Closure Properties of Locally Finite Omega Languages

CLOSURE PR OPER TIES OF LOCALL Y FINITE ω -LANGUA GES ⋆ Olivi er Fink el ∗ Equip e de L o gique Math´ ematique CNRS et Universit ´ e Paris 7, U.F.R. de M ath ´ ematiques 2 Plac e Jussieu 75251 Paris c e dex 05, F r anc e. Dedicated t o Denis Richard for his 60 th Birthday Abstract Lo cally finite ω -languages, defined via second order qu an tifications follo w ed by a first order lo cally finite sen tence, w ere in tro du ced b y Ressa yre in [Res88]. They enjo y very nice prop erties and e xtend ω -languages accepted b y fi n ite automata or defined b y monadic second order se n tences. W e study here closure prop erties of th e family LO C ω of lo cally fin ite omega languages. In particular w e sho w that the class LO C ω is n either closed u nder in tersection nor under complemen tation, giving an answ er to a question of Ressayre [Res89]. Key wor ds: F ormal languages; logical definability; infinite w ord s; lo cally finite languages; closure p rop erties. 1 In t r oduct ion In the sixties J.R. B ¨ uc hi w as the first to study ω -langua ges recognized b y finite auto mata in order t o pro ve the decidability of the monadic second order theory of one successor o v er the inte gers [B ¨ uc62]. In the course of his pro of he sho w ed that an ω -langua ge, i.e. a set of infi nite w ords o ver a finite alphab et, is accepted b y a finite automaton with the now called B ¨ uchi acceptance condition if and only if it is defined b y an (existen tia l) monadic second order sen tence. ⋆ P artially su pp orted b y Intas 00-447. ∗ Corresp onding auth or Email addr ess: finkel@ logique.juss ieu.fr (Olivier Fink el). Preprint submitted to Elsevier Algorithms hav e been found to giv e suc h an automaton from the monadic second order sen tence and con verse ly . Thus the ab o v e cited decision problem is reduced to the decidabilit y o f the emptiness problem f or B ¨ uchi automata whic h is easily sho wn to b e decidable. The equiv alence b etw een definability b y monadic second order sen tences and acceptance by finite a utomata holds also for lang uages of finite w or ds [B ¨ uc60], and has b een extended t o languages of words of length α , where α is a countable ordinal ≥ ω [BS73]. The researc h a rea, no w called “ desc riptiv e complexit y”, found its origin in the ab o v e cited work of B ¨ uc hi as w ell as in the fundamen tal result of F ag in who prov ed that the class NP is c haracterized b y existen tial second order form ula s, [F ag7 4]. Since then, a lot of w ork has b een ac hiev ed ab out the log ical definabilit y of classes of formal languages of finite or infinite w ords, or of relational structures lik e graphs, see [F ag93] [Pin96] [Tho96 ] [Imm99] for a surv ey ab out this field of researc h. Sev eral extensions of existen tial monadic second order logic o v er w or ds hav e b een studied. Lauteman, Sc h wen tic k and Therien pro v ed that con text free languages are c haracterized by existen tial second order formulas in whic h the second order quan tifiers b ear only on matc hings, i.e. pairing relations without crosso v er, [LST94]. P arigo t and Pelz , and mor e recen tly Y amasaki, extended monadic second order logic with tw o second or der relation sym b ols and c haracterized classes of P etri net ( ω )-languages [PP85] [Pe l87] [Y am99]. Eiter, Gottlob a nd Gurevic h studied the relationship b et ween monadic second order logic and syn ta ctic fragments of existen tia l second order logic o ver (fi- nite) words [EGG00]. Distinguishing prefix classes, they determined whic h of them define only regular languages a nd whic h o f them hav e the same expres- siv e p ow er as monadic second order logic. Another extension has b een in tro duced b y R ess a yre, in order to apply some p o we rful t o ols of mo del theory to the study of formal languages, [Res88 ]. He defined lo cally finite sen tences (firstly called lo cal). A lo cally finite sen tence ϕ is a first order sen tence whic h is equiv alent to a univ ersal o ne and whose mo dels satisfy simple structural prop erties: closure under functions tak es a finite num b er n ϕ of steps. These syntaxic and seman tic restrictions allow a meaningful use of the notion of indiscernables and lead to b eautiful stretc hing theorems connecting the existence of some w ell ordered infinite mo dels o f ϕ with the existence of some finite mo dels generated b y indisce rnables [FR9 6]. Lo cally finite languages are defined b y second order fo rm ulas in the fo rm 2 ∃ ¯ R ∃ ¯ f ϕ where ϕ is a lo cally finite sen tence and ¯ R (resp ectiv ely , ¯ f ) represen t the relatio n (resp ectiv ely , function) sym b ols in the signature of ϕ . These second order quan tifications are m uc h more general than the monadic ones as the follow ing results show: • Eac h regula r language is lo cally finite, [Res88], and man y contex t free as w ell as non context free languages are lo cally finite [Fin01]. • Eac h regular ω -language is a lo cally finite ω -language, [Fin01] [Fin89], and there exist many more lo cally finite ω -languages as w e shall see b elo w. • This result is extended t o lang uages of transfinite length w ords: if α is an ordinal < ω ω , each regular α -la nguage is also lo cally finite [Fin01]. But a pumping lemma, f ollo wing fr om a stretc hing theorem, ma k es lo cally finite ω -languages k eep imp ortan t prop erties of re gular ω -languag es, [Res88] [FR96]. It is an analogue for each lo cally finite ω -langua ge of the pr operty: “ A regular ω -language is non empt y if and only if it contains an ultimately p erio dic w o rd ”. This lemma implies in a similar manne r the decidabilit y of the em ptiness prob- lem f or lo cally finite ω -lang uages. Moreo ver fo r eac h coun table ordinal α < ω ω , the decidabilit y of the emptiness problem fo r lo cally finite α -la nguages follo ws from similar a rgumen ts, [FR 96]. Other decidability results, as the decidabilit y of the problem: “is a giv en finitary lo cally finite language infinite?” follow f rom stretc hing theorems of [Res88][FR96]. These inte resting prop erties of lo cally finite lang uages naturally lead to the question o f the ric hness of the class of locally finite languages: how large is this class? What are it s closure prop erties? The study of lo cally finite languages of finite w ords w as b egun by Ressa yre in [Res88] a nd con tinue d in [F in01]. W e fo cus in t his pap er o n the class LO C ω of lo cally finite ω - languages and study classical closure prop erties f or this class. In particular, w e sho w that LO C ω is neither closed under in tersection, nor under complemen tation. The pro of uses the notio n o f rationa l cone of finitary languages whic h is imp ortant in formal lang uage theory and the notion of indiscernables in a structure, often used in mo del theory . This giv es an answ er to a question of Ressay re, [Res89]. Of course w e w ould ha ve preferred a p ositiv e answ er to t his question whic h would hav e provided a useful class of sen tences for specification and v erification of prop erties of non-terminating systems. But this leav es still op en, for further study , the p ossibilit y to find suc h a useful class of sen tences as a sub class of t he class of lo cally finite sen tences. In section 2, we giv e the first definitions and some examples of lo cally finite 3 ω -langua ges. In section 3, closure prop erties for ω -langua ges are in ves tigated. W e sho w that the class LO C ω is not closed under in tersection with regu- lar ω -languages th us LO C ω is neither closed under intersec tion, nor under complemen tation (b ecause LO C ω is closed under union). Then w e prov e that LO C ω is closed under λ -fr ee morphism and λ - free substitution of lo cally finite (finitary) languages. 2 Definitions and examples 2.1 Definitions W e briefly indicate no w some basic facts ab out first order lo gic and mo del theory . See for example [CK73] for more bac kground on this sub ject. W e consider here formu las of first order log ic. The language of first o rder logic con tains (first order) v ariables x, y , z, . . . ranging ov er elemen ts of a structure, log ical symbols: the connectiv es ∧ (and), ∨ (or), → (implication), ¬ (negation) , and the quan tifiers ∀ (for all) , and ∃ (t here exists), and also the binary predicate sym b ol of iden tity =. A signature is a set of constan t, relation ( differen t f rom = ) and function sym b ols. w e shall consider here only finite signatures. Let S ig b e a finite signature. W e define firstly the set of terms in the signature S ig whic h is built inductiv ely as follows: (1) A v ariable is a term. (2) A constan t sym b ol is a term. (3) If F is a m-ary function sym b ol and t 1 , t 2 , . . . , t m are terms, then F ( t 1 , . . . , t m ) is a term. W e then define the set of atomic formulas which are in the form given b elo w: (1) If t 1 and t 2 are terms, then t 1 = t 2 is an atomic form ula. (2) If t 1 , t 2 , . . . , t m are terms and R is a m-ary r elation sym b ol, then R ( t 1 , . . . , t m ) is an atomic form ula. Finally the se t of form ulas is built inductive ly from atomic form ulas as f ollo ws: (1) An atomic form ula is a form ula. (2) If ϕ and ψ are formulas, then ϕ ∧ ψ , ϕ ∨ ψ , ϕ → ψ and ¬ ϕ are formulas. (3) If x is a v ariable and ϕ is a formu la, then ∀ xϕ and ∃ xϕ are f orm ulas. An op en formula is a formula with no quantifie r. 4 W e assume the reader to kno w the notion of free and b ound o ccurrences of a v a riable in a fo rm ula. Then a sen tence is a form ula with no free v ariable. A sen t ence in prenex nor mal form is in the fo rm ϕ = Q 1 x 1 . . . Q n x n ϕ 0 ( x 1 , . . . , x n ), where each Q i is either the quantifier ∀ or t he quan tifier ∃ and the formula ϕ 0 is an op en fo rm ula. It is w ell kno wn that ev ery sen tence is equiv alen t to a sen tence written in prenex nor mal form. A sen tence is said to b e univers al if it is in prenex normal form and each quan tifier is the univ ersal quan tifier ∀ . W e then r ecall the notion of a structure in a signature S ig : A structure is in the fo rm: M = ( | M | , ( a M ) a ∈ S ig ) Where | M | is a set called the univ erse o f the structure, and for a ∈ S ig , a M is the in terpretation of a in M : If f is a m-ary function sym b ol in S ig , then f M is a function: M m → M . If R is a m-ary r elation sym b ol in S ig , then R M is a relation: R M ⊆ M m . If a is a constant sym b ol in S ig , then a M is a distinguished elemen t in M . In order to simplify the notations w e shall sometimes write a instead o f a M when the meaning is clear f rom the contex t. When M is a structure and ϕ is a sente nce in the same signature S ig , w e write M | = ϕ fo r “ M is a mo del of ϕ ”, whic h means that ϕ is satisfied in the structure M . A detailed exp osition o f these no tions ma y b e found in [CK73]. When M is a structure in the signature S ig and S ig 1 is another signature such that S ig 1 ⊆ S ig , then the reduc tion of M to the signature S ig 1 is denoted M | S ig 1 . It is a structure in the signature S ig 1 whic h has same univ erse | M | as M , and the same in terpretations for sym b ols in S ig 1 . Con v ersely an expansion of a structure M in the signature S ig 1 to a structure in the signature S ig has same univ erse as M and same interpretations fo r sym b ols in S ig 1 . When M is a structure in a signature S ig and X ⊆ | M | , w e define: cl 1 ( X , M ) = X ∪ S { f n − ary function of S ig } f M ( X n ) ∪ S { a c onstan t of S ig } a M cl n +1 ( X , M ) = cl 1 ( cl n ( X , M ) , M ) for an integer n ≥ 1 and cl ( X , M ) = S n ≥ 1 cl n ( X , M ) is the closure of X in M . Let us no w define lo cally finite sen tences. W e shall denote S( ϕ ) the signature of a first o rder sen tence ϕ , i.e. the set of non logical sym b ols app earing in ϕ . Definition 2.1 A first o r der sentenc e ϕ is lo c al ly fini te if and only if: a) M | = ϕ and X ⊆ | M | imp l y cl ( X , M ) | = ϕ b) ∃ n ∈ N such that ∀ M , i f M | = ϕ and X ⊆ | M | , then cl ( X, M ) = cl n ( X , M ) (closur e in m o dels of ϕ takes less than n steps). 5 Notation. F or a lo cally finite sen tence ϕ , w e shall denote b y n ϕ the smallest in teger n ≥ 1 satisfying b) of the ab ov e definition. Remark 2.2 Be c ause of a) of Definition 2.1 , a lo c al ly finite sentenc e ϕ is always e quivalent to a universal se n tenc e. W e now in tro duce some notations for finite or infinite w ords. Let Σ be a finite alphab et whose elemen ts are called letters. A finite w ord o ver Σ is a finite sequence of letters: x = a 0 . . . a n where ∀ i ∈ [0; n ] a i ∈ Σ. W e shall denote x ( i ) = a i the i + 1 th letter of x and x [ i ] = x (0) . . . x ( i ) f or i ≤ n . The length of x is | x | = n + 1. The empt y w ord will b e denoted by λ and ha s 0 letter. Its length is 0. The set of finite w ords ov er Σ is denoted Σ ⋆ . Σ + = Σ ⋆ − { λ } is the set o f non-empt y w ords o ve r Σ. A (finitary) language L ov er Σ is a subset of Σ ⋆ . Its complemen t (in Σ ⋆ ) is L − = Σ ⋆ − L . The usual concatenation pro duct of u and v will b e denoted b y u.v or just uv . The set of non negative in tegers is denoted b y N . F or V ⊆ Σ ⋆ , w e denote V ⋆ = { v 1 . . . v n | ∀ i ∈ [1; n ] v i ∈ V } ∪ { λ } . The first infinite ordinal is ω . An ω -w ord o v er Σ is an ω -sequence a 0 a 1 . . . a n . . . , where ∀ i ≥ 0 a i ∈ Σ. F or σ ∈ Σ ω , σ ( n ) is the n + 1 th letter of σ and σ [ n ] = σ (0) σ (1 ) . . . σ ( n ). The set of ω -w ords ov er the alphab et Σ is denoted b y Σ ω . An ω -languag e o ver Σ is a subset of Σ ω . The ω -p o w er of a finita ry language V ⊆ Σ ⋆ is the ω -langua ge V ω = { σ = u 1 . . . u n . . . ∈ Σ ω | ∀ i ≥ 1 u i ∈ V } . F or a subset A ⊆ Σ ω , the complemen t of A (in Σ ω ) is Σ ω − A denoted A − . The concatenatio n pro duct is extended to the pro duct of a finite w ord u and an ω -word v : the infinite word u.v is then t he ω -word satisfying: ( u.v )( k ) = u ( k ) if k < | u | , and ( u.v )( k ) = v ( k − | u | ) if k ≥ | u | . A word o v er Σ ma y be considered as a structure in the usual ma nner: Let Σ b e a finite alphab et. F or eac h letter a ∈ Σ we denote P a a unary predicate and Λ Σ the signature { <, ( P a ) a ∈ Σ } . The length | σ | o f a non-empt y finite w ord σ ∈ Σ ⋆ ma y b e written | σ | = { 0 , 1 , . . . , | σ | − 1 } . σ is identifie d to the structure ( | σ | , < σ , ( P σ a ) a ∈ Σ ) o f signature Λ Σ where P σ a = { i < | σ | | the i + 1 th letter of σ is an a } . In a similar manner if σ is an ω -w ord o v er the alphab et Σ, then ω is the length of the w ord σ and w e ma y write | σ | = ω = { 0 , 1 , 2 , 3 , . . . } . σ is iden tified to the structure ( ω , < σ , ( P σ a ) a ∈ Σ ) o f signature Λ Σ where P σ a = { i < ω | the i + 1 th letter of σ is an a } . Definition 2.3 L et Σ b e a finite alphab et and L ⊆ Σ ⋆ . Then [ L is a lo c al ly finite language ] ← → ther e exists a lo c al ly finite sentenc e ϕ in a s i gnatur e Λ ⊇ Λ Σ such that σ ∈ L 6 iff ∃ M , M | = ϕ and M | Λ Σ = σ ]. We then den o te L = L Σ ( ϕ ) a n d, if ther e is no amb iguity, L = L ( ϕ ) the lo c al ly finite l a nguage define d by ϕ . The class of lo c al ly finite languages wi l l b e denote d LO C . The empt y w ord λ has 0 letters. It is represen ted b y the empty structure. Recall that if L ( ϕ ) is a lo cally finite language then L ( ϕ ) − { λ } and L ( ϕ ) ∪ { λ } are a lso lo cally finite [Fin01]. Definition 2.4 L et Σ b e a finite alphab et and L ⊆ Σ ω . Then [ L is a lo c al ly finite ω -lang uage ] ← → [ ther e exists a lo c al ly finite sentenc e ϕ in a sign a tur e Λ ⊇ Λ Σ such that ∀ σ ∈ Σ ω σ ∈ L iff ∃ M , M | = ϕ and M | Λ Σ = σ ]. We then denote L = L Σ ω ( ϕ ) , and, if ther e is no ambiguity, L = L ω ( ϕ ) the lo c al ly fi nite ω -lan guage define d by ϕ . The class of lo c al ly finite ω -languages wi l l b e denote d LO C ω . Remark 2.5 The notion of lo c al ly finite ( ω )-lan guage is very differ ent of the usual notion of lo c al ( ω )-language which r epr esents a sub class of the class of r a- tional ( ω )-language s . But fr om now on, as in [F i n 01], things b eing wel l defi n e d and pr e cis e d, we shal l c a l l si m ply lo c al ( ω )-languages (r esp e ctively, lo c al se n - tenc es) the lo c a l ly finite ( ω )-la n guages (r esp e ctively, lo c al ly finite sentenc es). 2.2 Examples o f lo c al ω -languages The following example should not b e skipp ed b ecause it is crucial to Theorem 2.9 b elow . Example 2.6 The ω -language which c ontains only the wor d σ = abab 2 ab 3 ab 4 . . . (wher e the i -th o c curr enc e of a is fo l lowe d by the factor b i a ) i s a lo c a l ω - language over the alphab et { a, b } . Pro of. Let the signature S ( ϕ ) = { P a , P b , <, p, p ′ , f } , where p and p ′ are unary function sym b ols, f is a binary function sym b ol. And let ϕ b e the follow ing sen tence, conjunction o f: (1) ∀ xy z [( x ≤ y ∨ y ≤ x ) ∧ ( x ≤ y ∧ y ≤ x ↔ x = y ) ∧ ( x ≤ y ∧ y ≤ z → x ≤ z )], (2) ∀ x [ P a ( x ) ↔ ¬ P b ( x )], (3) ∀ xy [( x < y ∧ P a ( x ) ∧ P a ( y )) → P b ( f ( xy ))], (4) ∀ xy [ x ≥ y → f ( xy ) = x ], (5) ∀ xy [ ¬ P a ( x ) ∨ ¬ P a ( y ) → f ( xy ) = x ], (6) ∀ x [ P a ( p ( x )) ∧ P a ( p ′ ( x ))], (7) ∀ x [ ¬ P a ( x ) → p ( x ) < p ′ ( x )], (8) ∀ x [ P a ( x ) → p ( x ) = p ′ ( x ) = x ], 7 (9) ∀ x [ ¬ P a ( x ) → x = f ( p ( x ) p ′ ( x ))], (10) ∀ xx ′ y ∈ P a [ x < x ′ < y → f ( x ′ y ) < f ( xy ) < y ], (11) ∀ xy y ′ ∈ P a [ x ≤ y ′ < y → y ′ < f ( xy ) < y ]. Ab o v e sen tence (1) means that “ < is a linear order ”. The sen tence (2 ) ex- presses that P a , P b form a partition in an y mo del M o f ϕ . The sen t ence (3) sa ys that f is a function from { ( x, y ) | x < y ∧ P a ( x ) ∧ P a ( y ) } in to P b while sen tences ( 4)-(5) express that the function f is t rivially defined elsewhere. Sen tences (6)-(7) say that p and p ′ are functions defined fro m ¬ P a = P b in to P a and ( 8) states that p and p ′ are trivially defined on P a . The pro jections p and p ′ are used to say that the function f is surjectiv e fr om { ( x, y ) | x < y ∧ P a ( x ) ∧ P a ( y ) } onto P b ; this is implied by sen tence (9). The 1 0 th and 1 1 th conjunctions are used to order the elemen ts of f ( P a × P a ) in order to obtain the w ord σ = abab 2 ab 3 ab 4 . . . when the reduction to the signature of w ords is considered. Notice that (10)-(11) imply also that f is injectiv e hence is in fact a bijection from { ( x, y ) | x < y ∧ P a ( x ) ∧ P a ( y ) } into P b .  Remark 2.7 We have define d the functions f and p , p ′ , in a trivial manner (like f ( xy ) = x or p ( x ) = x ) wher e they wer e not useful for de fi ning the l o c al ω -language { σ } , (se e the c onj unc tion s (4 ) , (5) and (8) ). This wil l imply that closur e in mo dels of ϕ takes at most a finite numb e r o f steps. This metho d wil l b e applie d in the c onstruction of other lo c al sentenc es in the se quel of this p ap er. W e can easily c hec k that ϕ is equiv a len t to a univ ersal form ula and that closure in its mo dels tak es at most n ϕ = 2 steps: one tak es closure under the functions p, p ′ then b y f . Hence ϕ is a lo cal sen tence a nd b y construction: L { a,b } ω ( ϕ ) = { abab 2 ab 3 . . . } . Let us giv e some examples o f closure in a mo del M of ϕ suc h that M | Λ { a,b } = σ = abab 2 ab 3 ab 4 . . . Let X n ⊆ | M | b e the segmen t of M corresp onding to the segmen t ab n a of σ . Then the closure of X n under the functions p, p ′ is the set X n ∪ Z n where Z n corresp onds to the set of the ( n − 1) first letters a of σ . cl ( X n , M ) is the closure of X n ∪ Z n under f and it is the initial segmen t of M corresp onding to the initial segmen t abab 2 ab 3 ab 4 . . . ab n a of σ . Let now Y ⊆ | M | b e the segmen t of M corresp onding to the three last letters b 2 a of the segmen t ab n a of σ , for some in teger n ≥ 3. Then the closure of Y under the functions p, p ′ is Y ∪ Z wh ere Z correspo nds to the set of the tw o first letters a of σ . The closure of Y ∪ Z under f is the set cl ( Y , M ) which induces the w ord abab 2 a but whic h is not a segmen t of M b ecause it con tains the t w o first letters a and the ( n + 1)- th letter a o f σ but not an y other letter a of σ . 8 W e are going now to get more examples of lo cal ω -languages. Recall first the follo wing: Definition 2.8 T he ω -Kle ene closur e of a family L of fin i tary languages is : ω -KC ( L ) = {∪ n i =1 U i .V ω i | ∀ i ∈ [1 , n ] U i , V i ∈ L} This notion of ω -Kleene closure app ears in the c haracterization of the class RE G ω of regular ω -languages (resp ectiv ely , of the class C F ω of context free ω - languages) whic h is the ω -Kleene closure of the family RE G of regular finitary languages (r espectiv ely , o f the family C F of con text free finitary languag es), [Tho90] [PP02] [Sta 97]. A natural question ar ises : do es a similar c haracterization hold for lo cal lan- guages? The answ er is giv en b y the following: Theorem 2.9 T he ω - Kle en e closur e of the class LO C of finitary lo c al l a n- guages is strictly include d into the class LO C ω of l o c al ω -languages. Pro of. W e hav e already pro v ed tha t ω - K C ( LO C ) ⊆ LO C ω in [Fin01]. In order to sho w that the inclusion is strict, remark that if an ω -la nguage L b elongs to ω - K C ( LO C ), then L con tains a t le ast an ultimately p erio dic w ord, i.e. a w ord in the form u.v ω where u and v ar e finite words. Now w e can easily c hec k that the lo cal ω - language giv en in example 2.6 do es not con tain a n y ultimately p erio dic w o rd b ecause its single w ord is not ultimately p erio dic.  A first consequence of Theoreme 2.9 is tha t ev ery regular ω -langua ge is a lo cal ω -language, i.e . RE G ω ⊆ LO C ω , because ev ery finitary regular langua ge is lo cal [Res88]. W e ha d shown in [Fin01] that man y con t ext free languages are lo cal th us C F ω = ω - K C ( C F ) implies that many con text free ω -langua ges a re lo cal. The problem to kno w whether eac h con text free language is lo cal is still op en but by Theorem 2.9, C F ⊆ LO C w o uld imply that C F ω ⊆ LO C ω . The ω -language giv en in example 2.6 is lo cal but non contex t free b ecause ev ery con text free ω -language con tains at least one ultimately p erio dic w ord. W e prov ed in [F in01] that the finitary language U = { a n b n 2 | n ≥ 1 } ⊆ { a, b } ⋆ is lo cal. Thus the ω -language U.c ω ⊆ { a, b, c } ω is lo cal but U.c ω is not con text free b ecause U / ∈ C F , [CG77]. These tw o examples show that the inclusion LO C ω ⊆ C F ω do es not hold. 9 3 Closure prop erties of lo cally finite omega languages Theorem 3.1 T he class of lo c al ly finite ome ga lan guages is not close d under interse ction wi th a r e gular ω -language in the f o rm L.a ω wher e L i s a r ational language, L ⊆ Σ ⋆ and a / ∈ Σ . Henc e LO C ω is n either close d under interse ction nor und e r c omp l e mentation. T o prov e this t heorem, w e shall pro ceed b y successiv e lemmas. W e shall as- sume that ev ery la nguage considered here is constituted of w o rds ov er a finite alphab et included in a giv en countable set Σ D . W e firstly define the family I of finitary languages by: for a finitary languag e L ⊆ Σ ⋆ , where Σ ⊆ Σ D is a finite alphab et, L ∈ I if and only if L.a ω ∈ LO C ω whenev er a is a letter of Σ D − Σ. It is easy to see that if L.a ω is a lo cal ω -language and a ∈ Σ D − Σ, then f or all b ∈ Σ D − Σ, it holds that L.b ω ∈ LO C ω . It suffices to replace the predicate P a b y P b in the sen tence defining L.a ω . Lemma 3.2 I is close d under inverse al p hab e tic morphism. Pro of. Let L ∈ I , i.e. L ⊆ Γ ⋆ for some finite alphab et Γ, a / ∈ Γ and L.a ω = L Γ ∪{ a } ω ( ϕ ) for a lo cal sen tence ϕ . Let h be the alphab etic morphism: Σ ⋆ → Γ ⋆ , defined b y h ( c ) ∈ Γ ∪ { λ } for c ∈ Σ, where λ is the empt y w ord. And let Σ ′ = { c ∈ Σ | h ( c ) = λ } . W e assume, p ossibly c ha nging a , that a / ∈ Σ. W e first replace in ϕ the letter pr edicates ( P c ) c ∈ Γ b y ( Q c ) c ∈ Γ . The language h − 1 ( L ) .a ω is then defined b y the follow ing sen tence ψ , in the signature S ( ψ ) = { P , A, ( P c ) c ∈ Σ , P a } ∪ S ( ϕ ), where S ( ϕ ) con t ains the letter predicates Q c for c ∈ Γ ∪ { a } , P is a unary predicate sym b ol and A is a constan t sym b ol. ψ is the conjunction o f: • ( < is a linear order ), • (( P c ) c ∈ Σ , P a ) form a partition, • ∀ x 1 ...x n ∈ P [ ϕ 0 ( x 1 ...x n ) ∧ V c ∈ Γ ( Q c ( x 1 ) ↔ W d ∈ h − 1 ( c ) P d ( x 1 )) ∧ ( Q a ( x 1 ) ↔ P a ( x 1 ))], where ϕ = ∀ x 1 ...x n ϕ 0 ( x 1 ...x n ) with ϕ 0 an op en form ula, • ∀ x 1 ...x k [( W 1 ≤ j ≤ k ¬ P ( x j )) → g ( x 1 ...x k ) = x 1 ] , for eac h k-ary function g of S ( ϕ ), • ( P c ) c ∈ Σ ′ form a partition of ¬ P , • P ( B ), for eac h constan t B of S ( ϕ ), • ∀ xy [ ¬ P a ( y ) ∧ P a ( x ) → y < x ], • P a ( A ). 10 ψ is equiv alent to a univ ersal sen t ence and closure in its mo dels takes at most n ϕ + 1 steps. Hence ψ is lo cal and b y construction L Σ ∪{ a } ω ( ψ ) = h − 1 ( L ) .a ω .  Lemma 3.3 I is close d under non er asing alphab etic morphism . Recall that an alphab etic morphism h : Σ ⋆ → Γ ⋆ , defined b y h ( c ) ∈ Γ ∪ { λ } , for c ∈ Σ is said to b e non erasing if ∀ c ∈ Σ , h ( c ) ∈ Γ. Pro of. Let L ∈ I , L ⊆ Σ ⋆ . Let a / ∈ Σ a nd L.a ω = L Σ ∪{ a } ω ( ϕ ) for a lo cal sen tence ϕ . Let h b e a no n erasing alphab etic morphism given b y h : Σ → Γ. Moreo ve r we assum e that a / ∈ Γ (p ossibly c hang ing a ). Then the language h ( L ) .a ω is defined by t he follow ing f orm ula ψ , in the signature S ( ψ ) = S ( ϕ ) ∪ { ( Q c ) c ∈ Γ } . The sen tence ψ is the conjunction of: • ϕ , • ∀ x [ V c ∈ Σ ( P c ( x ) → Q h ( c ) ( x ))], • [( Q c ) c ∈ Γ , ( P a )] form a partition. ψ is lo cal and if the predicates ( Q c ) c ∈ Γ , P a , a re the letter predicates, ψ defines the ω -language L Γ ∪{ a } ω ( ψ ) = h ( L ) .a ω .  Lemma 3.4 I c on tains the finitary lo c al languages. Pro of. LO C ω con tains the omega Kleene closure of the class LO C of finitary lo cal languages, and for each letter a the language { a } is lo cal.  Recall now the definitions o f the Antidyc k language and o f a ratio nal cone of languages. Definition 3.5 T he A ntidyck language over two sorts of p ar entheses is the language Q ′ ⋆ 2 = { v ∈ ( Y ∪ ¯ Y ) ⋆ | v → ⋆ λ } , wher e Y = { y 1 , y 2 } , ¯ Y = { ¯ y 1 , ¯ y 2 } and → ⋆ is the tr ansitive closur e of → define d in ( Y ∪ ¯ Y ) ⋆ by: ∀ y ∈ Y y v 1 ¯ y v 2 → v 1 v 2 if a n d o nly if v 1 ∈ Y ⋆ . The Antidy c k language Q ′ ⋆ 2 ma y b e seen as the language con taining w o rds with t wo sorts of paren theses, suc h that: “the first parenthes is to b e op ened is the first to b e closed” Definition 3.6 ([Ber79]) A r ational c on e is a cl a ss of languages which is close d under morphism , inverse morphism, and interse ction wi th a r ational language. (Or, e quivalently to these thr e e pr op erties, clo se d under r ational tr a nsduction). The notion of rational cone has been m uch studied. In particular the An tidyc k language Q ′ ⋆ 2 is a generator of the rational cone of the recursiv ely en umerable languages, [F ZV80]. 11 On the other hand Niv at’s Theorem states that a class of languages whic h is closed under alphab etic morphism, in ve rse alphab etic mor phism, and in ter- section with a rationa l language, is a rational cone, [Ber79 ]. Moreo ver ev ery rational transduction t is in the form t ( u ) = g [ h − 1 ( u ) ∩ R ], where g et h are alphab etic morphism s and R is a rational langua ge. Th us ev ery recursiv ely en umerable language ma y b e written in the form g [ h − 1 ( Q ′ ⋆ 2 ) ∩ R ], where R is a rationa l language, g and h are alphab etic mor phisms . This result will b e used here b ecause the language Q ′ ⋆ 2 is lo cal, [Fin01]. Return now to the pro of of Theorem 3.1 and supp ose that LO C ω w ere closed under in tersection with the languages R.a ω , where R ⊆ Σ ⋆ is a rational lan- guage and a / ∈ Σ. Claim 3.7 I wo uld b e close d under interse ction with a r ational language. Pro of. Let L ∈ I , L ⊆ Σ ⋆ , R ⊆ Σ ⋆ b e a rational language and a / ∈ Σ. L.a ω is a lo cal ω -language because L ∈ I and ( L.a ω ) ∩ ( R.a ω ) = ( L ∩ R ) .a ω w ould b e a lo cal ω - language, hence b y definition of I , L ∩ R w o uld b elong to I .  Claim 3.8 I would c ontain every language in the form g [ h − 1 ( Q ′ ⋆ 2 ) ∩ R ] , wher e h is an alphab etic morphism , g is a non er asing alphab etic m o rphism and R is a r ational lan g uag e . Pro of. It follows from the lemmas 3.2, 3.3, 3.4, the fact that Q ′ ⋆ 2 is lo cal and Claim 3.7 .  Claim 3.9 Th e r e would ex i s t an er as i n g alphab etic morphism h and L ∈ I such that { 0 n 1 p | p > 2 n } = h ( L ) , wher e an alphab etic morphism is said to b e er asing if it is in the form h : Σ → Σ ∪ { λ } , with h ( c ) = c if c ∈ A and h ( c ) = λ if c ∈ Σ − A , for some s ubset A ⊆ Σ . Pro of. W e know that ev ery recursiv ely en umerable langua ge ma y b e written in the form g [ h − 1 ( Q ′ ⋆ 2 ) ∩ R ], where R is a ra tional language, g and h are alphab etic morphisms. But ev ery alpha betic morphism ma y b e obtained as comp osed firstly by a non erasing alpha betic morphism follow ed b y an erasing alphab etic morphism. Th us it follo ws from Claim 3.8 that eac h recursiv ely en umerable language, and in par ticular the languag e { 0 n 1 p | p > 2 n } , w ould b e the image by an erasing alphab etic morphism of a la nguage of I .  Let then h b e an erasing morphism Σ ⋆ → Σ ⋆ where { 0 , 1 } ⊆ Σ , h (0) = 0 and h (1) = 1 and h ( c ) = λ if c ∈ Σ − { 0 , 1 } and let L ⊆ Σ ⋆ b e a language suc h that h ( L ) = { 0 n 1 p | p > 2 n } . Assume that L b elongs to I , so if a / ∈ Σ, L.a ω is a lo cal ω -langua ge and there exists a lo cal sen tence ϕ suc h that L.a ω = L Σ ∪{ a } ω ( ϕ ). F or all n ≥ 1 let M n b e a mo del of ϕ o f order t yp e ω suc h that M n | Λ Σ ∪{ a } = 12 σ n .a ω , where σ n ∈ L and the n um b er of o ccurrences of 0 in σ n is n and the n umber of o ccurrences of 1 in σ n is p n > 2 n . Let us now set the following definition in view of next lemma. Definition 3.10 L et X b e a set include d in a structur e M and P ⊆ | M | . X is a set of in d i sc erna b les ab ove P for the atomic formulas of c omple xity ≤ k , i.e . whose terms r e s ult by at most k applic ations of function symb ols, for k ∈ N , if a n d o nly if: i) X is line arly or der e d by < . ii) Wheneve r ¯ x and ¯ y ar e some n-tuples of ele m ents of X which ar e isomor- phic for the or der of ( X , < ) , ¯ x and ¯ y satisfy the same a tom i c formulas of c omplexity ≤ k , w i th p ar ameters in P . Lemma 3.11 I n the ab ove c onditions wher e M n is define d for every inte ger n ≥ 1 , ther e exists in M n an infinite set X n of indisc ernabl e s ab ove P M n 0 for the atomi c formulas of c omplexity ≤ n ϕ , w i th X n ⊆ P M n a Pro of. Let m ( ϕ ) b e the maxim um num b er of v ariables of the atomic form ulas of complexit y ≤ n ϕ i.e. whose terms result b y at mo st n ϕ applications of function sym b ols. These terms fo rm a finite set T ϕ . F or a ll strictly increasing sequences ¯ x and ¯ y of length m ( ϕ ) of P M n a , let us set ¯ x ∼ ¯ y if and only if ¯ x and ¯ y satisfy in M n the same ato mic form ulas with parameters in P M n 0 and of complexit y ≤ n ϕ . P M n 0 is a finite set of cardinal n , hence the set of atomic f orm ulas with param- eters in P M n 0 and of complexit y ≤ n ϕ is also finite. Then applying the infinite Ramsey Theorem, w e can find X n ⊆ P M n a homoge- neous for ∼ and infinite. This is the set w e are lo oking for.  W e return no w to the pro of of Theorem 3.1 and consider in | M n | the subse t X n ∪ P M n 0 = Y n . This subset is infinite hence it is of order ty p e ω in M n and it generates in M n a mo del of order t yp e ω to o, whic h will b e denoted b y M n ( Y n ) = A n . This mo del of ϕ induces a word u n .a ω of L.a ω , suc h that there are in u n : n o ccurrences of the letter 0 and q n ≤ p n o ccurrences of the letter 1. But u n .a ω ∈ L.a ω implies tha t 2 n < q n also holds. A n is generated from Y n b y the use of only a finite se t T ϕ of terms of les s than k ϕ v a riables. If n is big enough with regard to k ϕ and card( T ϕ ), b e- cause q n > 2 n , there exist parameters a 1 , . . . , a k , elemen ts of P M n 0 , and some indiscernables x 1 , . . . , x j , and y 1 , . . . , y j of X n , suc h that x 1 < . . . < x j and y 1 < . . . < y j and ¯ x 6 = ¯ y and a term t ∈ T ϕ suc h that t ( a 1 , . . . , a k , x 1 , . . . , x j ) < 13 t ( a 1 , . . . , a k , y 1 , . . . , y j ) and these tw o elemen ts b eing in P M n 1 . But then w e could find in X n a sequence ( ¯ x i ) 1 ≤ i ≤ N , with N a rbitrarily large, suc h that for each i , 1 ≤ i ≤ N , ¯ x i ¯ x i +1 is of the order t yp e of ¯ x ¯ y . Then for all in tegers i suc h that 1 ≤ i ≤ N , P M n 1 ( t ( a 1 , . . . , a k , ¯ x i )) and t he terms t ( a 1 , . . . , a k , ¯ x i ) a re distinct t w o b y t w o. This would imply that, for all in tegers N ≥ 1, card( P M n 1 ) ≥ N . So there w ould b e a contradiction with card( P M n 1 ) = p n and w e ha ve prov ed that L do es not b elong to I . Th us w e can infer Theorem 3.1 from Claim 3.9. The non closure under comple- men tation of the class LO C ω can be deduced from the non closure under inter- section and the fact that LO C ω is closed under union (see next Theorem) or from an example, deduced fr om preceding pro o f, of a lo cal ω -language which complemen t is not lo cal (see next remark).  Remark 3.12 The abov e pro of sho ws in particular that the ω -lang uage A = { 0 n 1 p 2 ω | p > 2 n } is not lo cal. F rom whic h w e can easily deduce that the lo cal ω -langua ge { 0 n 1 p 2 ω | p ≤ 2 n } = L has a complemen t whic h is not a lo cal ω -langua ge. (This ω -languag e L is lo cal b ecause { 0 n 1 p | p ≤ 2 n } is a lo cal finitary la nguage [Fin01]). Indeed if its complemen t w as ω -lo cal, w e w ould deduce, fro m a lo cal sen tence ϕ suc h that L ω ( ϕ ) = L − , a nother lo cal sen tence ψ suc h that L ω ( ψ ) = A . F or example the sen tence ψ , conjunction of : • ϕ , • ∀ xy [( P 0 ( x ) ∧ P 1 ( y )) → x < y ] , • ∀ xy [( P 1 ( x ) ∧ P 2 ( y )) → x < y ] , • ∀ xy [( P 0 ( x ) ∧ P 2 ( y )) → x < y ] , • P 2 ( c ) , wher e c is a new c onstant symb ol. No w w e establish that LO C ω is closed under sev eral op erations. Theorem 3.13 T he class LO C ω is close d under union, left c onc atenation with lo c al (fini tary) lang uag e s , λ -f r e e substitution of lo c al (finitary) languages, λ -fr e e morphism. Pro of. Closure under union. Let ϕ 1 and ϕ 2 b e t wo lo cal sen tences defining lo cal ω -languages L ω ( ϕ 1 ) and L ω ( ϕ 2 ) o ver a finite alphab et Σ. L et us define a new lo cal sen tence ϕ 1 ∪ ϕ 2 whic h defines the lo cal ω - language L ω ( ϕ 1 ∪ ϕ 2 ) = L ω ( ϕ 1 ) ∪ L ω ( ϕ 2 ): W e may a ssum e that S ( ϕ 1 ) ∩ S ( ϕ 2 ) = Λ Σ . Then S ( ϕ 1 ∪ ϕ 2 ) will b e S ( ϕ 1 ) ∪ S ( ϕ 2 ). And the sen tence ϕ 1 ∪ ϕ 2 is the following sen tence: 14 [ ϕ 1 V n − ary function symbol f ∈ S ( ϕ 2 ) ( ∀ x 1 , . . . , x n f ( x 1 , . . . , x n ) = min ( x 1 , . . . , x n ))] W [ ϕ 2 V n − ary function symbol g ∈ S ( ϕ 2 ) ( ∀ x 1 , . . . , x n g ( x 1 , . . . , x n ) = min ( x 1 , . . . , x n ))] Closure under left concatenation b y a lo cal finitary language. Consider a finitary lo cal lang uage L ( ϕ ) and a lo cal ω -language L ω ( ψ ) o ver the same alphab et Γ. W e ma y easily as sume that L ω ( ϕ ) is e mpt y , p ossibly adding a constan t sym b ol c to the signature of ϕ and adding the conjunction ∀ x [ x ≤ c ] to the sente nce ϕ (this means that ev ery mo del of ϕ has a greatest elemen t). W e ma y also assume that S ( ϕ ) ∩ S ( ψ ) = { < , ( P a ) a ∈ Γ } = Λ Γ . Let then P b e a new unary predicate sym b ol not in S ( ϕ ) ∪ S ( ψ ), and let ϕ.ψ b e the following sen tence in the signature S ( ϕ ) ∪ S ( ψ ) ∪ { P } , whic h is the conjunction of : • ( < is a linear order ), • (( P a ) a ∈ Γ form a partitio n), • ∀ xy [ P ( x ) ∧ ¬ P ( y ) → x < y ], • ∀ x 1 , . . . , x j ∈ P [ ϕ 0 ( x 1 , . . . , x j )], where ϕ = ∀ x 1 , . . . , x j ϕ 0 ( x 1 , . . . , x j ) with ϕ 0 an op en formula, • ∀ x 1 , . . . , x m ∈ P [ f ( x 1 , . . . , x m ) ∈ P ], for each m-ary function f o f S ( ϕ ), • ∀ x 1 , . . . , x m [ W 1 ≤ i ≤ m ¬ P ( x i ) → f ( x 1 , . . . , x m ) = min ( x 1 , . . . , x m )], for each m-ary function f o f S ( ϕ ), • P ( c ), for eac h constant c of S ( ϕ ), • ∀ x 1 , . . . , x j ∈ ¬ P [ ψ 0 ( x 1 , . . . , x j )], where ψ = ∀ x 1 , . . . , x j ψ 0 ( x 1 , . . . , x j ) with ψ 0 an op en formula, • ∀ x 1 , . . . , x m ∈ ¬ P [ f ( x 1 , . . . , x m ) ∈ ¬ P ], for each m-ary function f o f S ( ψ ), • ∀ x 1 , . . . , x m [ W 1 ≤ i ≤ m P ( x i ) → f ( x 1 , . . . , x m ) = min ( x 1 , . . . , x m )], for each m-ary function f o f S ( ψ ), • ¬ P ( c ), for eac h constan t c of S ( ψ ), This s en tence ϕ.ψ is equiv alen t to a univ ersal form ula and closure in its m o dels tak es at most max ( n ϕ , n ψ ) steps, hence it is a lo cal sen tence and by construc- tion it holds that L ( ϕ.ψ ) = L ( ϕ ) .L ( ψ ). Moreo v er when ω -w ords are considered L ω ( ϕ.ψ ) = L ( ϕ ) .L ω ( ψ ) holds b ecause b y hypothesis L ω ( ϕ ) is empty . Closure under λ -free substitution of lo cal languages. The pro o f is v ery similar to our pro of of the closure of the class LO C under substitution b y lo cal finitary languages in [F in01]. W e recall it no w. 15 Recall first the notion of substitution: A substitution f is defined b y a mapping Σ → P (Γ ⋆ ), where Σ = { a 1 , ..., a n } and Γ are t wo finite alphab ets, f : a i → L i where ∀ i ∈ [1; n ], L i is a finitary language ov er the alphab et Γ. The subs titution is said to be λ -free if ∀ i ∈ [1; n ], L i do es not contain the empt y w ord λ . It is a ( λ -f ree) morphism when ev ery language L i con tains only one (nonempt y) word. No w this mapping is extended in the usual manner to finite w o rds and to finitary languages: for some letters x (0), . . . , x ( n ) in Σ, f ( x (0) x (1) . . . x ( n )) = { u 0 u 1 . . . u n | ∀ i ∈ [0; n ] u i ∈ f ( x ( i )) } , and for L ⊆ Σ ⋆ , f ( L ) = ∪ x ∈ L f ( x ). If the substitution f is λ -free, w e can extend this to ω -w ords and ω -languages: f ( x (0) x (1) . . . x ( n ) . . . ) = { u 0 u 1 . . . u n . . . | ∀ i ≥ 0 u i ∈ f ( x ( i )) } and for L ⊆ Σ ω , f ( L ) = ∪ x ∈ L f ( x ). Let then Σ = { a 1 , . . . , a n } b e a finite alphab et and let f b e a λ - free substitution of lo cal languages: Σ → P (Γ ⋆ ), a i → L i where ∀ i ∈ [1; n ], L i is a lo cal language defined b y the sen tence ϕ i , o v er the alphabet Γ. W e ma y assume that L ω ( ϕ i ) is empt y , p ossibly adding a constant sym b ol c i to the signature of ϕ i and a dding the conjunction ∀ x [ x ≤ c i ] to the sen tence ϕ i (this means that ev ery mo del of ϕ i has a greatest elemen t). W e also a ssum e that the signat ures of the sen tences ϕ i v erify S ( ϕ i ) ∩ S ( ϕ j ) = { <, ( P a ) a ∈ Γ } for i 6 = j . Let now L ⊆ Σ ω b e a lo cal ω -language defined by a lo cal sen tence ϕ . W e shall denote Q a i the unary predicate of S ( ϕ ) whic h indicates the places of the le tters a i in a w ord of L , s o if a i ∈ Γ ∩ Σ for some indice i , t here will be tw o distinct pred icates Q a i and P a i . W e ma y a lso assume that ∀ i ∈ [1 , . . . , n ], S ( ϕ i ) ∩ S ( ϕ ) = { < } . Then we can construct a lo cal sen tence ψ (already give n in [Fin01 ]) suc h that L ω ( ψ ) = f ( L ). ψ is the conjunction of the follow ing sen tences, whic h meaning is explained b elo w: • “ < is a linear order ”, • ∀ xy [( I ( y ) ≤ y ) ∧ ( y ≤ x → I ( y ) ≤ I ( x )) ∧ ( I ( y ) ≤ x ≤ y → I ( x ) = I ( y ))], • ∀ x [ I ( x ) = x ↔ P ( x )], • P ( c ), for eac h constant c of S ( ϕ ), • ∀ x 1 , . . . , x k [ R ( x 1 , . . . , x k ) → P ( x 1 ) ∧ . . . ∧ P ( x k )], for each predicate R ( x 1 , . . . , x k ) o f S ( ϕ ), • ∀ x 1 , . . . , x j [( P ( x 1 ) ∧ . . . ∧ P ( x j )) → P ( f ( x 1 , . . . , x j ))], for each j-ary function sym b ol f of S ( ϕ ), • ∀ x 1 , . . . , x j [( W 1 ≤ i ≤ j ¬ P ( x i )) → f ( x 1 , . . . , x j ) = min ( x 1 , . . . , x j )], for each j-ary function sym b ol f of S ( ϕ ), • ∀ x 1 , . . . , x m [( P ( x 1 ) ∧ . . . ∧ P ( x m )) → ϕ 0 ( x 1 , . . . , x m )], where ϕ = ∀ x 1 , . . . , x m ϕ 0 ( x 1 , . . . , x m ) with ϕ 0 an op en formula, • ∀ x 1 , . . . , x j [ W i,k ≤ j ( I ( x i ) 6 = I ( x k )) → f ( x 1 , . . . , x j ) = min ( x 1 , . . . , x j )], for ev ery function f of S ( ϕ l ) f or an integer l ≤ n , • ∀ xy 1 , . . . , y j [( V 1 ≤ l ≤ j I ( y l ) = I ( x )) → I ( f ( y 1 , . . . , y j )) = I ( x )], for each j-ary function sym b ol f of S ( ϕ i ) f or an integer i ≤ n , 16 Finally , for eac h i ≤ n : • ∀ xy 1 , . . . , y p [( V 1 ≤ l ≤ p I ( y l ) = I ( x ) ∧ Q a i ( I ( x ))) → ϕ 0 i ( y 1 , . . . , y p ) ∧ { ( e j ( y 1 ) = I ( x ) ∧ f j ( y 1 , . . . , y p ) = y 1 ∧ ¬ R j ( y 1 , . . . , y p ) ∧ V e i ∈ S ( ϕ i ) e i ( y 1 ) = e i ( x ); where n ≥ j 6 = i , a nd e j , f j , R j run o v er the constan ts, functions, a nd predicates of S ( ϕ j ) } ], Ab o v e , 1 ) to eac h constan t e l of S ( ϕ l ) is asso ciated a new unary function e l ( y ) and 2) whenev er ϕ i = ∀ y 1 , . . . , y p ψ i ( y 1 , . . . , y p ) with ψ i an op en f orm ula, ϕ 0 i is ψ i in which ev ery constant e i has b een replaced b y the function e i ( y ). Construction of ψ : Using the function I whic h mar ks the first letters o f the sub w ords, w e divide an ω -word in to omega (finite) sub w ords (the function I is constan t on eac h sub w o rd and I ( x ) is the first letter of the sub word con taining x ). In ev ery mo del M of o rder t yp e ω of ψ , the set of the “first letters of sub w ords” , P M , grows ric her in a mo del of order type ω of ϕ (therefore will constitute an ω -word o f L ). Then we “ subs titute”: for each letter a i in P M , w e substitute a (finite) w ord of L i , using for that t he form ula ϕ i . If closure tak es a t most n ϕ (resp ectiv ely n ϕ i ) steps in ev ery mo del o f ϕ (re- sp ectiv ely of ϕ i ), then closure takes at most [1 + n ϕ + sup i ( n ϕ i )] steps in eac h mo del of ψ (one tak es closure under the function I , then under the functions of S ( ϕ ), and finally under the functions of S ( ϕ i ), 1 ≤ i ≤ n ). 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