Ngon ngir vet la mot trong cac me hlnh tot dg mo ta dang di~u ciia cac h~ turrng tanh, to g 0.6 m6i vet bigu dib mot qua trmh.. Dang chu an cua vet cho phiro'ng an toi U'U Mthirc hieri c
Trang 1T~p chi Tin iioc va D u khi€n hoc, T,17, S,l (2001), 72-77
HOANG CHI THAN
Abstract. The th ory of traces, originated by A, Mazurkiewicz in 1977, is an attempt to provide a mathematical description of the beh viour of con urrent systems, The normal form of a trace gives an optimal concurre t order to prform the process described by the trace,
After composing two c ncurrent systems, the synchronization ofcorrespon ing taces la guages d scribes the behaviour of the composed system whilst the syn hronizatio of corresp nding taces descrbes its processes The paper investigates the synchronizatio of traces a d points out algorithms for finding the normal form
of trace and synchronization tace,
The results not only show an optimal performa ce of processes of the compose systems but also aid in constructing whole beh viours of these systems
T OIll t { t Ngon ngir vet la mot trong cac me hlnh tot dg mo ta dang di~u ciia cac h~ turrng tanh, to g 0.6 m6i vet bigu dib mot qua trmh Dang chu an cua vet cho phiro'ng an toi U'U Mthirc hieri cac hanh d9ng trong qui trinh met each tro'ng tanh N9i dung bai bao la xay dung thuat to an d~ qui ngl{n gon dg tlrn dang chua'n cia vet, nghie cU'Usu'dong b9 ciia ca vet, dang bigu di€n v a dang chua ctiavet dong b9,
1. MC)' DAU
Vet va ngon n ir do A, Mazurkiewicz de xufi la mi?t cong cu tot Mmo hlnh hoa c ac qua trlnh
va dang di~ cua cac h~ tu-orig Lranh , Moi mot vet deu co duy nhfit mdt dang chuan, rna dan chuan nay chi ra each thirc hien toi U'U cho qua trinh dU'C?,Cmo d,bo'i vet, Ngoai nhfmg thu~t toan da co de' tlm dang chuan cti a vet [ 1 ] ch ung toi xay du'ng mi?t thuat toan d~ qui rat ngh gcn, giup tim nhanh dang chuari cua vet, Khai niern dong bi? vet la mot kh ai niern mang y nghia thu'c te rat quan trorig , d~c bi~t khi xfiydung h~ Ian hop thanh truc tiep ti dang dieu cu a c ac h~ thanh phan [31 , Bai bao ciing q,p trung nghien ciru cac thuat toan nhanh tim dang chuifn cu a vet do g b9, gop phan nghien cuu dang dieu v a cac tinh chat cua h h p thanh
Gia s1 'A la mot bang chir cai hiru h an
Djnh nghia 2 1
1) Mi?t quan h~ doi xirn va khong phar x~ p~ A x A diroc goi la mi?t quan h~ aqc 1 p tren A ,
2) Mi?t bdng c h ii: clii tu o r u ; t r a nh la mi?t c~p (A, p) trong do A la met ban ch ir cai va pIa rndt quan h~ di?c l~p tren A,
Ky hieu: FE = (A , p )
Bang chir cai ttro'ng tranh thiro'ng du'cc bi~u dien bo'i mot do thi va hutmg
Gia s11'FE = ( A, p) mi?t bang chir cai tu'o'ng tr anh
Quan h = p ~ A* x A* d ucc dinh nghia nhir sau:
Hai tv: du-oc goi la biing nh au theo p neu tir nay dtro c tao boi tu: kia bhg each giao hoan hai chir cai di?c l~ ding lien nh au Neu moi chir cai th~ hien mi?t hanh d9ng n ao do, moi tir th~ h ien m9t day cac hanh dong thl thir ur thu'c hien cac h anh di?ng di?c l~p dung lien nhau la khong quan
Trang 27 tro ng Do v ay chung co the' dtro'c thu'c hi~n mi?t each tu'o ng tranh vai nhau
Tir quan sat nay chung ta dua ra mi?t quan h~ tuo'ng dtrong nlnr sau:
Quan h~ = ' ~ A* XA* diro'c dinh nghia la quan h~ tu'o'ng dtro'ng nho nhat tren A * chua =1 "
V6- moi x, y EA*, x va y dU'C?,Cgoi la p - tuo ng dtro'ng n u va chi neu x =1'y ,
T a co:
1) =1' = (=1') * ,
2) x = 'y =:> I x l = I Y I,
Hai tir la p - nrong dtro ng se co cung di?dai vl cac chir cai trong chung la nhir nhau
Dmh nghia 2.2 Gii sti:FE= (A, p) la mot bang chir cai tiro'ng tranh
1) M6i mot lap tu'o ng du'o'ng cu a quan h~ tu'o ng du'o'ng = 1'd c goi lit mot v t tren FE, Di? dai
cu a vet t duo'c xac dinh bO'i di? dai cila tir dai dien cua no, Neu t = [ x ) 1' thl: I tI = [z]
2) Mi?t t~p cac vet duoc goi la mot tiqon ngit vet tren FE,
Nlnr v~y, m6i vet bao gom tat d, cac each thuc hien tuan t~' co the' cii a mot day cac h anh dong duo'c bie' u di~n bo'i d ai dien cu a vet nay,
Cac phep toan tren vet:
1 Neu t1, t 2 lit cac vet, tl = [ x l va t2 = [ X2 ) p thl ho p th anh cu a t v a t2 la:
t 1 ' t2 = [ Xl, X 2)1' ,
2, Neu T1, T 2 la cac ngon ngir vdt thl hop th anh cua T, va T 2 la:
T, ' T 2 ={t 1 't2 ItIE T; 1\ T 2 E Td ,
2.1 Dang chua'n cua vet
Nho' phep toan ho'p th anh vet, m6i mot vet co nhie u each ph an ra th anh ho'p th anh cii ami?t so
vet kh ac Chung ta quan tam tai mot ph an ra d~c bi~t nhir sau:
V"0'1 moxi. veAt t = 'I)w p an ra:hf - t = tl ,t 2 ' ,t-«, m >_ 0 ma:'
1) t khOn r6ng,
2) t , = [U i , m6i chir cai trong U i chi xuat hi~n mot ran va hai chir c ai trong Ui la di?c l~p voi
nhau (giao hoan du'oc]
3) Neu t, = [ Ui ) , ti + l = [ Ui + l thl m6i chii' cai trong Ui + l khcng doc l~p vo'i mc;Jt chir cai nao do
trong u; (di?c l~p C~'Cdai]
Phan ra nay diro-c goi la dq,ng chua!n cii a vet t,
2.2 Y nghia cua dang chua'n cua vet
Gi<l,sU' vet t treri mot bang chir cai tirong tranh FE co dang chuifn la:
t = [a cf ), [d ) [ab ) [eb) 1 '
Dang chuifn cu a vet cho phio'ng an thu'c hien mot each tu'o g tranh toi U'U cua qua trlnh dutrc
bie'u di~n boi vet nay,
Trang 3Dinh ly 2.3 VO'i moi vet t , d aru; c hud ' n c iia no a e u t o n tai va duy nluii,
2.3 Cac thuat toan tirn d ang ehua'n ctia vet
Thu~t toari 2.4 I ]
aJ x at hien thl no se du'oc ghep vao tir u (r( J )) Mang cac co to d u'oc dip nhat de' dim bao rang
mai ch ii: cai kho ng d c lap vo'i a J neu xu at h ien thl chi co the' d~t vao trong cac khoi sau do
1 Vo'i moi 1:::::z :::::n, r i ) : = 1(n Ii so c ac chir cai trong bhg chir cai A).
2 Vci moi 1:::::k :::::I, u(k) : = ) (Ila d~ dai cu a vet t )
3 k: = O
4 k : =k +1
5 D~t J m a u ( k ) = ai'
6 u (r( J )) := u (r(J)) ai ·
8 r (J) :=r( J ) +1
9 Neu k <1thi chuye n len biro'c 4
10 Ch n m rna u (m) i' ) va
hoac la m < 1 nlurng u ( m + 1) = ) hoac m = I
Thuat t oan 2.5
Truoc het ch ung ta nhic lai khai niern sau day:
Do thi phu thuoc cu a vet t = Iw]lit do thi phu thucc cua tir w , ky h ieu boi D(w) , dtroc dinh
nghi a d~ qui nhir sau:
- D ( A ) Ii do thi rang
- D( w a ) Ii do thi dtro'c tao b i D (w ) them mot dinh moi a va cac canh hu ng toi dlnh moi
nay ti c ac dinh cu a D( w ) m a chung phu thuoc vao a.
V6'i vet t chung ta xay dung do thi ph u thuoc cua no
Tir do thi phu t.huoc lay r a tap cac dlnh cuc ti~u bao gom cac c1lnh khorig c6 can h nao di t6'i,
do chlnh la t.han h phan Ul
Loai bo t~p cac dinh C I) ·C ti u cling c c canh di ra tir chung , ta dtroc mot do thi phu thuc moi,
lai l<iy t~p cac dinh CI) ·C ti u cuad thi nay ta diroc th anh phan U 2.
L~p l cac buo'c tren ch aen khi nh~n dtro'c m9t do t h i ran , chung t a se rihan du'o'c tat dc ac
thanh p an tro g dang chuiin cua vet
T'ir chinh dinh nghia cua dang chuan cii a vet, chung ta dua r a rn t thuat tcan d~ qui sau day
'I'huat toan 2.6 (D~ qui)
Cia sti.·t = Iw] v IU1].lu2] lurn]a d chuiin can tim c a vet t.
Khi do t = lu ·u2] lum ] = IUlU2' " urn ] = lud·lu2 urn ] ·
Nh ir vay cac chii' cai to g Ul duo'c "dich chuye n" tir ph ia phai ve dau tir d ai dien w, hay noi
cach khac: cac chir cai trong Ul phai "giao hoan drroc" voi tat d cac chir cai ben trai no
Tim d u'o'c Ul, ta co:
t = IU1J.lw']' vo'i w ' la.tir n hfin duoc tir w sau khi loai bo cac chir c ai trong Ul di.dich chuye n
duoc ve phia tr ai
Sau do ta l~p lai b cc lam & tren de' tlm U 2 gom cac chir cai trong w' m a ch ung "giao hoan
d o'c" v i tat dcac chii cai ben tr ai, '
Lap lai cac bu'oc tren cho den khi n h an diro'c tli' d ai dieri rang, ta se nhan diroc tat d c ac thanh
p fin to g d ang chuan cua t
So v i hai thuat toan o · tr en, Th uat toan 2.6 co cling de? phirc tap nhir Thuat toan 2.5 la.OW)
rihtrng thuat toan nay do gian hem rat nhieu
Trang 4c c TH U AT ToAN TIM DA NG C H UAN CU A V ET vA VE T DONG BO 75
t = [a dcf a e bb ] 1 '
Theo Th uat toan 2.4 chung ta xay dung mang cac con t3 r nhtr sau:
d
T v: do ta tirn duoc dang chuifn cua vih t = [ a cf [ d ] [ a b ] [eb ]
Theo Thuat toan 2.5, d thi ph u th uoc cua vet t va cac t~p dlnh cuc tiifu sau moi buoc lo ai bo
nlur sau:
/
fa
I
I I
I
I
I
\
f
< , ,
• - "
b \
\
1 I
el ,_/
Tir cac t%p dinh ClrCti~ , ch ung ta trm dircc dang chuari cu a vet t giong nlnr ket qua cua Thuat toan 2.4
Theo Thuat toan 2.6, cac chir cai c v a f co thif chuyfin sang tr ai, ta diro'c th anh phfin dau tien
Ul = acf. Kho g co phan tll' nao co thf chuyen sang trai df ket ho'p voi d , v ay U2 = d Bay gia b
co thif chuye n sang tr ai d~ Ht http vo'i a , nen U3 = a b, con lai U 4 = e b
a dcfa e b + acf l d l aebb + acf l d l ab le b
Dang chuifn cii a vet dtro'c iing dung nh ieu trong c c bai tcan thiet ke, dieu khi~ ching han trong viec dieu khi~ cac co's3' duoli~u phan tan, trong day chuyen san xufit
3 DONG B9 CUA cAc QuA TRINH
Gi<l.sll' It = ( A, p ) la m ot bang chir cai tu'ong tranh Quan h~ d = A x A \ p Ia ffi9t quan h~
p an xa, doi xu'ngv a du'o c goi Ia quan h~ ph,!! thuqc tren It
Vi~c su dung quan h~ d9CI~ phay quan h~ phu thuoc d theo dung n hia cu a chung trong hau
het cac tru'ong hop la n htr nhau
Trang 5HoANG CHi THAN H
Moi vet ten FEbi~u dien mot qua trinh va m9t ngon ng ir vet tren FEbi~ dien m9t ho at aqng.
G i3.sti·
FEl = (Al,Pl) va FE2= (A2,P2 ) la hai bang chir cai tuo'ng tranh
Vfiy dl = Al XAl \ Pl va d 2 = A2 XA 2 \ P2 la hai quan h~ phu thuoc tu'ong img
3.1 P'hep chieu vet
Dirih nghia 3.1 Gi3.su t la mot vet tre F Khi d6 hinh chisu cu a t tren d 2 dtro'c dinh nghia nhir
sau:
[ >' dll d = [ > ct nd 2
(t a J ,t, ) l d = { (t l d ) · [ a ] d, n d2
t l1
neu a EA 2
neu a EAi \ A2
3.2 P'hep ho'p t.hanh trro'ng tranh cua cac qua trinh
Bhg chir cai ttro'ng tranh If.= (Al UA2, p ) , v6i P = Pl UP2 la bang chir cai turmg tanh ho p
th anh tir FEl va F E 2
Dinh nghia 3.2 Gilt suot la mot vet tren FEl vat2 la m9t vet tren FE 2 Phep hop t h nh tuotu ;
t r an h ci a tl va t 2 , kf hieu tl # t2 la vet t tren If ma
t d , = t va tl d = t2 Neu vet t ton tai thl:
1) Moi mot h anh d9ng xujit hien trong t phaila mdt hanh d9ng cua t hoac t2 vakho g c6 hanh dong n ao kh ac xufit hi~n tron t
2) SV·xufit hien cua cac h anh d9ng sip xep to g t 1 va t 2 ciing diro'c sip xep giong h~t nhir trong t
3) So ran xuat h ien cu amot hanh d9ng n ao d6 trong tl hoac trong t2 ciing gion h t n hir trong t
Do d6 t diroc goi la aong bq cii a t 1 va t - z
D~ tim dang chuari cua don b tJ;'U"<JChet phai tim tu: dai dien ch vet dong b9, sau d6 ap dung
cac Thuat to an 2.4 - 2.6 dii trlnh bay trong phan 2 d€ tim dang chua:n cho vet dong b9 nay
4.1 P'hep tci ho'p song song cua cac ngon ngir
Gi<i suoA la mot bang chir cai va B ~ A
h t s : A * - > B * la aong cau xo a diroc dinh nghia nlur sau:
Va E A, ho ( a ) = { : neu a EB ,
neu atI - B
Neu x = a l a2 ak EA * thi
h ( x ) = h (adh n ( a2 ) ·hn(ak)
va thtro'ng diro'c viet la: xl n - hlnh chieu cua x tren B.
V6-i moi ngon ng ir L, ki h ieu L la bang chir c ai nho nhfit xay dung len L
Djnh nghia 4.1 Gia su Ll va L2 la hai ngon ngir, Toh o p song song cua hai ngon ngi Ll, L 2 la
mot ngon ngir, ki hieu la Lt IIL diro'c dinh ng ia nhir sau:
Trang 6CAC THUAT TOAN TiM DANG CHUAN CUA VET vA VET DONG 80 7
D!nh Iy 4.2. Gid sJ: tl = [WI)dl va t2 = [W2)d" neu aong bq t c d a tl va t2 to tq,i thi W = Wlllw2
G·ia" S Uo tl = [ )U < , t2 = [)'V'c.t, va ton tai d"o g
[a la ak ) dl n d,. veriaI, a2, , ak E Al nA2.
dieu cua cac h~ hop th anh tir chinh dang dieu cu a cac h~ thanh phan,
TAI LI:¢U THAM KHA 0
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Nh n b ai nga y 9 -7-2000
Tru oru; Do: hoc Khoa hoc t1F nhien