Synchronization of regular automata

Transcription

1 Synhrniztin f rgulr utmt Diir Cul IM CNRS Univrsité Pris-Est ul@univ-mlv.fr strt. Funtinl grph grmmrs r finit vis whih gnrt th lss f rgulr utmt. W rll th ntin f synhrniztin y grmmrs, n fr ny givn grmmr w nsir th lss f lngugs rgniz y utmt gnrt y ll its synhrniz grmmrs. Th synhrniztin is n utmtn-rlt ntin: ll grmmrs gnrting th sm utmtn synhrniz th sm lngugs. Whn th synhrnizing utmtn is unmiguus, th lss f its synhrniz lngugs frms n fftiv ln lgr lying twn th lsss f rgulr lngugs n unmiguus ntxt-fr lngugs. W itinlly prvi suffiint nitins fr suh lsss t ls unr ntntin n its itrtin. Intrutin n utmtn vr sm lpht n simply sn s finit r untl st f lll rs tgthr with tw sts f initil n finl vrtis. Suh n utmtn rgnizs th lngug f ll wrs llling n pting pth, i.. pth ling frm n initil t finl vrtx. It is wll-knwn tht finit utmt rgniz th rgulr lngugs. By pplying si nstrutins t finit utmt, w tin th ni lsur prprtis f rgulr lngugs, nmly thir lsur unr ln prtins, ntntin n its itrtin. Fr instn th synhrniztin prut n th trminiztin f finit utmt rsptivly yil th lsur f rgulr lngugs unr intrstin n unr mplmnt. This i n xtn t mr gnrl lsss f utmt. In this ppr, w will intrst in th lss f rgulr utmt, whih rgniz ntxtfr lngugs n r fin s th (gnrlly infinit) utmt gnrt y funtinl grph grmmrs [C 07]. Rgulr utmt f finit gr r ls prisly ths utmt whih n finitly mps y istn, s wll s th rgulr rstritins f trnsitin grphs f pushwn utmt [MS 85], [C 07]. Evn thugh th lss f ntxt-fr lngugs s nt njy th sm lsur prprtis s rgulr lngugs, n n fin sulsss f ntxt-fr lngugs whih, using th ntin f synhrniztin. Th ntin f synhrniztin ws first fin twn grmmrs [CH 08]. grmmr S is synhrniz y grmmr R if fr ny pting pth µ f (th grph gnrt y) S, thr xists n pting pth λ f R with th sm ll u suh tht λ n µ r synhrniz: fr vry prfix v f u, th prfixs

2 f λ n µ lll y v l t vrtis f th sm lvl (whr th lvl f vrtx is th miniml numr f rwriting stps nssry fr th grmmr t pru it). lngug is synhrniz y grmmr R if it is rgniz y n utmtn gnrt y grmmr synhrniz y R. funmntl rsult is tht tw grmmrs gnrting th sm utmtn yil th sm lss f synhrniz lngugs [C 08]. This wy, th ntin f synhrniztin n trnsfrr t th lvl f utmt: fr rgulr utmtn, th fmily Syn() is th st f lngugs synhrniz y ny grmmr gnrting. By xtning th v-mntin nstrutins frm finit utmt t grmmrs, n n stlish svrl lsur prprtis f ths fmilis f synhrniz lngugs. Th sum f tw grmmrs n th synhrniztin prut f grmmr with finit utmtn rsptivly ntil th lsur f Syn() unr unin n unr intrstin with rgulr lngug fr ny rgulr utmtn. Th (lvl prsrving) synhrniztin prut f tw grmmrs yils th lsur unr intrstin f Syn() whn is unmiguus i.. whn ny tw pting pths f hv istint lls. Nrmlizing f grmmr int grmmr nly ntining rs n thn th (lvl prsrving) trminiztin yils, fr ny unmiguus utmtn, th lsur f Syn() unr mplmnt rltiv t L(). This nrmliztin ls llws us t xprss Syn() in th s f n infinit gr utmtn, y prfrming th - lsur f Syn(H) fr sm finit gr utmtn H using n xtr ll. finl usful nrmliztin nly llws th prsn f initil n finl vrtis t lvl 0. It yils suffiint nitins fr th lsur f lsss f synhrniz lngugs unr ntntin n its itrtin. In Stin, w rll th finitin f rgulr utmt. In th nxt stin, w summriz knwn rsults n th synhrniztin f rgulr utmt [C 06], [NS 07], [CH 08], [C 08]. In th lst stin, w prsnt simplr nstrutin fr th lsur unr mplmnt f Syn() fr unmiguus [C 08] n prsnt nw rsults, spilly suffiint nitins fr th lsur f Syn() unr ntntin n its itrtin. Rgulr utmt n utmtn is lll rint simpl grph with input n utput vrtis. It rgnizs th st f wrs llling th pths frm n input t n utput. Finit utmt r utmt hving finit numr f vrtis, thy rgniz th lss f rgulr lngugs. Rgulr utmt r th utmt gnrt y funtinl grph grmmrs, thy rgniz th lss f ntxt-fr lngugs. ky rsult, riginlly u t Mullr n Shupp, intifis th rgulr utmt f finit gr with th utmt finitly gnrt y istn. n utmtn vr n lpht (finit st f symls) T f trminls is just st f rs lll vr T ( simpl lll rint grph) with initil n finl vrtis. W us tw symls n t mrk rsptivly th initil n finl vrtis. Mr prisly n utmtn is fin y T V V {, } V

3 whr V is n ritrry st suh tht th fllwing st f vrtis V = { s V (, s) (, s) T t V (, s, t) (, t, s) } is finit r untl. ny tripl (, s, t) is n r lll y frm sur s t gl t it is intifi with th lll trnsitin s t r irtly s t if is unrst. ny pir (, s) is lur vrtx s y {, } ls writtn s. vrtx is initil (rsp. finl) if it is lur y (rsp. ) i.. s (rsp. s ). n xmpl f n utmtn is givn 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 } n is rprsnt (up t ismrphism) lw. Figur. n utmtn. n utmtn is thus simpl vrtx- n r-lll grph. hs finit gr if fr ny vrtx s, th st { t (s t t s) } f its jnt vrtis is finit. Rll tht (s 0,, s,..., n, s n ) fr n 0 n s 0 s... s n n s n is pth frm s 0 t s n lll y u =... n u u w writ s 0 = s n r irtly s 0 = s n if is unrst. n pting pth is pth frm n initil vrtx t finl vrtx. n utmtn is unmiguus if tw pting pths hv istint lls. Th utmtn f Figur. is unmiguus. Th lngug rgniz y n utmtn is th st L() f ll lls f its pting pths: L() = { u T s, t (s = u t s, t ) }. Nt tht ε L() if thr xists vrtx s whih is initil n finl: s, s. Th utmtn f Figur. rgnizs th lngug L() = { m n 0 < n m } { n n n > 0 } { n n 0 }. Th lngugs rgniz y finit utmt r th rgulr lngugs vr T. W gnrliz finit utmt t rgulr utmt using funtinl grph grmmrs. T fin grph grmmr, w n t xtn n r (rsp. grph) t hyprr (rsp. hyprgrph). lthugh suh n xtnsin is nturl, this my xplin why funtinl grph grmmrs r nt vry wispr t th mmnt. But w will s in th lst stin tht fr ur purps, w n rstrit t grmmrs using nly rs. Lt F st f symls rnk y mpping : F IN ssiting t h f F its rity (f) 0 suh tht F n = { f F (f) = n } is untl fr vry n 0 with T F n, F. hyprgrph is sust f n 0 F n V n whr V is n ritrry st. ny

4 tupl (f, s,..., s (f) ), ls writtn fs...s (f), is hyprr f ll f n f sussiv vrtis s,..., s (f). W th nitin tht th st f vrtis V is finit r untl, n th st f lls F is finit. n r is hyprr fst lll y f F n is ls nt y s f t. Fr n, hyprr fs...s n is pit s n rrw lll f n sussivly linking s,..., s n. Fr n = n n = 0, it is rsptivly pit s ll f (ll lur) n vrtx s n s n islt ll f ll nstnt. This is illustrt in th nxt figurs. Fr instn th fllwing hyprgrph: = {4, 5, 5, 5 3, 6 3, 4, 6, 456} with, F n F 3, is rprsnt lw Figur. finit hyprgrph. (lur) grph is hyprgrph whs lls r nly f rity r : F F F. n utmtn vr th lpht T is grph with st f lls F T {, }. W n nw intru funtinl grph grmmrs t gnrt rgulr utmt. grph grmmr R is finit st f ruls f th frm fx...x (f) H whr fx...x (f) is hyprr f ll f ll nn-trminl jining pirwis istint vrtis x... x (f) n H is finit hyprgrph. W nt y N R th st f nn-trminls f R i.. th lls f th lft hn sis, y T R = { f F N R H Im(R), f F H } th trminls f R i.. th lls f R whih r nt nn-trminls, n y F R = N R T R th lls f R. W us grmmrs t gnrt utmt hn in th fllwing, w my ssum tht T R T {, }. W rstrit ny hyprgrph H t th utmtn [H] f its trminl rs n lur vrtis: [H] = H (T V H V H {, } V H ). Similrly t ntxt-fr grmmrs (n wrs), grph grmmr hs n xim: n initil finit hyprgrph. T init this xim, w ssum tht ny grmmr R hs nstnt nn-trminl N R F 0 whih is nt ll f ny right hn si th xim f R is th right hn si H f th rul f : H F K fr ny K Im(R). Strting frm th xim, w wnt R t gnrt uniqu utmtn up t ismrphism. S w finlly ssum tht ny grmmr R is funtinl mning tht thr is nly n rul pr nn-trminl: if (X, H), (Y, K) R with X() = Y () thn (X, H) = (Y, K). Fr ny rul fx...x (f) H, w sy tht x,..., x (f) r th inputs f f, n V H [H] is th st f utputs f f. T wrk with ths grmmrs, it is simplr t ssum tht ny grmmr R is trminl-utsi [C 07]: ny trminl r r lur in right hn si links t

5 t lst n nn input vrtx: H (T V X V X {, } V X ) = fr ny rul (X, H) R. In prtiulr n input is nt initil n nt finl. W will us uppr-s lttrs, B, C,... fr nn-trminls n lwr-s lttrs,,... fr trminls. Hr is n xmpl f (funtinl grph) grmmr R : B B 3 Figur.3 (funtinl grph) grmmr. Fr th prvius grmmr R, w hv N R = {,, B} with th xim n () = (B) = 3, T R = {,,, } n,, 3 r th inputs f n B. ivn grmmr R, th rwriting rltin is th inry rltin twn R hyprgrphs fin s fllws: M rwrits int N, writtn M R N, if w n hs nn-trminl hyprr X = s...s p in M n rul x...x p H in R suh tht N n tin y rpling X y H in M: N = (M X) h(h) fr sm funtin h mpping h x i t s i, n th thr vrtis f H injtivly t vrtis utsi f M this rwriting is nt y M R, X f hyprr X is xtn in n vius wy t th rwriting R, E f nn-trminl hyprrs. Th mplt prlll rwriting = R rwriting ring t th st f ll nn-trminl hyprrs: M= R N. Th rwriting R, X f ny st E is simultnus N if M N whr E is th st f ll nn-trminl hyprrs f M. W pit lw th first thr stps f th prlll rivtin f th prvius grmmr frm its nstnt nn-trminl : = = B = Figur.4 Prlll rivtin fr th grmmr f Figur.3. n utmtn is gnrt y R (frm its xim) if lngs t th fllwing st R ω f ismrphi utmt: R ω = { n 0 [H n] H 0 =... H n = H n+... }. R R R Nt tht in ll gnrlity, w n t nsir hyprgrphs with multipliitis. Hwvr using n pprprit nrml frm, this thnility n sfly mitt [C 07]. Fr instn th utmtn f Figur. is gnrt y th grmmr f Figur.3. rgulr utmtn is n utmtn gnrt y (funtinl grph) grmmr. Nt tht rgulr utmtn hs finit numr f nn-ismrphi nnt mpnnts, n hs finit numr f istint vrtx grs. nthr xmpl is givn y th fllwing grmmr: R, E

6 whih gnrts th fllwing utmtn: rgnizing th lngug { uũ u {, } + } whr ũ is th mirrr f u. Th lngug rgniz y grmmr R is th lngug L(R) rgniz y its gnrt utmtn: L(R) = L() fr (ny) R ω. This lngug is wll-fin sin ll utmt gnrt y givn grmmr r ismrphi. grmmr R is n unmiguus grmmr if th utmtn it gnrts is unmiguus. Thr is nnil wy t gnrt th rgulr utmt f finit gr whih llws t hrtriz ths utmt withut th xpliit us f grmmrs. This is th finit mpsitin y istn. Th invrs f n utmtn is th utmtn tin frm y rvrsing its rs n y xhnging initil n finl vrtis: = { t s s t } { s s } { s s }. S rgnizs th mirrr f th wrs rgniz y. Th rstritin I f t sust I f vrtis is th sugrph f inu y I : I = (T I I {, } I). Th istn I (s) f vrtx s t I is th miniml lngth f th unirt pths u twn s n I : I (s) = min{ u r I, r = s } with min( ) = +. W tk nw lur # F {, } n fin fr ny intgr n 0, D # n(, I) = { s I(s) n } { #s I (s) = n }. In prtiulr D # 0 (, I) = { # s s I }. W sy tht n utmtn is finitly mpsl y istn if fr h nnt mpnnt C f thr xists finit nn mpty st I f vrtis suh tht n 0 D# n(c, I) hs finit numr f nn-ismrphi nnt mpnnts. Suh finitin llws th

7 hrtriztin f th lss f ll utmt f finit gr whih r rgulr. Thrm.5 n utmtn f finit gr is rgulr if n nly if it is finitly mpsl y istn n it hs nly finit numr f nn ismrphi nnt mpnnts. Th prf is givn in [C 07] n is slight xtnsin f [MS 85] (ut withut using pushwn utmt). Rgulr utmt f finit gr r ls th trnsitin grphs f pushwn utmt rstrit t rgulr sts f nfigurtins n with rgulr sts f initil n finl nfigurtins. In prtiulr, rgulr utmt f finit gr rgniz th sm lngugs s pushwn utmt. Prpsitin.6 Th (rsp. unmiguus) rgulr utmt rgniz xtly th (rsp. unmiguus) ntxt-fr lngugs. This prpsitin rmins tru if w rstrit t utmt f finit gr. W nw us grmmrs t xtn th fmily f rgulr lngugs t ln lgrs f unmiguus ntxt-fr lngugs. 3 Synhrniztin f rgulr utmt W intru th i f synhrniztin twn grmmrs. Th lss f lngugs synhrniz y grmmr R r th lngugs rgniz y grmmrs synhrniz y R. W shw tht ths fmilis f lngugs r ls unr unin y pplying th sum f grmmrs, r ls unr intrstin with rgulr lngug y fining th synhrniztin prut f grmmr with finit utmtn, n r ls unr intrstin (in th s f grmmrs gnrting unmiguus utmt) y prfrming th synhrniztin prut f grmmrs. Finlly w shw tht ll grmmrs gnrting th sm utmtn synhrniz th sm lngugs. T h vrtx s f n utmtn R ω gnrt y grmmr R, w ssit nn ngtiv intgr l(s) whih is th miniml numr f rwritings ppli frm th xim nssry t rh s. Mr prisly fr = n 0 [H n] with H 0 =...H n = H n+..., th lvl l(s) f s V, ls writtn l R (s) R R R t spify n R,is l(s) = min{ n s V Hn }. W pit lw th lvls f sm vrtis f th rgulr utmtn f Figur. gnrt y th grmmr f Figur.3. This utmtn is rprsnt y vrtis f inrsing lvl: vrtis t sm lvl r lign vrtilly.

8 Figur 3. Vrtx lvls with th grmmr f Figur.3. W sy tht grmmr S is synhrniz y grmmr R writtn S R, r quivlntly tht R synhrnizs S writtn R S, if fr ny pting pth µ ll y u f th utmtn gnrt y S, thr is n pting pth λ ll y u f th utmtn gnrt y R suh tht fr vry prfix v f u, th prfixs f λ n µ lll y v l t vrtis f th sm lvl: fr (ny) R ω n (ny) H S ω n fr ny t 0 t... n t n with t 0, t n H, H H thr xists s 0 s... n s n with s 0, s n n l R (s i) = l S H (t i) i [0, n]. Fr instn th grmmr f Figur.3 synhrnizs th fllwing grmmr: B B Figur 3. grmmr synhrniz y th grmmr f Figur.3. In prtiulr fr S R, w hv L(S) L(R). Nt tht th mpty grmmr {(, )} is synhrniz y ny grmmr. Th synhrniztin rltin is rflxiv n trnsitiv rltin. W nt th i-synhrniztin rltin: R S if R S n S R. Nt tht i-synhrniz grmmrs R S my gnrt istint utmt: R ω S ω. Fr ny grmmr R, th img f R y is th fmily (R) = { S R S } f grmmrs synhrniz y R n Syn(R) = { L(S) S R } is th fmily f lngugs synhrniz y R. Nt tht Syn(R) is fmily f lngugs inlu in L(R) n ntining th mpty lngug n L(R). Nt ls tht Syn(R) = Syn(S) fr R S. Stnr prtins n finit utmt r xtn t grmmrs in rr t tin lsur prprtis f Syn(R). Fr instn th synhrniztin prut f finit utmt is xtn t ritrry utmt n H y H = { (s, p) (t, q) s t p q } { (s, p) s p H } { (s, p) s p H } whih rgnizs L( H) = L() L(H). Th synhrniztin prut f rgulr utmtn, gnrt y grmmr R, with finit utmtn K rmins rgulr: it is gnrt y grmmr R K tht w fin [CH 08]. Lt {q,..., q n } th vrtx st f K. T h N R, w ssit nw syml (, n) f rity () n xpt tht (, 0) = H

9 , n t h hyprr r...r m with m = (), w ssit th hyprr (r...r m ) K = (, n)(r, q )...(r, q n )...(r m, q )...(r m, q n ). Th grmmr R K ssits t h rul (X, H) R th fllwing rul: X K [H] K { (BY ) K BY H B N R }. Exmpl 3.3 Lt us nsir th fllwing grmmr R : s t gnrting th fllwing (rgulr) utmtn : n rgnizing th rstrit Dyk lngug D vr th pir (, ) [B 79] : L(R) = L() = D. W nsir th fllwing finit utmtn K : rgnizing th st f wrs vr {, } hving n vn numr f. S R K is th fllwing grmmr: p q (s,p) (, ) (s,q) (,p) (, ) (,q) (,p) (,q) (t,p) (, ) (t,q) gnrting th utmtn K : whih rgnizs D rstrit t th wrs with n vn numr f (r ). Th synhrniztin prut f grmmr R with finit utmtn K is synhrniz y R i.. R K R n rgnizs L(R K) = L(R) L(K). Prpsitin 3.4 Fr ny grmmr R, th fmily Syn(R) is ls unr intrstin with rgulr lngug. Prpsitins.6 n 3.4 imply th wll-knwn lsur prprty f th fmily f ntxt-fr lngugs unr intrstin with rgulr lngug. s R K is unmiguus fr R unmiguus n K trministi, it ls fllws Thrm 6.4. f [H 78] : th fmily f unmiguus ntxt-fr lngugs is ls

10 unr intrstin with rgulr lngug. nthr si prtin n finit utmt is th isjint unin. This prtin is xtn t ny grmmrs R n R. Fr ny i {, }, w nt R i = R ( i { i i T } {i, i} ) in rr t istinguish th vrtis f R n R. Fr (, H ) R n (, H ) R, th sum f R n R is th grmmr R + R = {(, H H )} (R {(, H )}) (R {(, H )}). S (R + R ) ω = { R ω R ω V V = } hn L(R + R ) = L(R ) L(R ). In prtiulr if S R n S R thn S + S R + R. Prpsitin 3.5 Fr ny grmmr R, Syn(R) is ls unr unin. Th synhrniztin prut f rgulr utmt n nn rgulr. Furthrmr fr th rgulr utmtn :,,,,,, th lngugs { m m n m, n 0 } n { m n n m, n 0 } r in Syn() ut thir intrstin { n n n n 0 } is nt ntxt-fr lngug. Th synhrniztin prut f grmmr with finit utmtn is xtn fr tw grmmrs R n S fr gnrting th lvl synhrniztin prut R,SH f thir gnrt utmt R ω n H S ω whih is th rstritin f H t pirs f vrtis with sm lvl: R,SH = ( H) P fr P = { (s, p) V V H l R (s) = ls H (p) }. This prut n gnrt y grmmr R S tht w fin. Lt (, B) N R N S ny pir f nn-trminls n E [, ()] [, (B)] inry rltin vr inputs suh tht fr ll i, j [, ()], if E(i) E(j) thn E(i) = E(j), whr E(i) = {j (i, j) E} nts th img f i [, ()] y E. Intuitivly fr pir (, B) N R N S f nn-trminls, rltin E [, ()] [, (B)] is us t mmriz whih ntris f n B r ing synhrniz. T ny suh, B n E, w ssit nw syml [, B, E] f rity E (whr [,, ] is ssimilt t ). T h nn-trminl hyprr r...r m f R ( N R n m = ()) n h nn-trminl hyprr Bs...s n f S (B N S n n = (B)), w ssit th hyprr [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, n ε thrwis. Th grmmr R S is thn fin y ssiting t h (X, P) R, h (BY, Q) S, n h E ( [ ()] [ (B)], ) th rul f lft hn si [X, BY, E] n f right hn si [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 ) n E = { (i, j) [ (C)] [ (D)] (U(i), V (j)) E }.

11 Exmpl 3.6 Lt us illustrt th lvl synhrniztin prut f tw grmmrs. W tk first grmmr R : B B x s t 3 3 B gnrting grph : sn grmmr S is th fllwing: J I y p J q K K J r gnrting grph H : Th lvl synhrniztin prut R,SH f th prvius tw grphs is th grph: This grph is gnrt y th fllwing grmmr R S rstrit t th ruls ssil frm : (x,y) U U (,) (,) (s,p) V (,) (,) (3,) V (,) (,) (3,) W (t,q) (,) (,) (,) (3,) W (3,) X (t,r) (3,) X (3,) W (t,q)

12 with U = [, I, {(, )}] V = [B, J, {(, ), (, ), (3, )}] W = [B, K, {(, ), (3, )}] X = [B, J, {(, ), (3, )}]. Nt tht R S is synhrniz y R n S, n is i-synhrnniz with S fr S R. Furthrmr R S gnrts R,SH fr R ω n H S ω hn rgnizs sust f L(R) L(S). Hwvr fr grmmrs S n S synhrniz y n unmiguus grmmr R, w hv L(S S ) = L(S) L(S ). Prpsitin 3.7 Fr ny unmiguus grmmr R, th fmily Syn(R) is ls unr intrstin. By xtning si prtins n finit utmt t grmmrs, it pprs tht grph grmmrs r t ntxt-fr lngugs wht finit utmt r t rgulr lngugs. W will ntinu ths xtnsins in th nxt stin. Lt us prsnt funmntl rsult nrning grmmr synhrniztin, whih stts tht Syn(R) is inpnnt f th wy th utmtn R ω is gnrt. Thrm 3.8 Fr ny grmmrs R n S suh tht R ω = S ω, w hv Syn(R) = Syn(S). Prf skth. By symmtry f R n S, it is suffiint t shw tht Syn(R) Syn(S). Lt R R. W wnt t shw tht L(R ) Syn(S). W hv t shw th xistn f S S suh tht L(S ) = L(R ). Nt tht it is pssil tht thr is n grmmr S synhrniz y S n gnrting th sm utmtn s R (i.. S S n S ω = R ω ). Lt R ω = S ω. ny vrtx s f hs lvl l R (s) ring t R n lvl l S (s) ring t S. Lt H R ω n lt K = ( lh) P th utmtn tin y lvl synhrniztin prut f with H n rstrit t th st P f vrtis ssil frm n -ssil frm. Th rstritin y ssiility frm n -ssiility frm n n y i-synhrniz grmmr [C 08]. By finitin f R R, th utmtn K n gnrt y grmmr R i-synhrniz t R with l R K (s, p) = lr (s) = lr H (p) fr vry (s, p) V K. In prtiulr L(K) = L(R ). Lt us shw tht K is gnrt y grmmr synhrniz y S. W giv th prf fr R ω f finit gr. In tht s n fr = N R (), l R (s) lr (t). (s, t) fr vry s, t V. Furthrmr K is ls f finit gr. W shw tht K is finitly mpsl nt y istn ut ring t l S K (s) fr th vrtis (s, p) f K.

13 Lt n 0 n C nnt mpnnt f K {(s,p) VK l S (s) n }. S C is fully trmin y its frntir : Fr K (C) = V C V K C its intrf : Int K (C) = { s t {s, t} Fr K (C) }. C Lt (s 0, p 0 ) Fr K (C) n D th nnt mpnnt f { s l S (s) n } ntining s 0. It rmins t fin un inpnnt f n suh tht (s, p) lr K (t, q) fr vry (s, p), (t, q) Fr K(C). l R K Fr ny (s, p), (t, q) Fr K (C), w hv s, t Fr (D) hn D (s, t) is un y th intgr = mx{ S ω ()(i, j) < + N S i, j [, ()] } whs S ω () = { n 0 [H n]... () = H 0 = S thus it fllws tht l R K... H n = S H n+... } (s, p) lr K (t, q) = lr (s) lr (t) (s, t) D (s, t). Fr f infinit gr n y Prpsitin 4.9, w n xprss Syn() s n ε-lsur f Syn(H) fr sm rgulr utmtn H f finit gr using ε- trnsitins. Thrm 3.8 llws t trnsfr th npt f grmmr synhrniztin t th lvl f rgulr utmt: fr ny rgulr utmtn, w n fin Syn() = Syn(R) fr (ny) R suh tht R ω. Th synhrniztin rltin is ls xtn twn rgulr utmt. rgulr utmtn H is synhrniz y rgulr utmtn, n w writ H r H, if thr xists grmmr S gnrting H whih is synhrniz y grmmr R gnrting : S R, H S ω n R ω. Lt us illustrt ths is y prsnting sm xmpls f wll-knwn sufmilis f ntxt-fr lngugs tin y synhrniztin. Exmpl 3.9 Fr ny finit utmtn, Syn() is th fmily f rgulr lngugs inlu in L(). Exmpl 3.0 Fr th fllwing rgulr utmtn : Syn() is th fmily f input-rivn lngugs [M 80] with pushing, ppping n intrnl. s th initil vrtx is nt sur f n r lll y, Syn() s nt ntin ll th rgulr lngugs. Exmpl 3. W mplt th prvius utmtn y ing n -lp n th initil vrtx t tin th fllwing utmtn :,

14 Th st Syn() is th fmily f visily pushwn lngugs [M 04] with pushing, ppping n intrnl. Exmpl 3. Fr th fllwing rgulr utmtn : th st Syn() is th fmily f ln lngugs [BB 0] with, pushing with thir rrspning ppping lttrs,, n is intrnl. Exmpl 3.3 Fr th fllwing rgulr utmtn : th fmily Syn( ) is th st f lngugs gnrt frm I y th fllwing linr ntxt-fr grmmrs: I = P + m m with m 0 n P {,..., m m } = Q + n n with n > 0 n Q {,..., n n }. Exmpl 3.4 Fr th fllwing rgulr utmtn : th fmily Syn( ) is th st f lngugs gnrt frm I y th fllwing linr ntxt-fr grmmrs: I = P + m m with m 0 n P {,..., m m } = Q + n n with n > 0 n Q {,..., n n }. Exmpl 3.5 Fr th fllwing unmiguus rgulr utmtn :

15 w hv Syn() = { L L L Syn( ) L Syn( ) } fr th rgulr utmt n f th prvius Exmpls 3.3 n 3.4. Exmpl 3.6 Th rgulr utmtn : synhrnizs th rgulr utmtn: whih rgnizs th lngug gnrt y th fllwing ntxt-fr grmmr: I = + + B = + B B = + B Mr gnrlly Syn() is th fmily f lngugs gnrt y th linr ntxt-fr grmmrs: I = L 0 + n0 + n0 BM 0 = L + n + n BM B = L + n B n fin fr n 0 0 n n > 0, n fr I 0, J 0, K 0 [0, n 0 [ n I, J, K [0, n [ suh tht fr vry k {0, }, L k = { i+ i+ j i I k j J k j i [j, i[ K k = } M k = { n k j j J k [j, n k [ K k = } L = { i+ i+ i I [0, i[ K = }. Intuitivly, th intgr n 0 (rsp. n ) is th lngth f th s (rsp. f th pri ) n fr ny k {0, }, I k, J k, K k r th susts f [0, n k [ suh tht I k is th st f th gls f th -ignls, J k is th st f th psitins f th utputs, n K k is th st f th nn llw psitins: thr r n gl f -hrizntl. Fr h rgulr utmtn mng th prvius xmpls, Syn() is ln lgr ring t L() n, fr th Exmpls 3.9, 3.0 n 3., is ls ls unr ntntin n its itrtin. W nw nsir nw lsur prprtis f synhrniz lngugs fr rgulr utmt.

16 4 Clsur prprtis W hv sn tht th fmily Syn() f lngugs synhrniz y rgulr utmtn is ls unr unin n unr intrstin with rgulr lngug, n unr intrstin whn is unmiguus. In this stin, w nsir th lsur f Syn() unr mplmnt rltiv t L() n unr ntntin n its trnsitiv lsur. T tin ths lsur prprtis, w first pply grmmr nrmliztins prsrving th synhrniz lngugs. Ths nrmliztins ls llw us t ε-rs t ny rgulr utmtn t gt rgulr utmtn f finit gr with th sm synhrniz lngugs. First w put ny grmmr in n quivlnt nrml frm with th sm st f synhrniz lngugs. s in th s f finit utmt, w trnsfrm ny utmtn int th pint utmtn whih is lngug quivlnt L( ) = L(), with uniqu initil vrtx V whih is gl f n r n n finl, n with uniqu nn initil n finl vrtx V whih is sur f n r: = ( {, } V ) {, } { s (s, s ) } { t s (s t s ) } { s t (s t t ) } { s, t (s t s, t ) }. Fr instn, th finit gr rgulr utmtn f Figur. is trnsfrm int th fllwing infinit gr rgulr utmtn : Figur 4. pint rgulr utmtn. Nt tht if is unmiguus, rmins unmiguus. Th pint trnsfrmtin f rgulr utmtn rmins rgulr utmtn whih n gnrt y n 0-grmmr : nly th xim hs initil n finl vrtis. Lt R ny grmmr n, tw symls whih r nt vrtis f R. Lt R ω with, V. W fin n 0-grmmr R gnrting n prsrving th synhrniz lngugs: Syn(R ) = Syn(R). First w trnsfrm R int grmmr R in whih w mmriz in th nntrminls th input vrtis whih r link t initil r finl vrtis f th gnrt utmtn. Mr prisly t ny N R n I, J [, ()], w ssit nw syml I,J f rity () with =,. W fin th grmmr

17 R ssiting t h (X, H) R n I, J [, ()] th fllwing rul: I,J X [H] { B I,J Y BY H B N R } with I = { i Y (i) I Y (i) H } n J = { j Y (j) J Y (j) H } n w rstrit th ruls f R t th nn-trminls ssil frm. Nt tht th st L(R) T f lttrs rgniz y R n trmin 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] n ε L(R) H Im( R) s (s, s H). T ny N R {} n ny I, J [, ()], w ssit nw syml I,J f rity () +, n w fin th grmmr R ntining th xim rul H, {, } { ε L(R) } { L(R) T } fr (, H) R, n fr ny ( I,J X, H) R with, w tk in R th rul I,J X H I,J suh tht H I,J is th fllwing hyprgrph: 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) } n w put R int trminl-utsi frm [C 07]. [H] [H] Exmpl 4. Lt us nsir th fllwing grmmr R : B B C C gnrting th fllwing utmtn (with vrtx lvls): First this grmmr is trnsfrm int th fllwing grmmr R :,, B, B, C, C,, In prtiulr ε,, L(R). Thn R is trnsfrm int th grmmr R :

18 ,,, B, B, C, C,, tht w put in trminl-utsi frm:,,, B, B,, C, C,,,, S R gnrts :,,,,,, Th grmmrs R n R synhrniz th sm lngugs. Prpsitin 4.3 Fr ny rgulr utmtn with, V, th pint utmtn rmins rgulr n Syn( ) = Syn(). It fllws tht, in rr t fin fmilis f lngugs y synhrniztin y rgulr utmtn, w n rstrit t pint utmt. strngr nrmliztin is t trnsfrm ny grmmr R int grmmr S suh tht Syn(S) = Syn(R) n S is n r-grmmr in th fllwing sns: S is n 0-grmmr whs ny nn-trminl N S {} is f rity, n fr ny nn xim rul st H, thr is n r in H f gl s r f sur t : fr ny p q, w hv p t n q s. H W n trnsfrm ny 0-grmmr R int i-synhrniz r-grmmr R. W ssum tht h rul f R is f th frm... () H fr ny N R.

19 W tk nw syml 0 (nt vrtx f R) n nw ll i,j f rity fr h N R n h i, j [, ()] in rr t gnrt pths frm i t j in R ω (... ()). W fin th splitting f ny F R -hyprgrph withut vrtx 0 s ing th grph: = [] { X(i) i,j X(j) X N R i, j [ ()] } n fr p, q V n P V with 0 V, w fin p,p,q = ( { s t t p s q s, t P } ) fr p q I p,p,p = ( { s with I = { s p = t t p s, t P } { s 0 s p } ) J s = q } n J = { s p = s = 0 }. This llws t fin th splitting R f R s ing th fllwing r-grmmr: H i,j h i,j ( (H ) i,[ ()] {i,j},j ) fr h NR n i, j [, ()] whr h i,j is th vrtx rnming fin y h i,j (i) =, h i,j (j) =, h i,j (x) = x thrwis, fr i j h i,i (i) =, h i,i (0) =, h i,i (x) = x thrwis. Thus R n R r i-synhrniz, n R is unmiguus whn R is unmiguus. Nt tht w n put R int ru frm y rmving ny nn-trminl i,j suh tht R ω ( i,j ) is withut pth frm t. Exmpl 4.4 Th fllwing 0-grmmr R : B 3 B 3 gnrts th fllwing utmtn : Th splitting R f R is th fllwing grmmr:,,,, B, B,,, B,3 B,3, gnrting th fllwing utmtn:

20 s R R, w hv Syn(R) = Syn( R ). T stuy lsur prprtis f Syn(R) fr ny grmmr R, w n wrk with its nrml frm R whih is n r-grmmr gnrting pint utmtn. This nrmliztin is rlly usful t stuy th lsur prprty f Syn(R) unr mplmnt rltiv t L(R), unr ntntin n its itrtin. W hv sn tht Syn(R) is nt ls in gnrl unr intrstin, hn it is nt ls unr mplmnt ring t L(R) sin fr ny L, M L(R), L M = L(R) [(L(R) L) (L(R) M)]. Fr R unmiguus, Syn(R) is ls unr intrstin, n this rmins tru unr mplmnt ring t L(R) [C 08]. W giv hr simplr nstrutin. s R rmins unmiguus, w n ssum tht R is n r-grmmr. Lt S R. W wnt t shw tht L(R) L(S) Syn(R). S S is n 0-grmmr n S is lvl-unmiguus s fin in [C 08]: fr ny pting pths λ, µ with th sm ll u n fr vry prfix v f u, th prfixs f λ n µ lll y v l t vrtis f th sm lvl i.. fr (ny) S ω, s 0 s... n s n t 0 = l S (s i) = l S (t i) i [0, n]. t... n t n s 0, t 0, s n, t n Thus S is lvl-unmiguus r-grmmr. W tk nw lur F {, } n fr ny grmmr S, w nt S (rsp. S ) th grmmr tin frm S y rpling th finl lur y (rsp. y ). S R + S is n r-grmmr n (R + S ) is lvl-unmiguus. It rmins t pply th grmmr trminiztin fin in [C 08] n givn lw, t gt th grmmr R/S = Dt(R + S ) suh tht (R/S) is unmiguus n i-synhrniz t (R+ S ). Finlly w kp in R/S th finl vrtis whih r nt lur y t tin grmmr synhrniz y R n rgnizing L(R) L(S). Thrm 4.5 Fr ny unmiguus rgulr utmtn, th st Syn() is n fftiv ln lgr ring t L(), ntining ll th rgulr lngugs inlu in L(). S w n i th inlusin L(S) L(S ) fr tw grmmrs S n S synhrniz y mmn unmiguus grmmr. Furthrmr fr grmmrs R n R suh tht R + R is lvl-unmiguus, Syn(R + R ) = { L L L Syn(R ) L Syn(R ) } is ln lgr inlu in L(R ) L(R ),

21 ntining Syn(R ) n Syn(R ). Th utmt f Exmpls 3.9 t 3.6 r unmiguus hn thir fmilis f synhrniz lngugs r ln lgr. This rgulr utmtn : is -miguus: thr r tw pting pths fr th wrs n n n with n > 0 n uniqu pting pth fr th thr pt wrs. But Syn() is nt ls unr intrstin sin { m m n m, n 0 } n { m n n m, n 0 } r lngugs synhrniz y. Lt us giv th Dt prtin ppli n ny r-grmmr. s fr th lvl synhrniztin prut, th stnr pwrst nstrutin t trminiz grph is nly n lvl prsrving. Th lvl-trminiztin f ny grmmr R is Dt(R ω ) := { K R ω, K ismrphi t Dt() } whs th lvl-trminiztin Dt() f ny R ω is fin y Dt() := { P Q P, Q Π Q Su (P) q Su (P) Q, Q {q} Π } { P P Π p P p q (q q P = P {q} Π) } { P P Π F {} p P p } rstrit t th vrtis ssil frm n suh tht Π is th st f susts f vrtis with sm lvl: Π := { P P V p, q P, l(p) = l(q) } n Su (P) is th st f sussrs f vrtis in P Π y F F : Su (P) := { q p P (p q) }. Cntrry t th lvl synhrniztin prut, Dt s nt prsrv th rgulrity. Hwvr Dt(R ω ) n gnrt y grmmr whn R is n r-grmmr. Lt R ny r grmmr with R ω ssil frm. W nt H th right hn si f th rul f N R. T ny N R {}, w ssit nw syml f rity n w fin th grmmr R tin frm R y ing th ruls H fr ll N R {}, n thn y rpling in th right hn sis ny nn-trminl r s B y s B :

22 R := { (, H ) } { (, (H N R V H ) { Bs B N R Bs H } ) N R {} } { (, (H N R V H ) { Bs B N R Bs H } ) N R {} }. W tk linr rr < n N R {} f smllst lmnt ( s nt ppr in th right hn si f R). T h P N R {}, w ssit nw syml P f rity P hyprr <P> = P p...p m with {p,..., p m } = P n p <... < p m n w tk grph H P suh tht { P } {} = R n fr P =, w fin < > = n H = H. T h P N R {}, w pply n H P th lvl-trminiztin t gt th grph H P := Dt(H P)[ /{}] { } whs th vrtx lvl l is fin y l() = 0 P N R l() = P N R l(s) = s V HP (P {}). Nt tht th lvl l() f is nt signifint us thr is n r f gl in H P. T h P N R {}, w ssit th fllwing rul: <P> [H P ] { <Q>[U E/E] E Q U V H P Q } with Q := { N R s U, s } H P U := U U E := { t s U E, s t } fr ny E Q. H P Nt tht fr R unmiguus, w n rstrit <P > t <P> = P p...p m with {p,..., p m } = P. By tking ll th ruls ssil frm, w gt grmmr Dt(R). Lt us illustrt th nstrutin f Dt(R) t th fllwing r grmmr R : H P B B B gnrting th fllwing grph : W hv th fllwing prlll rwriting:

23 B = B t B B s p q Tking l() = l(b) = n l(s) = l(t) = l(p) = l(q) =, th right hn si H,B givs y lvl-trminiztin th fllwing grph Dt(H,B ): {} {} {B} {,B},, B {p,s} {t} {q} {q,t} n th fllwing grmmr Dt(R): {, B} {} {B} {,B} {} {, B} {, B} {B} {,B} gnrting Dt(): similr xmpl is givn y th fllwing r grmmr R : B B B B gnrting th fllwing grph :

24 W tin th fllwing grmmr Dt(R): {, B} {} {B} {,B} {} {, B} {, B} {B} {,B} gnrting Dt(): Fr ny rgulr utmtn, th lsur f Syn() unr ntntin (rsp. unr its trnsitiv lsur + ) s nt rquir th unmiguity f. s L() Syn(), nssry nitin is t hv L().L() Syn() (rsp. L() + Syn()). Nt tht this nssry nitin implis tht L() is ls unr (rsp. + ). In prtiulr Syn() is nt ls unr n + fr th utmt f Exmpls 3. t 3.6. But this nssry nitin is nt suffiint sin th fllwing rgulr utmtn :,

25 rgnizs L() = ε + M( + ) fr M = { n n n > 0 }, hn L().L() = L() = L() + ut M Syn() n M.M, M + Syn(). Lt us giv simpl n gnrl nitin n grmmr R suh tht Syn(R) is ls unr n +. W sy tht grmmr is itrtiv if ny initil vrtx is in th xim n fr (ny) R ω n ny pting pth s 0 s... n s n with s 0, s n n fr ny finl vrtx t i.. t, thr xists pth t t... n t n with t n suh tht l(t i ) = l(t) + l(s i ) fr ll i [, n]. Fr instn th utmtn f Exmpl 3.0 n gnrt y n itrtiv grmmr. n ny 0-grmmr gnrting rgulr utmtn hving uniqu initil vrtx whih is th uniqu finl vrtx, is itrtiv. Stnr nstrutins n finit utmt fr th ntntin n its itrtin n xtn t itrtiv grmmrs. Prpsitin 4.6 Fr ny itrtiv grmmr R, th fmily Syn(R) is ls unr ntntin n its trnsitiv lsur. Hwvr th utmtn f Exmpl 3. nnt gnrt y n itrt grmmr ut Syn() is ls unr n + [M 04]. W n ls tin fmilis f synhrniz lngugs whih r ls unr n + y sturting grmmrs. Th sturtin + f n utmtn is th utmtn + = { s r r t (s t t ) } rgnizing L( + ) = (L()) +. Nt tht if is rgulr with infinit sts f initil n finl vrtis, + n nn rgulr (ut is lwys prfix-rgnizl). If is gnrt y n 0- grmmr R, its sturtin + n gnrt y grmmr R + tht w fin. Lt (, H) th xim rul f R n r,..., r p th initil vrtis f H w n ssum tht r,..., r p r nt vrtis f R {(, H)}. T h N R {} n I [, ()], w ssit nw syml I f rity () + p n w fin R + with th fllwing ruls: [H] + { { i X(i) H } Xr...r p X H N R } I Xr...r p K I fr h (X, K) R n n I [, ()] whs K I is th utmtn tin 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 synhrniz y R + n + (R + ) ω fr R ω. T hrtriz Syn(R + ) frm Syn(R), w fin th rgulr lsur Rg(E) f ny lngug fmily E s ing th smllst fmily f lngugs ntining E n ls unr,, +. K Prpsitin 4.7 Fr ny 0-grmmr R, Syn(R + ) = Rg(Syn(R)). By Prpsitins 4.3, 4.6 n 4.7, th fllwing rgulr utmtn :

26 ,,,,,,, hs th sm synhrniz lngugs thn th utmtn f Exmpl 3.0: Syn() is th fmily f input-rivn lngugs (fr pushing, ppping n intrnl). By ing n -lp n th initil (n finl) vrtx f, w tin n utmtn H suh tht Syn(H) is th fmily f visily pushwn lngugs hn y Prpsitin 4.7, is ls unr n +. Exmpl 4.8 nturl xtnsin f th visily pushwn lngugs is t rst lttrs. Fr pushing, ppping n intrnl, w rst lttr t fin th fllwing rgulr utmtn :,,, ny lngug f Syn() is visily pushwn lngug tking s n intrnl lttr, ut nt th nvrs: { n n n 0 } Syn(). By Thrm 4.5, Syn() is ln lgr. Furthrmr th fllwing utmtn H :,,,,,,,,,,,, stisfis Syn(H) = Syn() n H + = H hn y Prpsitin 4.7, Syn() is ls ls unr n +. Nt tht th utmt f th prvius xmpl hv infinit gr. Furthrmr fr ny utmtn f finit gr hving n infinit st f initil r finl vrtis, th pint utmtn is f infinit gr. Hwvr ny rgulr utmtn f infinit gr (in ft ny prfix-rgnizl utmtn) n tin y ǫ-lsur frm rgulr utmtn f finit gr using ε-trnsitins. Fr instn lt us tk nw lttr T (inst f th mpty wr) n lt us nt π th mrphism rsing in th wrs vr T {}: π () = fr ny T n π () = ε, tht w xtn y unin t ny lngug L (T {}) : π (L) = { π (u) u L }, n y pwrst t ny fmily P f lngugs: π (P) = { π (L) L P }. Th fllwing rgulr utmtn K :

27 ,,, is f finit gr n stisfis π (Syn(K)) = Syn() fr th utmtn f Exmpl 4.8. Lt us giv simpl trnsfrmtin f ny grmmr R t grmmr R suh tht R ω is f finit gr n π (Syn(R )) = Syn(R). s Syn(R) = Syn( R ), w rstrit this trnsfrmtin t r-grmmrs. Lt R n r-grmmr. W fin R t n r-grmmr tin frm R y rpling h nn xim rul st H y th rul: st ( [H] {s s, t t} h(h [H]) ) P with s, t nw vrtis n h th vrtx mpping fin fr ny r V H y h(r) = r if r {s, t}, h(s) = s n h(t) = t, n P is th st f vrtis ssil frm s n -ssil frm t. Fr instn th r-grmmr R is trnsfrm int th fllwing r-grmmr R : Fr ny rul f R, th inputs r sprt frm th utputs (y -trnsitins), hn R ω is f finit gr. Furthrmr this trnsfrmtin prsrvs th synhrniz lngugs. Prpsitin 4.9 Fr ny r-grmmr R, Syn(R) = π (Syn(R )). S fr ny R, Syn(R) = π (Syn( R )) n ( R ) ω is f finit gr. ll th nstrutins givn in this ppr r nturl gnrliztins f usul trnsfrmtins n finit utmt t grph grmmrs. In this wy, si lsur prprtis ul lift t su-fmilis f ntxt-fr lngugs. Cnlusin Th synhrniztin f rgulr utmt is fin thrugh vis gnrting ths utmt, nmly funtinl grph grmmrs. It n ls fin using pushwn utmt with ε-trnsitins [NS 07] us Thrm 3.8 ssrts tht th fmily f lngugs synhrniz y rgulr utmtn is inpnnt f th wy th utmtn is gnrt it is grph-rlt ntin. This

28 ppr shws tht th mhnism f funtinl grph grmmrs prvis nturl nstrutins n rgulr utmt gnrlizing usul nstrutins n finit utmt. This ppr is ls n invittin t xtn th ntin f synhrniztin t mr gnrl su-fmilis f utmt. knwlgmnts Mny thnks t rnu Cryl n ntin Myr fr hlping m prpr th finl vrsin f this ppr. Rfrns [M 04] R. lur n P. Mhusun Visily pushwn lngugs, 36 th STOC, CM Prings, L. Bi (E.), 0 (004). [B 79] J. Brstl Trnsutins n ntxt-fr lngugs, E. Tunr, pp. 78, 979. [BB 0] J. Brstl n L. Bssn Bln grmmrs n thir lngugs, Frml n Nturl Cmputing, LNCS 300, W. Brur, H. Ehrig, J. Krhumäki,. Slm (Es.), 3 5 (00). [C 06] D. Cul Synhrniztin f pushwn utmt, 0 th DLT, LNCS 4036, O. Irr,. Dng (Es.), 0-3 (006). [C 07] D. Cul Dtrministi grph grmmrs, Txts in Lgi n ms, mstrm Univrsity Prss, J. Flum, E. räl, T. Wilk (Es.), (007). [C 08] D. Cul Bln lgrs f unmiguus ntxt-fr lngugs, 8 th FSTTCS, Dgstuhl Rsrh Onlin Pulitin Srvr, R. Hrihrn, M. Mukun, V. Viny (Es.) (008). [CH 08] D. Cul n S. Hssn Synhrniztin f grmmrs, 3 r CSR, LNCS 500, E. Hirsh,. Rzrv,. Smnv,. Slissnk (Es.), 0 (008). [H 78] M. Hrrisn Intrutin t frml lngug thry, isn-wsly (978). [M 80] K. Mhlhrn Pling muntin rngs n its pplitin t DCFL rgnitin, 7 th ICLP, LNCS 85, J. Bkkr, J. vn Luwn (Es.), 4 43 (980). [MS 85] D. Mullr n P. Shupp Th thry f ns, pushwn utmt, n sn-rr lgi, Thrtil Cmputr Sin 37, 5 75 (985). [NS 07] D. Nwtk n J. Sr Hight-trministi pushwn utmt, 3 n MFCS, LNCS 4708, L. Kur,. Kur (Es.), 5 34 (007).