B ( x, 2 − k ) ∩ Dom ( F T z )] ⊆ w · Γ ω , for some finite word w ∈ Γ ⋆ suc h th at | w | = n + 1. As w e ha v e j ust seen the relation S ( x, k, n, z ) is computable, i.e. a ∆ 0 1 relation. On the other han d , we ha v e x ∈ X k ,n ( z ) ⇐ ⇒ ( x ∈ D om ( F T z ) and S ( x, k, n , z )) But D om ( F T z ) is a regular ω -language accepted b y a B¨ uc hi automaton A whic h can b e constructed effectiv ely from T z and hence from the index z . And the relation ( x ∈ L ( A )) is kno wn to b e an arithmetical Π 0 3 (and also a Σ 0 3 ) relation, see [L T94]. Thus “ x ∈ X k ,n ( z )” can b e expressed also by a Π 0 3 (and also a Σ 0 3 ) formula b ecause th e relation S is a ∆ 0 1 relation. No w we h a v e the follo wing equiv alences: C ( F T z ) 6 = ∅ ⇐ ⇒ ∃ x [ x ∈ C ( F T z )] ⇐ ⇒ ∃ x [ ∀ n ≥ 1 ∃ k ≥ 1 x ∈ X k ,n ( z )] 13 Clearly the formula ∃ x [ ∀ n ≥ 1 ∃ k ≥ 1 x ∈ X k ,n ( z )] is a Σ 1 1 -form ula where there is a second order quantificat ion ∃ x follo w ed by an arithmetical Π 0 5 -form ula in whic h the quan tifications ∀ n ≥ 1 ∃ k ≥ 1 are first ord er quan tifications on in tegers.  End of Pro of of Theorem 4.6. T o pro v e the completeness p art of the theorem we use some ideas of [CFS08] bu t w e s hall m o dify the constructions of [CFS08] in ord er to use the Σ 1 1 -completeness of the ω -PCP(Reg) in s tead of the u ndecidabilit y of the PC P . W e are no w going to redu ce the ω -PCP in a regular ω -language to the n on-emptiness of the cont inuit y set of an ω -rational fu nction. Let then I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) b e an instance of the ω -PCP(Reg ), where the x i and y i are words o v er an alphab et Γ . W e can construct an ω - rational fun ction F in the follo wing wa y . Firstly , th e domain D om ( F ) will b e a set of ω -w ords o v er the alphab et { 1 , . . . , n } ∪ { a, b } , wh ere a and b are new letters not in { 1 , . . . , n } . F or x ∈ ( { 1 , . . . , n } ∪ { a, b } ) ω w e denote x ( / { a, b } ) the (finite or infinite) w ord o v er the alphab et { 1 , . . . , n } obtained from x when remo ving ev ery o ccurren ce of the letters a and b . And x ( / { 1 , . . . , n } ) is the (fin ite or infi n ite) word o v er the alph ab et { a, b } obtained from x when r emo ving ev ery o ccurrence of the letters 1 , . . . , n . Then D om ( F ) is the set of ω -words x o v er the alphab et { 1 , . . . , n } ∪ { a, b } suc h that x ( / { a, b } ) ∈ L ( A ) (so in particular x ( / { a, b } ) is infinite) and x ( / { 1 , . . . , n } ) is infinite. It is cle ar that this domain is a regular ω -language. Secondly , for x ∈ D om ( F ) suc h that x ( / { a, b } ) = i 1 i 2 · · · i k · · · ∈ L ( A ) w e set: • F ( x ) = x i 1 x i 2 · · · x i k · · · if x ( / { 1 , . . . , n } ) ∈ ( { a, b } ⋆ · a ) ω , and • F ( x ) = y i 1 y i 2 · · · y i k · · · if x ( / { 1 , . . . , n } ) ∈ { a, b } ⋆ · b ω . The ω -language ( { a, b } ⋆ .a ) ω is the set of ω -w ords o ver the alphab et { a, b } ha ving in finitely many let ters a . The ω -language { a, b } ⋆ .b ω is the com- plemen t in { a, b } ω of the ω -language ( { a, b } ⋆ .a ) ω : it is the s et of ω -words o v er the alphab et { a, b } con taining on ly fin itely man y letters a . Th e t w o ω -languages ( { a, b } ⋆ .a ) ω and { a, b } ⋆ .b ω are ω -regular, and one can easily construct B ¨ uc hi automata accepting th em. Then it is easy to see that the function F is ω -ratio nal and that one can construct a B ¨ uchi transd ucer T ac- cepting the graph of the function F . Moreo v er one can construct an injectiv e computable fun ction ψ : N → N such that T = T ψ ( ¯ I ) and so F = F T ψ ( ¯ I ) . 14 W e now p ro v e that if x ∈ D om ( F T ψ ( ¯ I ) ) is a p o int of con tin uit y of the function F T ψ ( ¯ I ) then th e ω -PCP(Reg) of ins tance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) has a solution i 1 i 2 · · · i k · · · , i.e. an ω -word i 1 i 2 · · · i k · · · ∈ L ( A ) suc h that x i 1 x i 2 · · · x i k · · · = y i 1 y i 2 · · · y i k · · · . T o simplify the notati ons we denote b y F the fun ction F T ψ ( ¯ I ) . W e n o w distinguish tw o cases. First Case. Assume firstly that x ( / { 1 , . . . , n } ) ∈ ( { a, b } ⋆ · a ) ω and that x ( / { a, b } ) = i 1 i 2 · · · i k · · · ∈ L ( A ). Th en b y definition of F it holds that F ( x ) = x i 1 x i 2 · · · x i k · · · . W e denote z = x ( / { 1 , . . . , n } ). Notice that there is a sequence of elemen ts z p ∈ { a, b } ⋆ · b ω , p ≥ 1, such that the sequence ( z p ) p ≥ 1 is conv ergen t and l im ( z p ) = z = x ( / { 1 , . . . , n } ). This is due to the fact that { a, b } ⋆ .b ω is dens e in { a, b } ω . W e call t p the infin ite wo rd o v er the al ph ab et { 1 , . . . , n } ∪ { a, b } such that, for eac h in teger i ≥ 1, w e ha v e t p ( i ) = x ( i ) if x ( i ) ∈ { 1 , . . . , n } and t p ( i ) = z p ( k ) if x ( i ) is the k th letter of z . Then th e sequen ce ( t p ) p ≥ 1 is con v ergen t and l im ( t p ) = x . But b y definition of F it holds that F ( t p ) = y i 1 y i 2 · · · y i k · · · for eve ry in teger p ≥ 1 while F ( x ) = x i 1 x i 2 · · · x i k · · · . T h us if x is a p oint of con tin uity of the function F then it holds that x i 1 x i 2 · · · x i k · · · = y i 1 y i 2 · · · y i k · · · and th e ω -PCP(Reg) of instance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) has a solution i 1 i 2 · · · i k · · · . Second Case. Assume now that x ( / { 1 , . . . , n } ) ∈ { a, b } ⋆ · b ω and th at x ( / { a, b } ) = i 1 i 2 · · · i k · · · ∈ L ( A ). Notice that ( { a, b } ⋆ .a ) ω is also dense in { a, b } ω . Th en reasoning as in the firs t case we can prov e that if x is a p oint of con tin uit y of F then x i 1 x i 2 · · · x i k · · · = y i 1 y i 2 · · · y i k · · · and the ω -PC P(Reg) of instance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) has a solution i 1 i 2 · · · i k · · · . Con v ersely assu me that the ω -PC P in a regular ω -language of instance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) has a solution i 1 i 2 · · · i k · · · , i.e. an ω -word i 1 i 2 · · · i k · · · ∈ L ( A ) such that x i 1 x i 2 · · · x i k · · · = y i 1 y i 2 · · · y i k · · · . W e now sho w that eac h x ∈ D om ( F ) s uc h that x ( / { a, b } ) = i 1 i 2 · · · i k · · · is a p oin t of con tin uity of the fun ction F . C onsider an infinite sequence ( t p ) p ≥ 1 of ele- men ts of D om ( F ) suc h that l im ( t p ) = x . It is easy to see that the sequence ( t p ( / { a, b } )) p ≥ 1 is conv ergent and that its limit is the ω -wo rd x ( / { a, b } ) = i 1 i 2 · · · i k · · · . This implies ea sily that the sequence F ( t p ) p ≥ 1 is con v ergen t and th at its limit is the ω -w ord F ( x ) = x i 1 x i 2 · · · x i k · · · = y i 1 y i 2 · · · y i k · · · . Th us x is a p oin t of conti nuit y of F and this ends the p r o of.  W e consider no w the con tin uity set of an ω -rational function and its p ossible complexit y . The follo wing u n decidabilit y result was pr ov ed in [CFS08]. 15 Theorem 4.8 (see [CFS08]) One c annot de cide whether the c ontinuity set of a given ω -r ational f u nction f is a r e gular (r esp e ctively, c ontext-fr e e) ω -language. W e can n o w giv e the exact complexity of the first ab o ve undecidable p rob- lem. Theorem 4.9 It is Π 1 1 -c omp lete to determine whether the c ontinuity set C ( f ) of a give n ω -r ational function f is a r e gular ω -language. In other wor ds th e set { z ∈ N | C ( F T z ) is a r e gular ω -language } is Π 1 1 -c omp lete. W e fi rst pro ve the follo wing lemma. Lemma 4.10 The set { z ∈ N | C ( F T z ) is a r e gular ω -language } is in the class Π 1 1 . Pro of. Recall that A z denotes the B ¨ uc hi automaton of index z . W e can express the sen tence “ C ( F T z ) is a regular ω -language” by the sen tence: ∃ z ′ C ( F T z ) = L ( A z ′ ) . On the other han d we ha v e seen in the pro of of Lemma 4.7 that x ∈ C ( F T z ) ⇐ ⇒ [ ∀ n ≥ 1 ∃ k ≥ 1 x ∈ X k ,n ( z )] and then th at x ∈ C ( F T z ) can b e expressed b y an arithmetical Π 0 5 -form ula. W e can no w expr ess C ( F T z ) = L ( A z ′ ) by: ∀ x [( x ∈ C ( F T z ) and x ∈ L ( A z ′ )) or ( x / ∈ C ( F T z ) and x / ∈ L ( A z ′ ))] whic h is a Π 1 1 -form ula b ecause there is one u niv ersal second order quan tifi- cation ∀ x follo we d b y an arith m etical form ula (recall that x ∈ L ( A z ′ ) ca n b e expressed by an arithm etical Π 0 3 -form ula). Finally the sente nce ∃ z ′ C ( F T z ) = L ( A z ′ ) can b e exp ressed by a Π 1 1 -form ula b ecause the quantificati on ∃ z ′ is a first- order q u an tification b earing on inte gers and the formula C ( F T z ) = L ( A z ′ ) can b e expressed by a Π 1 1 -form ula.  End of Pro of of Theorem 4.9. T o pro v e the completeness p art of the theorem w e redu ce the ω -PCP in a regular ω -language to the problem of the non-regularit y of th e con tin uit y set of an ω -rati onal function. As in the p ro of of the ab o ve Theorem 4.8 in [CFS08], we sh all use a particular instance of Po st Corresp ond ence Problem. F or t wo letters c, d , let PCP 1 16 b e the Post Corresp ondence Problem of instance (( t 1 , t 2 , t 3 ) , ( w 1 , w 2 , w 3 )), where 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 sequences of ind ices in { 1 i · 2 i · 3 i | i ≥ 1 } ∪ { 3 i · 2 i · 1 i | i ≥ 1 } . In p articular, this language o v er the alph ab et { 1 , 2 , 3 } is not con text- free and this will b e u seful in the sequel. Let then I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) b e an instance of the ω -PCP(Reg ), where the x i and y i are words o v er an alphab et Γ . W e can construct an ω - rational fun ction F ′ in the follo wing w a y . Let D = { d 1 , d 2 , d 3 } s u c h that D and { 1 , . . . , n } ∪ { a, b } are disjoin t. Th e domain D om ( F ′ ) will b e a set of ω -words in D + · D om ( F ), where, as in the pro of of T heorem 4.6, D om ( F ) is the set of ω -wo rds x o ve r the alphab et { 1 , . . . , n } ∪ { a, b } suc h that x ( / { a, b } ) ∈ L ( A ) (so in particular x ( / { a, b } ) is infinite) and x ( / { 1 , . . . , n } ) is infin ite. It is clear that the domain D om ( F ′ ) is a regular ω -la nguage. No w, for x ∈ D om ( F ′ ) suc h that x = d j 1 · · · d j p · y with y ∈ D om ( F ) and y ( / { a, b } ) = i 1 i 2 · · · i k · · · ∈ L ( A ) we set: • F ′ ( x ) = t j 1 · · · t j p x i 1 x i 2 · · · x i k · · · if y ( / { 1 , . . . , n } ) ∈ ( { a, b } ⋆ · a ) ω , and • F ′ ( x ) = w j 1 · · · w j p y i 1 y i 2 · · · y i k · · · if y ( / { 1 , . . . , n } ) ∈ { a, b } ⋆ · b ω . Then it is ea sy to see that the function F ′ is ω -rational and that o ne can construct a B ¨ uchi transd ucer T ′ accepting the graph of th e function F ′ . Moreo v er one can construct an injectiv e computable fu n ction Θ : N → N suc h that T ′ = T Θ( ¯ I ) and so F ′ = F T Θ( ¯ I ) . Reasoning as in the pr eceding pro of we can prov e that the function F ′ is con tin uous at p oin t x = d j 1 · · · d j p · y , where y ∈ D om ( F ) , if and only if the the sequence j 1 , . . . , j p is a s olution of the Post Corresp ondence Problem PCP 1 and y ( / { a, b } ) = i 1 i 2 · · · i k · · · is a solution of the ω -PCP(Reg) of instance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ). Th us if the ω -PCP (Reg) of instance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) has no solution, then the contin u it y set C ( F ′ ) is empt y , hen ce it is ω -regular. On the other hand assume that the ω -PCP(Reg) of in stance I = (( x 1 , . . . , x n ), ( y 1 , . . . , y n ) , A ) has some solutions. In that case the cont inuit y set C ( F ′ ) is in the form T · R where T = { d i 1 · d i 2 · d i 3 | i ≥ 1 } ∪ { d i 3 · d i 2 · d i 1 | i ≥ 1 } and R is a set of in finite words o ver th e alphab et { 1 , . . . , n } ∪ { a, b } . In that case the con tin uit y set C ( F ′ ) ca n not b e ω -regular b ecause otherwise the language T 17 should b e regular (since D = { d 1 , d 2 , d 3 } and { 1 , . . . , n } ∪ { a, b } are disjoint ) and it is not even con text-free. This sho ws that the ω -PCP(Reg) of instance I = (( x 1 , . . . , x n ) , ( y 1 , . . . , y n ) , A ) has a solution if and only if the con tin uity set C ( F ′ ) is not ω -regula r. This ends the pro of.  It is n atural to ask w hether th e set { z ∈ N | C ( F T z ) is a con text-free ω - language } is also Π 1 1 -complete. Bu t one cannot extend d ir ectly Lemma 4.1 0, replacing regular b y conte xt-free. If we replace the B ¨ u c hi automaton A z of index z b y the B ¨ uchi pushd o wn automaton B z of index z , we get only that the set { z ∈ N | C ( F T z ) is a con text-free ω -language } is in the class Π 1 2 b ecause the “ x ∈ L ( B z ′ )” can only b e expr essed b y a Σ 1 1 -form ula. On the other h and, the second p art of the pro of of Theorem 4.9 prov es in the same w a y that the set { z ∈ N | C ( F T z ) is a con text-free ω -language } is Π 1 1 -hard. Th us we can no w state the follo wing result. Theorem 4.11 The set { z ∈ N | C ( F T z ) is a c ontext-fr e e ω -language } is Π 1 1 -har d and in the class Π 1 2 \ Σ 1 1 . 5 Concluding remarks W e ha v e give n a complete pro of of the Σ 1 1 -completeness of the ω -PCP in a regular ω -la nguage, also denoted ω -PCP(Reg). Then we hav e app lied this result and ob tained the exact complexit y of several highly undecidable problems ab out in finitary rational relati ons and ω -rational fu nctions. 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