An omega-power of a context-free language which is Borel above Delta^0_omega

Stefan Bold , Bened ikt L ¨ owe , Thoralf R ¨ asch , Johan van Benthem ( eds. ) Infinite Games Papers o f the conference “ F oundations of the F orma l Sciences V ” held in Bonn, Nov ember 26-29, 2004 An ω -power of a context-f r e e language which is Bor el above ∆ 0 ω Jacques Dup arc and Olivier Finkel Univ ersit ´ e de Lausanne, Information Systems Institute, and W estern Swiss Center for Logic, History and Philosophy of Sciences B ˆ atiment Provence CH-1015 Lausanne and Equipe Mod ` eles de Calcul et Complexit ´ e Laborato ir e de l’Informatique du P arall ´ elisme ⋆⋆ CNRS et Ecole Normale Sup ´ erieure de L yon 46, All ´ ee d’Italie 69 364 L yon Cede x 07, France. E-mail: jacques.dupa rc@unil.ch and Olivier.Finkel@ ens-lyon.fr Abstract. W e use erasers-like basic operations on words to construct a set that is both Borel and abo ve ∆ 0 ω , built as a set V ω where V is a language of finite words accepted by a pushdo wn automaton. In particular , this gi v es a first e xample of an ω -power of a conte xt free language which is a Borel set of infinite rank. 1 Pr eliminaries Giv en a set A (called the a lphabet) we writ e A ∗ , and A ω , for the sets of finite, and infinit e w ords over A . W e denote the empt y word by ǫ . In ⋆⋆ UMR 5668 - CNRS - ENS L yon - UCB L yon - INRIA LIP Research Report RR 2007 -17 Recei ved : ...; In re vised version : ...; Accepted by the editors : .... 2000 Mathe matics Subje ct Classification. PRIM ARY SECOND AR Y . c  2006 Kluwer Academic Publishers. Printed in The Netherlands, pp. 1– 14. 2 J A CQUES DUP ARC AND OLIVIER FINKEL order to facilitate t he reading, we use u, v , w for finite words, and x , y , z for infinite words. Giv en two w o rds u and v (respectiv ely , u and y ), we write uv (respecti vely , uy ) for the concatenation of u and v (respecti vely , of u and y ). Let U ⊆ A ∗ and Y ⊆ A ∗ ∪ A ω , we set: U Y = { uv , u y : u ∈ X ∧ v , y ∈ Y } . W e recall that, given a language V ⊆ A ∗ , the ω - power of this lan- guage is V ω = { x = u 1 u 2 . . . u n . . . ∈ A ω : ∀ n < ω u n ∈ V \ { ε }} A ω is equipped with the usual topolog y . i .e. the product of the discrete topology on the alphabet A . So th at e very open set is of the form W A ω for any W ⊆ A ∗ . Or , to say it di f ferently , e very closed set is defined as the set of all infinite branches of a t ree o ver A . W e work within the Borel hierarchy of sets which is the strictly incre asing (for inclusion) sequence of class es of sets (Σ 0 ξ ) ξ <ω 1 - together with the dual class es ( Π 0 ξ ) ξ <ω 1 and the ambiguous ones (∆ 0 ξ ) ξ <ω 1 - which reports how many operations of countable unions and i ntersections are necessary to produce a Borel set on the basis of the open ones. A reducti on relation between sets X , Y is a partial ordering X ≤ Y which expresses that the problem of knowing wh ether any element x be- longs to X is at most as complicated as deciding whether f ( x ) belongs to Y , for s ome giv en simple function f . A very natural reducti on relation between sets of infinite words (closely related to reals), has been th or- oughly stud ied by W adge in the se venties. From the topo logical point of view , simple m eans con tinuous, therefore the W adge ordering com pares sets of infinite sequences with respect to their fine topological comp lex- ity . Associated with determinacy , thi s partial ordering becomes a pre- wellordering wit h anti-chains of lengt h at most two. The so called W adge Hierarchy it induces i ncredibly r efines the old Borel Hierar chy . Determi- nacy makes i t way through a representation of continuous functions in terms of strategies for player II in a sui table two-player gam e: the W adge game W ( X , Y ) . In thi s game, players I and II, t ake turn playi ng letters of the alphabet corresponding to X for I, and lett ers of t he alphabet cor - responding to Y f or II. In order t o get the right correspondence between a strategy for player II and a continuous function, player II i s allowed to skip, whereas I is not. Howe ver , II must play infinitely man y letters. As usual, reduction relations induce the noti on of a com plete set: a set that bot h belongs to some cl ass, whose mem bers it also reduces. In the A BOREL ω -POWER OF A CONTEXT -F REE LANGU A GE ABO VE ∆ 0 ω 3 context of W adge reducibil ity , a set is complete if it belongs to s ome class closed by inv erse image of conti nuous functions , and reduces everyone of i ts members. A class which admits a comp lete set is called a W adge Class. As a matter of f act, all Σ 0 ξ , and Π 0 ξ , are W adge class es, whereas ∆ 0 ξ ( ξ > 1 ) are not. For in stance, t he set of all infinite sequences th at con tains a 1 is Σ 0 1 - complete, the one that contain s i nfinitely many 1 s is Π 0 2 -complete. As a matt er of f act, reaching c omplete sets for upper levels of the Borel hierarchy , requires other means which we introduce in next sections. 2 Erasers For climbing up along the finit e leve ls of the Borel hierarchy , we use erasers-like mov es, see [Dup01 ]. For simplicity , imagine a player (either I or II) playing a W adge game, in char ge of a set X ⊆ A ω , with the extra possibili ty to delete any terminal part of her last mov es. W e recall the definition of the operation X 7→ X ≈ over sets of infinite words. It was first introdu ced in [Fin0 1] by the s econd author , and is a simple v ariant of the first author’ s operation of e xponentiation X 7→ X ∼ which first appeared in [Dup01]. W e denote | v | the lengt h o f any finite word v . If | v | = 0 , v is the empty word. If v = v 1 v 2 . . . v k where k ≥ 1 and each v i is in A , then | v | = k and we writ e v ( i ) = v i and v [ i ] = v (1) . . . v ( i ) for i ≤ k ; so v [0] = ǫ . The prefix relation i s denoted ⊑ : the finite word u is a prefix of the finite w o rd v (denoted u ⊑ v ) if and only if there exists a (finite) word w such t hat v = u w . the finite word u is a prefix of the ω -word x (denoted u ⊑ x ) iff there e xists an ω -word y such that x = uy . Giv en a finite alphabet A , we write A ≤ ω for A ∗ ∪ A ω . Definition 2.1. Let A be any finite al phabet, և / ∈ A , B = A ∪ { և } , and x ∈ B ≤ ω , then x և is inductively defined by: ǫ և = ǫ , and for a finit e wor d u ∈ ( A ∪ { և } ) ∗ : ( ua ) և = u և a , if a ∈ A , ( u և ) և = u և with its last letter r emoved if | u և | > 0 , ( u և ) և is undefined if | u և | = 0 , and for u infinite: 4 J A CQUES DUP ARC AND OLIVIER FINKEL ( u ) և = lim n ∈ ω ( u [ n ]) և , wher e, given β n and v in A ∗ , v ⊑ lim n ∈ ω β n ↔ ∃ n ∀ p ≥ n β p [ | v | ] = v . W e now make easy th is definition to understand by describing it in - formally . For x ∈ B ≤ ω , x և denotes the string x , on ce e very և o ccurring in x has b een “ev aluated” t o the back space operation (the one familiar to your computer!), proceeding fr om lef t to right inside x . In other w ords x և = x from whi ch ev ery in terva l of the form “ a և ” ( a ∈ A ) is re- moved. By con vention, we assume ( u և ) և is u ndefined when u և is t he empty sequence. i.e. when the last letter և cannot b e used as an eraser (because ev ery letter of A in u has already been erased by some eraser և placed in u ). W e remark that the result ing word x և may be finite or infinite. For instance, – if u = ( a և ) n , for n ≥ 1 , or u = ( a և ) ω then ( u ) և = ǫ , – if u = ( ab և ) ω then ( u ) և = a ω , – if u = bb ( և a ) ω then ( u ) և = b , – if u = և ( a և ) ω or u = a ևև a ω or u = ( a ևև ) ω then ( u ) և is undefined. Definition 2.2. F or X ⊆ A ω , X ≈ = { x ∈ ( A ∪ { և } ) ω : x և ∈ X } . The following result ea sily foll ows from [Dup01] and w as a pplied in [Fin01,Fin04] to study the ω -powers of finitary conte xt free languages. Theor em 2 .3. Let n be an inte ger ≥ 2 and X ⊆ A ω be a Π 0 n -complete set. Then X ≈ is a Π 0 n + 1 -complete subset of ( A ∪ { և } ) ω . Next remarks will be essential later . Remark 2.4. Consider the following function: f : x ∈ ( A ∪ { և } ) ω 7→ y ∈ A ω defined by: – y = 0 ω if x և is finite or undefined, – y = x և otherwise. A BOREL ω -POWER OF A CONTEXT -F REE LANGU A GE ABO VE ∆ 0 ω 5 It is clearly Borel. In f act a qu ick computation shows t hat the in verse image of any basic clopen set is Borel of lo w finite rank. Remark 2.5. Let X b e any subset of the Cantor space { 0 , 1 } ω , and f as in remark 2.4. If 0 ω 6∈ X , then for any x ∈ { 0 , 1 , և } ω x ∈ X ≈ ⇐ ⇒ f ( x ) ∈ X In other words, X ≈ = f − 1 X . In particular , if X is Borel, so is X ≈ 3 Incr easing sequences of erasers The following construction has been partly used by the second aut hor in [Fin04] to construct a Bore l set o f infinite rank which is an ω -power , i.e. in the form V ω , wh ere V is a set of finite words over a finite alphabet Σ . W e iterate th e operation X 7→ X ≈ finitely many times, and take the limit. More precisely , Definition 3.1. G iven any set X ⊆ A ω : – X ≈ 0 k = X , – X ≈ 1 k = X ≈ , – X ≈ 2 k = ( X ≈ 1 k ) ≈ , – X ≈ ( k ) k = ( X ≈ ( k − 1) k ) ≈ , wher e we appl y k times the operation X 7→ X ≈ with differ ent new letters և k , և k − 1 , . . . , և 2 , և 1 , in such a w ay that we have successively: • X ≈ 0 k = X ⊆ A ω , • X ≈ 1 k ⊆ ( A ∪ { և k } ) ω , • X ≈ 2 k ⊆ ( A ∪ { և k , և k − 1 } ) ω , • X ≈ ( k ) k ⊆ ( A ∪ { և 1 , և 2 , . . . , և k } ) ω . – W e set X ≈ ( k ) = X ≈ ( k ) k . X ≈∞ ⊆ ( A ∪ { և n : 0 < n < ω } ) ω is defined by x ∈ X ≈∞ ⇐ ⇒ def – for each inte ger n, x n = x և 1 ... և n − 1 և n is defined, infinite, and – x ∞ = lim n<ω x n is defined, infinite, and belongs to X . Remark 3.2. Consider the following sequence of functions: 6 J A CQUES DUP ARC AND OLIVIER FINKEL – f 0 ( x ) = x ( f 0 is the identity), – f k +1 : ( A ∪ { և n : k < n < ω } ) ω 7− → ( A ∪ { և n : k + 1 < n < ω } ) ω defined by: • f k +1 ( x ) = x և k +1 if x և k +1 is infinite, • f k +1 ( x ) = 0 ω if x և k +1 is finite or undefined, By ind uction on k , on e shows that every functio n f k is Borel - and e ven Bore l of finite rank. Moreover , since Borel functions are closed under taking the limits [Kur61], the follo wi ng function is B orel. f ∞ : ( A ∪ { և n : 0 < n < ω } ) ω 7− → A ω defined by: – f ∞ ( x ) = lim n<ω f n ( x ) if lim n<ω f n ( x ) is defined, and infinite, – f ∞ ( x ) = 0 ω otherwise. Remark 3.3. Let X ⊂ { 0 , 1 } ω with 0 ω 6∈ X , then for any x ∈ ( { 0 , 1 } ∪ { և n : 0 < n < ω } ) ω x ∈ X ≈∞ ⇐ ⇒ f ∞ ( x ) ∈ X In o ther words, X ≈∞ = f ∞ − 1 ( X ) , which shows that whenev er X is Borel, X ≈∞ is Borel too. In fact, with tools described in [Dup01], and [Dup0?], it is p ossible to show that gi ven any Π 0 1 -complete set Y , the set Y ≈∞ belongs to Π 0 ω +2 . If X is the set of infinite words ove r the alphabet { 0 , 1 } which contains an infinite nu mber of 1 s, t hen it is also possible to show that X ≈∞ is Bore l by completely di f ferent m ethods in volving decompositions of ω -powers [FS03,Fin04]. Pr oposition 3.4. Let X be th e set of i nfinite wor ds over { 0 , 1 } that con- tain infinitely many 1 s, X ≈∞ ∈ ∆ 1 1 \ ∆ 0 ω A BOREL ω -POWER OF A CONTEXT -F REE LANGU A GE ABO VE ∆ 0 ω 7 Pr oof. The fact X ≈∞ is Borel i s Remark 3.3. As for X ≈∞ / ∈ ∆ 0 ω , i t is a consequence of t he fact that t he operation Y 7− → Y ≈ is strictl y increasing (for the W adge ordering) inside ∆ 0 ω (see [Dup01][Dup0?]). In other words, for any Y ∈ ∆ 0 ω the relation Y < W Y ≈ holds ( < W stands for the s trict W adge ordering). But, as a m atter of fact, ( X ≈∞ ) ≈ ≤ W X ≈∞ holds which forbids X ≈∞ to belong to ∆ 0 ω . Indeed, to see that ( X ≈∞ ) ≈ ≤ W X ≈∞ holds, it is e nough to describe a winni ng strategy for player II in the W adge gam e W  ( X ≈∞ ) ≈ , X ≈∞  . In this game, player II uses ω many different erasers: և 1 , և 2 , և 3 , . . . whose st rength is oppos ite to their indices ( և k erases all erasers և j for any j > k but no և i for i ≤ k ). Wh ile player I uses the same erasers as player II does, plu s an extra one ( և ) whi ch is stronger than all t he other ones. The winni ng strategy for II deri ves from ordinal arithmeti c: 1 + ω = ω . It consi sts in copying I’ s run with a shift on the indices of erasers: – if I plays a letter 0 or 1 , then II plays the sam e letter , – if I plays an eraser և n , II plays t he eraser և n + 1 . – if I plays the eraser և (the first o ne that will be taken in to account when the erasing process starts), then II plays և 0 . This strategy is clea rly winning. 4 Simulating X ≈∞ by the ω - power of a context-free language It was alrea dy kno wn t hat there e xists an ω -power of a finitary language which is Borel of infinite rank [Fin04]. But the questi on was l eft op en whether such a finitary language could be context fr ee . This article provides effecti vely a context free l anguage V such that V ω is a Borel set of infinite rank, and uses infinite W adge games to sho w that this ω -power V ω is located above ∆ 0 ω in the Borel hierarchy . The idea is to have X ≈∞ , where X stands for the set of all infinite words over { 0 , 1 } that contain infinitely m any 1 s to be of the form V ω for some language V re cognized by a (non determini stic) Pushdown Au- tomaton. W e first recall the no tion of pushdown automaton [Ber79,ABB96]. Definition 4.1. A p ushdown automaton (PD A) is a 7-tuple M = ( Q, A, Γ , δ, q 0 , Z 0 , F ) 8 J A CQUES DUP ARC AND OLIVIER FINKEL wher e – Q is a finite set of st ates, – A is a finite input alpha bet, – Γ is a finit e pushdown alphabet, – q 0 ∈ Q is the initi al state, Z 0 ∈ Γ is the star t symbol, – δ is a mappin g fr om Q × ( A ∪ { ε } ) × Γ to finite subsets of Q × Γ ∗ . – F ⊆ Q is the set of final stat es. If γ ∈ Γ + describes the p ushdown st or e content, the leftmost symbol of γ will be assumed to be on “top” of t he stor e. A configurati on of a PD A is a pair ( q , γ ) wher e q ∈ Q and γ ∈ Γ ∗ . F or a ∈ A ∪ { ε } , γ , β ∈ Γ ∗ and Z ∈ Γ , if ( p, β ) is in δ ( q , a, Z ) , then we write a : ( q , Z γ ) 7→ M ( p, β γ ) . 7→ ∗ M is the transitive and r eflexive closur e of 7→ M . Let u = a 1 a 2 . . . a n be a fini te wor d over A . A finite sequence of configurations r = ( q i , γ i ) 1 ≤ i ≤ p is called a run of M on u , s tarting in configuration ( p, γ ) , iff: 1. ( q 1 , γ 1 ) = ( p, γ ) 2. for each i , 1 ≤ i ≤ p − 1 , ther e exists b i ∈ A ∪ { ε } sa tisfying b i : ( q i , γ i ) 7→ M ( q i +1 , γ i +1 ) such that a 1 a 2 . . . a n = b 1 b 2 . . . b p − 1 . This run is si mply called a r un of M on u if it starts f r om configuration ( q 0 , Z 0 ) . The language ac cepted by M is L ( M ) = { u ∈ A ∗ : ther e is a run r of M on u ending in a final state } . For instance, the set 0 ∗ 1 ⊂ { 0 , 1 } ∗ is trivially context-free. Pr oposition 4.2 (Finkel). Let L n be the maximal subset of { 0 , 1 , և 1 , և 2 , . . . , և n } ∗ such that L և 1 n և 2 ... և n = 0 ∗ 1 , L n is context-fr ee This was first noticed by the second author in [Fin01]. T o be more precise, by u ∈ L n we mean: we start with som e u , then we ev aluate և 1 as an eraser , and obtain u 1 (providing that we must nev er use և 1 to erase the emp ty sequence, i.e. e very occurrence of a և 1 symbol does erase a letter 0 or 1 or an eraser և i for i > 1 ). Then we A BOREL ω -POWER OF A CONTEXT -F REE LANGU A GE ABO VE ∆ 0 ω 9 start again with u 1 , this time we e valuate և 2 as an eraser , wh ich yields u 2 , and so on. When there is no more symbol և i to be e valuated, we are left with u n ∈ { 0 , 1 } ∗ . W e define u ∈ L n iff u n ∈ 0 ∗ 1 . T o m ake a PD A recognize L n , the idea is to ha ve it guess (non deter- ministi cally), for each singl e l etter t hat it reads, whether this l etter wi ll be erased later or not. Moreover , the PD A should also guess for each eraser it encounters, wheth er this eraser shoul d be used as an eraser or whether it sho uld not - for the only reason t hat it wil l be erased later on by a str onger er aser . Durin g the reading, the stack should be used t o accumulate all pendant g uesses, in o rder to verify later o n that they are fulfilled. W e would very mu ch like to prove that L ∞ = [ n<ω L n is context- free. U nfortunately , we canno t g et such a result. Howe ver , we are able to show that a sli ghtly more comp licated set (stri ctly containing L ∞ ) is indeed context-free. Of course, t he first problem t hat comes to min d when working with L ∞ , is to hand le ω m any different erasers with a finite alph abet. This implies that erasers must be coded by finite words. This was done by the second author in [Fin0 3b]. Roughly speaking, th e eraser և n is coded by the word α B n C n D n E n β with new letters α, B , C, D , E , β . It is a little b it tricky , but the PD A m ust really be able to read the num ber n identifying the eraser four times. The very definiti on of t he s ets L n , requires the erasing operations to be executed in an increasing order: in a word that contains only the erasers և 1 , . . . , և n , one must cons ider first the eraser և 1 , then և 2 , and so on. . . Therefore this erasing process satisfy the following properties: (a) An eraser և j may only erase letters c ∈ { 0 , 1 } or erasers և k with k > j . (b) As sume that in a word u ∈ L n , there is a sequence cv w where c i s either in { 0 , 1 } or in the set { և 1 , . . . , և n − 1 } , and w is (the code of) an eraser և k which erases c once th e erasing process is achiev ed. If there is i n v (the code of) an eraser և j which erases e , wh ere e ∈ { 0 , 1 } or e is (the code of) another eraser , then e must b elong to v (it is between c and w in the word u ) ; moreove r th e erasing - by the eraser և j - has been achie ved before the other one with th e eraser 10 J A CQUES DUP ARC AND OLIVIER FINKEL և k . This implies j ≤ k . T hus the integer k m ust satisfy: k ≥ max { j : an eraser և j was used inside v } The essential dif ference with the ca se studied in [ Fin03b] is that here an e raser և j may only erase letters 0 or 1 or erasers և k for k > j , while in [Fin03b] an eraser և j was ass umed to be only able t o erase letters 0 or 1 or erasers և k for k < j . So the above inequality was re placed by: k ≤ min { j : an eraser և j was used inside v } Howe ver , with a slight m odification, we can const ruct a PD A B which, among words where lett ers α, β , B , C , D , E are only used to code erasers of the form և j , accepts exactly the words which belong to the l anguage L ∞ . W e now explain the behavior of this PD A. ( For simpli city , we some- times talk about the eraser և j instead of its code αB j C j D j E j β .) Assume that A is a finit e auto maton accepting (by final state) the finitary language 0 ∗ 1 over the alphabet A = { 0 , 1 } . W e can in formally describ e the b eha v ior of the PD A B when reading a word u such that the letters α, B , C , D, E , β are onl y used in u to code the erasers և j for 1 ≤ j . B si mulates the automaton A until it guesses (non determi nistically) that it begins to read a segment w which contains erasers which really erase and some letters of A or some other e rasers which are e rased when the operations of erasing are achiev ed in u . Then, still non deterministically , when B reads a l etter c ∈ A it may guess that this letter wil l b e erased and push it in the p ushdown store, keeping in memory the current state of the automaton A . In a similar manner when B reads the code և j = αB j C j D j E j β , it may guess that thi s eraser will be erased (by another eraser և k with k < j ) and then may push in the store the finite word γ E j ν , where γ , E , ν are in the pushdown a lphabet of B . But B may also gu ess th at the eraser և j = αB j C j D j E j β will really be u sed as an eraser . If it guesses that the code of և j will be used as an eraser , B has to pop from th e top of the pus hdown st ore eit her a l etter c ∈ A or the code γ E i .ν of another eraser և i , with i > j , which is erased by և j . A BOREL ω -POWER OF A CONTEXT -F REE LANGU A GE ABO VE ∆ 0 ω 11 In this case, it is easy for B to check whether i > j when reading the initial segment αB j of և j . But as we remarked in ( b ) , the PD A B must also check that the integer j is greater than or equal to ev ery int eger p such t hat an eraser և p has been used since th e lett er c ∈ A or th e code γ E i .ν was push ed in t he store. Then, after having pushed some lett er t ∈ A or the code t = γ E i .ν of an eraser in th e pushdown s tore, and b efore popping it from th e top of t he s tack, B must keep track of the following integer in the mem ory stack. k = ma x [ p / s ome eraser և p has been used since t was pushed in the stack ] For that purpose B pushes the finite w o rd L 2 S k L 1 in the pushdown store ( L 1 is pushed first, then S k and the letter L 2 ), with L 1 , L 2 and S are new letters added to the pushdown alphabet. So, wh en B gu esses that և j = αB j C j D j E j β will be really u sed as an eraser , there is on top of t he stack either a letter c ∈ A or a code γ E i .ν of an eraser which will be erased or a code L 2 S k L 1 . The behavior of B is then as follows. Assume first there is a code L 2 S k L 1 on top of t he stack. Then B firstly checks t hat j ≥ k holds by reading the segment α B j C of the eraser αB j C j D j E j β . If j ≥ k h olds, t hen usi ng ǫ -transitions, B completely pop s the w ord L 2 S k L 1 from the top of the stack. ( B has already checked it is allowed to use the eraser և j ). Then, in each case, the top of the stack contains either a letter c ∈ A , or the code γ E i ν of an eraser which should be erased later . B pops this letter c or th e code γ E i .ν (having checked that j < i after reading the segment αB j C j of the eraser αB j C j D j E j β ). A this point, we must ha ve a look at the top stack symbols. Ther e are three cases: 1. The top stack symbo l is the bot tom sym bol Z 0 . In which case, the PD A B , after having compl etely rea d the era ser և j , may pursue the simulatio n of the automaton A or guess that it begins to read another segment v which will be erased. Hence the next letter c ∈ A or the next code αB m .C m .D m .E m .β of the word wil l be erased. Then B pushes the l etter c ∈ A or the code γ E m .ν of և m in the pu shdown store. 12 J A CQUES DUP ARC AND OLIVIER FINKEL 2. If the top stack symbol is eith er a letter c ′ ∈ A or a code γ E m .ν , then B pushes the cod e L 2 S j L 1 in the pushdown store ( j is t hen the maximum of the set of int egers p such th at an eraser և p has b een used since the letter c ′ or t he cod e γ E m .ν has been pus hed i nto t he stack). 3. If the t op s tack symbo ls are a code L 2 S l L 1 , t hen th e PD A B must compare t he integers j and l , and replace L 2 S l L 1 by L 2 S j L 1 in case j > l . B achiev es this task while reading the segment D j E j β of the eraser αB j C j D j E j β . The PD A B pops a letter S for each letter D it rea ds. Then it checks whether j ≥ l is satisfied. If j ≥ l t hen it pushes L 2 S j L 1 while it reads the segment E j β of t he eraser և j . In case j < l , after i t reads D j , the part S l − j L 1 of the code L 2 S l L 1 remains in the stack. The PD A then pushes again j letters S and a letter L 2 while reading E j β . When again t he st ack on ly contains Z 0 - the initial stack symbol - B resumes the simulation of the automaton A or it guesses that it begins to read a new segment which will be erased later . W e are confronted wi th the fact B will also accept some words wh ere the letters α , β , B , C, D , E are not us ed to code erasers. How can we make sure that this PD A is not m isled by such wrong codes of erasers ? 5 Wrong codes of erasers and the righ t ω -power In fact, one cannot m ake sure that a PD A notices the discrepancy be- tween right codes of the form αB j C j D j E j β and wrong ones (of the form αB b C c D d E e β where b, c, d, e are not all the same integer for i nstance). Howe ver , t here is a satisfactory solut ion: instead of ha ving a PD A reject these wrong codes, sim ply let it accept all of them. Accepting a word if it contains a wrong code of an eraser is trivial for a non deterministic PD A. So instead of a PD A B that accepts p recisely L ∞ (up to the cod ing of erasers), we set Pr oposition 5.1. Ther e exists a PD A B s.t. L ( B ) = L ∞ ∪ W A BOREL ω -POWER OF A CONTEXT -F REE LANGU A GE ABO VE ∆ 0 ω 13 where W stands for t he s et o f all finite words which host a wrong code, L ∞ really is L ∞ where erasers are replaced by their correct codes, and L ( B ) is th e language recognized by B . E verything is ready for the m ain result. Theor em 5 .2. The ω -power Y = L ( B ) ω of the context-fr ee langu age L ( B ) described above satisfies Y ∈ ∆ 1 1 r ∆ 0 ω Pr oof. T o begin with, the set Y is the d isjoint uni on of t hree different sets: Y = Y 0 ∪ Y ∞ ∪ Y ∗ , where Y 0 is the set of all infinite sequences in Y wit h no wrong code i n them , Y ∞ the set of all i nfinite sequences with infinitely many wrong codes, and Y ∗ the set o f infinite sequ ences with finitely many wrong codes (a t least one). W e remark that: – Y 0 is W adge equiv alent t o th e set X ≈∞ as defined in 3.4. i.e. the set of all ω -words that, after takin g care of the erasing process, ultimately reduce to w ords with infinitely many 1 s . T o be more precise, it is this very same set up to a renaming of the erasers. So Y 0 belongs to ∆ 1 1 . – Y ∞ is W adg e equiv alent to X , so it is Π 0 2 -complete. – Y ∗ is more complicated. Ho wev er , it is of the form Y ∗ = W Y 0 , where W i s the set of all fi n ite words with at least a n occurrence of a wrong code. So Y ∗ is a countable union of sets, each of wh ich is W adge equiv alent to Y 0 . Hence, Y ∗ is a countable union of Borel sets, there- fore Y ∗ is Borel too. All three c ases put together show that Y = Y 0 ∪ Y ∞ ∪ Y ∗ , is a finite un ion of Borel sets, hence it Borel too. It remain s to prov e that Y / ∈ ∆ 0 ω . Th is, in fact, is im mediate from Proposition 3.4 which st ated that X ≈∞ / ∈ ∆ 0 ω . Because th ere is an ob- vious w inning strate g y for player II in the W adge game W ( X ≈∞ , Y ) . It consists in never playing a wrong code, and cop yi ng I’ s run up t o the re- naming of the erasers. Since X ≈∞ is clearly W adge equiv alent to Y 0 this strategy w orks perfectly well and shows that Y / ∈ ∆ 0 ω . This quick stu dy giv es an example o f how an infinite game theoretical approach leads to intriguing results i n Theoreti cal Computer Science. On one hand, the notio n of erasers is highly related to the dynami c beha vior of players in gam es. 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