Decision Problems for Recognizable Languages of Infinite Pictures

Decision Problems for Recognizable Languages of Infinite Pictures Oli vier Finkel Equipe de Logique Math ´ ematique CNRS et Uni versit ´ e P aris Diderot P aris 7 UFR de Math ´ ematiques case 7012, site Chev aleret, 75205 P aris C edex 13, F rance. finkel@logique.jussieu.fr Abstract Altenber nd, Thomas and W ¨ ohrle ha ve consid ered in [A TW03] accepta nce of lang uages of infini te two-dimens ional words (in finite pictures) by finite tiling systems, with the usual acceptanc e condit ions, such as the B ¨ uchi and Muller ones , firstly use d for infinite wor ds. Many classical decision pro b- lems are studie d in for mal language theory and in a utomata theory and ari se no w natural ly about recogniz able langua ges of infinite pict ures. W e first re view in this pa per some recent results of [Fin09b] where we gav e the ex- act de gree of numero us undecidabl e prob lems for B ¨ uchi-reco gnizable lan- guage s of infinite pictures , which are actually locate d at the first or at the second le vel of the ana lytical hiera rchy , and “hi ghly undecidab le”. Then we pro ve here some more (high) undec idability re sults. W e first sho w that it is Π 1 2 -complet e to dete rmine whether a gi ven B ¨ uchi-re cognizable language s of infinite pictures is unambigu ous. Then w e in ves tigate cardinal ity pr oblems. Using recent results of [FL09], we prov e that it i s D 2 (Σ 1 1 ) -comple te to de- termine whether a giv en B ¨ uchi-reco gnizable la nguage of infinite pictures is counta bly infinite, an d that it is Σ 1 1 -complet e to determine wheth er a gi ven B ¨ uchi-rec ognizable langua ge of infinite pictures is unco untable. Next we consid er complements of recog nizable langu ages of infinite pic tures. Using some results of Set Theory , we sho w that the cardina lity of the comple- ment o f a B ¨ uchi-r ecognizabl e lang uage of infinite pictures may depend on the model of the axiomat ic system ZF C . W e pr ove that the problem to de- termine whether the complement of a gi ven B ¨ uchi-r ecognizabl e language of infinite pic tures is countab le (respecti vely , uncou ntable) is in the class Σ 1 3 \ (Π 1 2 ∪ Σ 1 2 ) (respe ctiv ely , i n the class Π 1 3 \ (Π 1 2 ∪ Σ 1 2 ) ). 1 Ke ywords: Languag es of infinit e picture s; recogn izability by tilin g systems; decision proble ms; unambiguity problem; cardinali ty problems; highly und ecidable prob lems; an- alytica l hierarchy ; models of set theory; indepen dence from the axiomatic system ZF C . 1 Introduction Languages of infinite words accepted by finite automata were first studied by B ¨ uchi to prove the decidability of the monadic second order theory o f one s uc- cessor over the int egers. Since th en regular ω -languages ha ve been mu ch studi ed and man y applications have been foun d f or specification and verification of non- terminating systems , see [ Tho9 0, PP04] for many results and references. Altenbernd, Thomas and W ¨ ohrl e ha ve considered in [A TW03] acceptance of lan- guages of infinite two-dimensional words (infinite pictures) by finite til ing sys- tems, with the usual acceptance con ditions, such as the B ¨ uchi and Mul ler on es, firstly used f or infinite w ords. Th is w ay they extended b oth the classi cal theory of ω -regular languages and t he classical theory of recognizable lang uages o f finite pictures, [GR97], to the case of infinite pictures. Many classical decision problems are studied in formal l anguage theory and i n automata t heory and a rise no w n aturally about re cogni zable lang uages of infinite pictures. In a recent paper , we ga ve the exact degree of numerous undecidable problems for B ¨ uchi-recognizable languages of infinite pictures. In particul ar , th e non-empti ness and the infinit eness problem s are Σ 1 1 -complete, and t he un iv ersality prob lem, th e inclusion problem, the equi valence probl em, the complementability problem, and the determinizability problem, are all Π 1 2 -complete. These decision probl ems are then located at t he first or at t he second lev el of the analyti cal hierarchy , and “highly undecidable”. This ga ve ne w natural examples of decision problems lo- cated at the first or at the second lev el of the analytical hierarchy . Here we first revie w s ome of these results, and we study new decision problems, obtaining new results of high undecidability . W e first consider the not ion of unambig uous B ¨ uchi tiling system, and of unam- biguous B ¨ uchi-recognizable lang uage of infinite pictures. W e show that eve ry language of infinite pictures which is accepted by an unambiguous B ¨ uchi tili ng system is a Borel set. As a corollary thi s shows the existence of inherently am- biguous B ¨ uchi-recognizable language of infinit e pictures. Then we us e this result 2 to prov e that i t is Π 1 2 -complete to determine whether a given B ¨ uchi-recognizable language of infinite pictures is unambigu ous. Next we study cardinality problems. Using re cent results of Finkel and Lecom te in [FL09], we first show that it is D 2 (Σ 1 1 ) -complete to determin e whether a given B ¨ uchi-recognizable langu age of i nfinite pictures is countably in finite, where D 2 (Σ 1 1 ) is the class of 2 -dif ferences of Σ 1 1 -sets, i .e. the class of sets which are intersections of a Σ 1 1 -set a nd of a Π 1 1 -set. And it is Σ 1 1 -complete to determine whether a given B ¨ uchi-recognizable language of infinite pictures is uncountabl e. Then we consi der the complement s of B ¨ uchi-recognizable languages of infinite pictures. By usi ng some re sul ts of Set Theory , we show that the cardinality of the complement of a B ¨ uchi-recognizable language of infinite pictures may depend on the actual model of the axiom atic system ZFC . W e prove that on e can e ffec- tiv ely construct a B ¨ uchi tiling system T accepting a language L ⊆ Σ ω ,ω , whose complement is L − = Σ ω ,ω − L , such that: 1. Th ere is a model V 1 of ZFC in which L − is countable. 2. Th ere is a model V 2 of ZFC in which L − has cardinal 2 ℵ 0 . 3. Th ere is a model V 3 of ZFC in which L − has c ardinal ℵ 1 with ℵ 0 < ℵ 1 < 2 ℵ 0 . Then, usi ng the proof of this result and Schoenfield’ s A bsoluteness Theorem, we prove that the problem t o d etermine whether the complement of a given B ¨ uchi- recognizable language of infinite pictures i s countable (respectively , uncountable) is in the class Σ 1 3 \ (Π 1 2 ∪ Σ 1 2 ) (respectiv ely , in the class Π 1 3 \ (Π 1 2 ∪ Σ 1 2 ) ). This shows t hat natural cardinality problems are actually located at the th ird lev el of the analytical hierarchy . The paper is organized as follo ws. W e recall in Section 2 the not ions of tiling systems and of recognizable languages of pi ctures. I n sectio n 3 , we recall the definition of the analy tical hierarchy on subsets of N . The definitions of the Borel hierarchy and of analyt ical sets of a Cantor space, along with their ef fectiv e c oun - terparts, are given in Section 4 . Some notions of Set Theory , whi ch are useful i n the sequel, are exposed in Section 5 . W e st udy decision prob lems in Section 6 , proving ne w results. Some concludin g remarks are gi ven in Section 7 . 2 T iling Syste ms W e assum e the reader to be familiar with the theory of formal ( ω )-langu ages [Tho90, Sta97]. W e recall usu al notations of formal language theory . 3 When Σ is a finite alp habet, a non-empty finite wor d ove r Σ is any sequence x = a 1 . . . a k , w here a i ∈ Σ for i = 1 , . . . , k , and k i s an int eger ≥ 1 . The length of x is k , denoted by | x | . The empty wor d has no letter and is d enoted by λ ; its length is 0 . Σ ⋆ is the set of finite wor ds (including the empty word) o ver Σ . The first i nfinite or d inal is ω . An ω - wor d over Σ is an ω -sequence a 1 . . . a n . . . , where for all inte gers i ≥ 1 , a i ∈ Σ . When σ is an ω -w ord over Σ , we write σ = σ (1) σ (2) . . . σ ( n ) . . . , where for all i , σ ( i ) ∈ Σ , and σ [ n ] = σ (1) σ (2 ) . . . σ ( n ) for all n ≥ 1 and σ [0 ] = λ . The usual concatenation o f two finite words u and v is deno ted u.v (and som e- times just u v ). This product is extended to the product of a finite word u and 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 | . The set of ω - wor ds over t he alph abet Σ is denoted by Σ ω . An ω - language over an alphabet Σ is a subset of Σ ω . W e now define tw o-dim ensional words, i.e. pictures. Let Σ be a finite alphabet, let # be a letter not in Σ and let ˆ Σ = Σ ∪ { # } . If m and n are two positiv e i ntegers or if m = n = 0 , a picture o f size ( m, n ) ove r Σ is a function p from { 0 , 1 , . . . , m + 1 } × { 0 , 1 , . . . , n + 1 } into ˆ Σ s uch that p ( i, j ) = # if i ∈ { 0 , m + 1 } or j ∈ { 0 , n + 1 } and p ( i, j ) ∈ Σ o therwise. The empty picture is the only pi cture of size (0 , 0) and is denoted by λ . Pictures of size ( n, 0) or (0 , n ) , for n > 0 , are not defined. Σ ⋆,⋆ is the set of pictures ove r Σ . A picture l anguage L is a s ubset of Σ ⋆,⋆ . The r esearch on pictu re languages wa s firstly mot iv ated by the problems arising in pattern recognition and image processing, a survey on t he theory of picture languages may be found in [GR97]. An ω -picture over Σ is a functio n p from ω × ω into ˆ Σ such t hat p ( i, 0) = p (0 , i ) = # for all i ≥ 0 and p ( i, j ) ∈ Σ for i, j > 0 . Σ ω ,ω is the set of ω -pictures over Σ . An ω -picture language L is a subset of Σ ω ,ω . For Σ a finite alphabet we call Σ ω 2 the set of functions from ω × ω into Σ . So the set Σ ω ,ω of ω -pictures over Σ is a strict subset of ˆ Σ ω 2 . W e shall say that, for each integer j ≥ 1 , the j th row of an ω -picture p ∈ Σ ω ,ω is the infinite w ord p ( 1 , j ) .p (2 , j ) .p (3 , j ) . . . over Σ and the j th column of p is the infinite word p ( j, 1) .p ( j, 2) .p ( j, 3) . . . over Σ . As usual, one can imagi ne that, for inte gers j > k ≥ 1 , the j th column of p is on the right of th e k th column of p and that the j th row of p i s “a bove” the k th row of p . W e introdu ce no w (non deterministic) tiling systems as in the paper [A TW03]. A tiling system is a tuple A = ( Q, Σ , ∆) , where Q is a finite set of s tates, Σ is a finite alphabet, ∆ ⊆ ( ˆ Σ × Q ) 4 is a finite set of tiles. 4 A B ¨ uchi tiling system is a pair ( A ,F ) where A = ( Q, Σ , ∆) is a tiling sy stem and F ⊆ Q is the set of accepting states. A Muller tiling system is a pair ( A , F ) where A = ( Q, Σ , ∆) is a tilin g system and F ⊆ 2 Q is the set of accepting sets of states. T iles are denoted by  ( a 3 , q 3 ) ( a 4 , q 4 ) ( a 1 , q 1 ) ( a 2 , q 2 )  with a i ∈ ˆ Σ and q i ∈ Q, and in general, over an alphabet Γ , by  b 3 b 4 b 1 b 2  with b i ∈ Γ . A combination of tiles is defined by:  b 3 b 4 b 1 b 2  ◦  b ′ 3 b ′ 4 b ′ 1 b ′ 2  =  ( b 3 , b ′ 3 ) ( b 4 , b ′ 4 ) ( b 1 , b ′ 1 ) ( b 2 , b ′ 2 )  A run of a tili ng system A = ( Q, Σ , ∆) over a (finit e) pict ure p of size ( m, n ) over Σ i s a mapping ρ from { 0 , 1 , . . . , m + 1 } × { 0 , 1 , . . . , n + 1 } into Q s uch that for all ( i, j ) ∈ { 0 , 1 , . . . , m } × { 0 , 1 , . . . , n } w ith p ( i, j ) = a i,j and ρ ( i, j ) = q i,j we hav e  a i,j +1 a i +1 ,j +1 a i,j a i +1 ,j  ◦  q i,j +1 q i +1 ,j +1 q i,j q i +1 ,j  ∈ ∆ . A run of a tiling s ystem A = ( Q, Σ , ∆) over an ω -picture p ∈ Σ ω ,ω is a mapping ρ from ω × ω i nto Q such that for all ( i, j ) ∈ ω × ω with p ( i, j ) = a i,j and ρ ( i, j ) = q i,j we ha ve  a i,j +1 a i +1 ,j +1 a i,j a i +1 ,j  ◦  q i,j +1 q i +1 ,j +1 q i,j q i +1 ,j  ∈ ∆ . W e now reca ll acceptance of finite or infinite pictures by tiling syst ems: Definition 2.1 Let A = ( Q, Σ , ∆) be a tiling system, F ⊆ Q and F ⊆ 2 Q . • The pictur e language r ecognized by A is the set of pictures p ∈ Σ ⋆,⋆ such that ther e is some run ρ of A on p . • The ω -pictur e language B ¨ uchi-r ecognized by ( A ,F ) is the set of ω -pictur es p ∈ Σ ω ,ω such that ther e is some ru n ρ of A o n p and ρ ( v ) ∈ F for infinitely many v ∈ ω 2 . It is denoted by L B (( A ,F )) . • The ω -pictur e l anguage Muller- r ecognized by ( A , F ) is the set of ω -pictu r es p ∈ Σ ω ,ω such that ther e is some run ρ of A on p and I nf ( ρ ) ∈ F wher e I nf ( ρ ) is the set of s tates occurri ng infin itely often i n ρ . It is denoted by L M (( A , F )) . 5 Notice that an ω -picture language L ⊆ Σ ω ,ω is recognized by a B ¨ uchi t iling syst em if and only if it is recognized by a Muller tiling system, [A TW03]. W e shall denote T S (Σ ω ,ω ) the class of languages L ⊆ Σ ω ,ω which are recognized by some B ¨ uchi (or Muller) tiling syst em. 3 Recall of Known Basic Notions 3.1 The Analytical Hierarc hy The s et of nat ural nu mbers is deno ted by N and the set of all mappings from N into N will be denoted by F . W e assume the reader to be familiar with the arithmetical hierarchy on subsets of N . W e now recall the notions of analytical hierarchy and of complete sets for classes of this hierarchy which may be found in [Rog67]. Definition 3.1 Let k , l > 0 be some int e ger s. Φ is a p artial r ecursive fu nction of k function variables and l numb er variab les if ther e e xists 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 han d s ide is the output of the T uring machine with index z and oracles f 1 , . . . , f k over the input ( x 1 , . . . , x l ) . F or k > 0 a nd l = 0 , Φ is a par tial r ecursive function if, for some z , Φ( f 1 , . . . , f k ) = τ f 1 ,...,f k z (0) . The value z is called the G ¨ odel number or index for Φ . Definition 3.2 Let k, l > 0 be some int e ger s and R ⊆ F k × N l . The r elation R is said to be a rec ursive r elatio n of k functi on variables and l number variables if its characteristic f unction is r ecursive. W e now define ana ly tical subsets of N l . Definition 3.3 A subset R of N l is analytical i f it i s r ecursive or if t her e e xists a r ecursive set 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 eith er ∀ 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 . 6 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 called a pr edicate form for R . A quantifier appl ying o ver a function vari able is of type 1 , otherwise it is of type 0 . In a pr edicate form the (poss ibly empty) sequence of quan tifiers, indexed by their type, is called the pr efix of the form. The r educed pr efix is the sequence of quant ifiers obtain ed by suppr essing the quantifi ers of type 0 fr om the pr efix. The levels of the analytical hierarchy are distinguished b y consi dering the number of alternations in the reduced prefix. Definition 3.4 F or n > 0 , a Σ 1 n -pr efix is one whose r educed prefix be gins with ∃ 1 and has n − 1 alt ernations o f quantifiers. A Σ 1 0 -pr efix is on e whose r educed pr efix is empty . F o r n > 0 , a Π 1 n -pr efix is one w hos e re duced pr efix be gins with ∀ 1 and has n − 1 alternations of quant ifiers. A Π 1 0 -pr efix is one whose r educed p r efix is empty . A pr edicate f orm 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 can be e xpr essed in Σ 1 n -form (r espectively , Π 1 n -form) is denot ed by Σ 1 n (r espectively , Π 1 n ). The class Σ 1 0 = Π 1 0 is the class of arithmeti cal sets. W e now reca ll some well known results about the analytical hierar chy . Pr oposi tion 3.5 Let R ⊆ N l for som e int e ger l . Then R is an analytical set iff ther e is some inte ger n ≥ 0 such that R ∈ Σ 1 n or R ∈ Π 1 n . Theor em 3.6 F or each inte ger 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 complement is in the class Π 1 n . (c) Σ 1 n − Π 1 n 6 = ∅ an d Π 1 n − Σ 1 n 6 = ∅ . T ransformations of prefixes are often used, follo wing the rules gi ven by the ne xt theorem. Theor em 3.7 F or any pr edi cate form with the given pr efix, an equivalent pr edi - cate form with the new one can b e obtained, f ollowing the allo wed pr efix trans- formations given below : (a) . . . ∃ 0 ∃ 0 . . . → . . . ∃ 0 . . . , . . . ∀ 0 ∀ 0 . . . → . . . ∀ 0 . . . ; (b) . . . ∃ 1 ∃ 1 . . . → . . . ∃ 1 . . . , . . . ∀ 1 ∀ 1 . . . → . . . ∀ 1 . . . ; 7 (c) . . . ∃ 0 . . . → . . . ∃ 1 . . . , . . . ∀ 0 . . . → . . . ∀ 1 . . . ; (d) . . . ∃ 0 ∀ 1 . . . → . . . ∀ 1 ∃ 0 . . . , . . . ∀ 0 ∃ 1 . . . → . . . ∃ 1 ∀ 0 . . . ; W e can now define the not ion of 1-reduction and of Σ 1 n -complete (respecti vely , Π 1 n -complete) sets. Notice that we give the definition for subsets of N b ut one can easily extend this definition to the case of subsets of N l for some integer l . Definition 3.8 Given two sets A, B ⊆ N we say A is 1-r educible to B and writ e A ≤ 1 B if ther e exists a tot al computable inj ective function f fr om N t o N s uch that A = f − 1 [ B ] . Definition 3.9 A set A ⊆ N i s said to be Σ 1 n -complete (r espectively , Π 1 n -complete) iff A is a Σ 1 n -set (r esp ectively , Π 1 n -set) and for each Σ 1 n -set ( r espectively , Π 1 n -set) B ⊆ N it holds that B ≤ 1 A . For each int eger n ≥ 1 th ere exists some Σ 1 n -complete set E n ⊆ N . The com- plement E − n = N − E n is a Π 1 n -complete set. T hese set s are prec is ely defi ned in [Rog67] or [CC89]. 3.2 Borel Hie rarch y a nd Analytic Sets W e assume now the reader to b e familiar with basic notions of to pology which may be found in [Mos80, L T94, Kec 95, Sta97, PP04]. There is a natural metric on the set Σ ω of infinite words ov er a finite alphabet Σ contain ing at least two letters whi ch is called the pr efix metri c and defined as follows. For u, v ∈ Σ ω and u 6 = v let δ ( u, v ) = 2 − l pref ( u ,v ) where l pref ( u,v ) is the first inte ger n su ch that the ( n + 1) st letter of u is dif ferent from the ( n + 1) st letter of v . T his metric induces on Σ ω the usual Cantor to pology for which open subsets of Σ ω are in t he form W . Σ ω , where W ⊆ Σ ⋆ . A set L ⊆ Σ ω is a closed set iff its complement Σ ω − L is an open set. No w let define the Bor el Hierar chy of subsets of Σ ω : Definition 3.10 F or a non-null countab le or dinal α , t he classes Σ 0 α and Π 0 α of the Bor el Hierar chy on the topological space Σ ω ar e defined as follows: Σ 0 1 is the class of open subsets of Σ ω , Π 0 1 is the class of closed subsets of Σ ω , and for any countable or dinal α ≥ 2 : Σ 0 α is the class of countable unions of subsets of Σ ω in S γ <α Π 0 γ . Π 0 α is the class of countable intersections of subsets of Σ ω in S γ <α Σ 0 γ . 8 For a countable ordinal α , a subset of Σ ω is a Borel set of rank α iff it is in Σ 0 α ∪ Π 0 α but not in S γ <α ( Σ 0 γ ∪ Π 0 γ ) . There are also som e subsets o f Σ ω which are not Borel. Indeed there exists an- other hierarchy beyond t he Borel hierarchy , which is called the projecti ve hier- archy and which is ob tained from the Borel hierarchy by successiv e appl ications of operations of projection and compl ementation. The first level of the projective hierarchy is formed by t he class of analytic s ets a nd the c lass of co-analytic set s which are complements of analytic sets. In particular the class of Borel subsets of Σ ω is st rictly included in to the class Σ 1 1 of analyti c sets whi ch are obtain ed by projections of Borel sets. Definition 3.11 A subset A of Σ ω is in the class Σ 1 1 of analytic sets iff ther e exist a finite set Y and a Bor el subset B of (Σ × Y ) ω such that [ x ∈ A ↔ ∃ y ∈ Y ω ( x, y ) ∈ B ] , wher e ( x, y ) is the in finite wor d over the alphabet Σ × Y such that ( x, y )( i ) = ( x ( i ) , y ( i )) for each inte ger i ≥ 1 . W e now define completeness with re gard to reduction by continuous functions. For a countable ordinal α ≥ 1 , a set F ⊆ Σ ω is said to be a Σ 0 α (respectiv ely , Π 0 α , Σ 1 1 )- complete set iff for any set E ⊆ Y ω (with Y a finite alphabet): E ∈ Σ 0 α (respectiv ely , E ∈ Π 0 α , E ∈ Σ 1 1 ) i f f there e xists a continuous function f : Y ω → Σ ω such that E = f − 1 ( F ) . Σ 0 n (respectiv ely Π 0 n )-complete sets, with n an integer ≥ 1 , a re thoroug hly charac terized in [Sta86]. In particular R = (0 ⋆ . 1) ω is a well known e xample of a Π 0 2 -complete subset of { 0 , 1 } ω . It is the set of ω -words o ver { 0 , 1 } having infinitely m any occurrences of the letter 1 . Its complement { 0 , 1 } ω − (0 ⋆ . 1) ω is a Σ 0 2 -complete subset of { 0 , 1 } ω . W e recall now the definition of the arithmeti cal hierarchy of ω -languages which form the effec tive ana lo gue to the hierarchy of Borel sets of finite ranks. Let X be a finit e alphabet. An ω -l anguage L ⊆ X ω belongs to the class Σ n if and only if there exists a recursi ve relation R L ⊆ ( N ) n − 1 × X ⋆ such that L = { σ ∈ X ω | ∃ a 1 . . . Q n a n ( a 1 , . . . , a n − 1 , σ [ a n + 1]) ∈ R L } where Q i is one of t he quantifiers ∀ o r ∃ (not necessarily in an alt ernating order). An ω -language L ⊆ X ω belongs t o the class Π n if and only if its complement X ω − L belongs to the class Σ n . The inclus ion relation s t hat hold between the classes Σ n and Π n are the same as for the corresponding class es o f the Borel hierarchy . The classes Σ n and Π n are included in the respectiv e classes Σ 0 n and Σ 0 n of the Borel hi erarchy , and cardinality ar gument s suf fice t o show that these inclusions are strict. 9 As in the case of the Borel hierarchy , projections of ar it hmetical set s (of the sec- ond Π -class) lead beyond the arithmetical hierarchy , to t he analytical hierarchy of ω -lang uages. The first class of this hierarchy is the (lightface ) class Σ 1 1 of ef- fective a nalytic sets w hich are obt ained by projection of arithm etical sets. An ω -lang uage L ⊆ X ω belongs to the class Σ 1 1 if and on ly if t here e xist s a recursiv e relation R L ⊆ N × { 0 , 1 } ⋆ × X ⋆ such that: L = { σ ∈ X ω | ∃ τ ( τ ∈ { 0 , 1 } ω ∧ ∀ n ∃ m (( n, τ [ m ] , σ [ m ]) ∈ R L )) } Then an ω -language L ⊆ X ω is in the class Σ 1 1 iff it is t he projection of an ω - language over the alphabet X × { 0 , 1 } which is in the class Π 2 . The (lig htface) class Π 1 1 of effective co-analyti c sets is simp ly the class of complements of effec- tiv e analytic sets. W e deno te as usual ∆ 1 1 = Σ 1 1 ∩ Π 1 1 . Recall t hat an ω -language L ⊆ X ω is in t he class Σ 1 1 iff it is accepted by a non de- terministic T uring machine (reading ω -w ords ) with a B ¨ uchi or Mul ler acce pt ance condition [CG78, Sta97]. For Γ a finite alphabet h a vin g at least two letters, th e set Γ ω × ω of fun ctions from ω × ω into Γ is us ually equi pped wit h the product topol ogy of the discrete topology on Γ . This topology may be defined by the foll owing dist ance d . Let x and y in Γ ω × ω such that x 6 = y , then d ( x, y ) = 1 2 n where n = min { p ≥ 0 | ∃ ( i, j ) x ( i, j ) 6 = y ( i, j ) and i + j = p } . Then the to pological s pace Γ ω × ω is homeomorphic to the topolo gical space Γ ω , equipped wi th the Cantor to pology . Borel subsets of Γ ω × ω are defined f rom open subsets as in the case of the topological space Γ ω . Analytic subsets of Γ ω × ω are obtained as projecti ons on Γ ω × ω of B orel subsets of the product space Γ ω × ω × Γ ω . The set Σ ω ,ω of ω -pictures over Σ , considered a top ological subspace of ˆ Σ ω × ω , is easily seen to be homeomorphic to the topological sp ace Σ ω × ω , via the mappi ng ϕ : Σ ω ,ω → Σ ω × ω defined by ϕ ( p )( i, j ) = p ( i + 1 , j + 1) for all p ∈ Σ ω ,ω and i, j ∈ ω . 3.3 Some Results of Set Theory W e now recall some basic notions of set theory w hich will be useful in the s equel, and which are exposed in an y textbook on set theory , like [Jec02]. The usual axiomati c system ZFC is Zermelo-Fraenkel system ZF plus the axiom of cho ice A C . A model ( V , ∈ ) of the axiom atic sys tem ZFC is a collection V of 10 sets, equi pped with the mem bership relation ∈ , where “ x ∈ y ” means that t he set x i s an element of the set y , which satisfies the axioms of ZFC . W e sh all often say “ the model V ” instead of “the model ( V , ∈ ) ”. The axioms of ZFC express s ome natu ral facts that we cons ider to hold i n the univ erse of sets. For instance a natural fact is that two sets x and y are equal if f they hav e t he same elem ents. This is expressed by th e Axiom of Extensionality . Another natural axiom is the P airin g Axiom which states that for all sets x and y there exists a set z = { x, y } whose element s are x and y . Similarly the P owerset Axiom states t he existence of the set of sub sets of a set x . W e refer t he reader to any textbook on set theory , like [Jec02], for an expositio n of the other axioms of ZFC . The infinite cardinals are usually denoted by ℵ 0 , ℵ 1 , ℵ 2 , . . . , ℵ α , . . . The cardinal ℵ α is also denoted by ω α , as usual when it is considered an ordinal. The continuum hypothesis CH s ays that the first uncountable cardinal ℵ 1 is equal to 2 ℵ 0 which is the cardinal of the continu um. G ¨ odel and Cohen prov ed that the continuum hypot hesis CH is independent from the axiomatic syst em ZFC : providing ZFC is consi stent, there exist som e models o f ZFC + CH and also some models of ZFC + ¬ CH , where ¬ CH d enotes the negation of t he con tinuum hypothesis, [Jec02]. Let ON be the class of all ordinals. Recall that an ordinal α is said to be a successor ordinal i f f there exists an ordinal β s uch that α = β + 1 ; otherwise the ordinal α is said to be a li mit ordinal and in th at case α = sup { β ∈ ON | β < α } . The class L of constructible sets in a model V of ZF is defined by L = [ α ∈ ON L ( α ) where the sets L ( α ) are constructed by induction as follows: 1. L (0) = ∅ 2. L ( α ) = S β <α L ( β ) , for α a limit ordinal, and 3. L ( α + 1) is t he set of subsets of L ( α ) which are definable from a finite number of elements of L ( α ) by a first-order formula relati vi zed to L ( α ) . If V is a m odel of ZF and L is t he class of constr uctible s ets of V , then th e class L form s a model of ZFC + CH . Notice that the axiom ( V=L ) means “e very s et is constructible” and that it is consistent with ZFC . 11 Consider now a mo del V of the axiom atic system ZFC and the class of con- structible sets L ⊆ V which form s another model of ZFC . It is known that the ordinals of L are also the ordinals of V . But the cardinals in V may be different from the cardinals in L . In particular , the first uncountable cardinal in L i s denoted ℵ L 1 . It is in fact an ordinal of V which is denoted ω L 1 . It is k nown that this ordinal satisfies the in- equality ω L 1 ≤ ω 1 . In a model V of the axiomatic system ZFC + V=L the equality ω L 1 = ω 1 holds. But in som e oth er models of ZFC the inequality may be strict and then ω L 1 < ω 1 . This is explained in [Jec02, page 202]: one can start from a model V of Z FC + V=L and construct by forcing a generic e xtension V[G] in which the cardinals ω and ω 1 are collapsed; in this extension the inequality ω L 1 < ω 1 holds. W e now reca ll the notion of a perfect set. Definition 3.12 Let P ⊆ Σ ω , wher e Σ is a finite alphabet havi ng at l east two letters. The set P i s said to be a perfect subset of Σ ω if and only if : (1) P is a non-empty closed set, and (2) for every x ∈ P a nd every open set U conta ining x ther e is an element y ∈ P ∩ U such that x 6 = y . So a perfec t subset of Σ ω is a non-empty closed set whi ch has no iso lated poin ts. It is well known that a perfect subs et of Σ ω has cardinality 2 ℵ 0 , see [M os80, page 66]. W e now reca ll the notion of thin subset of Σ ω . Definition 3.13 A set X ⊆ Σ ω is said to be thin iff it c ont ains no perfect subset. The follo wing impo rtant result was pro ved by K echris [Kec75] and i ndependently by Guaspari [Gua73] and Sacks [Sac76]. Theor em 3.14 (see [Mos80] page 247) ( ZFC ) Let Σ be a finite alphabet havin g at least two lett ers. Ther e exists a th in Π 1 1 -set C 1 (Σ ω ) ⊆ Σ ω which contains eve ry thin, Π 1 1 -subset of Σ ω . It is called the lar gest thin Π 1 1 -set in Σ ω . An important fact is that the cardinality of the lar gest thi n Π 1 1 -set in Σ ω depends on the model of ZFC . The foll owing result o n the cardinality of C 1 (Σ ω ) , was proved by Kechris and independently by Guaspari and Sacks, see also [ Kan97, page 171]. Theor em 3.15 ( ZFC ) The car dinal of the lar gest thin Π 1 1 -set in Σ ω is equal to the car dinal of ω L 1 . 12 This means that in a gi ven model V of ZFC the cardinal of the lar gest thin Π 1 1 -set in Σ ω is equal to the cardinal in V of the ordinal ω L 1 which plays the role of the cardinal ℵ 1 in the inner model L of constructible sets of V . W e can now state the following r esul t which will be useful in the sequel. Corollary 3.16 (a) Ther e is a mod el V 1 of ZFC in which the lar gest thin Π 1 1 -set in Σ ω has car dinal ℵ 1 , wher e ℵ 1 = 2 ℵ 0 . (b) Ther e is a mod el V 2 of ZFC in which the lar gest thin Π 1 1 -set in Σ ω has car dinal ℵ 0 , i.e. is countably infinite. (c) Ther e is a model V 3 of ZFC in which the lar gest thin Π 1 1 -set in Σ ω has car dinal ℵ 1 , wher e ℵ 0 < ℵ 1 < 2 ℵ 0 . Pr oof. (a). In the model L , the c ardinal of the lar gest thin Π 1 1 -set in Σ ω is equal to the cardinal of ω 1 . Moreover th e continuum hypothesis is satisfied t hus 2 ℵ 0 = ℵ 1 . Thus the largest thin Π 1 1 -set in Σ ω has the cardinality 2 ℵ 0 = ℵ 1 . (b). Let V be a model of ( ZFC + ω L 1 < ω 1 ). In this model ω 1 is the first un- countable ordi nal. Thus ω L 1 < ω 1 implies th at ω L 1 is a countable o rdinal in V . Its cardinal is ℵ 0 and it is also the cardinal of the largest t hin Π 1 1 -set in Σ ω . (c). It s uffi ces to show that there is a mo del V 3 of ZFC in which ω L 1 = ω 1 and ℵ 1 < 2 ℵ 0 . Such a model can b e constructed by Cohen’ s forcing. W e can start from a model V of ZFC + V=L (in wh ich ω L 1 = ω 1 ) and construct by forcing a generic extension V[G] in wh ich ℵ 2 subsets of ω are added. N otice that t he cardinals a re preserved un der this extension (see [Jec02, page 2 19]) and t hat the constructible sets of V[G] are also the constructible sets of V . Thus in the new model V[G] we still hav e ω L 1 = ω 1 but now ℵ 1 < 2 ℵ 0 .  4 Decision Pr oblems W e now stu dy dec is ion problems for recognizable languages o f infinite pictures. W e gave in [Fin09b] the exact degree of sever al natural decision probl ems. W e first recall some of these results. Castro and Cucker pro ved in [CC 89 ] that t he no n-emptiness probl em and the in- finiteness p roblem for ω -languages of T uring machines are both Σ 1 1 -complete. W e easily in ferred from thi s result a si milar resul t for recognizable languages of in fi- nite pictures. 13 From now on we shall denot e by T z the no n determin istic til ing system of index z , (accepting pictures over Σ = { a, b } ), equipped wi th a B ¨ uchi acceptance condition. Theor em 4.1 ([Fin09b]) The non-emptiness problem and th e infinit eness pr ob- lem for B ¨ uchi-r ecognizable l anguages of i nfinite pictur es ar e Σ 1 1 -complete, i.e. : 1. { z ∈ N | L B ( T z ) 6 = ∅} is Σ 1 1 -complete. 2. { z ∈ N | L B ( T z ) is i nfinite } is Σ 1 1 -complete. In a simil ar w ay , the univ ersality problem and the inclusion and the equiv alence problems, for ω -languages of T uring machines, have been proved to be Π 1 2 -complete by Castro and Cucker in [CC89], and we used these results to prove the following results in [Fin09b]. Theor em 4.2 ([Fin09b]) The un iversality pr oblem for B ¨ uchi-r ecognizable lan- guages of infinite pictur es is Π 1 2 -complete, i.e . : { z ∈ N | L B ( T z ) = Σ ω ,ω } is Π 1 2 -complete. Theor em 4.3 ([Fin09b]) The i nclusion and the equivalence pr obl ems for B ¨ uchi- r ecognizable languages of infinite pictur es ar e Π 1 2 -complete, i.e . : 1. { ( y , z ) ∈ N 2 | L B ( T y ) ⊆ L B ( T z ) } is Π 1 2 -complete. 2. { ( y , z ) ∈ N 2 | L B ( T y ) = L B ( T z ) } is Π 1 2 -complete. The class of B ¨ uchi-recognizable l anguages of infinite pi ctures i s not closed un- der complement [A TW03]. Thus the following question naturally ar is es: “can we decide whether the comp lement of a B ¨ uchi-recognizable language of infini te pic- tures is B ¨ uchi-recognizable?”. And what is the exact complexity of this decisi on problem, called the complement ability problem. Another classi cal problem is the determinizabil ity problem: “ can we decide w hether a gi ven recognizable language of in finite pictures is recognized by a determinist ic tiling system?”. Recall that a tiling sys tem is called determi nistic if on any pictu re it allows at m ost one tile covering t he origi n, th e s tate assi gned to position ( i + 1 , j + 1) i s uniquely determined by the stat es at positions ( i, j ) , ( i + 1 , j ) , ( i, j + 1) and the s tates at the border positi ons (0 , j + 1) and ( i + 1 , 0) are determined by th e state (0 , j ) , respectiv ely ( i, 0) , [A TW03]. As remarked in [A TW03], the hierarc hy proofs of the classi cal Landweber h ierar - chy defi ned using deterministic ω -auto mata “carry over without essential changes 14 to pi ctures”. In p articular , a l anguage of ω -pictures which is B ¨ uchi-recognized by a deterministic til ing system is a Π 0 2 -set and a langu age of ω -pictures which is Muller-recognized by a determinis tic tiling system is a boolean combination of Π 0 2 -sets, hence a ∆ 0 3 -set. These topol ogical properties ha ve been used in [Fin09b], alon g with a dichotomy property , to prove th e following result s. Theor em 4.4 ([Fin09b]) The determin izability p r oblem and the complementabil - ity pr oblem fo r B ¨ uchi-r ecognizable languages of infinite pictur es ar e Π 1 2 -complete, i.e. : 1. { z ∈ N | L B ( T z ) is B ¨ uchi-r ecognizable by a determini stic tiling system } is Π 1 2 -complete. 2. { z ∈ N | L B ( T z ) is Muller-r ecognizable by a deterministic tiling system } is Π 1 2 -complete. 3. { z ∈ N | ∃ y Σ ω ,ω − L B ( T z ) = L B ( T y ) } is Π 1 2 -complete. W e already m entioned t hat we us ed some results of Castro a nd Cucker in the proof of t he above results. Castro and Cucker stud ied degrees of decision problems for ω -languages accepted by T uring machines and prov ed that m any of them are highly undecidable, [CC89]. W e are going to use again some of their results to prove here new results about B ¨ uchi-recognizable lang uages of infinite pictures. W e firstly recall th e notio n o f acceptance of infinite words b y T uring machines considered by Castro and Cucker in [CC89]. Definition 4.5 A non d eterministic T uring machine M is a 5 -tupl e M = ( Q, Σ , Γ , δ , q 0 ) , wher e Q is a finite set of states, Σ is a finite input alphabet, Γ is a finite tape alpha- bet satisfying Σ ⊆ Γ , q 0 is the ini tial state , and δ is a mapp ing fr om Q × Γ t o sub- sets of Q × Γ × { L, R, S } . A configuration o f M is a 3 -t uple ( q , σ , i ) , wher e q ∈ Q , σ ∈ Γ ω and i ∈ N . An infin ite sequence of config urations r = ( q i , α i , j i ) i ≥ 1 is called a run of M on w ∈ Σ ω iff: (a) ( q 1 , α 1 , j 1 ) = ( q 0 , w , 1) , and (b) for each i ≥ 1 , ( q i , α i , j i ) ⊢ ( q i +1 , α i +1 , j i +1 ) , wher e ⊢ is the transit ion r elation of M defined as usual. The run r is said to be complete if the l imsup of the head pos itions is infinity , i.e. if ( ∀ n ≥ 1)( ∃ k ≥ 1)( j k ≥ n ) . The run r is said to be oscillat ing if the liminf of th e head pos itions is bounded, i.e. if ( ∃ k ≥ 1)( ∀ n ≥ 1)( ∃ m ≥ n )( j m = k ) . 15 Definition 4.6 Let M = ( Q, Σ , Γ , δ, q 0 ) be a non det erministic T uri ng machine and F ⊆ Q . The ω -language accepted by ( M , F ) is the set of ω -wor ds σ ∈ Σ ω such that t her e e xists a complete non oscillating run r = ( q i , α i , j i ) i ≥ 1 of M on σ such that, for all i, q i ∈ F . The above acceptance conditi on is denoted 1 ′ -acceptance in [CG78 ]. An other usual acce pt ance condition is the no w called B ¨ uchi acceptance condition w hich is also denoted 2 -acceptance in [CG78]. W e now recall its definition. Definition 4.7 Let M = ( Q, Σ , Γ , δ, q 0 ) be a non det erministic T uri ng machine and F ⊆ Q . The ω -language B ¨ uchi accepted by ( M , F ) i s t he set of ω -wor ds σ ∈ Σ ω such that ther e exists a compl ete non oscil lating run r = ( q i , α i , j i ) i ≥ 1 of M on σ and infini tely many inte gers i such that q i ∈ F . Recall that Cohen and Gold proved in [CG78, T heorem 8.6] that one can effec- tiv ely cons truct, from a given non determi nistic T uring m achine, ano ther equ iv a- lent non deterministic Turing machine, equipped with the same kind o f acceptance condition, and i n which e very run is complete n on oscill ating. Cohen and Gold proved also in [C G78, Theorem 8.2] th at an ω -language i s accepted by a non de- terministic T uring machine with 1 ′ -acceptance condition iff it is accepted by a non deterministic T uring machine with B ¨ uchi acceptance condition. From no w on , we shall denote M z the non determin istic T uring machine of index z , (accepting words over Σ = { a, b } ), equipped with a 1 ′ -acceptance condition. An important notion in autom ata t heory is th e not ion of ambi guity . It can be defined also in the context of acceptance by tiling s ystems, see [A GMR06] for the case of finite pictures. Definition 4.8 Let A = ( Q, Σ , ∆) be a tiling system, and F ⊆ Q . The B ¨ uchi tiling system ( A , F ) is una mbiguous iff every ω -pictur e p ∈ Σ ω ,ω has a t mos t an acc ept- ing run by ( A , F ) . Definition 4.9 A B ¨ uchi r ecognizable language L ⊆ Σ ω ,ω is u nambiguous iff ther e e xist s an unamb iguous B ¨ uchi ti ling s ystem ( A , F ) suc h that L = L ( A , F ) . Ot h- erwise the language L is said to be inher ently ambiguou s. W e can no w prove t he fol lowing result, which is very similar to a correspondin g result for recognizable tree languages proved i n [FS09]. Pr oposi tion 4.10 Let L ⊆ Σ ω ,ω be an unambiguou s B ¨ uchi r ecognizable language of infinite pictur es. Then L is a Bor el subset of Σ ω ,ω . 16 Pr oof. Let L ⊆ Σ ω ,ω be a language accepted by an u nambiguous B ¨ uchi tiling system ( A , F ) , where A = ( Q, Σ , ∆) , and let R ⊆ ( ˆ Σ × Q ) ω × ω be defined by: R = { ( p, ρ ) | p ∈ Σ ω ,ω and ρ ∈ i s an accepting run of ( A , F ) on the picture p } . The set R is easily seen to be a Π 0 2 -subset of ( ˆ Σ × Q ) ω × ω . Consider now t he projection PR OJ ˆ Σ ω × ω : ˆ Σ ω × ω × Q ω × ω → ˆ Σ ω × ω defined by PR OJ ˆ Σ ω × ω (( p, ρ )) = p for all ( p, ρ ) ∈ ˆ Σ ω × ω × Q ω × ω . This projection is a c on- tinuous function and i t is injective on the Borel set R because the B ¨ uchi tiling system ( A , F ) is unambiguous . Hence, by a Theorem of Lusin and Souslin , see [K ec95, Theorem 15.1 page 89], th e injective i mage of R by the continuous func- tion P ROJ ˆ Σ ω × ω is Borel. Thus the language L = PR OJ ˆ Σ ω × ω ( R ) is a Borel subset of ˆ Σ ω × ω . But Σ ω ,ω is a closed s ubset of ˆ Σ ω × ω and L ⊆ Σ ω ,ω . Thus L is also a Borel subset of Σ ω ,ω .  Corollary 4.11 Ther e e xist some inher ently ambi guous B ¨ uchi-r ecognizable lan- guages of infinite pictur es. Pr oof. The result follows direc tl y from the abov e propositio n b ecause we kno w that there e xist s ome B ¨ uchi-recognizable languages of infinit e pictures which are not Borel sets, see [Fin04, Fin09b].  W e can now state that the unambiguity prob lem for recognizable language of in- finite pictures is Π 1 2 -complete. Theor em 4.12 The unambiguity pr oblem fo r r ecognizable languages of infinite pictur es is Π 1 2 -complete, i.e. : { z ∈ N | L B ( T z ) is non ambiguous } is Π 1 2 -complete. Pr oof. T o prove that the u nambiguity problem for recognizable language of in- finite pictures is in th e class Π 1 2 , we reason as in th e case of the unamb iguity problem for ω -lang uages accepted by 1 -counter or 2 -tape automata, see [Fin09c]. Notice first, as i n [Fin09b], that, using a recursi ve b ijection b : ( N − { 0 } ) 2 → N − { 0 } , one can associate with each ω -word σ ∈ Σ ω a u nique ω -picture p σ ∈ Σ ω ,ω which is s imply defined by p σ ( i, j ) = σ ( b ( i, j )) for all integers i, j ≥ 1 . And we can identify a run ρ ∈ Q ω × ω with an elem ent of Q ω and fi nally with a codin g of this element over the alphabet { 0 , 1 } . So the run ρ can be id entified with its cod e ¯ ρ ∈ { 0 , 1 } ω . If a tiling system A = ( Q, Σ , ∆) is equipped wi th a set of accepting states F ⊆ Q , then for σ ∈ Σ ω and ρ ∈ { 0 , 1 } ω , “ ρ is a B ¨ uchi accepting run o f ( A , F ) over the 17 ω -pi cture p σ ” can be expressed by a n arith metical formu la, see [A TW03, Section 2.4] and [Fin09b]. W e can now fir st express “ T z is non ambiguous” by : “ ∀ σ ∈ Σ ω ∀ ρ, ρ ′ ∈ { 0 , 1 } ω [( ρ and ρ ′ are accepting runs of T z on p σ ) → ρ = ρ ′ ]” which is a Π 1 1 -formula. Then “ L B ( T z ) is non ambiguous” can be expressed by the following formula: “ ∃ y [ L B ( T z ) = L B ( T y ) and T y is non ambiguous ] ”. This is a Π 1 2 -formula because L B ( T z ) = L B ( T y ) can be expressed by a Π 1 2 -formula, and the quantification ∃ y is o f t ype 0 . Thus t he s et { z ∈ N | L B ( T z ) is non ambiguous } is a Π 1 2 -set. T o prove the completeness part of the th eorem, we sh all use the following di- chotomy result proved i n [Fin09b, proof of Theorem 5.11]. There exists an injecti ve computable function H ◦ θ from N into N such that: First case: If L ( M z ) = Σ ω then L B ( T H ◦ θ ( z ) ) = Σ ω ,ω . Second case: If L ( M z ) 6 = Σ ω then L B ( T H ◦ θ ( z ) ) is not a Borel set. In the first case L B ( T H ◦ θ ( z ) ) = Σ ω ,ω is obviously an unambiguous language. And in the second ca se t he language L B ( T H ◦ θ ( z ) ) cannot b e unambiguous because it is not a Borel subset of Σ ω ,ω . Thus, using the reduction H ◦ θ , we see that : { z ∈ N | L ( M z ) = Σ ω } ≤ 1 { z ∈ N | L B ( T z ) is non ambiguous } and the result foll ows from the Π 1 2 -completeness of the universality probl em for ω -lang uages of T uring machines prov ed by Castro and Cucker in [CC 89 ].  Notice that the same dichotomy result above with the reduction H ◦ θ was used in [Fin09b] to pro ve that topological properties of recognizable languages of infinite pictures are actually highly undecidable. Theor em 4.13 ([Fin09b]) Let α be a non-null countab le or dinal . Then 1. { z ∈ N | L B ( T z ) is i n the Bor el class Σ 0 α } is Π 1 2 -har d. 2. { z ∈ N | L B ( T z ) is i n the Bor el class Π 0 α } is Π 1 2 -har d. 3. { z ∈ N | L B ( T z ) is a Bor el s et } is Π 1 2 -har d. A n atural questi on is to stu dy sim ilar problem s by replacing Borel classes by the ef fective classes of the arithm etical hierarchy . This was not s tudied in [Fin09b], but a sim ilar problem was sol ved i n [Fin09c] for ω -languages accepted b y 1 - counter or 2 -tape B ¨ uchi automata. W e can reason in a sim ilar way for the case of recognizable languages of infinite pictures, and state the following re sul t. 18 Theor em 4.14 Let n ≥ 1 be an inte ger . Then 1. { z ∈ N | L B ( T z ) is i n the arithmetical class Σ n } is Π 1 2 -complete. 2. { z ∈ N | L B ( T z ) is i n the arithmetical class Π n } is Π 1 2 -complete. 3. { z ∈ N | L B ( T z ) is a ∆ 1 1 -set } is Π 1 2 -complete. W e do not give th e com plete proof here. It is actually very similar to the case of ω -languages accepted by 1 -counter or 2 -tape B ¨ uchi automata i n [Fin09 c]. A key ar gum ent, to p rove that { z ∈ N | L B ( T z ) is i n the arithmetical class Σ n } (re- spectiv ely , { z ∈ N | L B ( T z ) is in the arithmetical class Π n } ) is a Π 1 2 -set, is the existence of a universal set U Σ n ⊆ N × Σ ω ,ω (respectiv ely , U Π n ⊆ N × Σ ω ,ω ) for the class of Σ n -subsets of Σ ω ,ω , (respectively , Π n -subsets of Σ ω ,ω ), [M os80, p. 172]. Notice also t hat the completeness part follows easily from the dichotom y result obtained with the reduction H ◦ θ . W e now come t o cardinality problems. W e already know that it is Σ 1 1 -complete to determine whether a gi ven recognizable language of infinit e pi ctures i s empty (respectiv ely , infini te). Rec all that every recognizable language o f infinite pictures is an analytic set. On t he o ther h and, e very analytic set is either c oun table or has the ca rdinali ty 2 ℵ 0 of t he con tinuum. Then s ome questi ons natu rally arise. What are the complexities of the following decision problems: “Is a given recognizable language of infini te pictures countable? Is it coun tably i nfinite? Is it uncount- able?”. Notice that s imilar questions were ask ed by C astro and Cuck er in the case of ω -languages of T uring machines and have bee n so lved v ery recently by Finkel and Lecomte in [FL09]. W e can now state the following result for recognizable languages of infinite pictures. Belo w D 2 (Σ 1 1 ) denotes the class of 2 -dif ferences of Σ 1 1 -sets, i.e. the class of sets which are int ersections of a Σ 1 1 -set and of a Π 1 1 -set. Theor em 4.15 1. { z ∈ N | L B ( T z ) is cou ntable } is Π 1 1 -complete. 2. { z ∈ N | L B ( T z ) is u ncountable } is Σ 1 1 -complete. 3. { z ∈ N | L B ( T z ) is cou ntably infinite } is D 2 (Σ 1 1 ) -complete. Pr oof. (1). W e can first prove that { z ∈ N | L B ( T z ) is countable } is in th e class Π 1 1 in the same way as in the case of ω -languages of T uring machines in [FL09]. W e know th at a recognizable language of i nfinite pictures L B ( T z ) is a Σ 1 1 -subset of Σ ω ,ω . But it is kno wn that a Σ 1 1 -subset L of Σ ω ,ω is countable if and onl y if for every x ∈ L t he singleton { x } is a ∆ 1 1 -subset of Σ ω ,ω , see [Mos80, page 19 243]. Then, using a ni ce coding of ∆ 1 1 -subsets of Σ ω ,ω giv en i n [HKL90, Theorem 3.3.1], we can prove that { z ∈ N | L B ( T z ) is countable } is i n the class Π 1 1 , see [FL09] for more details. T o prove the com pleteness part of Item (1), we shall use the following two lemmas proved i n pre vious papers. For σ ∈ Σ ω = { a, b } ω we deno te σ a the ω -picture whose first row i s the ω -word σ and whos e other rows are labelled with the letter a . For an ω -l anguage L ⊆ Σ ω = { a, b } ω we denote L a the language of infinite pictures { σ a | σ ∈ L } . Lemma 4.16 ([Fin04]) If L ⊆ Σ ω is accepted b y some T uring machine (in which every run is complete non o scillating) with a B ¨ uchi acceptance condition, then L a is B ¨ uchi r ecognizable by a finite tiling system. Lemma 4.17 ([Fin09b]) Ther e is an i njective computable f unction K fr om N i nto N satisfying the following pr operty . If M z is t he non determinist ic T ur ing ma chine (equipped with a 1 ′ -acceptance condition) of index z , and if T K ( z ) is the tiling system (equipped with a B ¨ uchi acceptance condition) of index K ( z ) , th en L ( M z ) a = L B ( T K ( z ) ) On the oth er hand, we can easily see that the cardinal ity of the ω -language L ( M z ) is equal to the cardinalit y of the ω -picture language L ( M z ) a . Thus using the reduction K giv en in the abov e lemma we see that: { z ∈ N | L ( M z ) is countable } ≤ 1 { z ∈ N | L B ( T z ) is countable } Then the completeness part fol lows from the fact that { z ∈ N | L ( M z ) is cou ntable } is Π 1 1 -complete, proved in [FL09 ]. (2). The proof of Item (2) follows directly from Item (1). (3). W e already know that t he set { z ∈ N | L B ( T z ) is infinite } is in the cl ass Σ 1 1 . Thus the set { z ∈ N | L B ( T z ) is countably infinite } i s the intersection of a Σ 1 1 -set and of a Π 1 1 -set, i.e. it is in the class D 2 (Σ 1 1 ) . Using again the reduction K we see that: { z ∈ N | L ( M z ) is countably infinite } ≤ 1 { z ∈ N | L B ( T z ) is countably infinite } It wa s proved in [FL09] that { z ∈ N | L ( M z ) is countably infinite } i s D 2 (Σ 1 1 ) - complete. Th us the set { z ∈ N | L B ( T z ) is countably infinite } is also D 2 (Σ 1 1 ) - complete.  20 W e are n ow looking at complements of recognizable languages of infini te pictures. W e first state the foll owing re sul t which shows that actually the cardinalit y of the complement of a recognizable language of i nfinite pictures may depend on the models of set theory . W e denote L B ( T ) − the complement Σ ω ,ω − L B ( T ) of a B ¨ uchi-recognizable language L B ( T ) ⊆ Σ ω ,ω . Theor em 4.18 The car dinality of the complement o f a B ¨ uchi-r ecognizable lan- guage of infini te pictur es is no t determined by t he axiomat ic system ZFC . Indeed ther e is a B ¨ uchi tiling system T such that: 1. Ther e is a model V 1 of ZFC in which L B ( T ) − is countable. 2. Ther e is a model V 2 of ZFC in which L B ( T ) − has car dinal 2 ℵ 0 . 3. Ther e i s a model V 3 of ZFC in which L B ( T ) − has car dinal ℵ 1 with ℵ 0 < ℵ 1 < 2 ℵ 0 . Pr oof. Moschovakis ga ve in [M os80, page 248] a Π 1 1 -formula φ defining the set C 1 (Σ ω ) . Thus its compl ement C 1 (Σ ω ) − = { a, b } ω − C 1 (Σ ω ) is a Σ 1 1 -set defi ned by the Σ 1 1 -formula ψ = ¬ φ . Recall that o ne can construct, fr om the Σ 1 1 -formula ψ defining C 1 (Σ ω ) − , a B ¨ uchi T uring machine M accepting the ω -language C 1 (Σ ω ) − . On the other hand i t is easy to see that the language Σ ω ,ω − (Σ ω ) a of ω -pictures is B ¨ uchi recognizable. But the class T S (Σ ω ,ω ) is closed under finite union, so we get the following result. Lemma 4.19 ([Fin09b]) If L ⊆ Σ ω is accepted by some T uring machine with a B ¨ uchi acceptance cond ition, then L a ∪ [Σ ω ,ω − (Σ ω ) a ] is B ¨ uchi re cognizabl e by a finite tilin g system. Notice that t he con structions are ef fectiv e and that they can be achie ved in an in- jectiv e way . Thus we can const ruct, fr om the B ¨ uchi T uring machine M accepting the ω -language C 1 (Σ ω ) − , a B ¨ uchi tiling system T such t hat L B ( T ) = L ( M ) a ∪ [Σ ω ,ω − ( Σ ω ) a ] . It is then easy to see that: L B ( T ) − = (Σ ω − L ( M )) a = ( C 1 (Σ ω )) a . Thus the cardinality of L B ( T ) − is equal to t he cardinali ty of t he ω -language C 1 (Σ ω ) , and th en we can infer th e resul ts of the theorem from previous Corol- lary 3.16.  21 W e can no w use the proof of the above result to prove the fol lowing result which shows that natural cardinali ty problems are actually located at the third lev el o f the analytical hierarchy . Theor em 4.20 1. { z ∈ N | L B ( T z ) − is finite } is Π 1 2 -complete. 2. { z ∈ N | L B ( T z ) − is countable } is in Σ 1 3 \ (Π 1 2 ∪ Σ 1 2 ) . 3. { z ∈ N | L B ( T z ) − is uncountabl e } is in Π 1 3 \ (Π 1 2 ∪ Σ 1 2 ) . Pr oof. Item (1) was proved in [Fin09b]. T o prove Item (2), we first show that { z ∈ N | L B ( T z ) − is countable } is in the class Σ 1 3 . As in [Fin09b], using a recursi ve bijection b : ( N − { 0 } ) 2 → N − { 0 } , we can consider an infinit e word σ ∈ Σ ω as a countably infinite family of infinit e words over Σ : the fa m ily of ω -words ( σ i ) such that for each i ≥ 1 , σ i is defined by σ i ( j ) = σ ( b ( i, j )) for eac h j ≥ 1 . And one can associate with each ω -word σ ∈ Σ ω a unique ω -pict ure p σ ∈ Σ ω ,ω which is simply defined by p σ ( i, j ) = σ ( b ( i, j )) for all integers i, j ≥ 1 . W e can now e xpress “ L B ( T z ) − is countable ” by the formula: ∃ σ ∈ Σ ω ∀ p ∈ Σ ω ,ω [( p ∈ L B ( T z )) or ( ∃ i ∈ N p = p σ i )] This is a Σ 1 3 -formula because “ p ∈ L B ( T z ) ”, and hence also “ [( p ∈ L B ( T z )) or ( ∃ i ∈ N p = p σ i )]” , is expressed by a Σ 1 1 -formula. W e can now pro ve that { z ∈ N | L B ( T z ) − is countable } i s neither in the class Σ 1 2 nor in the class Π 1 2 , by usi ng Shoenfield’ s Absoluteness Theorem from Set Theory . Let T be the B ¨ uchi tili ng system obtained in Theorem 4.18 and let z 0 be its index so that T = T z 0 . Assume no w that V is a model of ( ZFC + ω L 1 < ω 1 ). In the mod el V , by the proofs of Theorem 4.18 and of Coroll ary 3.1 6, the integer z 0 belongs t o t he set { z ∈ N | L B ( T z ) − is countable } . But, by the proofs o f Theorem 4.18 and of Corollary 3.1 6, in the inner model L ⊆ V , the language L B ( T z 0 ) − has c ardinali ty 2 ℵ 0 . Thus t he i nteger z 0 does not belong to the set { z ∈ N | L B ( T z ) − is countable } . 22 On the other hand, Schoenfield’ s Absoluteness Theorem implies that e very Σ 1 2 -set (respectiv ely , Π 1 2 -set) is absolute for all inner models of (ZFC), see [Jec 02, page 490]. In particular , if the set { z ∈ N | L B ( T z ) − is countable } was a Σ 1 2 -set or a Π 1 2 -set then it could not be a dif ferent subset o f N in the models V and L consid ered above. Therefore, the s et { z ∈ N | L B ( T z ) − is countable } is neither a Σ 1 2 -set nor a Π 1 2 -set. Item (3) follows d irectly from Item (2).  5 Concludin g Remar ks Using the notion of lar gest ef fective coanalytic set, we ha ve proved in another pa- per that the topological com plexity of a recognizable language of infinite pictures is no t determi ned by t he axiom atic system ZFC . In parti cular , there is a B ¨ uchi tiling system S and model s V 1 and V 2 of ZFC such that: the ω -picture language L ( S ) id Borel in V 1 but not in V 2 , [Fin09a]. W e hav e p rove d i n thi s paper that { z ∈ N | L B ( T z ) − is countable } i s in Σ 1 3 \ (Π 1 2 ∪ Σ 1 2 ) . It remains open whether th is set is actually Σ 1 3 -complete. References [A GMR06] M. Ansel mo, D. Gi ammarresi, M. Madoni a, and A. 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