In this article wedescribe Croisot-lan uages an Dubreil-languages having a group as syntactic monoid.. We pro ide different characteristics of these lan uages.. Then, we describe the for
Trang 1T?-p chi Ti n hoc v Di eu khi e hoc, T.20, S.4 (2004), 343-350
LE o oc RAN , RO TIEN DUa N G
Kh oa T a , Truemg D ei h C Vinh
Abstract In this article wedescribe Croisot-lan uages an Dubreil-languages having a group as syntactic monoid We pro ide different characteristics of these lan uages Then, we describe the formsofgroup regular-lan uages
Tom Hit Tron bai bao nay, cluin toi khao sat cac ng6n ngir Kroaz6 va ng6n ngir Duybr ay co vi
n om c u ph ap l a mot n om. Chung t6i n an d ir oc nhieu d~c tnrng khac n au cu a c c lap ng6n ngir nay.Tir do, c hu ng t6i mo ta dir oc dang dieu c u a cac ng6n n ir nhom chinh qui tong quat
1 MCr DAU Gia su S la mra n 6m va H la mot t~p can cua s Ta xet quan h~ RH ~ S X S nhir sau
RH = {( x, y ) ES x Slxu EH {:} yu E H ,V u ES}.
Khi d R H la tuo nq clling phdi tren S va diroc goi la tu onq clling chinh phdi Duybray sinh
b di H t r on g S.
Bay gio ta xet tirong dan hai phia tren S nhir sau:
r H = { (x,y) ES x Sluxv EH {:} uyv EH, Vu,v E S}.
Khi d6 rH d oc goi la tuang cllin chinh hay iu anq clling cu p l uip c ' lia H va vi nhom thuan
S /r H diroc g i la vj nh om csi pl uip cua H trong S. Tap can H diroc goi la rei rac tro g S
neu tu ng dang vn la tuan dan d n nhat, n hia la (x, y) E r H {:} x =y.
Gia su X la mot bang chir c i hiru han, X* la vi nhorn tv do sin bo i X vo i dan vi la tir
A Khi d6 moi t~p can bat ki L cua X* duoc g i la mot ng6n nqii.
Gia su L la ngon n ir tren X, khi d6 vi nhorn ci phap cua L trong X* se duoc goi la vj
n om cii phcip c ' lia L , ki hieu ban f L ( L). Ngon ngir L diroc goi la ng6n nqii n om neu f L(L)
la mot n om
Thea [7], "L t o , mot ng6n ngu co vj nhom cu pluip f L(L) clling ciiu vai mot nhom G neu to' n tai to an c i u rp : X* + G, sao c h L = r p -l ( H ) , t rong clo H t o , ttip con rai rac cua idiom
G ". Bai baa nay trinh bay viec xet ngon ngir L ln voiH la lap ghep theo mot nhorn can roi rac cua G Lap n on ngir nay thirc sir chira lap ng6 n ir diroc dira ra boi Anixinov [1]
Gia su S la mra nhorn Khi d6 moi tap can H cua S diroc goi la mot iiip con ttumh , neu n6 thoa man dieu kien:
Va, b, x, yES, (ax, ay, b x E H keo theo by E H).
Trang 2344 LEo uoo HA N , HO TI EN DUO N
Ngon ngir L tren X diroc goi la ngon ngi - Duybray neu thoa man hai dieu kien sau:
l M9i tu thuoc X* 1 mot dean ban dau nao do cua mot tir thuoc L (tuc la Vu EX 3 vE X* sao ch u v E L
2 Neu ba trong bon tir ux, uy, vx, vy E L thi tir con lai cling thuoc L.
B6 d 1 ([2]) G i d su G la mot nh6m v a H la t4 c on k lu i c rong c u G Khi a c t c k h dng
ajnh sou l a t ucrn g a uc r n g:
(i) H truit i h th e o ngh f a - Duybray ;
(ii) h I, h a, h E H t h i hIh :; 1 h E H;
(iii) H la l ap g h ep p h di ( t rai ) th eo ttu i t nhom c on c ua G.
Chung minh
(i)::::}(ii) hI , h 2, h E H ::::} h h :; lh = h E H , h h : lh3 = h E H , hIh :; lh2 = hI E H H1 a
rnanh theo n hia Duybray, nen hIh:lh 3 E H (& day da SU-dung dinh nghia t~p can manh &
tren vo-i a = h h :; I , b =hIh :; I , X =h , Y = h ).
(ii)::::}(i) Cia su-a x, ay , b e E H. Khi do bx(ax) - Iay E H ::::} by E H :: ::} H la t~p con manh
cua G.
(ii)::::}(iii) VI H - 0 nen :3 gE H Cia su-K = {x E GIgx E H}. Do e E K nen K - 0. Neu
a, b E K thi ga , gb , 9 E H ::::} g.(ga) - I.gb E H hay g - Ib E H:: :: } a-Ib EK ::::} K la nhOIDcan cua G VI G la nh6m nen tir each xac dinh cua K ta c6 H =gK
(iii):: } (i) Cia SU-H = gK , tro g d K la nh6m con cua G va hI, h2, h E H. Khi d6
h I = gkl, h = gk2 ' h = gk3 voi kl, k2 , k3 E K Tir do hIh: lh3 = gkIk : ;lg-lgk3 =
gkIk:;lk 3 E gK = H , VI K 1 nh6m con cua G Do do hIh:;lh3 E H •
Dinh ly 1 G id su L la ng on ng i i r e ti X v a L = cp- I H) , trong ao c p :X* -+ G la i oa s i ca u
va H l a t 4 co n r o i rac c ' li a G Th e th i L la ngon ngii - Duybray khi va chi khi H la m¢t lap
gh ep th eo m 9 t t i hom c on ro i rac iuio ao cua G.
C h un g mi nh. Cia SU-L la ngon ngu Duybray va as, at, bs E H Khi do ton tai u, V , x, Y EX',
sao cho a = cp (u ) , b = cp ( v ) , s = c p( x ) , t = c p(y). The thi as = c p(u x ) E H, at = cp(u y ) E H , b s = cp ( vx ) E H , nen uX , uy , v E c p-I(H) = L VI L la ngon ngir Duybray
nen'vy E L , su ra cp(vy) E c p(L) = H :: } p(v) c p(y) = bt E H V~y H manh theo nghia
Duybray ::::}H = gK , tro g d K la nh6m con cua G. M~t khac r,JH = r JK va H 1 t~p
con rei rac nen K roi ra tro g G Ta clnrng minh r,J = r J K Cia SU-(a,b) E r J H , khi d6
sat E K { : } g sa t E gK = H { :} gsbt E H = gK {:} sbt E K, Vs, E G ::::} (a, b) E r,J K , d d
r,J H C r ,J K· Dao lai neu (a , b) E r JK: ::: } sat E H = gK { :} g - at E K {:} g-Isbt E K ¢ : ?sbt E
gK = H, Vs,t E G ::::} (a, b) E r JH, do do r J K C r ,J H· V~ r ,J K = r ,J ·
Dao lai neu H = gK tro g d K 13.nh6m con roi ra cua G thi VIr J K = r J H , nen H
la t~p con rei ra cua G Khi do L 13.ngon ngir nh6m va H 1a t~p con manh theo n hia Duybray cua G. Cia SU-u x , uy , v E L va c p(u) = a , c p(v) = b, c p(x) = s, c p(y) = t. Khi
do a s =cp (u) cp ( x ) = cp (ux) E c p(L) = H Tirong tv at, bs E H ma H J a t~p con rnanh theo
n hia Duy ray tro g G nen bt E H. Suy ra c p(vy) = cp(v)cp(y) = bt E H = cp(L) ::::}vy E
cp- I(H) = L Cia SU-u E X suy ra c p(u) E G VI G la nh6m nen :3bE G sao ch
c p(u).b = 9E H.
VI c pla toan cau nen :3v E X* sao cho cp(v) = b : ::: } p(u) c p(v) E H ::::}c p(uv) E H :: } uv E cp- (H) = L ::::} u la doan ban dau cua uv E L V~y L la ngon ngir Duybray •
Trang 3NGON N GU N HOM £)UYBRAy VA N GO N N GU N HO M KROAZO 345 Ngon n ir L tren X diroc goi la ngon ngii Kroa z o neu n6 thoa man hai dieu kien sau: (i) M9i tir thuoc X * la mot doan ban dau nao do cua mot tir thuoc L
(ii) Neu ba tro g bon tir x uy , xv y , z ut , zv t thuoc L thi tir con lai thuoc L
Dinh l y 2 Gid s ' ll L co vj nhom c ii plu i p l a fJ(L) adng d i u v6i n lurm G Th e th i L l a t qo r: ngii K roa z o khi va c h i k h i L = < - I(g) , trong a o i p : X* ~ G la i o i ui d i u v a 9 E G
Chung m i nh. Cia S l L la n on n ir Kr oazo v a L = < p-I( H) , ip : X* ~ G la toa n diu v a H
la t$,pcon rai r ac cua G Cia Slr h l, h E :H , khi do 3 , v E L sao cho <p (u) = hI, < p( v )= h 2.
Khi do tir A.u.A E L v a L la n o n ir Kro az o n e n xuy E L {=? xvy E L, V x, Y E X*.
Do d < p(x) < p(u) < p(y) E H { =? < ( x ) < ( v) < (y) E H , Vx, y E X * =? ( < p( u ) < ( v )) E P H hay
( h I h2) E P H =? hI =ba , d d6 I H I = l
Dao lai gia SlrL =< - I(g) , VIG la nh6m v a sp la t o a n ca u n e n voi m o i x E X* ,3 y E X *
sao cho c p(x) cp (y) = 9 = xy = cp - l (g) = L =? x la doan ban dau cua tu u = x E L M~t
khac, gia Slr xu y ,xvy , zut E L , khi do c p(xuy) = c p( xv y) = < ( z ut) = 9 = cp ( x ) cp (u) c p(y) =
c p ( x ) c p(v) c p(y) = c p( z) c p(u) < (t) = 9 = c p( u) = c p(v ) =? < p(z) c p ( v ) c p( t ) = 9 =? < p(zv t) = 9 =?
zv t E < - l ( g ) =L V$,yL la ngon ngir Kroazo •
Ta dinh nghia otomat doan nhan n on ngir L, ki hieu bKn w ( L) nhir sau: w( L) = X* IR D X , X ,6,{ u Iu E L }) ma tac dung X* len X* I R£ diroc xac dinh boi: u j =uj, ao =
Ala tran thai ban dau, con { u IuE L } la tap trang thai cudi, (; day, RL la tuang d n phai
t e n X* diroc xac din boi (u,v) E R L { =? (u x E L { = v E L , Vx E X*) , con u la R L - lap tuang dircng chira u. Ta thirorig ki hieu X* I R£ = A va {u Iu E L} = A' , con ham chuyen
tr a ng thai 6, c u the ta viet 6(a , f) = b thay ch u j = uj , tro g d a = u , b = uj Nhir v$,y otornat doan nhan L la w(L) = (A,X,ao,6,A'), tro g do L = {u E X * 1 6(ao,u) E A'}. R6
ran , moi tir u E X* xac dinh mot an xa 6 u : A ~ A , con tir A irng voi anh xa dong nhat
X H X
Tap hop tat d cac an xa 6 u la mot vi nh6m con cua vi nh6m cac phep bien doi cua A , ki
hieu T(A).
Be; d 2 ([4]) V 6i moi ngon ngii L tr e n X ta cof J(L) ~ T(A)
Chung min h. Xet anh x: ; ' /J : f J(L) ~ T(A) , tro g d [uJla P H -lap tuorig dirong clnra u Khi
[uJH6 u
d [UIJ= [U2J{=? (Ul , U 2 ) E P L {=? ( X UlY E L {=? XU2 YE L , Vx, y E X* ). Cia S l a E A , a= X.
Ta se chirng minh 8 U 1 (a) = 6 U2 (a) That vay, 6 U 1 (a) = 8 U 2 ( a ) { =?8( a, uI) = 8( a,u 2) { = X UI = XU2{ =? (XUI, X U2) E RL {=? ( X UlY E L { = XU 2 YE L ,V y E X *) Dieu nay dung Va E A , nghia
la d ng Vx E X*. Do do ' /J la d n anh Theo each xac dinh 6 ' /J, ta c6 6 ' /J la toan anh Do
do 6 ' / J la song anh Ta chimg minh 6 ' /J la d ng cau
Cia Slr [uJH 8 u, [vJH 8 v Khi do [uJ.[vJH 8 uv That v a y , ta c6 6v 0u(a) =6 v [ 8(a , u)J =
8v(x u ) = 6 (xu , u) = xuv = 8u v(x) = 6 uv (a), Va = x EA Do do 8 v°6u = 8u v V$,y'l/Jla d n
- '
cau
M9t ngon ngir L tren X duoc goi la ch i nh q ui neu n6 la n on ngir hiru han hoac thu duoc tir cac t$,p con hiru han nao do c u X * bang each ap dung mot so phep toan lap,
Cia Sl r L la ngon ngir tren X. Khi do otomat wL diroc goi la tach tiu o cneu Va, bE A ,
Trang 4346 L ouoc H N, HO T I EN DU aN G
tir J (a,x) = J ( b ,x), v iz nao do thuoc X , keo theo a = b Otomat w( L ) duoc goi la aay au ,
neu \: aE A, \: xE X , ::J bE A sao cho J ( b , x) =a
Dinh l y 3. Gid s u L l a ngo n n gu t h om ch in h q ui i r e n X , khi a o c t ic ai' e u k i ~ n so u lli tuang
au a ng :
(i) L l a ngon n g n ot t man l - D u y b ra y ;
(i) Gto m at toi tieu w( L ) = (A, X , ao , 15,A ' ) t ac h auq c va I A' I= 1;
(i ) G t o m at t oi t ieu w(L) = (A , X , ao,15,A' ) a y au va I A ' I = 1
Chung minh.
(i):::}(ii) Gia Slr J(a, x) = J(b , x) tro g d a = ii , b = V, tire la ta co: u x = v x, khi d6
(ux,vx) E R L :::: } : J E X * s a o cho u x E L v a v y E L 'Cia Sl r u z E L , VI Lm a nh theo
n hia Du bray n e n v E L TU' Cmg tv v E L : :::} u z E L , \:IzE X * V~y (u, v ) E RL => U= v
hay a = b , d do R L la tach dirc Gia S a , bE A' , trong do a = ii, b= v. Khi do u , vE L ,
do d ux E L ¢:}vx E L , \:IxE X * : : :: }(u , v ) E R L :::: } i = v::::} a = b , do do IA ' I = 1
(ii)::::}(i) Gia Sl r w( L ) la tach diroc, VIL la n go n n ir nh6m chinh qui n e n theo dinh lyKlene [4] ta co w ( L ) hiru han ::::}A hiru han
M~t khac, do w ( L ) tach dU ' C nen \ : IuE X* anh xa Ju : A r A Ia d n anh Vi A hiru
han nen Ju la toan an ::::}Ju la so g an Do d T ( A ) la vi nhom con cua nh6m G A tir A
len chinh n VI A hiru han nen G A hiru han su ra T ( A ) la nh6m con hiru han cua G A ,ma
T ( A) ~ f - L( L )nen f -L ( L )la mot nh6m hiru han
Vi I A' I = 1 nen A' = {a} trong do a = w. Gia Sl r ux,uy E L suy ra J(ao , ux) =
J(ao , vx ) ::::}J(i, x ) = J(v, x ) : : :: } il = v ( v l w( L ) la tach duoc) ma uy E L nen v E L Talai
co f- L( L )la mot nh6m nen \:I uE X*,::Jv E X * sao ch [ u ] [v]= [w ] ::::} [u v = [w] :: :: } u v =i (VI
PL c R c), ma w =w A E L nen u v E L V~y u la de n ban dau cua tr uv E L. Do d Ll a
n o n ir nh6m Du bray
(i):: }(iii) Gia S a E A v a x E X , a = il. Vi f - L(L)la mot nh6m nen ::JvE X* sao ch [ v] [ ] =
[u] :::: } vx] = [u] : :::}(vx, u) E P L ::: : } ( vx, u) E R L (vl P L c RL ) :::: } (ao , vx ) = J (ao , u) h y
J( b ,x) =a V~y w( L ) day dd Viec chirng minh I A ' I = 1tuan tv chirng minh (i) = (ii) (i )::::} (i) Tirorig tv nhir chirng minh (ii)::::}(i) nhimg ta thay lap luan w(L) tach d11<?,Cboi
w( L ) day dd w (L ) day dd ::::}Ju : A r A la toan c u, \:Iu E X* , ma A hiru han nen
H~ qu a , M oi n on nqi i c h i nh q ui - D u ybra y a e u l a ngon ngu n o m
C hung minh Gia Sl r L la ngon ngir chinh qui Duybray v a u E X* Khi d ::JvE X* , sao
ch u v E L (vl u la doan ban dau cua mot tir thuoc L) ::::}IA'I =I cP Gia S l a , b« A' , tro g
do a = il, b = v. Khi do u A = u E L , v A = vE L Vi L la ngon ngir Duybray, nen
ux E L ¢:}vx E L , \:IxE X * : ::: } il = v hay a = b : ::: }I A' I = 1 Gia Sl r J (a, x ) = J(b , x) , n hial a
ux = vx. Khi do (u x, v ) E R L Vi L la ngon ngir Duy ray nen : J y E X * , sao ch ux y EL
va vxy E L VI L la ngon n ir manh theo n hia Duy ray, nen u z E L ¢:}vz E L , \ : IzEX* : :} (ux, vx ) ER L : ::} il = v hay a = b : :: }w( L) tach duoc Do L la n on ngir chinh qui nen tird6
Otomat w (L) = ( A , X , ao,J , A') diroc goi la l e n th O ng neu \ : a,a' , ::Ju E X * san cho
J( a, u) = a' hoac J (a' , u) = a
Trang 5NCO N NClr N H OM £)U Y B A y vA NCON NCl r N HOM KROAZO 347
O tomat w( L ) = (A,X, a ,5,A') dircc goi la li en thOng truuih neu Va,a',-::Ju,v E X* sao
c h 5 (a, u ) = a ' va 5 (a ' , v) = a
Otomat w( L ) = (A,X,ao,5,A') dircc goi l a f n ajnh n e u tir 5(ao ,u) = 5(ao,v) s u r a
Dinh If 4 G i d s u L l a ngo n ngii tren X T e t h i L l a ngon n gii nh6m K ro zo khi va chi khi
C hu ng m in h. G ia s l 'r L la ngon n ii Kroazo Khi do Va , bE A (a = ii, b = v) -::Jx, Y E X*
u A E L => v A E L => vE L H an nira, u,v E L thi A.u.A E L,A.v.A E L v a L l a
n go n n gi r nh 6 m Kr oa z o nen A u.x E L <=> A.v.x E L, Vx E X* = (u, v) E RL. N htr v ~y
L gor n mot va c hi mot l ap tiro g d : 1n g R L. D o do IA' I = 1 Gia s l 'r A' = w voi w E L.
zui E L <=> zv t E L (VI L l a ngo n n ir Kroa z o) V~y (u, v) E PL : : ::}RL CPL T a 1 - i c 6
PL C R L => PL = R L. Khi do ne u 5 (ao, u) = 5(a o , v) => (u, v) E R L = (u, v) E PL n e n
Vx E X * ta c6 xu y E L <=> xv y E L , Vy E X* ::::}5(x, u) = 5(x, v). V ~ w(L) o dinh
Dao 1 - ,neu w( L ) = (A, X ,ao,5,A ' ) la otorn a t l e n thon g manh va o din T a s e clnrng minh L la ngon n ir Kroazo, VI w( L) la li e n thong manh nen Vu E X*, -::JvE X* sao
5 ( b ,A ) , V b E A=>( u v, A) E PL => [v] l a n h ich dao c u a [u] = f L( L ) l a mot nh6m Cia s l ' r
u E X*, w E L Khi do VIf L( L )lamot n h6m n e n -::JvE X* : [u] [v] = [w] => uv E L ::::}ula doan
b a n da u cua tir u v E L Ta 1 - ic6 IA ' I = 1 : :: : }A' = {a'} vci a' = W, w E X nen L gorn mot
v a chi mot R L ap. VI w(L) on dinh n e n n e u (u,v) E RL thI5(ao,u) = 5(ao,v) = 5(a,u) =
5 ( a, v ) , V a E A => (u, v) E PL. D o d R L CPL H i e n n ien PH C RL nen PL = R L. VI v~y ,
[ u ] = [v] (VI f L ( L ) 1 , mot n 6 m) => zu t E L = zv t E L,Vz , t E X* => L l a ngon n ir Kro az o.
Tnroc het, t a dira ra dieu kien can va du d e mot ngo ngir la n on n ir nh 6 m c hinh qui
Dinh I f 5. Noon ngii L za ngon nqi n 6m chinh qui khi va chi khi L chsi a ngon ngii nh6m
C h ung min h
* Di e u kien can: G i a s l 'r L la ngon n ir nh6m c h inh qu i , khi do L = < p-l(H), t o ng do ip
la toan cau tir X* len n 6m h ir u h a n G va H l a tap co n roi r a cua G G i a s l ' r g E H v a
M =cp- l ( g) Khi do M ~ H va f L(M) ~ G n en M la ngon ngir nh6m c hinh qui H a n nira,
Vu E X*,-::JWl'W2 E X* sao c ho < p(u)<.p(wd= <.p(W2)< p(U)= g VI <.pla toan cau va G l a mot
nhom S u y ra
Trang 6348 LEQuae HAN , HO TIEN DUON
< p(UW1 )= < p(W 2 U)= g:::: } UW1 , W2UE < - 1( g ) = M
Gi1i st'rUW,VW E M :::: } < p(uw) = <p(v w ) = g : :: } <p(u) < p(w ) = < p(v)<p(w ) :: : } <p(u) = <p ( v ) d d
neu x uy E M ::::} <p ( x uy) = 9 : :: } < p(x) < p(u) < p(y) = 9 ::::} < p(x) < p(v) < p(y) = 9 : :::} < p(xvy ) = 9=}'.
xvy E < - 1( g ) = M
* Dieu kien du: Vi M la ngon ngir nhom chinh qui nen f -L(M) hiru han Gi1ist'rUEX * , d (i) nen M = c Gi1ist'rW E M, khi d6 uw E X* nen theo (i) ::JvE X* sao ch v u w E M. Th e
(u , v) E vi That vay, gi1ist'rx uy E L, d (i) nen::Jw E X* sao cho xuyw E M Theo (iii) ta
diu cua X*/fiJ L= f L( M ) (xem [2, H~qua 1.6]) Vi f L(M) la nh6m hiru han nen f -L( L ) cling lit
nh6m hiru han Do do L la n on ngir nh6m chinh qui • Triroc khi dira ra ket qua mo t1idang dieu ngon ngir nhom chinh qui, tiro g tv Dinh ly
Myhill-Norode (xem [1 , tran 112]), ta hay clnrn minh bo oe sau day
B5 de 3 Lap ca c ng6n ngii n h 6m c h inh q ui k h ep k i n c l Oi iuri h iiu h en c ti c p h e p io dn Bun.
Chu ng m i nh Tnroc het ta neu ra khai niern ngon ngir tuan h an Ngon ngir L diroc goilit
n 6n ng ii nh 6m tu i in harm , neu f- L(L) la mot nhom tuan hoan, nghia la moi p an t'r cua J l( L )
Ta c6: giao c da hiiu I u i ct ic ng6n ngii nh6m tuan h arm la ng6n ngii n h 6m t uan h t ui
That vay, ta chi can chirng minh ch giao cua hai ngon ngir n om tuan hoan Gia su
-L1 va L la hai ngon ngir nhom tuan hoan tren X Khi do \l u E X*,::Jn , n2 E N sao ch
( u 1 , A) E gJ Ll va (un2,A) E fiJL2 : :::} (Un1n2 , A) E fiJLl va ( un1n2,A ) E f i J L 2' Tir do su ra
x u n1n2 y E L1 nL 2 ¢:} xun1n 2 y E L1 va x un 1 n2y E L2 ¢:} x y E L 1 va xy E L2 { : ? x E
L1 nL 2, \I x, y E X Do do (u n 1n2 , A) E gJ Ll n L2 :::: } [U] n n 2-1 la nghich dao cua lap [u]tro g
f-L(L1nL ) :: ::} f-L(L I nL2) la mot n om tuan hoan ::::}L nL la ngon ngir nh6m tuan hoan
Bay gio, ta chirng minh khan dinh cua Bo de 3
k
i=l
ngon ngir nhorn chinh qui [4]va moi L, la ng n ngir nhorn tuan hoan, theo nhan xet tren
ngir nhorn chinh qui, thi do vi = fi X * \L , nen f -L ( L) = f -L( X * \ L ) do do X * \ L cling lang6
n ir nh6m chinh qui M~t khac, VI UL , = nX* \ Li nen hop cua hiru han ngon ngir n 6m
i= l i= l
chinh qui cling la ngon ngir nhom chinh qui Gia st'rL va L la cac ngon ngir nhorn chinh
q i tren X , d L1 \ L2 = L1 n(X* \ L 2) nen L1 \ L2 cling la ngon ngir nh6m chinh qui •
hat: c tic ng6n ngii c hinh qui Duybray tr e n X.
Chu n g m i nh. Dieu kien du ducc suy ra tir H~ qua cua Dinh ly 3 va Bo de 3 Diroi day lit
chirng minh dieu kien can
Gi1i st'rL langon ngir nhom chinh qui du c doan nhan boi otornat w(L) = (A, X, aQ ,8,A')
Trang 7NCON NCU NHOM ElUYBRAy VA )ICO N N C U N HOM KROA Z O 349
hiru han va tach duoc, trong 0 A' = {aI, ( l,2, , am} (xem [7] Cia Slr Lk = {w E
X * I< 5(ao , w) = ak , k = 1 , 2 , , m} Khi 00 R L = R ' c il That v y gia Slr (u, v ) E RL :: ::}
(u x, vx ) E RL ,Vx E X * ::: :} ( < 5(ao , u x ) = a k ¢} < (ao , v ) = a k, Vx E X * ) :: } (u x E L k ¢ } VX E
(vI w(L) tach diroc) ::::}(u,v) E RL ::::} RLk = RL ma w(L) tach duoc va hiru han,
nen w (Lk) cling tach duoc va hiru han Theo [7] ta co Li la ngon ngir nh6m chinh qui
M~t khac, theo clnrng minh tren Vu E X*, 3 z E X * sao cho u z E Lk Han nira, neu
uX,vx,u y E L k thi < 5(ao , u x ) = < 5(ao ,vx ) = a k : :::} < 5(ao , u) = <5(ao ,v ). Vi w(L) ta h diroc su
ra < 5(ao , uy) = < 5(ao ,v y) , ma < 5(ao , uy) = ak ::::}< 5 ( ao ,v y) = ak : ::}v y E Li : V~y L k la ngon ngir
5 KET LU~N
Chung toi oa tim diroc oi'eu kien oe mot ngon ngir la ngon ngir nh6m Duybray, n on ngir nh6m Kroazo dong thai mo ta duoc dang dieu va otornat cua ca lap ngon ngir nay Tren c a
TAl Lr¢U THAM KHAO
[1] A.V Anixinov, ve n on ngir nh6m, Di' e u khi e n hoc , No.4 (1971) 18-24 (tieng Nga)
[2] A.H Cliphat va C B Prenstan, Ly th u y t Nua n 6 m (2 t~p), NXB 8q,i h9C va Trung
h9C chuyen nghiep, Ha N9i, 1979
[3] Phan Dinh Dieu, Ly thuy e t Ot om at va Th u i tot ui, NXB Dai h9C va Trung h9Cchuyen
n hiep, Ha N9i 1977
[4] S Eilenberg, Aut o mata , Languag e s and Ma c hines , Volum B, Academic Frees, New York, 1976
6 -69
[6] Le Quoc Han va Nguyen Thi Bich, Ngon ngir nhorn co lap, Top chi Tin h9C va Di 'e u
k h ien h9 C19 (2003) 101-109
[7] 'Iran Van Hao va Le Quac Han, Ng n ngir nh6m, T uye n t ~ Cotu ; t ri nh H 9i th do C(J sd
t in h 9C v a Bdo v~ t in , Vien Toan h9CViet Nam, Ha N9i, 46-49, 1987
[8] B Le Saec, Saturating right congruences, Theor e t al Informati c s and Appli c ation 24 (6)
[9] B Le Saec, Dare V R., and Seromony R Strong recognition of rational w-languages,
Int e rnat i onal Conf er enc e Math e m a t cal F o und a t on of Informat i c s, Hanoi, 1999
[10] J.B Pecuchet On the complecmentatio of Buchi automata, Th e or e t ic al C o mput e r Sci
-e n ce 47 (1986) 9 -98
Trang 8[11] A , Prasad Sistla, Y Moshe, and Pierre Wolper, The complementatio problem for Buchi
automata wih applicati ns to temporal logic, The or et cal Com p te r Scien ce 49 (198 )
217 - 237.
350 LE cuoc H N, H TI EN D O N G
N lu in 1 - isa u su a n gay 9-8-2 04