Tfp cht khoa hfc TrUbng Bfi hfc Quy Nhan - Sol, Tip VIII nam 2014
AUTOMAT MCK D^TA TREN D^I S 6 GIA Ttf VA Sir M 6 R p N G DESH LY MYHILL-NERODE
LE XUAN VffiT, LE XUAN VINH' I.M6BAU
Dya tren ca scr t^p ma do L.A. Zadeh de xuat, cau true toan hoc dau tien ve automat ma duac Wee [6] dua ra nam 1967 Tiep sau duac E.S. Santos [1] dinh nghia vai ten goi la maximin automata nam 1968. Lee va Zadeh [2] dua ra khai niem automat ma huu han trang thai nam 1969. Hang lo^t cac dinh nghia khac vejuzzy automat cung duac de xuat nhu [3, 4, 5, 9],... Cho den nay no van la van de dang duac nhieu nha nghien cuu quan tam. Trong bai bao nay, ngoai viec tong hop mpt so dgng automat ma con de xuat mot dang moi do la automat co phep chuyen trang thai bang ngon ngu dua tren dai so gia tu huu han va buac dau khao sat cac tinh chat cua no ma trong do cong viec chii yeu la tap trung vao viec mof rong dinh ly Myhill- Nerode.
2. M O T S 6 DANG AUTOMAT MCi
Noi dung duac trinh bay trong phan nay la neu torn tat dinh nghia cac dang automat ma cua mot so tac gia, qua do giup chiing ta thay duac tinh phong phii cua viec nghien cuu thupc ITnh vuc nay.
Dinh nghia 2.1. ([5]) Mpt automat ma la mpt bp gpm nam thanh phan, A = (J, V, Q, f, g) trong do:
/ la tUp hiiu han khac rong gom cac ky hieu dau vao, K la tap hihi han khac rong gom cac ky hieu dau ra, Q la tap hiiu han khac rong cac trang thai;
/ l a h a m t h u o c c i i a c a c t a p m a t r o n g Q x / x Q, f.Q^l^ Q^\^,\\
g la ham thupc oua cac tap ma trong K x / x Q , g: V^I^Q^ [0,1].
Vai I, la xau cac ky hieu dau vao (dp dai^), thi/i dupc dinh nghia nhu sau.
fA{q\,h 9r,)= max{min[/(9i,^,?o)./(?o.'2.?,)..../fe.',.?.)]}
Dinh nghia 2.2. ([1]) Mpt automat m a ^ la mpt hp nhu sau: A = (U, S, fi, F. Mid trpng do:
f/ la tap hOu han cac ky hieu vao;
LE XUAN Vlfir, L 6 XUAN VINH"
S la tap huu han cac trang thai;
//': 5 X [/ X s -> [0,1] la ham chuyen trang thai fl, s-s"
ft(s.A.s')=\
[0, s^s"
u\s, u'u, s')= imxmm[u*(s,u',s"),fj'{s",u,s')]
F la tap cac trgng thai ket;
/ih'. 5 -*• [0,1] la cac phan bo khoi dau.
Dinh nghia 2.3. ([2]) Mot automat huu ban dan dinh ma FA la mpt bp nam,' FA = (Q,£,S,q„.F)tiongd6:
Q la tap hiru han khac rong cac trang thai;
Xia tap hiiu han khac rong cac ky hieu vao;
^la ham chuyen trang thai, tJdi tii Q x 2"vao Q x [0,1]
a e X £^ Q X [0,1]
S(.q, a) = (IJ', ft) go la trang thai khoi dau;
F c-Q la tap trang thai k^ thiic.
Mot automat hiiu han khong dan dinh ma FA la mpt bp nam, FA = (Q, I, S, qo, F) trong do'
Q. £, qo, F dupc xac dinh nhu tren;
S\k ham chuyen trang thai, 5di tir g x Jvao 2^"''°'' .5: e X ^ - > 2e'"''U
Ma rpng ham chuyen S:
S(q X) = (?,1), S{,q. ax) = S(qux), S(q a) = (?,, ft).
Dinh nghia 2.4. ([3]) Mpt autpmat huu han ma A tren bang chii E la mpt bp bon, A= (Q. ;i; {f (0-) lo- e E}, rf), trong do:
2 = {?i, ?2, ...,?,} la tap hiiu han khac rong cac trang thai;
7!= (a-,,, ;r^, ..., ;r,„) la vector dong cac trgng thai khai dau <i<n^<\;
Gc Q\it$p trang thai ket thiic;
n = (.>J,i, Igz, . , »7y.) la vector cot (mo) n& ?, € G thi 7^ = 1, ngoai ra 17^= 0.
rf dupc gpi la tap trang thai ket thiic;
'^(f' = fA.</'^']J"i •.» '* "la trgn ma cap «, F{(f) dupc gpi la ma tr|n chuyen trang thai ciia .4.
AUTOMAT MQ DIJA TREN BAI s6 GIA Tif VA SlJ M 6 RQNG 19 Moi phan tu /^ ^^ cua ma tran F(d) dgng fiA{q„ c, Qj), voi q„ qj e Q, a e "L,
MA- Q '^^ ^ Q -> [0,1] la ham mo chuyen trang thai. Vcri moi x = ffiOi--- cr„ ^ S', s. t e Q, ta co:
MA{S,X, 0 = max min[^ji(s,<Ti,qi),MAiqh<^2,q2l-^MA(^m-h^m,0^
Q<.i<m-\
Voi x,y ^ I*, J, / e Q, ta co:
fl, s = t M{s,X,t)=\^
[0, s^t
fiA (s, XV, 0 = max min[//^ (s, x, q\ //^ (^, 7,0]
Dinh nghia 2.5. ([4] ve automat ma Mealy)
Mot automat mo Mealy Af= (.S*, X, U, F(u/x)) trong do:
S = {si, j'2> ••, •^v} goni V trgng thai;
X = {xu X2, ..., Xf^} gom // ky hi$u dau vao;
U = {ui, «2, • • •, «§} gom ^ ky hi?u dau ra;
F{ulx) cap vxv la ma tran chuyen ma, nghia la moi phan tu cua ma tran la mgt tgp mcf the hi^n muc dp khi chuyen tu ky hi^u .x trong X sang ky hieu u trong qua moi trgng thai thuoc 5.
Trong truong hop chi co mpt dau vao, ma trgn chuyen ma se co dang . F(«,/x,) = [^X%/A:,)],/i-l,...,f
fy{UhlXi) la ham thupc ma khi chuyen trang thai s, sang trang thai Sj win dir lieu dau vao JC, va dau ra «» ( v = 1, .., v),
f',j(ulx,) = max {f,j(uJx!),f,,(Milxd, ., Mu(/x,)}
Sir dung phep cpng dai so thay cho phep lay max, ta dupc:
f',j(.u/x,)=Mu,/x,)+Mu2/x,)+ .. +Mui/x,) TCr do:
F'(U/X,) = P(u,/xi) + i^(«2*i) + ... + F'iUf/xi) F*(u/x,) = Piui/xi) 0 F'(ujx^ 0 ... o i^(«|&,).
Chiing ta thay ring trong phan lan cac dinh nghia deu sii dung phep hap thanh max-min de chuyen trang thai. Dinh nghia 2.2 va Dinh nghia 2.3 co cau tnic tuong tv nhau, dieu khac biet chinh la xuat phat tgi mpt trang thai hay nhieu trang thai. Dmh nghia ve automat ma Mealy trong Dinh nghia 2,5 chi mang tinh tham khao. Cac Dinh nghTa 2.1 va 2.4 tuang doi phiic tap. Vi vay, trong phan ke tiep chiing toi se dua ra cac dinh nghia moi ve automat ma hiru hgn dan dinh va khong dem dinh dua tren dai so gia tli CO oau tnic ttra Dinh nghia 2.3 nhung xuat phat tii tgp trgng thai cho truoc.
20 LE XUAN VIEr, LE XUAN yiNH*
Dong thoi ciing chi ra su t6n tai trang thai trung gian r khi chuyen trang thai tii trgng thai p sang trgng thai q ciia automat thong qua viec quet mpt xau ky tu.
3. AUTOMAT MCi DUA TREN DAI SO GIA liS 3.1. Vai net ve dai so gia tit [7, 8]
Gia su X la mpt bien ngon ngu va mien gia tri ciia X la Dom (X). Mot dai so gia tiii4^tuang ling cua X la mpt bp bon thanh phan A ^ = (Dom {X), C,H, <) trong do C la tap cac phan tii sinh, H la tgp cac gia tii va quan he "<" la quan he cam sinh ngii nghia tren X. Dom (X) la mien tri ngon ngii thu dupc bang each tac dpng cac toan til tix H vao C. Trong mien tri Dom (X) ta quy uac co chiia cac gia tri dac bipt 0,1, W vai y nghia ia phan tii be nhat, phan tii lan nhat va phan tii trung hoa mpt each tuong ling (theo thii tu ngu nghia). Mpt gia tri ngon ngii la day lien tiep cac gia tii tac dong len phan tu sinh, chang hgn A„An.i... hic, h, G H (i = 1,..., n), c G C, ta ky hieu la y = OC ((7thay thS cho chuoi gia tu). Vi du nhu Jf la bien chan ly TRUTH thi Dom (X)
= {true, very true, possible true, very false, false,...] ^ {9,1,W},C= {true, false, 0, 1, W), va H- {very, more, possible, little}. Hai gia tri ngon ngii trong dgi so AX\i sanh dupc voi nhau bai quan he ngu nghia, dl thay very true > true, possibletrue <
true. Neu cac gia tri trong mien ngon ngii duac sip tuySn tinh thi AX ivtac gpi la dgi sp gia tli tuyen tinh va ta ky hieu A la phep toan lay min va v la phep toan lay max thep thii tu ngii nghia.
3.2. Djnh nghia automat ma dva tren DSGT
Dinh nghia 3.1. Mpt autpmat huu han don dinh ma dua tren DSGT (Deterministic Finite Automata based on Hedge Algebras-vik tJt la DFAm) la mpt bp gpm 5 thanh phan: M = (Q, I, 5.1, F) trong do:
Q la tap hiiu hgn khac rong cac trang thai;
Xia tap huu han khac rong cac ky hieu vao;
^ la ham chuyen trgng thai, <5 di tir tich Decac Q x 2' x Q vao dai so gia ti tuyen tinh y4X cua bien chan ly, .5: g x 2'x g -^AX
<5(?. a,p) = ac = y.
IcQ la tap trang thai khcri dau;
F C g la tap trang thai ket thiic.
Ma rpng ham chuyen iJcho xau ky tu dau vao:
^•.Q'^I^xQ^AX
i) ^(q, /!,?) = 1; g'(q, A. p) = 0 niu p i= q; vdi JI la xau rong;
ii) g(q, xa,p) = v^ {g(q x, r)rsg(r,a, p)}.
Djnh nghta 3 J . Mot automat hiiu han khong dan dinh ma (Nondelerministtc Finile Automata based on Hedge Algebras-iu„c viit tit la AK4^) la mpt bp nam thanh
AUTOMAT M(S DUA TREN BAI sg GIA TLf VA SU MC) RQNG 21 phan My = (Q, Z, S^ I,, Fj) trpng do g, Z dupc xac dinh nhu trong dinh nghia ciia
DFAfjji, Iy la cac trang thai trong g thupc tap trang thai khoi diu / a miic /, vcri gia tri Y e i4^va Fy la cac trgng thai trong g thupc tgp trgng thai ket thiic F a miic ;', can Sy la ham chuyen trang thai dupc xac dinh:
Sy-.Q^t ^2^
i)Sy(q,X)={q],yqBQ;
ii) Sy(q, a) = {p\ S(q, a,p)>ry,
iii)Sy(q,xa) = Sy(Sy(q,x).a)= u Sy(p,a).
Dinh nghia 3.3. Ngcn ngii ciia autpmat ma khong don dinhiV/^ ky hieu L(My), L(Md
= {x e E* I S(q, a,p)>r,q^Iy,p(E Fy).
Menh de3.1. Cho automat m& huu hgn khong dan dinh My= (Q, £, Sy Iy F^. Khi do tqp trgng thdi Sy (q, xy) khdc rdng niu vd chl niu ton tqi trgng thdi r thuoc S^q,x) sao cho tgp 5y(r, y) cung khdc rong.
Chung minh. De chimg minh menh de tren ta can chi ra:
Sy(q, xji-) jt 0 <=> 37- e Sy(q, x): Sy(r, y) + <Z (1) (<=) Dya vao Dinh nghia 3.2 hien nhien ta co ket qua.
(=>) Gia sii Sy(q, xy) # 0 . Ta se chiing minh quy nap theo dp dai tiry.
+ Cff so quy nap: 1 y I = 0, tuang img >' = A thi theo dinh nghia ham chuyen trang thai Sy suy ra(l) dung.
+ Budc quy nop: Gia sil (1) dung voi mpi tir z thoa I z I < t . Ta se chiing minh (1) dung voi tir;'= 0102 ..f^fc. lyl = A. That vay.
14
Sy(q, xy) = Sy(q, xa^ai.. a^.^a^) = Sy(Sy(q, xa^a^ .. at,), at).
Neu Sy(q, xa^a^... flt-i) = 0 thi suy ra Sy(Sy(q, xa^a^... ai,.j). a,,) = Sy(0, at) = 0 hay Sy(q, xy) = 0 , mau thuan vdi gia thiet. Do do bat buoc Sy(q, xa^a^... at.i) i^ 0 . Theo gia thiet quy nap thi luon ton tai r e Sy(q, x) sao cho Sy(r, 0,02. • "^.i) ^ 0 . Khi do chic chin ton tai r e 5^ (q, x) va ton tai p e ^^ (''• "i"! •• "'-i) *^ ^r fe "*) # 0 vi neu khong thi u 5y(p,at)=<S, Vr G Sy (q, x) hay S, (r. a^Oi. atiOk) = Sy (r,;;) = 0 Vr e Sy(q, x). Suy ra 5y(q xy) = 0 (vo ly). -
Tir Menh de 2-1 ta co hp qua sau:
Hf qua 3.1. Cho automat md hm hgn khong don dinh My = (Q, Z, Sy Iy F,). Khi do vdip, q G Q,vd aja2... at^ ^ ta co:
p 6 Sy(q, a,a2... at) o 3 q,,q2,., qt.i e g: (q\^Sy(q, aO) & (qi e Sy(qi, ai)) &
&(pGSy(qt.,.at)).
Ching minh. De dang thu dupc ket qua dua vao mpnh de tren.B
22 LE XUAN VIEr. Lfe XUAN VINtf 4. M6 RQNG DJNH LY MYHnX-NERODE
Dinh ly 4.1 Cho S la mot monoid v&i phdn tir dom vi XvaLld tap con ma cua S. Cac khang dinh sau day la tuang duang:
i) L la ngon ngir chinh quy ma;
ii) L la hap ciia mot sd lop tuang ducmg cua quan he tuang duang bdt bien phdi CO chi so hihi han L = L, KJL, ^...^L„ vai L., =^[x\y> i = l,...,t
Hi) Gid su ta co quan he Ri nhu sau:
\^, y e S. X Ri.y o (Vz e S. ye AX, Lixz)>y ^ L (yz)>y) Khi do Rl la quan he tuong duang bdt bien phdi co chi so hiht hgn.
Chung mink. De tien trong vi^c trinh bay chung ta su dung cac ky hieu sau:
L(M) la ngon ngii cua automat M\ L(M)(x) la do thuoc cua tu x vao ngon ngu L{M);
I(p), F(p) la dp thupc cua trang thai p vao cac tap I va F tuang ung; con L(x) la dp thupc ciia tu x vao ngon ngu L.
Ta se chung minh i) ^ ii) ^ iii) => i).
i) => ii). Vi L la ngon ngu chinh quy ma nen ton tai automat hiju hgn khong dan dinh mcr M chap nhan no (L = L{Af)). Gia su ta c6 y e AX va My = (Q, H 6^ Iy, Fy), trong do:
S^: Q X I ' -> 2^, S^(q. x)-{peQ\ S(q, x,p)>y hay S,{q. p) > y}
^={P^Q \lip)'^'i),Fi={p^Q I F{p)>y} voi/(p) chi dp thupc ciia trgng thai p vao tgp trgng thai khai dau / va Fip) chi dp thupc ciia trang thai p vao t ^ tr^ng thai ket thuc F
Goi L, la tap gom cac phan tu thupc L vcri miic lan hem hay biing y, Ta se chung minh rang L^ = L{My).
Truac het, ta chiing minh vm x e L^ thi ton tai /? G /^ va 9 e Fy sao cho S(p, X, q) > y. That vgy:
X eLyhay loS^oF > y, tiic la v^ [{S^oF)ip) A I(p)] > y. Do (4oF)0?) A /(p) > y voi/J nao do nen IQ?) > y hay p e Iy. Han nua (^oF)(p) > y suy ra v [S^.q) A F(q)] S y.
tli do ta CO F(q) > y hay qe Fy.
Chiing ta cung de dang chung minh ring ton tai;? e /, sao cho Sj.(p, x)nFy^0 thi JC € L{My) hay Ly c L{My).
Nguiyc lgi, voi mpi x e L{My) suy ra t6n t^ p e Iy sao cho SJ^, x)nFy^ 0.
Lkyqe S^,x) oFy. Vipely nenI{p) >y,qeFy nenF{q) >y,vaqe S^.x) nen S^,q) > y. Do vgy S^,q) n F{q) > y, hon nua (4oF)(p) = v [S^,r) A Fir)] S y ^
AUTOMAT MO DUA TREN B^l s6 014 TC VA SlJ M 6 RONG 23 loSfiF = V [I(p) A (4oF)(p)] > y. Vi L(M,)(x) = i(;() > y chiing to ;c e i , . Vgy L(M,)
Cuoi cung ta co L(My) = Ly.
Bay gicr gia sii ta c6 cac gia tri yi, y2, ..., yk e A £ duac sap theo thii tu khong giam, tlic la yi < y2 <... < yt. Tu chung minh tren ta thu ducrc L^^ =L{M^^)v\ L^.
dupfc c h ^ nhan boi automat hiiu han (dgng ro) nen theo dinh ly Myhill-Nerode se ton tgi quan he tuong duong bSt bien phai c6 chi so hiru hgn va L^.^ = ^[x\_ vcri [x]j, la lap tuong duong chiia :x cua quan he do. Dieu nay diing vcri mpi ; = 1,2,..., A.
Tiep theo, ta ky hi^u:
Chox e 5vagiasui(;c) =Yi, nhuvay yi<y2S ... <y,.i <i(x) = y,<yj+i<... <
Vk. Ta lgi co ((y,)L u (y2)L u... u (y,)L)(^) = (VOLC^) V (y2)L(^) v . v (yM') =
•yivy2V... vy, = y,. Suy ra L = (YOL U (y2)L u... u (yt)L.
ii) => iii). Tg se chi ra Ri la quan he tuong duang bat bien phai co chi so hiiu hgn. That vgy, dl thay Ri thoa tinh chat phan xa va doi xiing, bay gia ta chimg minh i?i th6a tinh bac cau. Gia sii X i?i >> va y i?i z, V r e S:
L(xl) > y thi L(yt) > y (do ;c Ri, y).
Kyt) > y thi i(zf) > y (do j - Ri z).
Vay neu L(xt) > y thi L(zt) a y, suy ra x Ri z.
De chiing minh Ri_ la quan he bit bien phai ta gia su :t fl^ y va z £ 5 can suy ra xzRiyz. Taco V? e 5, y e A ^ , dgt « = zr. Vi i flj,>> nen L(ra) >y o i(y«) > y hay L(xzt) a y O Uyzt) > y, Vr e S hay A:Z RL yz.
Cuoi ciing ta chi ra Ri co chi so hiiu hgn.
Gpi R, (i =1,..., *) la quan he tuang duong co chi so hiiu hgn tren I , . Lay R = R^nR.ir\...r\Rt, suy ra R la quan he tucmg duang co chi so hiiu hgn. Can chi ra neu xRy thi x Ri y, hay R la quan he mm hoa cua i?i, tiic la fli co chi so hiru han.
GiisiixRy.yz ^ S,y €AX,vkxz e i,. Ta chimg minh yz e L,. Do L(xz) = y, a y hay xz e i c I-r Theo dinh nghia cua fi ta c6 x fi, y. Vi fi^ la quan hp tuong duang bat bien phai co chi so hiru ban nen xz Rj yz va L^ la hpp ciia mpt sp Ipp tuong duang ciia quan he R,, xz thupc mpt trong so cac lop tuomg duong nao dc nen yz cung thupc lop dc. Dieu dp chimg to yz e L^ Vi L^ c L, nen yz e i,. Vgy Ri c6 chi so hihi hgn.
24 L6 XUAN vijr, LE XUAN vanf iii) => i). Tir quan he RL chung ta se xay dung automat m a M = (Q,£ IF) sao choi(AO = i .
Ta ky hieu [x] la lop tuong duong cua quan he fit, Vx e S. Chpn g = {[x] I x e S}.
Do fii CO chi s6 hihi han nen g hihi han. Ta dinh nghia:
I.Q^AX, /([x]) = 1 vai [x] = [X], I([x]) = 0 trong cac truong hpp khac;
F:g->AS;F([x])=i(x);
S.QxSxQ^AX
5([x],j,[z]) = lvcri[z] = [xy], 8 ([x], y, [z])= 0 trong cac truong hop khac.
Ta chimg minh M duac xay dung nhu tren la dimg dan. Hay [x] = [y] thi L(x) = L(y), mpt each tupng duang la niu I(x) = y thi I(y) = y. Vi [x] = [)-] nen xRi,y do da i(x) = L(xXi > y o !()') = L(yX) > y.
Gia su I(y) = f > y, ta lay y < Tl < t Vi X fii y nen L(y) > q suy ra i(x) >^>y (vo ly). Tir mau thuan tren bit buoc Ky) = y.
Van de con lgi la chimg minh i = L(M) hay Vx e S, L(x) = UM)(x). Ta co:
L(M)(x) = IoSfiF= v { / ( [ y ] ) A ( * . c F M
feoFJtv] -vfeWizjAFlIz])} ^jvf^Oyl.x.tDAFdz])} = ^Vj{5(b'],x,[z])Ai(z)} = L(yx) (Luu y ring 8 ({y], x, [z]) = 1 vai [z] = [)«]).
L(M)(x) = jyj(/(b [y]=[X]). (dpcm)
L(M)(x) = yJ/lLyBAfecF)^]} = ,v{/(b])Ai(yx)} = L(Ax) = i(x) (do I([y]) = 1 niu W \y\
5. KET LUAN
Automat mo da va dang dupc nhieu nguoi quan tam nghien ciiu. Trong bai bao nay chiing toi da tong ket mpt so dang automat mo va da dua ra cac dinh nghia ve automat hihi hgn co phep chuyen trgng thai dang ngon ngu, cy the la DFAHA va NFAHA- Dinh ly MyhiU-Nerode trong ly thuyet ngon ngii kinh dien dup'c ma rpng va dupc chiing minh mot each day du. Voi dgng automat ma da neu, chiing ta hoan toan CO the xem xet mpt so tinh chat khac cua no dong thai cung c6 the ling dyng trong nhieu ITnh vvc chang han nhu dieu khien mcr, kiem dinh phin mem,...
AUTOMAT MCJ D U A TREN DAI s 6 GIA TLf VA SU ud R 6 N G TAI LIEU THAM K H A O
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[3] M. Mizumoto, J. Toyoda, K Tanaka, Some considerations onjuzzy atomata, J. Compt, Syst, Sci, (3): 409-422, (1969).
[4] K. Asai, S. Kitajima, A method for optimazing control of multimodal systems usmg fuzzy automata, Inform, Sci, (1): 343-353, (1971).
[5] W. G. Wee, K. S. Fu, A formulation of automata and its application as a model of learning systems, IEEE Transaction on systems science and Cybemetic, SSC-5, (3):
215-223,(1969).
[6] W. G. Wee, On generalizations of adaptive algorithm and application of the fuzzy sets concept to pattem classifications, Th.D Thesis, Purdue University, (1967).
[7] N. C. Ho, V. N. Lan, Hedges Algebras: An algebraic approach to domains of linguistic variables and their applicability, ASEAN J. Sci, Technol, Development, 23 (1), pp. 1- 18, (2006).
[8] N. C. Ho, A topological completion of refined hedge algebras and a model offuzziness of linguistic terms and hedges. Fuzzy Sets and Systems 158 (4), pp 436-451, (2007).
[9] H. A. Girijamma and H. A.Y.Ramasv/amy, An extension ofMyhill Nerode Theorem for Fuzzy Automata, Advances in Fuzzy Mathematics (4), 41-47, (2009).
This research is Junded by Viemam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.01- 201106.
SUMMARY
FUZZY AUTOMATA BASED ON HEDGE ALCKBRAS AND AN EXTENSION OF MYHILL-NERODE THEOREM
Le Xuan Viet, Le Xuan Vinh In this paper, we proposed the fiizzy automata based on hedge algebras besides the collection of a number of fuzzy automata. We first studied some properties of NFAHA and then extended the Myhill-Nerode theorem. In the future, we will consider their applications and study fiizzy pushdown automata that accepts the fuzzy context free languages.
'Khoa Cong ngh? thong tin, Tniong D ^ h(?c Quy Nhon.
'T'hong Dao tjio, Tnrcmg D^i hpc Quy Nhon.
Ngay nh|n bai: 13/5/2013; Ngay nh^n dang. 24/10/2013.
