On Winning Conditions of High Borel Complexity in Pushdown Games

Fundamenta Informaticae (2005) 1–22 1 IOS Pr ess On Wi nning Conditions of High Borel Complexity in Pushdo wn Gam es Olivie r Finkel Equipe de Logique Math ´ ematique U.F .R. de Math ´ ematiques, Universi t ´ e P aris 7 2 Place J ussieu 7525 1 P aris cedex 05, F rance. finkel@logique.jussieu.fr Abstract. In a recent paper [19, 20] Serre has presented some decidable winning condition s Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 of arbitrarily high finite Borel complexity for games on fi nite grap hs or on push- down graphs. W e answer in this paper se veral questions which were raised by Serre in [19, 20]. W e study classes C n ( A ) , defined in [ 20], an d show that th ese classes are in cluded in the class o f non-am biguou s context free ω -langu ages. Moreover from the study of a larger class C λ n ( A ) we infer that the complemen ts of lang uages in C n ( A ) are also non-amb iguous con text free ω -lan guages. W e conclud e the study of classes C n ( A ) by showing that they are neither clo sed unde r union nor under intersection. W e prove also tha t ther e exists pushdown games, equip ped w ith winning co nditions in th e form Ω A 1 ⊲ A 2 , where the winnin g sets are n ot determ inistic context free languag es, giving examples of winning sets which a re non -determin istic non- ambiguo us co ntext free lan guages, inheren tly am- biguou s context free langua ges, or e ven non context f ree language s. Keywords: Pushdown automata; infinite two-play er games; pushdown games; winn ing co nditions; Borel comp lexity; context fre e ω -lan guages; closure under boolean o perations; set of winnin g posi- tions. 1. Introd uction T wo-player infini te games ha ve b een much stu died in set theory and in pa rticular in Desc ripti ve Set The- ory . Martin’ s T heorem st ates that e very Gale S te wart game G ( A ) , wher e A is a Borel set, is determined, i.e. that one of the two play ers has a winning strate gy [14]. Address for correspond ence: E Mail: finkel@logiqu e.jussieu.fr 2 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games In Computer Science, the con ditions of a Gale Stew art game may be seen as a spe cification of a reacti v e system, where the two players are respec tiv ely a non terminatin g reacti ve program and the “en viron- ment”. Then the probl em of the synthe sis of winning strate gies is of great prac tical interes t for the prob- lem of program synthesis in reacti ve systems. B ¨ uchi-Lan dweber Theorem states that in a Gale Ste wart game G ( A ) , where A is a re gular ω -langu age, one can decide who is the winner and c ompute a winning strate gy giv en by a finite state transd ucer . In [23, 16] Thomas asked for an extensio n of this result to games played on pushdo wn graphs. W alukie wicz firstly sho wed in [25] that one can eff ectiv ely construc t winning strate gies in parity games played on pushd own graphs and that these strategi es can be computed by pushdo wn transdu cers. Sev eral autho rs ha ve then studied pushdo wn games equipped with other decida ble winning conditio ns, [4, 5, 17, 11 ]. C achat, D uparc and Thomas ha ve presented the first decidable winni ng condi tion at the Σ 3 le vel of the Borel hi erarchy [6]. Bouquet, Serre and W alukie wicz ha ve studied winnin g conditi ons which are boolean combinat ions of a B ¨ uchi condition and of the unbou ndedness conditio n which requires the stack to be unboun ded, [3]. Recently Serre has giv en a family of decidabl e winning conditio ns of arbitra rily high finite Borel rank [19, 20]. A game between two players Adam and E ve on a pushdo wn graph, is equipped with a win- ning condition in the form Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 , where A 1 , . . . , A n are deterministi c pushdo wn automata, the stack alphab et of A i being the input alphab et of A i +1 , and A n +1 is a determini stic pushdo wn automato n with a B ¨ uchi or a parity acceptance condition . Then an infinite play is won by Eve iff during this play the stack is strictly unboun ded , that is con ver ges to an infinite word x and its limit x ∈ L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) , where L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) is an ω -languag e defined as foll ows. A wo rd α 0 is in L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) if f: for all 1 ≤ i ≤ n , when A i reads α i − 1 its st ack is stri ctly unbou nded and the limit of the stack contents is an ω -word α i ; and A n +1 accept s α n . Serre pro ved that for these w inning conditions one can decide the winner in a pushdo wn game and that the winning strate gies are ef fecti ve. W e solv e in this paper sev eral questions which are raised in [19, 20]. W e first study the classes C n ( A ) which contain lan guages in the f orm L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) , where A is the inp ut alphab et of A 1 . W e sho w that these classes are included in the class of non-ambiguo us conte xt free ω -languages. Moreov er from the study of a larger class C λ n ( A ) we infer that the complements of langua ges in C n ( A ) are also non-amb iguous conte xt free ω -languages . W e conclu de the stu dy of classes C n ( A ) by sh owing that they are neither closed under union nor unde r intersectio n. For all pre viously st udied de cidable winnin g con ditions for pushdo wn ga mes th e s et of w inning positio ns for any player had been sho wn to be regular . In [19, 20] Serre prov ed that e very deterministic context free language may occur as a w inning set for E ve in a pushdo wn game equippe d with a winning con- dition in the form Ω B , where B is a determinis tic pushdo wn automato n. The exact nature of these sets remains op en and the qu estion is raised in [19 , 20] whe ther there exi sts a push down game e quipped with a winning con dition in the form Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 such that th e set of winni ng positions for Eve is no t a determin istic contex t free language . W e gi ve a pos itiv e answer to this question , givin g examples of win- ning sets which are non-determin istic non-ambi guous context free languages, or inherentl y ambiguou s conte xt free langua ges, or ev en non contex t free languages . The pape r is or ganized as follo w s. In section 2 we recall definitions and results about push down au- tomata, conte xt free ( ω )-langu ages, push down games, and winning condit ions in the form Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 . In secti on 3 are studied the classes C n ( A ) . Results on sets of winning positio ns are presented in Section 4. Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 3 2. Recall of pr evious defin itions and results 2.1. Pushdown automata W e assume the reader to be familiar with the theory of formal ( ω )-langu ages [22, 21, 12]. W e shall use usual nota tions of formal language theory . When A is a finite alpha bet, a non-empty finite wor d over A is any sequ ence x = a 1 . . . a k , where a i ∈ A for i = 1 , . . . , k , and k is an integer ≥ 1 . The length of x is k , denoted by | x | . The empty wor d has no letter and is denoted by λ ; its length is 0 . For x = a 1 . . . a k , we write x ( i ) = a i and x [ i ] = x (1) . . . x ( i ) for i ≤ k and x [0] = λ . A ⋆ is th e set of finite wor ds (includ ing the emp ty word) over A and A + = A ⋆ − { λ } . The first infinit e or dinal is ω . A n ω - wor d ov er A is an ω -sequ ence a 1 . . . a n . . . , where for all integers i ≥ 1 , a i ∈ A . When σ is an ω -word ov er A , we write σ = σ (1) σ (2) . . . σ ( n ) . . . , where for all i , σ ( i ) ∈ A , and σ [ n ] = σ (1) σ (2) . . . σ ( n ) for all n ≥ 1 and σ [0] = λ . The pr efix rela tion is denoted ⊑ : a fi nite word u is a pr efix of a finite word v (respe ctiv ely , an infinite word v ), denoted u ⊑ v , if and only if there exis ts a fi nite word w (respecti ve ly , an infinite word w ), such that v = u.w . The set of ω - wor ds ov er the alphabe t A is denote d by A ω . An ω - languag e over an alphab et A is a subs et of A ω . In [19, 20] deterministic pushd own automata are defined with two restrictio ns. It is supposed that there are no λ -transi tions, i.e. the automata are r eal time . Moreov er one can push at most one symbol in the pushd own stack using a single transitio n of the automaton . W e no w define pushd own automata, keep ing this second restrict ion b ut allo wing the existe nce of λ - transit ions; and we define also the non determinist ic versio n of pushdo w n automata. A pushdown automaton (PD A) is a 6-tuple A = ( Q, Γ , A, ⊥ , q in , δ ) , where Q is a finite set of states, Γ is a finite pushdo wn alphabet, A is a fi nite input alphabet, ⊥ is the bottom of stack symbol, q in ∈ Q is the i nitial state, and δ is the transitio n relation which is a mapping from Q × ( A ∪ { λ } ) × Γ to subset s of { sk ip ( q ) , p op ( q ) , p us h ( q , γ ) | q ∈ Q, γ ∈ Γ − {⊥}} The bott om symbol a ppears only a t the bottom of the stack an d is nev er poppe d thus for all q , q ′ ∈ Q and a ∈ A , it holds that pop ( q ′ ) / ∈ δ ( q , a, ⊥ ) . The push down automat on A is determin istic if for all q ∈ Q , a ∈ A and Z ∈ Γ , the set δ ( q , a, Z ) contai ns at most one element; moreov er if for some q ∈ Q and Z ∈ Γ , δ ( q , λ, Z ) is non-empty then for all a ∈ A the set δ ( q , a, Z ) is empty . If σ ∈ Γ + descri bes the pushdo wn store content, the rightmost symbol will be assumed to be on “ top” of the stor e . A configuration of the pushdo wn automaton A is a pair ( q , σ ) where q ∈ Q and σ ∈ Γ ⋆ . For a ∈ A ∪ { λ } , σ ∈ Γ ⋆ and Z ∈ Γ : if ( sk ip ( q ′ )) is in δ ( q , a, Z ) , then we w rite a : ( q , σ.Z ) 7→ A ( q ′ , σ.Z ) ; 4 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games if ( pop ( q ′ )) is in δ ( q , a, Z ) , then we w rite a : ( q , σ.Z ) 7→ A ( q ′ , σ ) ; if ( push ( q ′ , γ )) is in δ ( q , a, Z ) , then we write a : ( q , σ.Z ) 7→ A ( q ′ , σ.Z.γ ) . 7→ ⋆ A is the transiti ve and reflexi ve closure of 7→ A . (The subscript A will be omitted whene ver the meaning remains clear). Let x = a 1 a 2 . . . a n be a finite word over A . A finite sequence of configura tions r = ( q i , γ i ) 1 ≤ i ≤ p is called a run of A on x , starting in configuration ( q , γ ) , if f: 1. ( q 1 , γ 1 ) = ( q , γ ) 2. for each i , 1 ≤ i ≤ ( p − 1) , there exis ts b i ∈ A ∪ { λ } satis fying b i : ( q i , γ i ) 7→ A ( q i +1 , γ i +1 ) 3. a 1 a 2 . . . a n = b 1 b 2 . . . b p − 1 A run r of A on x , starting in configu ration ( q in , ⊥ ) , will be simply called “a run of A on x ”. Let x = a 1 a 2 . . . a n . . . be an ω -word ove r A . An infinite sequenc e of configuratio ns r = ( q i , γ i ) i ≥ 1 is called a run of A on x , starting in configuration ( q , γ ) , if f: 1. ( q 1 , γ 1 ) = ( q , γ ) 2. for each i ≥ 1 , there exist s b i ∈ A ∪ { λ } satis fying b i : ( q i , γ i ) 7→ A ( q i +1 , γ i +1 ) 3. eithe r a 1 a 2 . . . a n . . . = b 1 b 2 . . . b n . . . or b 1 b 2 . . . b n . . . is a finite prefix of a 1 a 2 . . . a n . . . The run r is said to be complete when a 1 a 2 . . . a n . . . = b 1 b 2 . . . b n . . . A complet e run r of A on x , starting in configuratio n ( q in , ⊥ ) , will be simply called “a run of A on x ”. If the pushd own automaton A is eq uipped with a set of final states F ⊆ Q , the finitary langu age accepted by ( A , F ) is : L f ( A , F ) = { x ∈ A ⋆ | the re exists a run r = ( q i , γ i ) 1 ≤ i ≤ p of A on x such that q p ∈ F } The class C F L of conte xt fr ee languag es is the class of finitary languag es which are accepted by push- do wn automata by final states. Notice that other accepting condition s by PDA ha ve bee n sho wn to be equi valen t to the acceptan ce con- dition by final s tates. Let us cite , [1 ]: (a) accep tance by empty sto rage, (b) accep tance by final sta tes and empty storage, (c) acceptan ce by topmost stack letter , (d) acceptan ce by final states and topmost stack letter . The clas s D C F L of dete rministic contex t fr ee langua ges is the class of finitary languag es which are accept ed by deterministi c pushdo wn automata (DP D A) by final states . Notice that for DP D A, accept ance by final states is not equi valen t to acceptan ce by empty storag e: this is due to the fact that a langua ge accepted by a DPDA by empty storage must be pre fix-fr ee while this is not necess ary in the case of acceptance by final states [1]. Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 5 The ω -language B ¨ uch i accepted by ( A , F ) is : L ( A , F ) = { x ∈ A ω | there ex ists a run r of A on x such tha t I n ( r ) ∩ F 6 = ∅} where I n ( r ) is the set of all states entered infinitely often during run r . If instead the pushdo w n automaton A is equipped w ith a set of accepting sets of state s F ⊆ 2 Q , the ω -language Muller accepted by ( A , F ) is : L ( A , F ) = { x ∈ A ω | the re exists a run r of A on x such that I n ( r ) ∈ F } The class C F L ω of conte xt fr ee ω - langua ges is the class of ω -langu ages which are B ¨ uchi or Muller accept ed by pushdo wn automata . Another usual acceptance co ndition for ω -words is th e pari ty con dition. In that ca se a pushdo wn automa- ton A = ( Q, Γ , A, ⊥ , q in , δ ) is equipped with a function col from Q to a finite set of colors C ⊂ N . T he ω -language accepted by ( A , col ) is: L ( A , col ) = { x ∈ A ω | the re exists a run r of A on x such that sc ( r ) is e ve n } where sc ( r ) is the smallest color appeari ng infinitely often in the run r . It is easy to see that a B ¨ uchi accept ance condit ion can be ex pressed as a parity acceptance condition which itself can be exp ressed as a M uller cond ition. Thus the class of ω -languages which are accepted by pushdo wn automata with a parity accept ance con- dition is still the class C F L ω . Consider now determinis tic pushdo wn automata. If A is a determinist ic pushdo w n automaton , then for e very σ ∈ A ω , there exists at most one run r of A on σ determined by the starting configuration. The pushd own automaton has the continui ty property iff for e very σ ∈ A ω , there exis ts a unique run of A on σ and this run is comple te. It is shown in [8] that each ω -language accep ted by a deterministi c B ¨ uchi (re- specti ve ly , Muller) pushdo wn automato n can be accepted by a determinis tic B ¨ uchi (resp ectiv ely , M uller) pushd own automaton with the continu ity property . T he same proof works in the case of determini stic pushd own automata with parity acceptanc e condition. The class of ω -languag es accepte d by determin istic B ¨ uchi pushdo wn automata is a strict subclass of the class D C F L ω of ω -languages accepted by deterministic pushdo wn automata with a Muller conditio n. One can easily sho w that D C F L ω is also the class of ω -languag es accepted by DPD A with a parity accept ance condition . Each ω -language in D C F L ω can be accepte d by a deterministic pushdo w n automaton having the con- tinuity pr operty with pari ty (or Muller) acceptan ce condition. One can then show t hat the class D C F L ω is close d under complementatio n. The notion of ambiguity for context free ω -langu ages has been firstly studied in [10]. A conte xt free ω - langua ge is non ambiguous iff it is accepted by a B ¨ uchi or Muller pushdo wn automaton such that e very ω -word on the input alphabet has at most one accepting run. Notice that we conside r here that two runs 6 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games are equal iff they go thro ugh the same infinite sequence of configurati ons and λ -tr ansitions occur at the same steps of the comput ations. The class N A − C F L ω is the class of non ambiguous conte xt free ω -languages . The inclusion DC F L ω ⊆ N A − C F L ω will be useful in the sequel. W e shall denote C o − N A − C F L ω the class of complement s of non ambiguous context free ω -languages. 2.2. Pushdown games Recall first that a pushdown pr ocess may be viewed as a PD A without input alpha bet and initial state. A pushd own proces s is a 4-tuple P = ( Q, Γ , ⊥ , δ ) , where Q is a finite set of states, Γ is a finite pushdo wn alphab et, ⊥ is the bottom of stack symbol, and δ is th e transit ion relation which is a mapping from Q × Γ to subs ets of { sk ip ( q ) , p op ( q ) , p us h ( q , γ ) | q ∈ Q , γ ∈ Γ − {⊥}} Configurati ons of a pushdo wn process are defined as for PD A. A con figuration of the pushd own process P is a pair ( q , σ ) where q ∈ Q and σ ∈ Γ ⋆ . T o a pushd own pro cess P = ( Q, Γ , ⊥ , δ ) is n aturally ass ociated a p ushdo wn graph G = ( V , → ) which is a directed graph. The set of vertices V is the set of configuration s of P . The edge relation → is defined as follo ws: ( q , σ ) → ( q ′ , σ ′ ) if f the configura tion ( q ′ , σ ′ ) can be reached in one transition of P from the configura tion ( q , σ ) . W e shall consid er in the sequel infinite games between two pl ayers named Eve and Adam on s uch push- do wn graphs. So we shall assu me that the set Q of stat es of a pus hdown process is partit ioned in two set s Q E and Q A . A configuration ( q , σ ) is in V E if f q is in Q E and it is in V A if f q is in Q A so ( V E , V A ) is a partition of the set of configura tions V . The game grap h ( V E , V A , → ) is calle d a pushdown game graph . A pl ay from a verte x v 1 of th is graph is d efined as follows. If v 1 ∈ V E , Ev e choose s a vertex v 2 such th at v 1 → v 2 ; otherwise Adam chooses such a verte x. If there is no such vertex v 2 the play stops. Otherwise the play may c ontinue. If v 2 ∈ V E , Eve chooses a v ertex v 3 such th at v 2 → v 3 ; oth erwise Adam chooses such a v ertex. If there is no su ch verte x v 3 the play stop s. O therwise the p lay continues in the same wa y . So a p lay starting from the v ertex v 1 is a fin ite or infinite sequence of ve rtices v 1 v 2 v 3 . . . such th at for all i v i → v i +1 . W e may assume, as in [19, 20], that in fact all pla ys are infinite. A winning conditi on for Eve is a set Ω ⊆ V ω . An infinite two-playe r pushd own game is a 4-tuple ( V E , V A , → , Ω ) , where ( V E , V A , → ) is a pushdo wn game graph and Ω ⊆ V ω is a winning conditio n for Eve. In a pushdo wn game e quipped with the winning condit ion Ω , Ev e wins a play v 1 v 2 v 3 . . . if f v 1 v 2 v 3 . . . ∈ Ω . A strat e gy for Eve is a parti al functi on f : V ⋆ .V E → V suc h that, for all x ∈ V ⋆ and v ∈ V E , v → f ( x.v ) . Eve use s the strate gy f in a play v 1 v 2 v 3 . . . iff for al l v i ∈ V E , v i +1 = f ( v 1 v 2 . . . v i ) . Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 7 A strategy f is a winning strat e gy for Eve from some position v 1 if f Eve wins all plays starting from v 1 and during which she uses the strate gy f . A ve rtex v ∈ V is a winning position for Eve if f she has a winning strate gy from it. The notion s of winning strate gy and winning position are defined for the other player A dam in a similar way . The set of winning positi ons for Eve and Adam will be respec tiv ely denoted by W E and W A . 2.3. Winn ing condition Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 W e first recall the definition of ω -langu ages in the form L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) which are used in [19, 20] to define the winnin g condition s Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 . W e shall need the notion of limit of an infinite sequenc e of finite words o ver some finite alphab et A . Let then ( β n ) n ≥ 0 be an infinite sequenc e of words β n ∈ A ⋆ . The finite or infinite word lim n ∈ ω β n is determin ed by the set of its (finite) prefixes: for all v in A ⋆ , v ⊑ lim n ∈ ω β n ↔ ∃ n ∀ p ≥ n β p [ | v | ] = v . Let now A = ( Q, Γ , A, ⊥ , q in , δ ) be a pushdo wn automaton reading words ov er the alphabet A and let α ∈ A ω . The pushdo wn stack of A is said to be strictly unbounded during a run r = ( q i , γ i ) i ≥ 1 of A on α iff lim n ≥ 1 γ n is infinite. W e define no w ω -languages L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) in a slightly more gene ral case than in [20], becaus e this w ill be useful in the next section. Notice that in [20], these ω -lang uages are only defined in the case where A 1 , . . . , A n , are r eal-time dete rministic pushdo wn automata , and A n +1 is a r eal-time determin istic pushdo wn automato n equipped w ith a pari ty or a B ¨ uchi acceptanc e condition. Let n be an inte ger ≥ 0 and A 1 , A 2 , . . . A n , be some determinis tic pushdo wn automata (in the case n = 0 the re are not any s uch automata) . Let ( A n +1 , C ) be a push down automaton equipped with a B ¨ uchi or a parity accept ance condition. The inp ut alphabet of A 1 is de noted A and we assume t hat, for each integer i ∈ [1 , n ] , the input alp habet of A i +1 is the stack alph abet of A i . W e define inducti vel y the ω -language L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) ⊆ A ω by: 1. If n = 0 , L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) = L ( A n +1 , C ) is the ω -language accep ted by A n +1 with accept ance condition C . 2. If n > 0 , L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) is the set of ω -word s α ∈ A ω such that: • When A 1 reads α , the stack of A 1 is strictly un bounded hence the seq uence of s tack cont ents has an infinite limit α 1 . • α 1 ∈ L ( A 2 ⊲ . . . ⊲ A n ⊲ A n +1 ) . Let now ( V E , V A , → ) be a pushd own game graph associa ted with a push down process P . An infinite play v 1 v 2 v 3 . . . , where v i = ( q i , γ i ) , is in the set Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 if f: 8 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games 1. The push down stack of P is strictly unbou nded during the play , i.e. lim n ≥ 1 γ n is infinite, and 2. lim n ≥ 1 γ n ∈ L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) . 3. Classes C n ( A ) 3.1. Classes C n ( A ) and context fr ee ω -languages For each integer n ≥ 0 and each finite alphabet A the class C n ( A ) is defined in [20] as the class of ω -langua ges in the form L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) , where A 1 , . . . , A n , are r eal-ti me determin - istic pushdo wn automata, the inp ut alpha bet of A 1 being A , and A n +1 is a r eal-time deterministic pushd own automaton equipped w ith a parity acceptan ce cond ition. It is easy to see that we obtain the same class C n ( A ) if we restrict the definition to the case of r eal-ti me determini stic pushdo w n automata A 1 , . . . , A n , A n +1 , ha ving the continuit y pr opert y . W e shall denote C λ n ( A ) the class obtained in the same way except that the deterministic pushdo wn au- tomata A 1 , . . . , A n , A n +1 , havi ng still the contin uity property , may ha ve λ -transiti ons, i.e. may be non real time. In th e sequel of this pap er w hen w e con sider languages in the fo rm L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) , we sh all alw ays implicitel y assume tha t the push down au tomata A 1 , . . . , A n , A n +1 , ha ve the co ntinuity pr opert y , and that, for each inte ger i ∈ [1 , n ] , the input alphabe t of A i +1 is the stack alph abet of A i . In order to prov e that classes C n ( A ) , C λ n ( A ) , are includ ed in the class of context free ω -langua ges we first state the follo wing lemma. Lemma 3.1. Let A 1 = ( Q 1 , Γ 1 , A 1 , ⊥ 1 , q 1 0 , δ 1 ) be a det erministic pushdo wn automaton and A 2 = ( Q 2 , Γ 2 , Γ 1 , ⊥ 2 , q 2 0 , δ 2 ) be a pushd own automato n equipped w ith a set of final states F ⊆ Q 2 . Then the ω -language L ( A 1 ⊲ A 2 ) is a context free ω -language. Pro of. L et A 1 = ( Q 1 , Γ 1 , A 1 , ⊥ 1 , q 1 0 , δ 1 ) be a determini stic pu shdo wn a utomaton and A 2 = ( Q 2 , Γ 2 , Γ 1 , ⊥ 2 , q 2 0 , δ 2 ) be a pushdo wn automaton equipped with a set of final states F ⊆ Q 2 . Recall that an ω -word α ∈ A ω 1 is in L ( A 1 ⊲ A 2 ) iff: • When A 1 reads α , the stack of A 1 is strictly unboun ded hence the sequence of stack contents has an infinite limit α 1 . • α 1 ∈ L ( A 2 , F ) . W e can decompo se the readin g of an ω -word α ∈ L ( A 1 ⊲ A 2 ) by the pushdo wn automaton A 1 in the follo wing way . When reading α , A 1 goes through the infinite sequence of configura tions ( q i , γ i ) i ≥ 1 . The infinite se- quenc e of stack contents ( γ i ) i ≥ 1 has limit α 1 thus for each inte ger j ≥ 1 , there is a smallest integ er n j such that, for all integ ers i ≥ n j , α 1 [ j ] = γ i [ j ] . Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 9 The word α can then be decompos ed in the form α = σ 1 .σ 2 . . . σ n . . . where for all inte gers j ≥ 1 , σ j ∈ A ⋆ 1 and σ j : ( q n j , α 1 [ j ]) 7→ ⋆ A 1 ( q n j +1 , α 1 [ j + 1]) = ( q n j +1 , α 1 [ j ] .α 1 ( j + 1)) Notice that n 1 = 1 , q 1 = q 1 0 and α 1 [1] = ⊥ 1 hence σ 1 : ( q 1 0 , ⊥ 1 ) 7→ ⋆ A 1 ( q n 2 , α 1 [2]) . Let no w , for each q , q ′ ∈ Q 1 and a, b ∈ Γ 1 , the langu age L ( q , q ′ ,a,b ) be the se t of wo rds σ ∈ A ⋆ 1 such that: σ : ( q , a ) 7→ ⋆ A 1 ( q ′ , a.b ) . This language of finite words ove r A 1 is accepted by the pushdo wn automaton A 1 with the foll owing mod ifications: the ini tial configurat ion is ( q , a ) and the accep tance is by final state q ′ and by fi nal stack content a.b . It is easy to see that this languag e is also accepted by a deterministic pushd own automaton by final states so it is in the class D C F L . Then each wor d σ j belong s to the deterministic conte xt free langua ge L ( q n j ,q n j +1 ,α 1 ( j ) ,α 1 ( j +1)) = { σ ∈ A ⋆ 1 | σ : (( q n j , α 1 ( j )) 7→ ⋆ A 1 ( q n j +1 , α 1 ( j ) .α 1 ( j + 1)) } In order to describ e the ω -langu age L ( A 1 ⊲ A 2 ) from the ω -language L ( A 2 , F ) and the determinis tic conte xt free langua ges L ( q , q ′ ,a,b ) , for q , q ′ ∈ Q 1 and a, b ∈ Γ 1 , we no w recall the notion of substituti on. A subst itution is a mapping f : Σ → 2 Γ ⋆ , where Σ and Γ ar e two finite alphabe ts. If Σ = { a 1 , . . . , a n } , then for all inte gers i ∈ [1; n ] , f ( a i ) = L i is a finitary langua ge over the alpha bet Γ . No w this mapping is exte nded in the usual manner to finite words: for all letters a i 1 , . . . , a i n ∈ Σ , f ( a i 1 . . . a i n ) = f ( a i 1 ) . . . f ( a i n ) , and to finitary languag es L ⊆ Σ ⋆ : f ( L ) = ∪ x ∈ L f ( x ) . If for eac h letter a ∈ Σ , the languag e f ( a ) does not contain the empty wor d, then the substituti on is said to be λ -free and the mapping f may be ex tended to ω -words : f ( x (1) . . . x ( n ) . . . ) = { u 1 . . . u n . . . | ∀ i ≥ 1 u i ∈ f ( x ( i )) } and to ω -language s L ⊆ Σ ω by setting f ( L ) = ∪ x ∈ L f ( x ) ⊆ Γ ω . If the substitu tion is not λ -free we can define f ( L ) in the same way for L ⊆ Σ ω b ut this time f ( L ) ⊆ Γ ⋆ ∪ Γ ω , i.e. f ( L ) may contain finite or infinite words. The substitution f is said to be a contex t free substitution iff for all a ∈ Σ the finitary language f ( a ) is conte xt free. Recall that Cohen and G old prove d in [7] that if L is a conte xt free ω -languag e and f is a conte xt free substi tution then f ( L ) ∩ Γ ⋆ and f ( L ) ∩ Γ ω are cont ext free. W e define no w a ne w alphabet ∆ = { L ( q , q ′ , a, b ) | q , q ′ ∈ Q 1 and a, b ∈ Γ 1 } and we cons ider the substituti on h : Γ 1 → 2 ∆ defined, for all b ∈ Γ 1 , by: h ( b ) = { L ( q , q ′ , a, b ) | q , q ′ ∈ Q 1 and a ∈ Γ 1 } 10 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games Applying this substit ution to the ω -language L ( A 2 , F ) ⊆ Γ ω 1 , we get h ( L ( A 2 , F )) . The substitut ion h is λ -free thus h ( L ( A 2 , F )) is a ω -languag e over ∆ . Moreov er for e ach b ∈ Γ 1 the set h ( b ) is finite hence conte xt free. Thus h ( L ( A 2 , F )) ⊆ ∆ ω is a context free ω -langua ge because L ( A 2 , F ) is a conte xt free ω -language and the substituti on h is a contex t free substituti on. Let now R ⊆ ∆ ω be the ω -langu age defined as follows. An ω -word x ∈ R has its first letter in the set { L ( q 1 0 , q ′ , ⊥ 1 , b ) | q ′ ∈ Q 1 and b ∈ Γ 1 } , and each letter L ( q , q ′ , a, b ) , for q , q ′ ∈ Q 1 and a, b ∈ Γ 1 , in x is follo wed by a letter in the set { L ( q ′ , q ′′ , b, c ) | q ′′ ∈ Q 1 and c ∈ Γ 1 } . The ω -langua ge R is re gular thus h ( L ( A 2 , F )) ∩ R ⊆ ∆ ω is a conte xt free ω -language bec ause the class C F L ω is close d under intersectio n with regu lar ω -languages [7]. Consider now the substit ution Θ : ∆ → 2 A ⋆ 1 defined, for all letters L ( q , q ′ , a, b ) ∈ ∆ , by Θ( L ( q , q ′ , a, b )) = L ( q , q ′ ,a,b ) . The substitut ion Θ is co ntext free thus Θ[ h ( L ( A 2 , F )) ∩ R ] ∩ A ω 1 is a conte xt free ω -langu age and s o is ⊥ 1 . ( Θ [ h ( L ( A 2 , F )) ∩ R ] ∩ A ω 1 ) . By c onstructio n this ω -languag e is L ( A 1 ⊲ A 2 ) .  W e can in fact obtain a refined result if the langua ge L ( A 2 , F ) is non ambiguous. Lemma 3.2. Let A 1 = ( Q 1 , Γ 1 , A 1 , ⊥ 1 , q 1 0 , δ 1 ) be a det erministic pushdo wn automaton and A 2 = ( Q 2 , Γ 2 , Γ 1 , ⊥ 2 , q 2 0 , δ 2 ) be a pushd own automat on equipped with a set of final states F ⊆ Q 2 . If the ω -language L ( A 2 , F ) is non ambiguous then L ( A 1 ⊲ A 2 ) ∈ N A − C F L ω . Pro of. L et A 1 = ( Q 1 , Γ 1 , A 1 , ⊥ 1 , q 1 0 , δ 1 ) be a determini stic pu shdo wn a utomaton and A 2 = ( Q 2 , Γ 2 , Γ 1 , ⊥ 2 , q 2 0 , δ 2 ) be a pushdo wn automaton equipped with a set of final states F ⊆ Q 2 . W e assume that L ( A 2 , F ) is non ambiguo us so we can assume, without loss of general ity , that the pushd own automaton A 2 itself is non ambiguo us. W e are going to explain informally the constru ction of a non ambiguous B ¨ uchi pushdo w n automaton A accept ing the ω -langu age L ( A 1 ⊲ A 2 ) . W e refer no w to the proof of the preceding lemma. W e ha ve consider ed the readi ng of an ω -word α ∈ L ( A 1 ⊲ A 2 ) by A 1 , and we ha ve sho wn that the word α can then be de composed in the form α = σ 1 .σ 2 . . . σ n . . . where for all inte gers j ≥ 1 , σ j belong s to the deterministic conte xt free langua ge L ( q n j ,q n j +1 ,α 1 ( j ) ,α 1 ( j +1)) = { σ ∈ A ⋆ 1 | σ : (( q n j , α 1 ( j )) 7→ ⋆ A 1 ( q n j +1 , α 1 ( j ) .α 1 ( j + 1)) } W e can see that the inte gers n j were defined in a unique way . Howe ver there may exist sev eral decom- positi ons of the ω -word α into words of langua ges L ( q , q ′ ,a,b ) . In order to ensure a unique decompositio n we are going to slightly m odify the definition of these lan- guage s. For eac h q , q ′ ∈ Q 1 and a, b ∈ Γ 1 , the languag e U ( q , q ′ ,a,b ) is the set of word s σ ∈ A ⋆ 1 such that: Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 11 (a) σ : ( q , a ) 7→ ⋆ A 1 ( q ′ , a.b ) and (b) If for some σ ′ ⊏ σ and s ∈ Q , σ ′ : ( q , a ) 7→ ⋆ A 1 ( s, a.b ) then there is a word u ∈ A ⋆ 1 and a state t ∈ Q , such that σ ′ .u ⊑ σ and u : ( s, a.b ) 7→ ⋆ A 1 ( t, a ) . (c) If there is a run ( q i , γ i ) 1 ≤ i ≤ p of A 1 on σ such that ( q 1 , γ 1 ) = ( q , a ) and ( q p , γ p ) = ( s, a.b ) for some s ∈ Q , s 6 = q ′ , then either there is an integer p ′ < p such that ( q i , γ i ) 1 ≤ i ≤ p ′ is a run of A 1 on σ and ( q p ′ , γ p ′ ) = ( q ′ , a.b ) or it holds that λ : ( s, a.b ) 7→ ⋆ A 1 ( s ′ , a ) for some s ′ ∈ Q and λ : ( s ′ , a ) 7→ ⋆ A 1 ( q ′ , a.b ) . It is easy to see that the languag es U ( q , q ′ ,a,b ) are also in the class D C F L and that, for each q , q ′ ∈ Q 1 and a, b ∈ Γ 1 , it holds that U ( q , q ′ ,a,b ) ⊆ L ( q , q ′ ,a,b ) . W e can see that, in the ab ove decompositi on α = σ 1 .σ 2 . . . σ n . . . of the ω -word α , for all inte gers j ≥ 1 , the wor d σ j belong s in fact to the deterministi c conte xt free langua ge U ( q n j ,q n j +1 ,α 1 ( j ) ,α 1 ( j +1)) . The rest of the proof of Lemma 3.1 can be pursued, replacing language s L ( q , q ′ ,a,b ) by languages U ( q , q ′ ,a,b ) . But no w we ha ve a unique decompositio n of α in the for m α = σ ′ 1 .σ ′ 2 . . . σ ′ n . . . where for all intege rs j ≥ 1 , the word σ ′ j belong s to some language U ( s j ,t j ,a j ,b j ) satisfy ing: (1) s 1 = q 1 0 , a 1 = ⊥ 1 , (2) for all integ ers j ≥ 1 , t j = s j +1 and b j = a j +1 . This unique decompositio n is crucial in the constru ction of the non ambiguous B ¨ uchi PD A A accepting L ( A 1 ⊲ A 2 ) . W e shall exp lain informal ly the behavi our of this automaton. For each q , q ′ ∈ Q 1 and a, b ∈ Γ 1 , the langu age U ( q , q ′ ,a,b ) is accepted by a determin istic pushd own automato n B ( q , q ′ ,a,b ) whose stack alphab et is denoted Γ ( q , q ′ ,a,b ) . W e can assume that all these alphabets are disjoi nt and that they are also disjoint from Γ 1 , the stack alphabet of A 1 . The stack alphab et of A will be Γ A = Γ 1 ∪ [ q ,q ′ ∈ Q 1 and a,b ∈ Γ 1 Γ ( q , q ′ ,a,b ) When reading an ω -word α ∈ L ( A 1 ⊲ A 2 ) the pushd own aut omaton A will guess, using the non determin ism, the uniqu e dec omposition of α in th e form α = σ ′ 1 .σ ′ 2 . . . σ ′ n . . . where for all intege rs j ≥ 1 , the word σ ′ j belong s to some language U ( s j ,t j ,a j ,b j ) satisfy ing: (1) s 1 = q 1 0 , a 1 = ⊥ 1 , (2) for all integ ers j ≥ 1 , t j = s j +1 and b j = a j +1 . In addit ion A will simu late the reading of the ω -word α 1 = a 1 a 2 a 3 . . . by the PD A A 2 . During a run of A the stack content is alway s a word in the form ⊥ .u.v where ⊥ is the bottom symbol of A , u ∈ (Γ 1 − {⊥} ) ⋆ and v is in (Γ ( q , q ′ ,a,b ) ) ⋆ for some q , q ′ ∈ Q 1 and a, b ∈ Γ 1 . After hav ing read the initial seg ment σ ′ 1 .σ ′ 2 . . . σ ′ j of α , the content of the stack of A is equal to the conten t of the stack of A 2 after ha ving read a 1 a 2 . . . a j . 12 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games Then A guesses that the nex t word in the decompositi on of α belongs to some U ( s j +1 ,t j +1 ,a j +1 ,b j +1 ) . It uses the stack alphabet Γ ( s j +1 ,t j +1 ,a j +1 ,b j +1 ) on the top of the stack to simulate the reading of σ ′ j +1 by B ( s j +1 ,t j +1 ,a j +1 ,b j +1 ) . Then w hen it has guessed that it has completel y read the word σ ′ j +1 , it erases letters of Γ ( s j +1 ,t j +1 ,a j +1 ,b j +1 ) from the stack , and simulates the reading of the letter a j +1 by A 2 , and so on. A B ¨ uchi acce ptance condition is then used to simulate the acceptanc e of α 1 by A 2 . The B ¨ uchi PD A ( A 2 , F ) is non ambiguou s and the abov e cited decompositi on of α is unique so there is a uniqu e accepting run of the B ¨ uchi PD A A on α . Finally we ha ve prov ed that L ( A 1 ⊲ A 2 ) ∈ N A − C F L ω .  Pro position 3.3. Let n be an inte ger ≥ 1 , A 1 , A 2 , . . . A n , be some deterministi c pushdo w n automata and ( A n +1 , C ) be a pushdo wn automaton equipped with a B ¨ uchi acceptance con dition. The inp ut alphab et of A 1 is denoted A and we assume that, for each inte ger i ∈ [1 , n ] , the input alphabet of A i +1 is the stack alphab et of A i . Then L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) ∈ C F L ω . Moreov er if L ( A n +1 , C ) is non ambiguous then L ( A 1 ⊲ . . . ⊲ A n ⊲ A n +1 ) ∈ N A − C F L ω . Pro of. W e reason by inductio n on the inte ger n . For n = 1 the result is stated in the abov e Lemmas 3.1 and 3.2. Assume no w that the result is true for some integer n ≥ 1 . Let A 1 , A 2 , . . . A n , A n +1 , be some deterministi c pushdo wn automata and ( A n +2 , C ) be a pushdo wn automato n equipped w ith a B ¨ uchi acceptan ce condition such that the language L ( A 1 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) ⊆ A ω is well defined . By induction h ypothesis the language L ( A 2 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) is a context free ω -language accep ted by a B ¨ uchi pushd own automaton ( A , F ) . But by definition of the langu age L ( A 1 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) it holds that L ( A 1 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) = L ( A 1 ⊲ A ) thus Lemma 3.1 implies that L ( A 1 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) ∈ C F L ω . Assume no w that L ( A n +1 , C ) is non ambiguous . R easoni ng as abov e bu t applying Lemma 3.2 instead of Lemma 3.1 we infer that L ( A 1 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) is in N A − C F L ω .  In parti cular , P roposi tion 3.3 implies the follo wing result. Cor ollary 3.4. For ea ch integ er n ≥ 0 , the follo wing inclusi ons hold: C n ( A ) ⊆ C λ n ( A ) ⊆ N A − C F L ω W e shall later get a stronger result (see Corollar y 3.8) from the study of closure prope rties of classes C n ( A ) , C λ n ( A ) . Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 13 3.2. Closure p roperties of classes C n ( A ) , C λ n ( A ) W e first state the follo wing lemma. Lemma 3.5. The class C λ 1 ( A ) is closed under complementat ion. Pro of. L et A 1 = ( Q 1 , Γ 1 , A 1 , ⊥ 1 , q 1 0 , δ 1 ) be a deterministi c push down automaton and ( A 2 = ( Q 2 , Γ 2 , Γ 1 , ⊥ 2 , q 2 0 , δ 2 ) , col 2 ) be a d eterministic pushdo w n automaton equipped with a parity acceptance condition. Recall that an ω -word α ∈ A ω 1 is in L ( A 1 ⊲ A 2 ) if f: w hen A 1 reads α , the stack of A 1 is strictly unbou nded and the sequence of stack contents has an infinite limit α 1 ∈ L ( A 2 , col 2 ) . Thus an ω -word α ∈ A ω 1 is in the complement of L ( A 1 ⊲ A 2 ) iff one of the two followin g conditi ons holds: (1) When A 1 reads α , the stack of A 1 is str ictly unbound ed and the limit α 1 of st ack contents is in the complemen t of L ( A 2 , col 2 ) . (2) When A 1 reads α , the stack of A 1 is not strict ly unbound ed . The clas s D C F L ω is closed unde r complement ation thus the comple ment of L ( A 2 , col 2 ) is equa l to L ( A 3 , col 3 ) , for some determini stic pushdo wn automaton A 3 equipp ed with a parity accepta nce condi- tion. The language L ( A 1 ⊲ A 3 ) is the set of ω -words α ∈ A ω 1 such that, when A 1 reads α , the stack of A 1 is strictly unbounde d and the limit α 1 of stack contents is in L ( A 3 , col 3 ) . So we see that, in order to get the complemen t of L ( A 1 ⊲ A 2 ) we hav e to add to L ( A 1 ⊲ A 3 ) the set B of all ω -word s α ∈ A ω 1 such that, when A 1 reads α , the stack of A 1 is not strict ly unbound ed . T o do this we are going first to modify the automaton A 1 in such a way that, when readin g ω -words in B , the stac k will be strictly unbound ed . W e no w e xplain informally the beha viour of t he ne w p ushdo wn automaton A ′ 1 . The s tack alphab et of A ′ 1 is Γ 1 ∪ Γ ′ 1 , where Γ ′ 1 = { γ ′ | γ ∈ Γ 1 } is ju st a copy of Γ 1 , such that Γ 1 ∩ Γ ′ 1 = ∅ . The essential idea is that A ′ 1 will simulate A 1 b ut it has the additional followin g beha viour . Using λ - transit ions it pushes in the stack letters of Γ ′ 1 , always keepi ng the information about the content of the stack of A 1 . More p recisely , if at some st ep while reading an ω -word α ∈ A ω 1 by A 1 the stac k content is a finit e word γ = γ 1 , γ 2 , . . . γ j , where each γ i is a letter of Γ 1 , then the corresp onding stack content of A ′ 1 will be in the form γ 1 .γ ′ n 1 1 γ 2 .γ ′ n 2 2 . . . γ j .γ ′ n j j , where n 1 , n 2 , . . . , n j , are positi ve intege rs. If when A 1 reads α the stack is strictly unbounded and the limit of the stack conten ts is an ω -word α 1 , then w hen A ′ 1 reads the same word α its stack w ill be also strictly unbou nded and the limit of the stack conten ts will be an ω -word α ′ 1 . Moreov er it will hold that ( α ′ 1 / Γ ′ 1 ) = α 1 , where ( α ′ 1 / Γ ′ 1 ) is the word α ′ 1 from which are remov ed all letters in Γ ′ 1 . On the other hand if when A 1 reads α the stack is not strictly unbounded , the limit of the stack contents being a finite word α 1 , then when A ′ 1 reads the same word α its stack will be strictly unbound ed and its limit will be an ω -word α ′ 1 such that ( α ′ 1 / Γ ′ 1 ) = α 1 . 14 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games Notice that the stack conte nt of A ′ 1 will alway s be in the form ⊥ 1 . ( ⊥ ′ 1 ) ⋆ or u.Z. ( Z ′ ) n for some u ∈ ⊥ 1 . (Γ 1 ∪ Γ ′ 1 ) ⋆ , Z ∈ Γ 1 , Z ′ being the copy of Z in Γ ′ 1 , and n ≥ 0 being an integ er . The beha viour of the determin istic pushdo wn automaton A ′ 1 , readi ng an ω -word, will be the s ame as the beha viour of A 1 b ut with the follo wing modifications . (a) Between an y two transitio ns of A 1 is ad ded a λ -tran sition of A ′ 1 which s imply pushes in the stack, when the topmost stack letter of A ′ 1 is Z ∈ Γ 1 or Z ′ ∈ Γ ′ 1 , an additi onal letter Z ′ . (b) Assume now that at some step of the readi ng of α by A ′ 1 and A 1 , and after the ex ecution of a λ -transi tion as explain ed in abov e item ( a ) , the topmost stack letter of A ′ 1 is some letter Z ′ ∈ Γ ′ 1 . Recall that the stack content of A ′ 1 will be in the form ⊥ 1 . ( ⊥ ′ 1 ) n (if Z ′ = ⊥ ′ 1 ) or u.Z . ( Z ′ ) n for some u ∈ ⊥ 1 . (Γ 1 ∪ Γ ′ 1 ) ⋆ , Z ∈ Γ 1 , Z ′ being the cop y of Z in Γ ′ 1 , and n ≥ 1 . Notice that the corres ponding stack conten t of A 1 will be ⊥ 1 or ( u/ Γ ′ 1 ) .Z . Suppose no w that A 1 reads a letter a ∈ A 1 or ex ecutes a λ -transit ion. If it pushes letter T in the stack then A ′ 1 would pus h the same letter T in its stack. If A 1 would ski p (its topmost stack letter being Z ), th en A ′ 1 also skips. But if A 1 , reading the letter a ∈ A 1 or ex ecuting a λ -transition , the topmost stack letter being Z , would pop the letter Z , then A ′ 1 pops the whole segment Z. ( Z ′ ) n at the top of the stack, using λ -transi tions. Notice that we do not detail here the set of states of A ′ 1 . It contains the set of states Q 1 of A 1 and is suf ficiently enriched, to achie ve the goal of simulatin g the behavi our of A 1 , addin g the modification s cited abo ve. Assume now that when A 1 reads α its stack is strictly unboun ded and the limit of the stack contents is an ω -word α 1 . Then when A ′ 1 reads th e same word α its stack is also strictly unbounded and the l imit of the stack conte nts will be an ω -word α ′ 1 such that ( α ′ 1 / Γ ′ 1 ) = α 1 . On the other hand if when A 1 reads α the stack is not strictl y unboun ded , then the limit of its stack conten ts is a finite word α 1 = α 1 (1) .α 1 (2) . . . α 1 ( | α 1 | ) . In that case when A ′ 1 reads the same word α its stack will be strictly unbounded and its limit will be an ω -word α ′ 1 in the form α ′ 1 = α 1 (1) . ( α 1 (1) ′ ) n 1 .α 1 (2) . ( α 1 (2) ′ ) n 2 . . . ( α 1 ( | α 1 | − 1) ′ ) n | α 1 |− 1 . ( α 1 ( | α 1 | ) . ( α 1 ( | α 1 | ) ′ ) ω for some inte gers n 1 , n 2 , . . . , n | α 1 |− 1 . In particula r it will hold that ( α ′ 1 / Γ ′ 1 ) = α 1 . It is now easy to modif y the pushdo wn auto maton A 3 in such a way that we obtain a determini stic pushd own automato n A ′ 3 equipp ed with parity acceptanc e conditio n col ′ 3 such that the input alphab et of A ′ 3 is Γ 1 ∪ Γ ′ 1 , an d an ω -word α ′ 1 ∈ (Γ 1 ∪ Γ ′ 1 ) ω is in L ( A ′ 3 , col ′ 3 ) iff [ ( α ′ 1 / Γ ′ 1 ) is a finite w ord o r ( α ′ 1 / Γ ′ 1 ) is infinite and is in L ( A 3 , col 3 ) ]. Thus the ω -languag e L ( A ′ 1 ⊲ A ′ 3 ) is the complement of L ( A 1 ⊲ A 2 ) and this ends the proof.  Pro position 3.6. For ea ch integer n ≥ 0 , the class C λ n ( A ) is closed under complementat ion. Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 15 Pro of. W e no w reason by induction on the inte ger n ≥ 0 . For n = 0 , C λ 0 ( A ) = D C F L ω is kno wn to be closed under complementatio n [21]. For n = 1 , C λ 1 ( A ) is closed under complementation by Lemma 3.5. Assume now that w e hav e proved that for ev ery positi ve inte ger k ≤ n the class C λ k ( A ) is closed under complemen tation. Let A 1 , A 2 , . . . A n , A n +1 , be some determinis tic pushdo wn automata a nd ( A n +2 , col ) be a d eterministic pushd own automaton equipped with a parity accepta nce condition such that the language L ( A 1 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) ⊆ A ω 1 is well defined. An ω -word α ∈ A ω 1 is in the complement of L ( A 1 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) iff one the two follo wing condit ions holds: (1) When A 1 reads α , the stack of A 1 is str ictly unbound ed and the limit α 1 of st ack contents is in the complemen t of L ( A 2 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) (2) When A 1 reads α , the stack of A 1 is not strict ly unbound ed . By induct ion hypothesis the complement of the ω -langu age L ( A 2 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) is in C λ n ( A ) so it is in the form L ( A ′ 2 ⊲ . . . ⊲ A ′ n +1 ⊲ A ′ n +2 ) . W e can do similar m odificatio ns as in the case n = 1 , replac ing A 1 , whose stac k alphabet is Γ 1 , by anothe r deterministi c pushd own automaton A ′ 1 , whose alphabe t is Γ 1 ∪ Γ ′ 1 where Γ ′ 1 is a cop y of Γ 1 . If w hen A 1 reads α the limit of its stack content s is a finite or infinite word α 1 then when A ′ 1 reads the same word α the limit of its stack content s is an ω -word α ′ 1 such that ( α ′ 1 / Γ ′ 1 ) = α 1 . It is no w easy to modify the l anguage L ( A ′ 2 ⊲ . . . ⊲ A ′ n +1 ⊲ A ′ n +2 ) in s uch a w ay that we get a language L ( A ′′ 2 ⊲ . . . ⊲ A ′′ n +1 ⊲ A ′′ n +2 ) of ω -words ove r Γ 1 ∪ Γ ′ 1 contai ning an ω -word α ′ 1 if and only if: either ( α ′ 1 / Γ ′ 1 ) is a fi nite wor d or ( α ′ 1 / Γ ′ 1 ) belongs to the ω -langua ge L ( A ′ 2 ⊲ . . . ⊲ A ′ n +1 ⊲ A ′ n +2 ) . Thus it hold s that L ( A ′ 1 ⊲ A ′′ 2 ⊲ . . . ⊲ A ′′ n +1 ⊲ A ′′ n +2 ) is the co mplement of L ( A 1 ⊲ . . . ⊲ A n +1 ⊲ A n +2 ) .  Remark 3.7. In [19, 20] Serre defined winn ing c onditions Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 for pushdo wn games using langua ges in classes C n ( A ) . He then showed that these winning conditions lead to decision procedur es to decide the winner in pushdo wn games. The questio n no w natural ly arises whether the proofs can be ext ended to winning conditions defined in the same way from classes C λ n ( A ) . Then the closure under complemen tation of these classes would be rele v ant from a game point of vie w . On the other hand this closur e property provid es also some more information about classes C n ( A ) , gi ven by nex t corolla ry , which is alread y important from a game point of vie w . Cor ollary 3.8. For each inte ger n ≥ 0 , the follo wing inclus ions hold: C n ( A ) ⊆ C λ n ( A ) ⊆ N A − C F L ω \ C o − N A − C F L ω 16 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games Pro of. It follo ws directly from Corollary 3.4 and Proposition 3.6.  W e now pro ve that the classes C n ( A ) , C λ n ( A ) , are not closed under other boolean operati ons. Pro position 3.9. For each intege r n ≥ 0 , the classes C n ( A ) and C λ n ( A ) are neither closed under union nor under intersec tion. Pro of. Notice first that for each integer n ≥ 0 , C n ( A ) ⊆ C n +1 ( A ) and C λ n ( A ) ⊆ C λ n +1 ( A ) . The ω -langua ges L 1 = { a n .b m .c p .d ω | n, m, p ≥ 1 and n = m } and L 2 = { a n .b m .c p .d ω | n, m, p ≥ 1 and m = p } , over the alphab et A = { a, b, c, d } , are in D C F L ω and they are in all classes C n ( A ) and C λ n ( A ) . But their intersec tion is L 1 ∩ L 2 = { a n .b n .c n .d ω | n ≥ 1 } . This ω -language is not conte xt free becaus e the finitary language { a n .b n .c n | n ≥ 1 } is not contex t free [1] and an ω -langua ge in the form L.d ω , with L ⊆ { a, b, c } ⋆ , is context free iff the fi nitary language L is contex t free [7]. Thus L 1 ∩ L 2 canno t be in any class C n ( A ) and C λ n ( A ) because these classes are included in C F L ω . On the othe r hand cons ider the ω -language s L 3 = { a n .b m .c p .d ω | n, m , p ≥ 1 an d n 6 = m } and L 4 = { a n .b m .c p .d ω | n, m, p ≥ 1 and m 6 = p } . T hese ω -langua ges are in D C F L ω and in e very class C n ( A ) or C λ n ( A ) . If the language L 3 ∪ L 4 was in some class C n ( A ) or C λ n ( A ) , then by P roposi tion 3.6 its co mplement L 5 would be als o in C λ n ( A ) and i t wou ld be a conte xt free ω -langua ge. This would imp ly that L 5 ∩ a + .b + .c + .d ω is conte xt free because the class C F L ω is closed under intersect ion with regula r ω -language s. But L 5 ∩ a + .b + .c + .d ω = { a n .b n .c n .d ω | n ≥ 1 } is not contex t free thus for each inte ger n ≥ 0 , the classes C n ( A ) , C λ n ( A ) are not closed under unio n. Notice that the union ∪ n ≥ 0 C λ n ( A ) is also neither closed under inter section nor under union.  4. Win ning set s in a pushd own g ame Recall that it is prov ed in [19] that ev ery determinist ic context free languag e may occur as a winning set for Eve in a pushdo wn game equipp ed with a winning condition in the form Ω B , where B is a determin- istic pushd own automaton. Serre asked also whether there exist s a pushdo wn game equippe d with a winning conditio n in the form Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 such that the set of winning positio ns for E ve is not a determinis tic conte xt free lan- guage . W e are going to pro ve in this section that such pushdo wn games exis t, givi ng exampl es of winning sets which are non-d eterministic non -ambiguous conte xt free languag es, or inheren tly ambigu ous context free langua ges, or ev en non contex t free languages . The exact form of the winning sets remains open. Serre conjectured in [18] that one could pro ve that, for n ≥ 0 , the winning sets for Eve in pushdo wn games equipp ed with a w inning conditio n in the form Ω A 1 ⊲ ... ⊲ A n ⊲ A n +1 , form a class of languages at lev el n , and that for n = 0 the winning sets could be determin istic contex t free language s. So we think that, in order to better understa nd what is the exact form of the w inning sets, it is useful to see dif ferent examples of winning sets of dif ferent comple xities, and not only of the greate st complexity we ha ve got, i.e. a non context free langua ge. Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 17 Moreo ver the techniq ues, in volvi ng Duparc’ s eraser operator , used to prov e Proposit ion 4.3 belo w , are interes ting by their own an d are useful to unders tand how the games go on. In order to present the first example we begin by recalling the oper ation x → x և which has bee n introd uced by Duparc in his study of the W adge hi erarchy [9], where it works als o on infinite w ords, and is also cons idered by Serre in [19]. For a finite word u ∈ (Σ ∪ { և } ) ⋆ , wher e Σ is a finite alph abet, the finite wor d u և is in ducti vely defined by: λ և = λ , and for a finite word u ∈ (Σ ∪ { և } ) ⋆ : ( u.c ) և = u և .c , if c ∈ Σ , ( u. և ) և = u և with its last letter remov ed if | u և | > 0 , i.e. ( u. և ) և = u և (1) .u և (2) . . . u և ( | u և | − 1) if | u և | > 0 , ( u. և ) և = λ if | u և | = 0 , Notice that for x ∈ (Σ ∪ { և } ) ⋆ , x և denote s the s tring x , on ce e very և occurin g in x , used as an e raser , has been “e valu ated” to the back space operation, proceeding from left to right inside x . In other words x և = x from w hich e very interv al of the form “ c և ” ( c ∈ Σ ) is remov ed. For a lang uage V ⊆ Σ ⋆ we set V ∼ = { x ∈ (Σ ∪ { և } ) ⋆ | x և ∈ V } . Lemma 4.1. Let L = { a n .b n | n ≥ 1 } . Then L ∼ is a non ambiguous contex t free language which can not be accep ted by any deter ministic pushdown automaton . Pro of. Let L be the co ntext free lang uage { a n .b n | n ≥ 1 } . The la nguage L is a d eterministic, hence non ambiguo us, context free language. Thus by T heorem 6.16 of [10] the language L ∼ is a non ambiguous conte xt free langua ge. It remains to sho w that L ∼ can not be accep ted by any deter ministic pushdown automaton . The idea of the proof is essentially the same as in the proof that the contex t free language { a n .b n | n ≥ 1 } ∪ { a n .b 2 n | n ≥ 1 } can not be accepted by any deterministic pushdown automaton . It can be found in [1, Proof of Propos ition 5.3] or in [12, Exercise 6.4.4 page 251]. T oward s a contradictio n assume that the languag e L ∼ is accepted by a deterministi c push down automaton A . All words a n .b n , for n ≥ 1 , are in the language L ∼ . T hen one could sho w that there exis ts a pair ( n, k ) , w ith n, k > 0 , such that the accepting configuration s of A reading a n .b n or a n + k .b n + k are the same. Consider now the word a n .b n . և 2 n .a.b . It belongs to L ∼ and the valid computation of A readin g a n .b n should be the begi nning of the vali d computat ion of A readi ng a n .b n . և 2 n .a.b . Thus the pushd own automaton A wou ld also accepts a n + k .b n + k . և 2 n .a.b which is clearly not in L ∼ .  Lemma 4.2. Let L ⊆ Σ ⋆ be a deterministic context free languag e. T hen there exists a pushdo wn proce ss P = ( Q, Γ , ⊥ , δ ) , a partition Q = Q E ∪ Q A , two determinis tic pushdo w n automata A 1 , A 2 , and a state q ∈ Q such that, in the induced pushdo w n game equipped with the winning condition Ω A 1 ⊲ A 2 , one has { u | ( q , u ) ∈ W E } = L ∼ . 18 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games Pro of. Let P = ( { p, q } , Γ = Σ ∪ {⊥ , և , # } , ⊥ , δ ) be a pushdo wn proces s where δ is defined by: push ( p, #) ∈ δ ( q , c ) for all letters c ∈ Σ ∪ {⊥ , և } and push ( p, #) ∈ δ ( p, #) . So the pushdo wn proces s P is deterministic and its beha viour is very similar to the behav iour of the pushd own process gi ven in the proof of P roposit ion 42 of [20 ]. It can only push the letter # on the top of a gi ven configurat ion. Q = Q E ∪ Q A is any pa rtition of Q . For each configurati on ( q , u.c ) , for c ∈ Σ ∪ {⊥ , և } and u ∈ Γ ⋆ , there is a unique infinite play starting from ( q , u.c ) , during w hich the pushdo wn stack of P is strictly unbounded , and the limit of the stack conten ts is u.c. # ω . The deterministic pushdo wn automaton A 1 reads words ov er the alphabet Γ = Σ ∪ {⊥ , և , # } and its stack alphabe t is Γ 1 = Σ ∪ {⊥ 1 } . Its beha viour is described as follo ws: Consider first the readin g of an ω -word in the fo rm ⊥ .u. # ω , where u ∈ (Σ ∪ { և } ) ⋆ . After hav ing read the botto m symbol ⊥ , the content of its stack is still ⊥ 1 . Then w hen the pushdo w n automato n A 1 reads a letter c ∈ Σ it pushes the same letter in the stack. But if A 1 reads the symbol և and the topmost stack symbol is not ⊥ 1 (so it is in Σ ) then it pops the letter at the top of its stack. So, after having read the initial segment ⊥ .u of ⊥ .u. # ω , the stack content of A 1 is ⊥ 1 .u և . Next the PD A A 1 pushe s a lette r # in the stack for each lette r # read. Thus, when A 1 reads the ω -word ⊥ .u. # ω , its stack is strictly unbou nded and the limit of the stack contents is ⊥ 1 .u և . # ω . In addit ion, it is easy to ensure that, when A 1 reads an ω -word which is not in ⊥ . (Σ ∪ { և } ) ⋆ . # ω ∪ ⊥ . (Σ ∪ { և } ) ω , then its stack is not strictly unboun ded . If there is a letter ⊥ after the fi rst letter of the word or if A 1 reads a letter in Σ ∪ { և } aft er some letter # , then the stack content remains undefinitel y uncha nged. On the other hand, A 2 is a deterministic pushd own automato n equipped w ith a parity acceptance condi- tion which accept s the ω -langu age ⊥ 1 .L. # ω . Consider now a gi ven configurat ion ( q , ⊥ .u ) of the pushdo wn process P for some u ∈ (Σ ∪ { և , # } ) ⋆ , the last letter of u being n ot # . There is a u nique infinite play s tarting from this po sition. The sta ck of P is strict ly unbounded during this play and the limit of stack contents is ⊥ .u. # ω . When A 1 reads the ω -word ⊥ .u. # ω its stack is strictly un bounded if f u ∈ (Σ ∪ { և } ) ⋆ and t hen th e limit of stack conten ts is ⊥ 1 .u և . # ω . The ω -word ⊥ 1 .u և . # ω is accepted by A 2 if f u և ∈ L . Thus the configuration ( q , ⊥ .u ) is a winning position for Eve in the induced pushdo wn game, equipp ed with the winning condit ion Ω A 1 ⊲ A 2 , if and only if u ∈ L ∼ .  W e can now state the follo w ing result which follo ws directly from Lemmas 4.1 and 4.2. Pro position 4.3. There exists a pushdo wn process P = ( Q, Γ , ⊥ , δ ) , a partitio n Q = Q E ∪ Q A , two determin istic pushdo w n automata A 1 , A 2 , and a state q ∈ Q such that, in the induced pushdo w n game equipp ed w ith the w inning condition Ω A 1 ⊲ A 2 , the set { u | ( q , u ) ∈ W E } is a non-de terministic non ambiguo us context free languag e. Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 19 Pro of. L et L be the language { a n .b n | n ≥ 1 } . The language L is a determini stic contex t free language, thus by L emma 4.2 there exists a pushdo wn proces s P = ( Q, Γ , ⊥ , δ ) , a partitio n Q = Q E ∪ Q A , two determin istic pushdo w n automata A 1 , A 2 , and a state q ∈ Q such that, in the induced pushdo wn game equipp ed with the winning conditio n Ω A 1 ⊲ A 2 , one has { u | ( q , u ) ∈ W E } = L ∼ . But by L emma 4.1 L ∼ is a no n ambiguous conte xt free lang uage which can not b e accepted by any det erministic pushdown automato n .  Remark 4.4. In the pushdo wn game gi ven in the proof of Lemma 4.2, there are some plays which are not infinit e. Howe ver it i s easy to find a push down game with th e same winning set for Eve but in which all plays are infinite. The same remark will hold for pushdo wn games giv en in the proo fs of the two follo wing propositio ns. W e are no w going to sho w that the set o f winning positio ns for Eve can a lso be an inhe rently ambiguou s conte xt free language. Recall that it is well kno wn that the languag e V = { a n .b m .c p | n, m, p ≥ 1 and ( n = m or m = p ) } is an inhere ntly ambiguous context free langua ge, [1, 12]. Pro position 4.5. T here exists a pushdo w n process P = ( Q, Γ , ⊥ , δ ) , a partition Q = Q E ∪ Q A , two determin istic pushdo w n automata A 1 , A 2 , and a state q ∈ Q such that, in the induced pushdo wn game equipp ed with the winning condit ion Ω A 1 ⊲ A 2 , the set { u | ( q , u ) ∈ W E } is an inherentl y ambiguou s conte xt free langua ge. Pro of. Let P = ( { q , q ′ , q ′′ , p } , Γ = {⊥ , a, b, c, # } , ⊥ , δ ) be a pushdo wn proces s where δ is defined by: { pop ( q ′ ) , sk ip ( q ′′ ) } ⊆ δ ( q , c ) , p op ( q ′ ) ∈ δ ( q ′ , c ) , p ush ( p, #) ∈ δ ( q ′ , b ) , push ( p, #) ∈ δ ( q ′′ , c ) , and push ( p, #) ∈ δ ( p, #) . W e set Q E = { q } and Q A = { q ′ , q ′′ , p } . Consider now an infinite play from a gi ven configuration ( q , ⊥ .u ) , for u ∈ { a, b, c, # } ⋆ . The topmost stack lette r of this initial configuratio n must be a letter c . T hen at most two c ases m ay happ en. 1. In the first one are push ed infinitely many l etters # on the top of the stac k. In this play the stack is strictl y unbounde d and the limit of the stack content s is ⊥ .u. # ω . 2. In the second case the letter c is popped and all next letters c are popp ed from the top of the stack until some letter b is on the top of th e s tack. F rom this moment infinitely many letters # are pushe d in the stack. Then the stack is strictly unboun ded and the limit of the stack contents is ⊥ .u ′ .b. # ω if u = u ′ .b.c k for some inte ger k > 0 . Notice that this second case can only occur if u is in the form u = u ′ .b.c k for some integ er k > 0 . The deterministic pushdo wn automaton A 1 reads words over the alphabe t {⊥ , a, b, c, # } and its stack alphab et is Γ 1 = {⊥ 1 , a, b, # } . It is easy to ensure that the stack of A 1 is not strictly unboun ded during the readin g of an ω -word which is not in W = ⊥ .a ω ∪ ⊥ .a + .b ω ∪ ⊥ .a + .b + . # ω ∪ ⊥ .a + .b + .c ω ∪ ⊥ .a + .b + .c + . # ω . Consider now the reading by A 1 of an ω -word which is in W . A fter ha ving read the bottom symbol ⊥ , the stack content of A 1 is still ⊥ 1 . Then it pushes a letter a or b each time it reads the correspond ing letter a or b . Then when A 1 reads an ω -word in the form ⊥ .a ω (respe ctiv ely , ⊥ .a n .b ω for n ≥ 1 ) then its stack is strictl y unbounde d and the limit of stack content s is ⊥ 1 .a ω (respe ctiv ely , ⊥ 1 .a n .b ω ). 20 Olivier Fin kel / On W inning Conditions of High Bor el Complexity in Pushdown Games If no w A 1 reads letters # then it pushes them in the stack. In this case the inpu t word is in the form ⊥ .a n .b m . # ω , and the limit of stack conten ts of A 1 readin g this ω -word is ⊥ 1 .a n .b m . # ω . If A 1 reads some letters c after an initial seg ment in the form ⊥ .a n .b m then it pops a letter b for each letter c read. If the number of c is equal to the number of b of the input word, then after havin g read the segment ⊥ .a n .b m .c m of the input word the stack content of A 1 is simply ⊥ 1 .a n . Nex t A 1 reads the final seg - ment # ω and it pushes it in the stack. S o the limit of stack cont ents of A 1 readin g the input ω -word ⊥ .a n .b m .c m . # ω is in the form ⊥ 1 .a n . # ω If the number of c is not equal to the number of b of the input word (the numbe r of c being finite or infinite), then, once this has been check ed, the stack content remains unchanged so the stack will not be strictl y unbounde d. One can define a determin istic push down automaton A 2 , equipped w ith a parity accep tance condition , which accept s the ω -langu age {⊥ 1 .a n .b n . # ω | n ≥ 1 } ∪ {⊥ 1 .a n . # ω | n ≥ 1 } . W e are now going to determine the winning positio ns ( q , ⊥ .u ) of Eve in the induce d pushdo w n game equipp ed with the winning condition Ω A 1 ⊲ A 2 . Let ( q , ⊥ .u ) be a gi ven configuration of the pushdo wn proce ss P for some u ∈ { a, b, c, # } ⋆ , the last letter of u being c . T here are one or two infinite plays starting from this position. When there are two such pl ays, they de pend on the first choice of Eve and the position ( q , ⊥ .u ) is a win ning position for Ev e if f one of the two possible infinite plays is winning for her . In the first play the stack is strict ly unbounded and the limit of the stack contents is ⊥ .u. # ω . There is a second play if u = u ′ .b.c k for some integer k > 0 . Then in this play the stack is strictly unbou nded and the limit of the stack contents is ⊥ .u ′ .b. # ω . When A 1 reads the ω -word ⊥ .u. # ω , its stack is strictly unbounded if f u is in the form a n .b m .c m for some n, m ≥ 1 (the number of c and of b in u are equal). Then the limit of stack contents is ⊥ 1 .a n . # ω and it is in L ( A 2 ) . S o ⊥ .u. # ω ∈ L ( A 1 ⊲ A 2 ) . If u = u ′ .b.c k for some int eger k > 0 and A 1 reads the ω -word ⊥ .u ′ .b. # ω then the stack of A 1 is strictly unbou nded iff u ′ is in the form a n .b m − 1 for some n, m ≥ 1 . In this case the limit of stack contents is ⊥ 1 .a n .b m . # ω and it is accepte d by A 2 if f n = m ≥ 1 . Thus the configuratio n ( q , ⊥ .u ) is a winning p osition for Ev e, wit h the winn ing co ndition Ω A 1 ⊲ A 2 , if and only if u is in the inheren tly ambiguo us conte xt free languag e V = { a n .b m .c p | n, m, p ≥ 1 and ( n = m or m = p ) } .  Pro position 4.6. There exists a pushdo wn process P = ( Q, Γ , ⊥ , δ ) , a partitio n Q = Q E ∪ Q A , two determin istic pushdo w n automata A 1 , A 2 , and a state q ∈ Q such that, in the induced pushdo w n game equipp ed with the winning cond ition Ω A 1 ⊲ A 2 , the set { u | ( q , u ) ∈ W E } is a non conte xt free language. Pro of. W e define the pushdo wn proces s P = ( Q, Γ , ⊥ , δ ) as in the proof of preceding Propositio n 4.5 exc ept that we set t his time Q A = { q } and Q E = { q ′ , q ′′ , p } . T he tw o deterministi c pushdo wn automata Olivier F inkel / On W inning Cond iti ons of High Bor el Complexity in Pushdown Games 21 A 1 , A 2 , are also defined in the same way . Consider no w a configuratio n in the form ( q , ⊥ .a n .b m .c p ) for some integ ers n, m, p ≥ 1 . T here are two infinite plays starting from this configurat ion b ut they depend this time on the first choice of the second player Adam . The positi on ( q , ⊥ .a n .b m .c p ) is w inning for Eve if f these two infinite plays are won by her . This implies that n = m and m = p . Thus it hold s that { u | ( q , u ) ∈ W E } ∩ ⊥ .a + .b + .c + = ⊥ . { a n .b n .c n | n ≥ 1 } This languag e is not contex t free because of the well kno wn non contex t freeness of the languag e { a n .b n .c n | n ≥ 1 } [1, 12]. This implies that the set { u | ( q , u ) ∈ W E } itself is not conte xt free. 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