Synchronization of regular automata

Transcription

1 Synhrniztin f regulr utmt Didier Cul IM CNRS Université Pris-Est ul@univ-mlv.fr strt. Funtinl grph grmmrs re finite devies whih generte the lss f regulr utmt. We rell the ntin f synhrniztin y grmmrs, nd fr ny given grmmr we nsider the lss f lnguges regnized y utmt generted y ll its synhrnized grmmrs. The synhrniztin is n utmtn-relted ntin: ll grmmrs generting the sme utmtn synhrnize the sme lnguges. When the synhrnizing utmtn is unmiguus, the lss f its synhrnized lnguges frms n effetive len lger lying etween the lsses f regulr lnguges nd unmiguus ntext-free lnguges. We dditinlly prvide suffiient nditins fr suh lsses t e lsed under ntentin nd its itertin. Intrdutin n utmtn ver sme lphet n simply e seen s finite r untle set f lelled rs tgether with tw sets f initil nd finl verties. Suh n utmtn regnizes the lnguge f ll wrds lelling n epting pth, i.e. pth leding frm n initil t finl vertex. It is well-knwn tht finite utmt regnize the regulr lnguges. By pplying si nstrutins t finite utmt, we tin the nie lsure prperties f regulr lnguges, nmely their lsure under len pertins, ntentin nd its itertin. Fr instne the synhrniztin prdut nd the determiniztin f finite utmt respetively yield the lsure f regulr lnguges under intersetin nd under mplement. This ide n e extended t mre generl lsses f utmt. In this pper, we will e interested in the lss f regulr utmt, whih regnize ntextfree lnguges nd re defined s the (generlly infinite) utmt generted y funtinl grph grmmrs [C 07]. Regulr utmt f finite degree re ls preisely thse utmt whih n e finitely dempsed y distne, s well s the regulr restritins f trnsitin grphs f pushdwn utmt [MS 85], [C 07]. Even thugh the lss f ntext-free lnguges des nt enjy the sme lsure prperties s regulr lnguges, ne n define sulsses f ntext-free lnguges whih d, using the ntin f synhrniztin. The ntin f synhrniztin ws first defined etween grmmrs [CH 08]. grmmr S is synhrnized y grmmr R if fr ny epting pth µ f (the grph generted y) S, there exists n epting pth λ f R with the sme lel u suh tht λ nd µ re synhrnized: fr every prefix v f u, the prefixes

2 f λ nd µ lelled y v led t verties f the sme level (where the level f vertex is the miniml numer f rewriting steps neessry fr the grmmr t prdue it). lnguge is synhrnized y grmmr R if it is regnized y n utmtn generted y grmmr synhrnized y R. fundmentl result is tht tw grmmrs generting the sme utmtn yield the sme lss f synhrnized lnguges [C 08]. This wy, the ntin f synhrniztin n e trnsferred t the level f utmt: fr regulr utmtn, the fmily Syn() is the set f lnguges synhrnized y ny grmmr generting. By extending the ve-mentined nstrutins frm finite utmt t grmmrs, ne n estlish severl lsure prperties f these fmilies f synhrnized lnguges. The sum f tw grmmrs nd the synhrniztin prdut f grmmr with finite utmtn respetively entil the lsure f Syn() under unin nd under intersetin with regulr lnguge fr ny regulr utmtn. The (level preserving) synhrniztin prdut f tw grmmrs yields the lsure under intersetin f Syn() when is unmiguus i.e. when ny tw epting pths f hve distint lels. Nrmlizing f grmmr int grmmr nly ntining rs nd then determinizing it yields, fr ny unmiguus utmtn, the lsure f Syn() under mplement reltive t L(). This nrmliztin ls llws us t express Syn() in the se f n infinite degree grph, y perfrming the e-lsure f Syn(H) fr sme finite degree utmtn H using n extr lel e. finl useful nrmliztin nly llws the presene f initil nd finl verties t level 0. It yields suffiient nditins fr the lsure f lsses f synhrnized lnguges under ntentin nd its itertin. In Setin, we rell the definitin f regulr utmt. In the next setin, we summrize knwn results n the synhrniztin f regulr utmt [C 06], [NS 07], [CH 08], [C 08]. In the lst setin, we present simpler nstrutin fr the lsure under mplement f Syn() fr unmiguus [C 08] nd present new results, espeilly suffiient nditins fr the lsure f Syn() under ntentin nd its itertin. Regulr utmt n utmtn is lelled riented simple grph with input nd utput verties. It regnizes the set f wrds lelling the pths frm n input t n utput. Finite utmt re utmt hving finite numer f verties, they regnize the lss f regulr lnguges. Regulr utmt re the utmt generted y funtinl grph grmmrs, they regnize the lss f ntext-free lnguges. key result, riginlly due t Muller nd Shupp, identifies the regulr utmt f finite degree with the utmt finitely generted y distne. n utmtn ver n lphet (finite set f symls) T f terminls is just set f rs lelled ver T ( simple lelled riented grph) with initil nd finl verties. We use tw symls nd t mrk respetively the initil nd finl verties. Mre preisely n utmtn is defined y T V V {, } V

3 where V is n ritrry set suh tht the set f verties V = { s V T t V (, s, t) (, t, s) } f is finite r untle. ny triple (, s, t) is n r lelled y frm sure s t gl t ; it is identified with the lelled trnsitin s t r diretly s t if is understd. ny pir (, s) is lured vertex s y {, } ls written s. vertex is initil (resp. finl) if it is lured y (resp. ) i.e. s (resp. s ). n exmple f n utmtn is given y = { n n + n 0 } { n x n n > 0 } { n y n n > 0 } { x n+ x n n > 0 } { y n+ y n n > 0 } {0, y} { x n n > 0 } { y n+ n 0 } nd is represented (up t ismrphism) elw. Figure. n utmtn. n utmtn is thus simple vertex- nd r-lelled grph. hs finite degree if fr ny vertex s, the set { t (s t t s) } f its djent verties is finite. Rell tht (s 0,, s,..., n, s n ) fr n 0 nd s 0 s... s n n s n is pth frm s 0 t s n lelled y u =... n ; u u we write s 0 = s n r diretly s 0 = s n if is understd. n epting pth is pth frm n initil vertex t finl vertex. n utmtn is unmiguus if tw epting pths hve distint lels. The utmtn f Figure. is unmiguus. The lnguge regnized y n utmtn is the set L() f ll lels f its epting pths: L() = { u T s, t (s = u t s, t ) }. Nte tht ε L() if there exists vertex s whih is initil nd finl: s, s. The utmtn f Figure. regnizes the lnguge L() = { m n 0 < n m } { n n n > 0 } { n n 0 }. The lnguges regnized y finite utmt re the regulr lnguges ver T. We generlize finite utmt t regulr utmt using funtinl grph grmmrs. T define grph grmmr, we need t extend n r (resp. grph) t hyperr (resp. hypergrph). lthugh suh n extensin is nturl, this my explin why funtinl grph grmmrs re nt very widespred t the mment. But we will see in the lst setin tht fr ur purpse, we n restrit t grmmrs using nly rs. Let F e set f symls rnked y mpping : F IN ssiting t eh f F its rity (f) 0 suh tht F n = { f F (f) = n } is untle fr every n 0 with T F nd, F. hypergrph is suset f n 0 F n V n where V is n ritrry set. ny tuple (f, s,..., s (f) ), ls written fs...s (f), is hyperr f lel f nd

4 f suessive verties s,..., s (f). We dd the nditin tht the set f verties V is finite r untle, nd the set f lels F is finite. n r is hyperr fst lelled y f F nd is ls dented y s f t. Fr n, hyperr fs...s n is depited s n rrw lelled f nd suessively linking s,..., s n. Fr n = nd n = 0, it is respetively depited s lel f (lled lur) n vertex s nd s n islted lel f lled nstnt. This is illustrted in the next figures. Fr instne the fllwing hypergrph: = {4, 5, 5, 5 3, 6 3, 4, 6, 456} with, F nd F 3, is represented elw. 3 Figure. finite hypergrph. (lured) grph is hypergrph whse lels re nly f rity r : F F F. n utmtn ver the lphet T is grph with set f lels F T {, }. We n nw intrdue funtinl grph grmmrs t generte regulr utmt. grph grmmr R is finite set f rules f the frm fx...x (f) H where fx...x (f) is hyperr f lel f lled nn-terminl jining pirwise distint verties x... x (f) nd H is finite hypergrph. We dente y N R the set f nn-terminls f R i.e. the lels f the left hnd sides, y T R = { f F N R H Im(R), f F H } the terminls f R i.e. the lels f R whih re nt nn-terminls, nd y F R = N R T R the lels f R. We use grmmrs t generte utmt hene in the fllwing, we my ssume tht T R T {, }. Similrly t ntext-free grmmrs (n wrds), grph grmmr hs n xim: n initil finite hypergrph. T indite this xim, we ssume tht ny grmmr R hs nstnt nn-terminl Z N R F 0 whih is nt lel f ny right hnd side; the xim f R is the right hnd side H f the rule f Z : Z H Z F K fr ny K Im(R). Strting frm the xim, we wnt R t generte unique utmtn up t ismrphism. S we finlly ssume tht ny grmmr R is funtinl mening tht there is nly ne rule per nn-terminl: if (X, H), (Y, K) R with X() = Y () then (X, H) = (Y, K). Fr ny rule fx...x (f) H, we sy tht x,..., x (f) re the inputs f f, nd V H [H] is the set f utputs f f. T wrk with these grmmrs, it is simpler t ssume tht ny grmmr R is terminl-utside [C 07]: ny terminl r r lur in right hnd side links t t lest ne nn input vertex: H (T R V X V X T R V X ) = fr ny rule (X, H) R. We will use upper-se letters, B, C,... fr nn-terminls nd lwer-se letters,,... fr terminls. Here is n exmple f (funtinl grph) grmmr R : 4 6 5

5 Z ; B ; B 3 Figure.3 (funtinl grph) grmmr. Fr the previus grmmr R, we hve N R = {Z,, B} with Z the xim nd () = (B) = 3, T R = {,,, } nd,, 3 re the inputs f nd B. iven grmmr R, the rewriting reltin is the inry reltin etween R hypergrphs defined s fllws: M rewrites int N, written M R N, if we n hse nn-terminl hyperr X = s...s p in M nd rule x...x p H in R suh tht N n e tined y repling X y H in M: N = (M X) h(h) fr sme funtin h mpping eh x i t s i, nd the ther verties f H injetively t verties utside f M; this rewriting is dented y M R, X f hyperr X is extended in n vius wy t the rewriting R, E f nn-terminl hyperrs. The mplete prllel rewriting = R rewriting rding t the set f ll nn-terminl hyperrs: M= R N. The rewriting R, X f ny set E is simultneus N if M N where E is the set f ll nn-terminl hyperrs f M. We depit elw the first three steps f the prllel derivtin f the previus grmmr frm its nstnt nn-terminl Z: Z = = B = Figure.4 Prllel derivtin fr the grmmr f Figure.3. iven grmmr R, we restrit ny hypergrph H t the utmtn [H] f its terminl rs nd lured verties: [H] = H (T V H V H {, } V H ). n utmtn is generted y R (frm its xim) if elngs t the fllwing set R ω f ismrphi utmt: R ω = { n 0 [H n] Z H 0 =... H n = H n+... }. R R R Nte tht in ll generlity, we need t nsider hypergrphs with multipliities. Hwever using n pprprite nrml frm, this tehnility n e sfely mitted [C 07]. Fr instne the utmtn f Figure. is generted y the grmmr f Figure.3. regulr utmtn is n utmtn generted y (funtinl grph) grmmr. Nte tht regulr utmtn hs finite numer f nn-ismrphi nneted mpnents, nd hs finite numer f distint vertex degrees. nther exmple is given y the fllwing grmmr: Z ; R, E

6 whih genertes the fllwing utmtn: regnizing the lnguge { uũ u {, } + } where ũ is the mirrr f u. The lnguge regnized y grmmr R is the lnguge L(R) regnized y its generted utmtn: L(R) = L() fr (ny) R ω. This lnguge is well-defined sine ll utmt generted y given grmmr re ismrphi. grmmr R is n unmiguus grmmr if the utmtn it genertes is unmiguus. There is nnil wy t generte the regulr utmt f finite degree whih llws t hrterize these utmt withut the expliit use f grmmrs. This is the finite dempsitin y distne. The inverse f n utmtn is the utmtn tined frm y reversing its rs nd y exhnging initil nd finl verties: = { t s s t } { s s } { s s }. S regnizes the mirrr f the wrds regnized y. The restritin I f t suset I f verties is the sugrph f indued y I : I = (T I I {, } I). The distne d I (s) f vertex s t I is the miniml length f the undireted pths u etween s nd I : d I (s) = min{ u r I, r = s } with min( ) = +. We tke new lur # F {, } nd define fr ny integer n 0, De # n(, I) = { s di(s) n } { #s d I (s) = n } In prtiulr De # 0 (, I) = { # s s I }. We sy tht n utmtn is finitely dempsle y distne if fr eh nneted mpnent C f there exists finite nn empty set I f verties suh tht n 0 De# n(c, I) hs finite numer f nn-ismrphi nneted mpnents. Suh definitin llws the hrteriztin f the lss f ll utmt f finite degree whih re regulr. Therem.5 n utmtn f finite degree is regulr if nd nly if it is finitely dempsle y distne nd it hs nly finite numer f nn ismrphi nneted mpnents.

7 The prf is given in [C 07] nd is slight extensin f [MS 85] (ut withut using pushdwn utmt). Regulr utmt f finite degree re ls the trnsitin grphs f pushdwn utmt restrited t regulr sets f nfigurtins nd with regulr sets f initil nd finl nfigurtins. In prtiulr, regulr utmt f finite degree regnize the sme lnguges s pushdwn utmt. Prpsitin.6 The (resp. unmiguus) regulr utmt regnize extly the (resp. unmiguus) ntext-free lnguges. This prpsitin remins true if we restrit t utmt f finite degree. We nw use grmmrs t extend the fmily f regulr lnguges t len lgers f unmiguus ntext-free lnguges. 3 Synhrniztin f regulr utmt We intrdue the ide f synhrniztin etween grmmrs. The lss f lnguges synhrnized y grmmr R re the lnguges regnized y grmmrs synhrnized y R. We shw tht these fmilies f lnguges re lsed under unin y pplying the sum f grmmrs, re lsed under intersetin with regulr lnguge y defining the synhrniztin prdut f grmmr with finite utmtn, nd re lsed under intersetin (in the se f grmmrs generting unmiguus utmt) y perfrming the synhrniztin prdut f grmmrs. Finlly we shw tht ll grmmrs generting the sme utmtn synhrnize the sme lnguges. T eh vertex s f n utmtn R ω generted y grmmr R, we ssite nn negtive integer l(s) whih is the miniml numer f rewritings pplied frm the xim neessry t reh s. Mre preisely fr = n 0 [H n] with Z H 0 =...H n = H n+..., the level l(s) f s V, ls written l R (s) R R R t speify nd R,is l(s) = min{ n s V Hn }. We depit elw the levels f sme verties f the regulr utmtn f Figure. generted y the grmmr f Figure.3. This utmtn is represented y verties f inresing level: verties t sme level re ligned vertilly Figure 3. Vertex levels with the grmmr f Figure.3.

8 We sy tht grmmr S is synhrnized y grmmr R written S R, r equivlently tht R synhrnizes S written R S, if fr ny epting pth µ lel y u f the utmtn generted y S, there is n epting pth λ lel y u f the utmtn generted y R suh tht fr every prefix v f u, the prefixes f λ nd µ lelled y v led t verties f the sme level: fr (ny) R ω nd (ny) H S ω nd fr ny t 0 t... n t n with t 0, t n H, H H there exists s 0 s... n s n with s 0, s n nd l R (s i) = l S H (t i) i [0, n]. Fr instne the grmmr f Figure.3 synhrnizes the fllwing grmmr: Z ; B ; Figure 3. grmmr synhrnized y the grmmr f Figure.3. In prtiulr fr S R, we hve L(S) L(R). Nte tht the empty grmmr {(Z, )} is synhrnized y ny grmmr. The synhrniztin reltin is reflexive nd trnsitive reltin. We dente the i-synhrniztin reltin: R S if R S nd S R. Nte tht i-synhrnized grmmrs R S my generte distint utmt: R ω S ω. Fr ny grmmr R, the imge f R y is the fmily (R) = { S S R } f grmmrs synhrnized y R nd Syn(R) = { L(S) S R } is the fmily f lnguges synhrnized y R. Nte tht Syn(R) is fmily f lnguges inluded in L(R) nd ntining the empty lnguge nd L(R). Nte ls tht Syn(R) = Syn(S) fr R S. Stndrd pertins n finite utmt re extended t grmmrs in rder t tin lsure prperties f Syn(R). Fr instne the synhrniztin prdut f finite utmt is extended t ritrry utmt nd H y H = { (s, p) (t, q) s t p q } { (s, p) s p H } { (s, p) s p H } whih regnizes L( H) = L() L(H). This llws us t define the synhrniztin prdut R K f grmmr R with finite utmtn K [CH 08]. Let {q,..., q n } e the vertex set f K. T eh N R, we ssite new syml (, n) f rity () n exept tht (Z, 0) = Z, nd t eh hyperr r...r m with m = (), we ssite the hyperr (r...r m ) K = (, n)(r, q )...(r, q n )...(r m, q )...(r m, q n ). The grmmr R K ssites t eh rule (X, H) R the fllwing rule: X K [H] K { (BY ) K BY H B N R }. Exmple 3.3 Let us nsider the fllwing grmmr R : Z H ; s t generting the fllwing (regulr) utmtn : B

9 nd regnizing the restrited Dyk lnguge D ver the pir (, ) [Be 79] : L(R) = L() = D. We nsider the fllwing finite utmtn K : regnizing the set f wrds ver {, } hving n even numer f. S R K is the fllwing grmmr: p q (s,p) Z (, ) (s,q) (,p) ; (, ) (,q) (,p) (,q) (t,p) (, ) (t,q) generting the utmtn K : whih regnizes D restrited t the wrds with n even numer f. The synhrniztin prdut f grmmr R with finite utmtn K is synhrnized y R i.e. R K R nd regnizes L(R K) = L(R) L(K). Prpsitin 3.4 Fr ny grmmr R, the fmily Syn(R) is lsed under intersetin with regulr lnguge. Prpsitins.6 nd 3.4 imply the well-knwn lsure prperty f the fmily f ntext-free lnguges under intersetin with regulr lnguge. s R K is unmiguus fr R unmiguus nd K deterministi, it ls fllws Therem 6.4. f [H 78] : the fmily f unmiguus ntext-free lnguges is lsed under intersetin with regulr lnguge. nther si pertin n finite utmt is the disjint unin. This pertin is extended t ny grmmrs R nd R. Fr ny i {, }, we dente R i = R ( i { i i T } {i, i} ) in rder t distinguish the verties f R nd R. Fr (Z, H ) R nd (Z, H ) R, the sum f R nd R is the grmmr R + R = {(Z, H H )} (R {(Z, H )}) (R {(Z, H )}). S (R + R ) ω = { R ω R ω V V = } hene L(R + R ) = L(R ) L(R ). In prtiulr if S R nd S R then S + S R + R. Prpsitin 3.5 Fr ny grmmr R, Syn(R) is lsed under unin.

10 The synhrniztin prdut f regulr utmt n e nn regulr. Furthermre fr the regulr utmtn :,,,,,, the lnguges { m m n m, n 0 } nd { m n n m, n 0 } re in Syn() ut their intersetin { n n n n 0 } is nt ntext-free lnguge. The synhrniztin prdut f grmmr with finite utmtn is extended fr tw grmmrs R nd S fr generting the level synhrniztin prdut lh f their generted utmt R ω nd H S ω whih is the restritin f H t pirs f verties with sme level: lh = ( H) P fr P = { (s, p) V V H l R (s) = ls H (p) }. This prdut n e generted y grmmr R ls tht we define. Let (, B) N R N S e ny pir f nnterminls nd E [, ()] [, (B)] e inry reltin ver inputs suh tht fr ll i, j [, ()], if E(i) E(j) then E(i) = E(j), where E(i) = {j (i, j) E} dentes the imge f i [, ()] y E. Intuitively fr pir (, B) N R N S f nn-terminls, reltin E [, ()] [, (B)] is used t memrize whih entries f nd B re eing synhrnized. T ny suh, B nd E, we ssite new syml [, B, E] f rity E (where [Z, Z, ] is ssimilted t Z). T eh nn-terminl hyperr r...r m f R ( N R nd m = ()) nd eh nn-terminl hyperr Bs...s n f S (B N S nd n = (B)), we ssite the hyperr [r...r m, Bs... s n, E] = [, B, E](r, s ) E... (r, s n ) E... (r m, s ) E...(r m, s n ) E with (r i, s j ) E = (r i, s j ) if (i, j) E, nd ε therwise. The grmmr R ls is then defined y ssiting t eh (X, P) R, eh (BY, Q) S, nd eh E ( [ ()] [ (B)], ) the rule f left hnd side [X, BY, E] nd f right hnd side [P] [Q] E {[CU, DV, E ] CU P C N R DV Q D N S } with E = { (X(i), Y (j)) (i, j) E } ( V P V X ) ( VQ V Y ) nd E = { (i, j) [ (C)] [ (D)] (U(i), V (j)) E }. Nte tht R ls is synhrnized y R nd S, nd is i-synhrnnized with S fr S R. Furthermre R ls genertes lh fr R ω nd H S ω hene regnizes suset f L(R) L(S). Hwever fr grmmrs S nd S synhrnized y n unmiguus grmmr R, we hve L(S ls ) = L(S) L(S ). Prpsitin 3.6 Fr ny unmiguus grmmr R, the fmily Syn(R) is lsed under intersetin. By extending si pertins n finite utmt t grmmrs, it ppers tht grph grmmrs re t ntext-free lnguges wht finite utmt re t regulr lnguges. We will ntinue these extensins in the next setin. Let us present fundmentl result nerning grmmr synhrniztin, whih sttes tht Syn(R) is independent f the wy the utmtn R ω is generted.

11 Therem 3.7 Fr ny grmmrs R nd S suh tht R ω = S ω, we hve Syn(R) = Syn(S). Prf sketh. By symmetry f R nd S, it is suffiient t shw tht Syn(R) Syn(S). Let R R. We wnt t shw tht L(R ) Syn(S). We hve t shw the existene f S S suh tht L(S ) = L(R ). Nte tht it is pssile tht there is n grmmr S synhrnized y S nd generting the sme utmtn s R (i.e. S S nd S ω = R ω ). Let R ω = S ω. ny vertex s f hs level l R (s) rding t R nd level l S (s) rding t S. Let H R ω nd let K = ( lh) P e the utmtn tined y level synhrniztin prdut f with H nd restrited t the set P f verties essile frm nd -essile frm. The restritin y essiility frm nd -essiility frm n de dne y i-synhrnized grmmr [C 08]. By definitin f R lr, the utmtn K n e generted y grmmr R i-synhrnized t R with l R K (s, p) = lr (s) = lr H (p) fr every (s, p) V K. In prtiulr L(K) = L(R ). Let us shw tht K is generted y grmmr synhrnized y S. We give the prf fr R ω f finite degree. In tht se nd fr = N R (), l R (s) lr (t).d (s, t) fr every s, t V. Furthermre K is ls f finite degree. We shw tht K is finitely dempsle nt y distne ut rding t l S K (s) fr the verties (s, p) f K. Let n 0 nd C e nneted mpnent f K {(s,p) VK l S (s) n }. S C is fully determined y its frntier : Fr K (C) = V C V K C its interfe : Int K (C) = { s t {s, t} Fr K (C) }. C Let (s 0, p 0 ) Fr K (C) nd D e the nneted mpnent f { s l S (s) n } ntining s 0. It remins t find und independent f n suh tht l R K (s, p) lr K (t, q) fr every (s, p), (t, q) Fr K(C). Fr ny (s, p), (t, q) Fr K (C), we hve s, t Fr (D) hene d D (s, t) is unded y the integer = mx{ d Sω ()(i, j) < + N S i, j [, ()] } whse S ω () = { n 0 [H n]... () = H 0 =... H n = H n+... } S S thus it fllws tht l R K (s, p) lr K (t, q) = lr (s) lr (t) d (s, t) d D (s, t). Fr f infinite degree nd y Prpsitin 4.9, we n express Syn() s n ε-lsure f Syn(H) fr sme regulr utmtn H f finite degree using ε- trnsitins. Therem 3.7 llws t trnsfer the nept f grmmr synhrniztin t the level f regulr utmt: fr ny regulr utmtn, we n define

12 Syn() = Syn(R) fr (ny) R suh tht R ω. Let us illustrte these ides y presenting sme exmples f well-knwn sufmilies f ntext-free lnguges tined y synhrniztin. Exmple 3.8 Fr ny finite utmtn, Syn() is the fmily f regulr lnguges inluded in L(). Exmple 3.9 Fr the fllwing regulr utmtn : Syn() is the fmily f input-driven lnguges [Me 80] with pushing, ppping nd internl. s the initil vertex is nt sure f n r lelled y, Syn() des nt ntin ll the regulr lnguges. Exmple 3.0 We mplete the previus utmtn y dding n -lp n the initil vertex t tin the fllwing utmtn :, The set Syn() is the fmily f visily pushdwn lnguges [M 04] with pushing, ppping nd internl. Exmple 3. Fr the fllwing regulr utmtn : the set Syn() is the fmily f lned lnguges [BB 0] with, pushing with their rrespnding ppping letters,, nd is internl. Exmple 3. Fr the grmmr R f Figure 3., Syn(R) is the fmily f lnguges generted y the fllwing liner ntex-free grmmrs: I = P + m ( m ) with m 0 nd P { i j j i m } = Q + n ( n ) with n > 0 nd Q { i j j i n }. Fr eh regulr utmtn mng the previus exmples, Syn() is len lger rding t L() nd, exept fr the lst tw exmples, is ls lsed under ntentin nd its itertin. We nw nsider new lsure prperties f synhrnized lnguges fr regulr utmt.

13 4 Clsure prperties We hve seen tht the fmily Syn() f lnguges synhrnized y regulr utmtn is lsed under unin nd under intersetin with regulr lnguge, nd under intersetin when is unmiguus. In this setin, we nsider the lsure f Syn() under mplement reltive t L() nd under ntentin nd its trnsitive lsure. T tin these lsure prperties, we first pply grmmr nrmliztins preserving the synhrnized lnguges. These nrmliztins ls llw us t dd ε-rs t ny regulr utmtn t get regulr utmtn f finite degree with the sme synhrnized lnguges. First we put ny grmmr in n equivlent nrml frm with the sme set f synhrnized lnguges. s in the se f finite utmt, we trnsfrm ny utmtn int the pinted utmtn whih is lnguge equivlent L( ) = L(), with unique initil vertex V whih is gl f n r nd n e finl, nd with unique nn initil nd finl vertex V whih is sure f n r: = ( {, } V ) {, } { s (s, s ) } { t s (s t s ) } { s t (s t t ) } { s, t (s t s, t ) }. Fr instne, the finite degree regulr utmtn f Figure. is trnsfrmed int the fllwing infinite degree regulr utmtn : Figure 4. pinted regulr utmtn. Nte tht if is unmiguus, remins unmiguus. The pinted trnsfrmtin f regulr utmtn remins regulr utmtn whih n e generted y n 0-grmmr : nly the xim hs initil nd finl verties. Let R e ny grmmr nd, e tw symls whih re nt verties f R. Let R ω with, V. We define n 0-grmmr R generting nd preserving the synhrnized lnguges: Syn(R ) = Syn(R). First we trnsfrm R int grmmr R in whih we memrize in the nnterminls the input verties whih re linked t initil r finl verties f the generted utmtn. Mre preisely t ny N R nd I, J [, ()], we ssite new syml I,J f rity () with Z = Z,. We define the grmmr

14 R ssiting t eh (X, H) R nd I, J [, ()] the fllwing rule: I,J X [H] { B I,J Y BY H B N R } with I = { i Y (i) I Y (i) H } nd J = { j Y (j) J Y (j) H } nd we restrit the rules f R t the nn-terminls essile frm Z. Nte tht the set L(R) T f letters regnized y R n e determined s { ( I,J X, H) R ( i I t, X(i) [H] t t H) ( j J s, s X(j) s H) ( s, t, s t s, t H) } [H] [H] nd ε L(R) H Im( R) s (s, s H). T ny N R {Z} nd ny I, J [, ()], we ssite new syml I,J f rity () +, nd we define the grmmr R ntining the xim rule Z H, {, } { ε L(R) } { L(R) T } fr (Z, H) R, nd fr ny ( I,J X, H) R with Z, we tke in R the rule I,J X H I,J suh tht H I,J is the fllwing hypergrph: H I,J = ([H] {, }) V H ) { B P,Q X B P,QX H B P,Q N R } { t i I (X(i) t) s (s H s t) } [H] [H] { s j J (s X(j)) t (t H s t) } nd we put R int terminl-utside frm [C 07]. [H] [H] Exmple 4. Let us nsider the fllwing grmmr R : Z ; B B C ; C generting the fllwing utmtn (with levels f sme verties): First this grmmr is trnsfrmed int the fllwing grmmr R : Z, ;, B, B, C, C ;,, In prtiulr ε,, L(R). Then R is trnsfrmed int the grmmr R :

15 Z,, ;, B, B, C, ; C,, tht we put in terminl-utside frm: Z,, ;, B, B,, ; C, C,,,, S R genertes :,,,,,, The grmmrs R nd R synhrnize the sme lnguges. Prpsitin 4.3 Fr ny regulr utmtn with, V, the pinted utmtn remins regulr nd Syn( ) = Syn(). It fllws tht, in rder t define fmilies f lnguges y synhrniztin y regulr utmtn, we n restrit t pinted utmt. strnger nrmliztin is t trnsfrm ny grmmr R int grmmr S suh tht Syn(S) = Syn(R) nd S is n r-grmmr in the fllwing sense: S is n 0-grmmr whse ny nn-terminl N S {Z} is f rity, nd fr ny nn xim rule st H, there is n r in H f gl s r f sure t : fr ny p q, we hve p t nd q s. H We n trnsfrmed ny 0-grmmr R int i-synhrnized r-grmmr R. We ssume tht eh rule f R is f the frm... () H fr ny N R.

16 We tke new syml 0 (nt vertex f R) nd new lel i,j f rity fr eh N R nd eh i, j [, ()] in rder t generte pths frm i t j in R ω (... ()). We define the splitting f ny F R -hypergrph withut vertex 0 s eing the grph: = [] { X(i) i,j X(j) X N R i, j [ ()] } nd fr p, q V nd P V with 0 V, we define p,p,q = ( { s t t p s q s, t P } ) fr p q I p,p,p = ( { s t t p s, t P } { s 0 s p } ) J with I = { s p = s = q } nd J = { s p = s = 0 }. This llws t define the splitting R f R s eing the fllwing r-grmmr: Z H Z i,j h i,j ( (H ) i,[ ()] {i,j},j ) fr eh NR nd i, j [, ()] where h i,j is the vertex renming defined y h i,j (i) =, h i,j (j) =, h i,j (x) = x therwise, fr i j h i,i (i) =, h i,i (0) =, h i,i (x) = x therwise. Thus R nd R re i-synhrnized, nd R is unmiguus when R is unmiguus. Nte tht we n put R int redued frm y remving ny nn-terminl i,j suh tht R ω ( i,j ) is withut pth frm t. Exmple 4.4 The fllwing 0-grmmr R : Z d ; B ; 3 B 3 genertes the fllwing utmtn : d d d The splitting R f R is the fllwing grmmr:,, Z, ;, B, ; B,,, d B,3 ; B,3, generting the fllwing utmtn:

17 d d d s R R, we hve Syn(R) = Syn( R ). T study lsure prperties f Syn(R) fr ny grmmr R, we n wrk with its nrml frm R whih is n r-grmmr generting pinted utmtn. This nrmliztin is relly useful t study the lsure prperty f Syn(R) under mplement reltive t L(R), under ntentin nd its itertin. We hve seen tht Syn(R) is nt lsed in generl under intersetin, hene it is nt lsed under mplement rding t L(R) sine fr ny L, M L(R), L M = L(R) [(L(R) L) (L(R) M)]. Fr R unmiguus, Syn(R) is lsed under intersetin, nd this remins true under mplement rding t L(R) [C 08]. We give here simpler nstrutin. s R remins unmiguus, we n ssume tht R is n r-grmmr. Let S R. We wnt t shw tht L(R) L(S) Syn(R). S S is n 0-grmmr nd S is level-unmiguus s defined in [C 08]. Thus S is level-unmiguus r-grmmr. We tke new lur F {, } nd fr ny grmmr S, we dente S (resp. S ) the grmmr tined frm S y repling the finl lur y (resp. y ). S R + S is n r-grmmr nd (R + S ) is levelunmiguus. It remins t pply the grmmr determiniztin in [C 08] t get the grmmr R/S = Det(R + S ) suh tht (R/S) is unmiguus nd i-synhrnized t (R+ S ). Finlly we keep in R/S the finl verties whih re nt lured y t tin grmmr synhrnized y R nd regnizing L(R) L(S). Therem 4.5 Fr ny unmiguus regulr utmtn, the set Syn() is n effetive len lger rding t L(), ntining ll the regulr lnguges inluded in L(). S we n deide the inlusin L(S) L(S ) fr tw grmmrs S nd S synhrnized y mmn unmiguus grmmr. Furthermre fr grmmrs R nd R suh tht R + R is level-unmiguus, Syn(R + R ) = { L L L Syn(R ) L Syn(R ) } is len lger inluded in L(R ) L(R ), ntining Syn(R ) nd Syn(R ). The utmt f Exmples 3.8 t 3. re unmiguus hene their fmilies f synhrnized lnguges re len lger. This regulr utmtn :

18 is -miguus: there re tw epting pths fr the wrds n n n with n > 0 nd unique epting pth fr the ther epted wrds. But Syn() is nt lsed under intersetin sine { m m n m, n 0 } nd { m n n m, n 0 } re lnguges synhrnized y. Fr ny regulr utmtn, the lsure f Syn() under ntentin (resp. under its trnsitive lsure + ) des nt require the unmiguity f. s L() Syn(), neessry nditin is t hve L().L() Syn() (resp. L() + Syn()). Nte tht this neessry nditin implies tht L() is lsed under (resp. + ). In prtiulr Syn() is nt lsed under nd + fr the utmt f Exmples 3. nd 3.. But this neessry nditin is nt suffiient sine the fllwing regulr utmtn :, regnizes L() = ε + M( + ) fr M = { n n n > 0 }, hene L().L() = L() = L() + ut M Syn() nd M.M, M + Syn(). Let us give simple nd generl nditin n grmmr R suh tht Syn(R) is lsed under nd +. We sy tht grmmr is itertive if ny initil vertex is in the xim nd fr (ny) R ω nd ny epting pth s 0 s... n s n with s 0, s n nd fr ny finl vertex t i.e. t, there exists pth t... n t n with t n suh tht l(t i ) = l(t) + l(s i ) fr ll i [, n]. t Fr instne the utmtn f Exmple 3.9 n e generted y n itertive grmmr. nd ny 0-grmmr generting regulr utmtn hving unique initil vertex whih is the unique finl vertex, is itertive. Stndrd nstrutins n finite utmt fr the ntentin nd its itertin n e extended t itertive grmmrs. Prpsitin 4.6 Fr ny itertive grmmr R, the fmily Syn(R) is lsed under ntentin nd its trnsitive lsure. Hwever the utmtn f Exmple 3.0 nnt e generted y n iterted grmmr ut Syn() is lsed under nd +. We n ls tin fmilies f synhrnized lnguges whih re lsed under nd + y sturting grmmrs. The sturtin + f n utmtn is the utmtn + = { s r r t (s t t ) } regnizing L( + ) = (L()) +.

19 Nte tht if is regulr with infinite sets f initil nd finl verties, + n e nn regulr (ut is lwys prefix-regnizle). If is generted y n 0- grmmr R, its sturtin + n e generted y grmmr R + tht we define. Let (Z, H) e the xim rule f R nd r,..., r p e the initil verties f H ; we n ssume tht r,..., r p re nt verties f R {(Z, H)}. T eh N R {Z} nd I [, ()], we ssite new syml I f rity () + p nd we define R + with the fllwing rules: Z [H] + { { i X(i) H } Xr...r p X H N R } I Xr...r p K I fr eh (X, K) R nd Z nd I [, ()] whse K I is the utmtn tined frm K s fllws: K I = [K] { s r j j [p] i I (s X(i)) } { B { j i I, Y (j)=x(i) } Y r...r p BY K B N R }. S R is synhrnized y R + nd + (R + ) ω fr R ω. T hrterize Syn(R + ) frm Syn(R), we define the regulr lsure Reg(E) f ny lnguge fmily E s eing the smllest fmily f lnguges ntining E nd lsed under,, +. Prpsitin 4.7 Fr ny 0-grmmr R, Syn(R + ) = Reg(Syn(R)). By Prpsitins 4.3, 4.6 nd 4.7, the fllwing regulr utmtn : K,,,,,,, hs the sme synhrnized lnguges thn the utmtn f Exmple 3.9: Syn() is the fmily f input-driven lnguges (fr pushing, ppping nd internl). By dding n -lp n the initil (nd finl) vertex f, we tin n utmtn H suh tht Syn(H) is the fmily f visily pushdwn lnguges hene y Prpsitin 4.7, is lsed under nd +. Exmple 4.8 nturl extensin f the visily pushdwn lnguges is t dd reset letters. Fr pushing, ppping nd internl, we dd reset letter d t define the fllwing regulr utmtn :,, d, d d d ny lnguge f Syn() is visily pushdwn lnguge tking d s n internl letter, ut nt the nverse: { n d n n 0 } Syn(). By Therem 4.5, Syn() is len lger. Furthermre the fllwing utmtn H :

20 ,,, d,,, d,,, d,,, d stisfies Syn(H) = Syn() nd H + = H hene y Prpsitin 4.7, Syn() is ls lsed under nd +. Nte tht the utmt f the previus exmple hve infinite degree. Furthermre fr ny utmtn f finite degree hving n infinite set f initil r finl verties, the pinted utmtn is f infinite degree. Hwever ny regulr utmtn f infinite degree (in ft ny prefix-regnizle utmtn) n e tined y ǫ-lsure frm regulr utmtn f finite degree using ε-trnsitins. Fr instne let us tke new letter e T (insted f the empty wrd) nd let us dente π e the mrphism ersing e in the wrds ver T {e}: π e () = fr ny T nd π e (e) = ε, tht we extend y unin t ny lnguge L (T {e}) : π e (L) = { π e (u) u L }, nd y pwerset t ny fmily P f lnguges: π e (P) = { π e (L) L P }. The fllwing regulr utmtn K :,, d, d e d d e e is f finite degree nd stisfies π e (Syn(K)) = Syn() fr the utmtn f Exmple 4.8. Let us give simple trnsfrmtin f ny grmmr R t grmmr R e suh tht Re ω is f finite degree nd π e(syn(r e )) = Syn(R). s Syn(R) = Syn( R ), we restrit this trnsfrmtin t r-grmmrs. Let R e n r-grmmr. We define R e t e n r-grmmr tined frm R y repling eh nn xim rule st H y the rule: st ( [H] {s e e s e, t e t} h(h [H]) ) P with s e, t e e new verties nd h the vertex mpping defined fr ny r V H y h(r) = r if r {s, t}, h(s) = s e nd h(t) = t e, nd P is the set f verties essile frm s nd -essile frm t. Fr instne the r-grmmr R Z ; is trnsfrmed int the fllwing r-grmmr R e : Z ; e e

21 Fr ny rule f R e, the inputs re seprted frm the utputs (y e-trnsitins), hene R ω e is f finite degree. Furthermre this trnsfrmtin preserves the synhrnized lnguges. Prpsitin 4.9 Fr ny r-grmmr R, Syn(R) = π e (Syn(R e )). S fr ny R, Syn(R) = π e (Syn( R e)) nd ( R e) ω is f finite degree. ll the nstrutins given in this pper re nturl generliztins f usul trnsfrmtins n finite utmt t grph grmmrs. In this wy, si lsure prperties uld e lifted t su-fmilies f ntext-free lnguges. Cnlusin The synhrniztin f regulr utmt is defined thrugh devies generting these utmt, nmely funtinl grph grmmrs. It n ls e defined using pushdwn utmt with ε-trnsitins [NS 07] euse Therem 3.7 sserts tht the fmily f lnguges synhrnized y regulr utmtn is independent f the wy the utmtn is generted; it is grph-relted ntin. This pper shws tht the mehnism f funtinl grph grmmrs prvides nturl nstrutins n regulr utmt generlizing usul nstrutins n finite utmt. This pper is ls n invittin t extend the ntin f synhrniztin t mre generl su-fmilies f utmt. knwledgements Mny thnks t rnud Cryl nd ntine Meyer fr helping me prepre the finl versin f this pper. Referenes [M 04] R. lur nd P. Mdhusudn Visily pushdwn lnguges, 36 th STOC, CM Preedings, L. Bi (Ed.), 0 (004). [Be 79] J. Berstel Trnsdutins nd ntext-free lnguges, Ed. Teuner, pp. 78, 979. [BB 0] J. Berstel nd L. Bssn Blned grmmrs nd their lnguges, Frml nd Nturl Cmputing, LNCS 300, W. Bruer, H. Ehrig, J. Krhumäki,. Slm (Eds.), 3 5 (00). [C 06] D. Cul Synhrniztin f pushdwn utmt, 0 th DLT, LNCS 4036, O. Irr, Z. Dng (Eds.), 0-3 (006). [C 07] D. Cul Deterministi grph grmmrs, Texts in Lgi nd mes, msterdm University Press, J. Flum, E. rädel, T. Wilke (Eds.), (007). [C 08] D. Cul Blen lgers f unmiguus ntext-free lnguges, 8 th FSTTCS, Dgstuhl Reserh Online Pulitin Server, R. Hrihrn, M. Mukund, V. Viny (Eds.) (008).

22 [CH 08] D. Cul nd S. Hssen Synhrniztin f grmmrs, 3 rd CSR, LNCS 500, E. Hirsh,. Rzrv,. Semenv,. Slissenk (Eds.), 0 (008). [H 78] M. Hrrisn Intrdutin t frml lnguge thery, ddisn-wesley (978). [Me 80] K. Mehlhrn Peling muntin rnges nd its pplitin t DCFL regnitin, 7 th ICLP, LNCS 85, J. de Bkker, J. vn Leeuwen (Eds.), 4 43 (980). [MS 85] D. Muller nd P. Shupp The thery f ends, pushdwn utmt, nd send-rder lgi, Theretil Cmputer Siene 37, 5 75 (985). [NS 07] D. Nwtk nd J. Sr Height-deterministi pushdwn utmt, 3 nd MFCS, LNCS 4708, L. Kuer,. Kuer (Eds.), 5 34 (007).