TRONG eAe HE.. We consider in this paper knowledge systems whose knowledge base consis of F-rules.. Its reasoning process is an iterative execution of a deductio o erator on F-rules.. A
Trang 1T -p chi Tin iioc v aDieu khdn tioc, T.17, S.1 (2001), 54-61
LAP LUAN TRONG eAe HE TRI 'r+urc F-LUAT
NGUYEN THANH THl1Y, PHAN DUONG HIEU
Abstract We consider in this paper knowledge systems whose knowledge base consis of F-rules Each rule allows us to find the truth probability interval of the consequence as a functio of the o es of premise Its reasoning process is an iterative execution of a deductio o erator on F-rules A knowledge system is called stable iff it is consistent and its reasoning process is stationary We have found a necessary and sufficient condito for a strongly mo otone knowledge system to be stable and proved that the reasoning process is stationary for knowledge systems with knowledge base represented by "cracked graph"
Torn tit Trong bai nay ta xet cac h~ tri thuc voi CO" so'tri th u:c gom cac Fvluat ,m luat ch ta qui t~c tinh khodrig xac sufit dung cua ket luan du'oi dang mot ham doi v oicric khoang x ac suat dung cd a cac ti'en
de Qui trlnh lQ-p luan mo ti viec thu'c h ien 11jptorin t1 ·suy di~n tr en cac F-Iu~t cii a h~ Mot h~ tri t h u'c dtro'c goi la 5 diuh khi no phi mau thu~ va qua trlnh IQ-pluan la dirng Chung tai dfi trm d u'o'c di'eu kien din va d11 d~ mot h~ tri th ire d n d ieu rnanh la 6n dinh va cling chtrng minh du'oc ding d i voi cc h tri thirc co do th i bi~u di~n co' sd-tri thirc "bi ran", qua trinh I~p luan la dirn
1 MO'DAU
Trong Iinh v u'c tr tu nhan tao, viec xay dung cac h~ tri thirc la mot tro g nh img van de trung tam drro'c nhieu t.ac gi quan tam ng hien ciu M<?th~ tri th irc gom mot co' s6' tri thu'c va mot co' che l$,p luan Trong th uc Uf, cac tri th irc thion la kho g chic ch C6 n hieu each bie'u di~n tri
thtic khorig cHc chan v6i nhirng phuorig p ap l$,p luan khac nhau [1,2,4] MQt c ach tiep c~n den cac tri thtrc dang nay, h~ tri tlnrc F-Iuat m a ta se xet diro'i day, da du'oc de xufit tro g [3,5] Van de quan trorig la nghien ciru tinh 5 dinh va tinh d ng cua h~ tri thtrc (M<?th~ tri thirc duo-c coi la dirng khi qua trlnh l~p luan se dung sau mot so hii'u han b cc l~p MQt he tri tlnrc la 5n dinh khi n6 phi mau thuan va dirng]
Trong [5]d a ra dih kien nhan biet tinh dung cua h tri tlurc F-Iu$,t la: i) hoac rnoi ham xufit
h ien trong c ac F-Iu$,t khong tang theo moi bien khoang cu a n6, ii) ho~c d t.hibie'u di~ tri th irc phi chu trtnh Hai tru'o'ng ho-p nay thu hep dan ke'ho ca CO" s6' tri tlurc trong cac bai toan thuc te Trong bai nay, chUng toi xet cac he tri thrrc don d ieu manh Dinh ly 1 dtra r a dieu kien can va
dti de' h~ tri thirc don dieu m anh la 5n dinh Mot nhan xet la tinh don dieu manh cua F-Iu~t phan anh sat thu'c v a tru'c quan quan h~ nhan qui giii'a tien de va ket luan theo nghia khi c6 nhieu th n tin hon ve tien de thl se c6 nhieu thong tin hon ve ket luan Khi xet h~ tri thirc bat k (gom d
F-Iuat khong tang va don dieu m anh}, cac tac gii da chirng minh dtro'c tinh dirng cua qua trln l~
lu an neu do thi bie'u dien CO" 5 ' t thirc cti a h~ hoac khorig c6 chu trlnh, hoac neu c6 thl moi chu trlnh dh chira cung ran [Dinh ly 2) C6 the' nhan tHy cac ket qui tong [5]a trtro n h 'p rien
cu a cac ket qui duo'c dua ra
Coi t~p cac khoan con cua [0,l a C [ O , I = { [ a,,B ]I 0 < a : ::; , B:::; I}
SV' kien: la mot c~p gom mot atom S v a mot khoan I E C[o, I] va diro'c ki hieu la (S,J ) v i ng ia rlng xac suSt dung cd a S n~m tron khoan I (ta n6i I la gia tr icua atom S ).
Tr i thirc d arig F-Iu~t [goid.t la F-Iuat) co dang sau:
Trang 2r: ( S l ,I d 1\ 1\ (Sn ,I n) - + (S, 1 = 1 (11 , ,In) ), (1)
tong do I lt ham cu a cac bien khoang I i'
Co' so· ri t.hirc F-Iu~t [kf hi~u lit 8) gom hai th anh phan: t~p cac su:kien 8, = { S , I} v a t~ cac F-Iudt 8r = {r ; }. Moi luat r; E 8r co dang:
r, = ( Ai" I , ) 1 \ 1\ ( Ai.: I,r n ) - + ( A i, I = I , ( 1 , • ,t, )
Ky hieu r lit t%Pcac atom xufit hien to g cac lu~t cua CO ' so' ti t.lnrc 8
Toan tli' suy di{\nt8 t.ren co' so'tri t.hirc 8: G9i J lit t~p cac anh x atir rv ao C [ O , I Moi I E J
duo c xem lit phep gan gia tri eh cac atom Khi do, t8 : J - + J diroc xac dinh nhtr sau:
t8 ( I )( A) = I (A) n ( n /(ii" ,I".,)), VAE r , ( 3 )
i E EA trong do: I E J va EA litt~p cac luat co ve ph ai chua atom A
H~ tri t.hirc F-Iu~t (ki hieu lit6.8 ) bao gom CO ' so' tri thtic 8 va toan tti: suy dien
t»-Gia tr! c ac atom d6i vo'i h~ tri t.hirc 6.8 :
+Phep gin tri ban dau eho cac atom I ~ E J:
I ~ ( A i = t,neu ( Ai , I E 8, va I ~ (Ai) = [0,I neu ngu'o:clai (4) +Phep gan tri cho cac atom sau bU'<1CHip thii' n (n ~ 1)
P'hari loai cac he tri t . htrc:
- H~6.8 lit phi mau thuan tai b o-c l~ thli' n k i VAE r :I~ = j :. 0
- He 6.8 lit phi mau thuan khi vo'i moi n,6.8 lith~ phi mau thuan tai bU'<1Cl~ thir n.
- H~ 6.8lit dirng tai btro'c l~p tlur n khi: V E r : I ~ = I ~ -
- H~ 6.8 lit dirn khi co n M6.8 lt h~ d irng tai buo'c l~ tlur n.
- He 6.8 li 6n dinh tai bucc l~p thu' n khi 6.8 vira lt h~ phi mau thuan vira l h~ dimg tai bu'o'c
l~p th u' n.
- H~ 6.8 lit 5n dinh khi co n de' 6.8 lith~ 5n dinh tai bu 'c l~p thu: n.
Mot sO'ki h ieu:
• Ta viet I ~ thay ch I ( A ), viet /; (I n ) thay cho I (I;~ , , I ~ n , ) (tro g d I lit gia tr icii a atom
Ai , sau buo'c l~p th irn).
• lefti, right, tu'o g ling la t%pcac atom xuat hien o' ve tr ai, ve ph ai cua luat r i.
• V6"i moi khoarig I = [ x, y ] E C [ o , 1],ta d~t: l (I ) = x , r(I) = y
2.2 D o th] co hrro'ng ttrcrng irrig vo'i co' so'tri t.hirc dang F-Iu~t
Do t.hi co huo-ng G tucng ling vo i h~ tri th irc 6.8 gom t~p dlnh rva t%p cung co hu'o'ng
Ki hieu dmax(A, B ) vo'i A, B E r lit d9 d ai diro'ng di xa nhfit tu' A toi B tong G t.hoa man moi
dlnh di qua toi da mot Ian
D9 sau cu a dlnh A E I':
Depth ( A) = maxdllax(X, A)
3.1 Mot sO'kha i ni~m rno' dau
• Voi A, X E r va so tu'nhien n ta dinh nghia cac tan tu' sau:
Trang 3GU YEN TH A H Tm l Y , PHAN DUONG HIJ!;U
t E T x n E
(Nghia la X tic dong lam gia tri c a A bi co trai (; biroc lap thir n ) ,
a, Cr ( A, n),
i E T x n EA
(Nghia la X tic d$ g lam gii tr ctia A bi co ph ai 6,bu'oc Hip thu' n ) ,
• V &iA E I', ta goi l-dm'l11g (ho~c r-ducn ] bfic n cu a A la mdt day Xl + X2 + , + Xn = A, v i
Xi E r tuo ng iing t.ho a man: Vi = 1, n - 1: actl(Xi' X i+l, i+1) (hoac actr(Xi, X i+l, i+1)), Khi do
v 'i 1::;k :: n ta co Xk - > , + Xn = A la mot I-du'(rng b%c n - k +1cua A,
• Diro'n d n la mot day Xl - > X2 - > - > Xn vo i Xi Er va Xi # - X J ' VI::; i# - J : :; n ,
3.2 Luat don di~u t.rai [phai]
Xet luat r : (Sl , 11) /\ ,/\ (Sn, In) + (S , I = f(Il, " In) trong CO' s6' tri thtrc 8,
r du-oc goi la d 'n di~u tr ai khi vo'i hai b9 gia tr i bat ky (I I'" I n, 1) v a (1[ , " I " J ' ) t.hoa man:
II s : I, Vi = ~ , tong do 1 =f(I1 , I n ) va I' = f (I " I; ) neu:
+ (:3i : S,Er v a I(Ii ) < I( I m thi I (J) < I (I') ,
r dU'<?,Cgoi la don dieu phai khi vo i hai b$ gia tr~ bat ky (II , ,, I n ) va (I; , " I ; " J') thoa man:
II s : I,Vi = ~, trong do 1 = f (I l, " I,, ) va J' = i (I ;, " I , neu:
+ (:3i S, E I', v a r (I i) < Tr I m thl I( J) <I(i'),
+ (Vi r(I ; = r«)) thl r( J ) = r(i ),
Co'so ' tri t.hirc 8 du'oc g i la C O ' s d- tri th u c don ai~u man h : khi moi lu%t cu a no vira la do n dieu
t ai vira la do'n dieu ph ai
H~ tri thirc ~ B du'oc goi la hi f tri tliiic don aii f u mo n l khi co's6' tri th irc cua n la don dieu manh
Vi d u 1 Xet co' s6' tri th irc 8 : A [ x, y ] + A [ ~, ~ ] H~~ B la d 'n dieu m anh
Ta thfiy h~ tri tlnrc ~ B khongd irng va d do, khcng 5n dinh
3.3 Tinh 5n din.h cua he tri t.hirc don dieu rnanh .
Dinh ly 1 Gid s, ); ~B Ia.hi f tri ih u:« d o aii f u manh, {Jat N max = T : Dept h( A ) +1 H i t r i thuc
~B Ia.5 di nh k hi va chi kh i n e « dinh tai ln c src lap thu: NI IIax '
Tru-ce het ta s e chung minh cac be; de sau:
B5 de 1 X et h e tri ths i c ~B doti aii f u iruinh; phi mau t h ud n , tu« co A E r va s o n > 2 sao c h
C I(A , n) ( C r( , n ) t u on q 1 i'ng) thi co X E r s ao cho C X, n - 1 ) va actl(X, A , n) (Cr(X, n - 1) va
a tr(X, A,n) tuoruj u'ng ) ,
C hU ' ng mi nh Ta xet C I( A , n ) , Theo dinh nghia (8) ta c6:
1 (I ~ - 1 ) < l( n fJ ( r - l )) = I(I ~ ) ,
J E EA
( 13 )
Luon co
Trang 4LAP LUAN TRONG cAcHE TRI rnuc F - LUA T 57
JEEA
(14)
Suy r a
(1 5 ) ( 1 6) Nlnr vay tir (14), (16) ta co:
JETxnEA
(17)
JETxnEA
(18)
B5 de 2 Xet h~ iri tlui c /::"8 doii aieu manh, phi rruiu thu r in Neu c6 A E r va s on ::::2 sao cho CI(A, n) (Cr(A, n) iu oru; u'ng) thi luo n ton to: l-iiuotu; (r-au'cmg tuon q ung) b~c n cii a A
Chu'ng minh , Ta xet truong hop CI(A, n).
B5 de3 Xet he tri thu'C /::"8 do - n a~eu manh, phi rruiu thuan Gid s - d' Xk - t - t X" la mot l-au'cJ"ng (r-au'cxng tuoru; u'ng) doti bac n - k +1 ciia A Khi 0.6 neu :3k o > k : CI(Xk' ko) (Cr(Xk' ko) tuon q
u ' ng) thi :3no> n: CI(Xn' no) (Cr(X", no) tuoruj u · ng).
Chu'ng minh , Ta xet tru o ng hop Xi; - t - t Xn Ii mot 1- aU ' cJ"ngdon bac n - k +1 cua A.
moi lu at deu la dun dieu m anh]
Tif (19), (20) , (21) rut r a:
hay
Nghia l a :3no = k n- k > k+( n - k ) = n : CI(X", no)
Trang 5Bo de4 Xit hi t r i thsic b B don ai~u mo.nh, p hi mau thuan Ne u co A Er s a cho ~ N >Depth(A)
th6a man Cl(A, N ) ( Cr ( A , N) iuaru; u'ng) thi ~ N* > N: Cl(A, N* ) ( Cr (A , N* ) tuoiiq u " ng)
C hu'ng minh Xet tru'o'ng hap ~N > Depth(A) thoa man Cl(A, N )
Cl(Xio,fo)
Truoru; hop 1 : l(ro+ 1 ) = 1 ( 1 ) )
X 1 + 1 x1 + 1
= I n f ) (PII)) > I( n f (r " )). (22)
iE T x n Ex
1 t o jETx,o nEx,o
X ' O + l
j E Tx, o n Ex ' lI
l x 1 + 1 ETx n Ex, o
Tir (22 ) , ( 23 ) , (24) rut ra: l (1 i " + l ) >I (Ii O +l )= l ( i " ).
X , n + 1 x1 + 1 X 1 + 1
Truon q h.p 2: I(I i ox+ 1 ) <I (I i o )
1 + 1 xt O + 1
Nhi i n zet: Tjr B5 de 4, taco th~ xac dinh tInh bat bien ve gia tri ciia atom A sau NA = Depth(A) +1
b cc
C h u' n minh Dinh 1111
~ N A : CI(A, NA) v a ' I n > N A -,Cl(A, n) (28)
Trang 6LAP LUAN TRO GcAc H E TRI TH U C F-L U AT 5
Theo Bo' ae 4: :.JN * >NA : CI(A, N ' ) Di'eu nay t rai voi(28)
Chieu nghich: Cia sli·h~ 6n din h tai buoc N 1l J ax
Khi d :
V E r r r;,u" = I :! ,u , ,, - => V] : f J (I Nu , , x +1 ) = f J (I N" " )
= VA E r : I " ,· x +1 = I;,u,x
Ch ung min t.io g tu: VA Er Vn 2' : N 1I1 a x : I ~ = I ~ - , tirc la he d irng
VI VA E r : I "' · x i 0= Vn 2': N1lJ a : I 'j i - 0 , tirc la he phi rnfu tIman
Tv : ( 29) , ( 3 ) su ra he 6 ! l din h
(29) ( 3 0)
o
3· bu'oc lip N l J1a X - 1nhtrng 6n djnh 3·bu'oc l~p N 1lJax•
Vi du 2 Xet CO"so' tri tlnrc 8:
A I 2, 1] A( x, y ) > B( x, y ) B( x, y ) > C ( x, y ) C ( x, y ) > A( 2' y )
C (x , y) > O ( x , y ) O ( x , y ) > D (x , y); D ( x , y ) > E ( 2 + 4 "' y );
E(x , y) > F( x, y) ; F ( x, y) > D( x , y)
Do thi t.uo'n in vo'i he tri thirc ll B:
Voi co so tri th trc tren ta thfiy N 1I1ax = 7
Sau day I gia tri c ac atom sau cac phep bien do'i
1
0, I ] 0, I ] 0 , I ] 0, I ] 0, I ] 0 , I
0 [2' I
1
7 (Dau "_" tro g bing ngfim hi~u gia tri cu a atom van gii n uyen n htr biroc liip trurrc]
T'ir bing ten ta thfiy atom F co mot Iau·o·ng b~c 5: E > F > D > E > F chua chu trinh va
6·bu'o c6 he khcng o'n d inh rihun a buoc lip thu· 7 ( = N he lai o'n din h
Trang 74 H:¢ TRI THlJ'C DUQ'C BIED nIEN B()] M9T DO TH~ B~ R~N
4.1 Cun r~n
X do thi G = (V , E ) bigu dien ch h~ tri thirc 6 ,
Ta n i ring ham 1 ( 11, "" I n ) khorg tang theo th anh phan J (can noi, theo thanh phan IJ ) neu
voi hai bo ( 11 , " I n ) va (I ~ " I: J th a man: I = I, (Vi = 1, n, i i= J ) va I c I J ta luon co
[Kh ai niern nay yeu hon nhieu khai niern kho g tan tro g [5]
Ta dinh n hia (X , Y) E E la mot cung ran khi voi moi lufit T ichira X Ii ve trai nhi thanh phfin
th irJ va Y o' ve ph ai thi ham f; kho g tang theo than h pha~ thtr J do
Do thi G = ( V , E ) duoc goi la bi ran khi n6 kho g chua chu trinh ho~c neu co thi m6i chu trinh
deu c6 it nhfi mot cu g r<).n
4.2 Tinh dirn cua h~ tri t.hirc co do th] bie'u dien co' sO-tri t.hirc b] r~n
D!nh ly 2 N e « aath i G bi e 'u u : « C ' s d ' t ri thu:c B b i r a n th i 68 lt i h d U ' ng ,
Bo de 5 Neu :3 A E I', n 2 2 : C (A < n ) , thi : 3 X Er :C (X , n ~ 1) va (X , A) k h ot u; l a cu ng Tq.n
C hu ' ng m i n h Vi C ( A , N) nen I ~ C I ~
-Do d6, :3 TiE BT sao cho A la ve p ai cua r va / 1 , - 1) c I ~ - ( 3 1)
Neu V X Elefti: - C (X ,n ~ 1 ) thi / (1 n - = / (1 n - 2 ) : 2 I ~ - Dieu nay mau thuan v i( 3 1)
Do vay: :3 Y E left, : C(Y , n ~ 1)
Gil sD:doi voi tat dYE left, sa cho Cry, n ~1) deu co (Y , A) la cung ran Ti; C rY , n ~1)suy
ra: I ~ - C I ~ - 2 Do ham I,kho ng tang theo cac tharih phan thay d5i Y , nen / (1n - :2 / (1, , - 2 ).
BO'n1111'a/ (I n-2 ) :2 I ~ - (theo ( 3 )) Su ra / (1 " - 1 ) : 2 I ~- l Dieu nay mau thuh v i ( 3 1)
Trrc la: :3 X E left, de' C (X , n ~ 1)va (X , A) kho g la cung r<).n, 0
C hU ' ng m in h Dinh I y 2
Gia s1 '6 k ho g la h dimg Ta se chirng minh ton tai mdt chu trinh khorig chua cung ran
Do he kho g dirng nen :3 A E r d e 'V n , :3 N > n : C (A , N) Ta lay truo'ng h 'p n = Depth(A) Theo B5 de 5, C(A , N) nen :3 A 1 : C (A N - 1 , N ~ 1) va (A N - I, A) khcng la cung r<).n
Tu'ong t.u, : 3 A i : C (A N - , N ~ i va (A N - i, A N - + d khorig la cung ran, i = n ~ 2,n ~ 1, 1
Xet du n A l - > A 2 - > - > A N - 1 - > A di qua N die'm Do N > Depth(A) n n : 3 i< J : A i =
A J.
Su ra A i - > Ai +1 - > , > AJ la mot elm trinh khong chtra cung r<).n
Th ufit toan sau xac din hxem mot d thi G c6 bi ran hay kho g nhu: sau:
'I'Iruat t.oan:
Vao: Do thi G (r , E )
Ra: Do thi G bi ran hay khorig
P'htro'ng phap:
E ' = E.
For m6i (X , Y) EE d
f khorig tan theo I i then E' = E' \ (X , Y).
If ( E ' kho g chua chu trinh) then Gbi ran else G khcrig bi ran
Vi du 3 Ch CO ' s6' tri th irc B :
A [ g' 1] D [ 4 ' ' 2 ; B[O, '2 ;
o
Trang 8LAP LUAN TRONG cAc Hit TRI THlrC F-LUAT 6
A [ x, y ] - + B [y 'x , ~ ] ; D [ Xl , Yl]/\ B [ 2, Y 2 ] - + G [ xd 1 - X2 ) , 2Yd1 - Y 2 ) ] ; G [ , y ] - + A [ x, y ].
Do thi tuo'ng trng vo'i h~tr thuc 68:
~ - ~
B~~O
Sau day I giri tri cac atom sau cac phep bien do'i
0 [9,1 I ,o ~J2
1 1
1
1 1
G
[ 0, I ]
1 1 [ 4' 2]
1 1 [ 4 ' 2 ]
4
Ta thfy do thi tren co chu t n ,co mot cung ran B G. Cac ham xufit hien trong cac luat th ir 1,3 la ham tang, ham 2 tang theo bien khoan cua atom D Sau biroc lap thir 3 he se dirng
Tren day chung toi dil n hien cuu tinh o'ndinh v a tinh dung cu a mot so h~ tri t.hirc Fvluat, Doi
vo i c ac h~ tri tlnrc don dieu m anh, Dinh Iy 1 kho g n h irng cho ta dieu kie n din v a du de' xet h~ l
o'n dinh ,ma can chi ra diro'c so b cc I~p Nl lax can thiet de' xac dinh qua trlnh lap lufin li.dirng hay khorig So Nl ll a Ii chung ch moi atom Tuy nh ien to g nhieu tru'o ng ho'p, khi chi quan tam den mot atom A cu the', B de 4 ch thfiy c6 the' xet so bu'oc it h n Nl ax mi~n la vu'ot qua Depth( A)
de' xac dinh ducc tin h bat bien cil a gia tr ictia A Do d6, co the' xay ra truo g h p h~ tri thtc tuy khcng dung nhung gia tri mot atom nao do lai xac dinh Ta thfiy r5.n trong cac he tri thirc do n dieu manh, do thi bie'u dien CO" s6' tr th irc tu n ung chi chua cac cu g khong "ran" Doi vo i c ac h~ tri thuc k ho g do n d ieu, ch ng toi dil xet tr u'o'n hop cac CO" s6'tri thirc d 'o'cbie'u dien b6'i "do thj bi ran" (m6i chu tr in h tro g do deu c6 it n hfit mot cung r an] Din h Iy 2 kh5.ng dinh d u'o'c tfn h
d ir g cua h t thtrc tro g truo g ho'p do
[ D Dubois an H Pru e, Po ss ibility T heo r y: an App r oa c h to Compute r ized Proce ss ing of
Un-ce r tainty , Plenum Press, New York and London, 1988
[2] L.A Zadeh, Fuzzy sets, I nfo r m and Cont r ol 8 (1965) 338~353
[3] N G Raymond and V.S Subrahman ian , Probabilstc logic programming, Inf o rmat ion and
C omp u tation 101 (1992) 150-20l
[4] P.D Dieu, On a Theory on Interval-Valued Probabilistic Logic, R es earch R e port (Vietnam CSR), 1991
[ 5 ] T D Que, From a Convergence to a reasoning with interval-valued probability, T op c h i T in
h oc va D ieu khi e n ho c 13(3) (1997) 1-9
Nluin b t ngay 4 -5-2 000 Nhiin lai s au kh» s J:a ngay 19 -2-2 001