DSGT khon thuan nhat diroc kf hieu la X = X, G,LH, ~, trong do G la t~p cac phan tli-sinh nguyen th y, LH gorn cac gia tl- va mot so toan tli- khac diroc dinh nghia tuan irng voi cac pha
Trang 1TIJ-pchi Tin h9Cva Dreu khien h9C, T.2 , S.4 (2004), 355-367
DlfA TREN £>1;\1 SO GIA Tlf KHONG THUAN NHAT
LE x uAN VINH
T r u atu ; D ei h 9C Quy Nhan
Abstract The method in linguistic reasoning which was introduced in [7,8], based on extended hedge algebras This article is aimed to establish some new inference rules being applicable in ling istic
re soning to the case that fuzzy clauses contain "Not so" Its basis is n n-homogeneous hedge algebras which have been investigated recent y [9- 12] Thanks to the approach such as bui ding the deductive system in classical logic, fuzzy deductive system and the consistency of the fuzzy knowledge base wil
be examinated.
Tom t it Phuo g phap l<%pluan trirc t ep tren ngon n ir da diroc trinh bay trong [7,8] dira tren dai so gia tu mer rong Muc dich cua bai bao nay la dira ra mot so qui tac suy dien moi nharn mer rong kha nang lap luan doi veri cac menh de rno chira gia tu "Not so" ma c aso cua no la Dai so gia
tu kho g thuan nhat da diroc n hien cirugan day { - 12] Bang each tiep can nhir viec xay dirnghe
su diEm trong logic kinh dien, h~ su dien mo va tinh phi mau thuan cua ca ser tri thirc mo cling dtroc quan tam nghien ciru
Lap luan ngon ngir la phuong phap tirn cac ket luan tir tap cac khang dinh, trong do ket luan va khang dinh aeu & dang n on ngir, b ng each st'rdung cac qui tac su dien Co 5&
cua n la logic gia tri ngon ngir Chung ta biet rli g moi loai logic deu co cc 5&dai so tiron irng, ch n han logic kinh dien co co 5&la dai so Bool logic da tri la dai so Lukasiewic Dai so gia tt'r (DSGT) co the dUQ'Cxem nhir la mot co 5&dai so cua logic gia tri ngon ngir
va dua tren DSGT rno ro g [5],cac tac gia d trinh bay mot phirong phap l<%pluan ngon
n ir [7,8] Dira tren nhirng tinh chat cua DSGT kho g thuan nhat trong nhirng n hien ciru gan day [9,12], chung toi se dira ra mot so qui tac suy dien moi de mo rong kha nang xt'rI cac menh de mo chira gia tt'r "Not so"
Chi tiet ve DSGT kho g thuan nhat co the xem tro g [9-12] Sau day chung ta chi trinh bay khai quat mot so khai niem, tinh chat coban lien q an den plnro g phap lap luan ngon ngir
Theo each tep can dai so, mien gia tri cua bien ngon ngir co the xem nhir mot dai so sinh tir cac khai niern ng yen thuy boi cac phep toan mot ngoi la cac gia tt'r Chan han
(iu ng, n i t (iung , kh On g (i u ng l iim, sa i, r a t s ai , k hO n sa i l iim la cac gia tri chan ly diroc sinh
ra tu khai niern (i un g , sai boi cac gia tt'r r at , khO ng Xet gia tri k h Ong (i u ng l iim trong tap cac gia tri chan Iftren Theo n ir nghia th ng thiro'n , khOng & day hoan toan k ong phai
la phep toan logic phu dinh ma no chi lam giam mire d9 khan dinh cua khai niem (iung
Trang 2356 LE XUAN VINH
x ong mot ft Nhu vay, kh6ng ro rang la mot gia tl-
M9t each tr c giac, khOn g dung lam, co th€ dung, gan dung co mire d9 nhir nhau va cling
yeu hen dung Hon nira, co the e rn nhan r~ng rat gan dung va rat co th€ dung manh hem gan dung, co th € dung nhung rat khOng dung lam lai yeu hon khOng dung lam Chinh tinh
chat "khong thuan nhat" nay goi y cho chung toi xay dirng cau true dai so moigoi la DSGT
kho g thuan nhat [9]
DSGT khon thuan nhat diroc kf hieu la X = (X, G,LH, ~), trong do G la t~p cac phan
tli-sinh nguyen th y, LH gorn cac gia tl- va mot so toan tli- khac diroc dinh nghia tuan
irng voi cac phan tli-trong dan phan phoi sinh tv do tir cac gia tli-cln mire cua t~p cac gia
tli-H, ~ la quan he thir tv b9 phan earn sinh tir quan h~ n ir nghia giira cac gia tri ngon
ngir trong X cling nhir giira cac gia trr Qui tree ket qua tac dong cua toan tli-h E LH len
gia trj x E X la hx thay VI h( x). Nhtr v~y, mot gia tri ngon ngir nao do trong X se co dan
x = hn h1a v i a E G va hi ELH, i= 1, n.
Trong thirc te, chung ta c hi tac dong hiru han Ian cac gia tli- len phan tli-sinh ng yen
thuy de nhan manh n ir nghia, tire la n ~ p vci p la mot so tv nhien co dinh va du Ian Khi do DSGT khong thuan nhat hiru han X = (X, G,LH,~) la mot dan, can tren dung va
can diroi dung cua hai phan tli-bat ki x, y E X la xU y va x ny diroc tfnh boi cac cong
thirc tron [11,1 ]
Han nira, cac bien ngon ngir thirorig co dung hai phan tli-sinh co n ir nghia n iro'c nhau
nhir dung - sai, nhanh - c lu i m , gia - tr~ Khi do G ={a+, a-} voia+ =1 = o > , goi a+ l a phan
t -sinh dirorig va a - la phan tli-sinh am Phan tli-y = hn h1a- diroc goi la phan tu doi
xirng cua x = b« h a+ va ngiroc lai DSGT kho g thuan nhat diroc goi la doi xirng neu
\/x EX, x co duy nhat mot phan t -doi xirng z "
Nhir vay, trong DSGT kho g thuan nhat hiru han doi xirng X ta da dinh nghia dUQ'Ccac
p ep toan u,n, - Them vao do, p ep keo theo diroc dinh nghia theo ea h th ng thuong
nhtr sau:
x: : ::} Y = -x Uy
Do do ta co the viet
X = (X,G,LH,~,U,n,-,::::}).
Voi pEN co dinh du Ian & tren, X co phan tli- 1 la VPa+ va phan tli-0 la V a-, & day
v ki hieu cho gia tl- Very. Phan tli-trung hoa W diroc dinh nghia la phan tli-thuoc X sao
cho LH(a-) < W ~ LH(a+).
M9t each day d
X = (X, G,LH,~, u, n , -,:::: } 0,1, W).
Tinh chat cac phep toan u, n, -,::::} tren X dUQ'Cphat bieu qua cac dinh ly sau day
Dinh ly 1 1 [12] Trang DSGT khOng thuiir: nhat hiiu Iuin doi xung X = (X, G, LH, ~), v6i moi x, Y EX, h E LH ta co:
1) -(hx) = h( -x),
2) - (-x) = x,
3) -(xUy) = -xn-y va - xny) = -xU-v,
5) x n-x ~ W ~ xU-x,
Trang 3M OT PHUO NG PHAp L A P L UAN NG O N NG l r DVA TRE N B A I s6 G I A T l J KHO N T H UAN N 1\T 357
6) -1 = 0, -0 = 1,-W = W ,
7 ) x > y k h i v a ch i k hi -x < - y o
Dinh 1y 1.2 [12 ] Tr an g f)SGT khOng ti i ui i t: n /i t hi iu luu i r16i x ung X = ( X , G , LH, : :; ) , v6i
1) x::::} Y = - y ::::} -x,
2) x : :::} (y ::: :} z) = Y ::: :} ( x = ? z),
3) Neu Xl ::; x2 th ix l =? y 2:X2 ::::} y ,
4) Neu YI 2:Y2 th i x =? Y1 2: x : ::: } Y 2 ,
6) 1 ::::}x = x v a x =? 1= 1, 0 ::::}x = 1v ax : :::}0= -x,
Cau true cua DSGT khong thuan nhat hiru han aoi xirng au manh a lam C (J sa cho logic gia tri ngon ngir Cac phep toan U,n, -,:: :: }se tirong irng voi cac phep tuyeri, h i phu dinh va keo theo trong logic
2 HINH THUe HOA eAe M~NH DE M<1v A HAM D~NH GIA
M~c du so tir ngir cua ID9t ngon ngir tv nhien la hiru han nhimg kha nan bieu dat cua
n on ngir tv nhien hau nhtr la vo han V &i ID9t vai tir giau tho g tin chung ta co the mo
ta v van trang thai cua sir vat Chang han mau "xa nh l o" cua bau troi ngay h rn nay va
n ay h rn qua chac chan la kho g giong nhau Do vay khi bieu dat tri thtrc cua minh ban
n on ngir tv nhien con ngiroi thircng su-dung chung va cac tir nhir the duoc goi la ca khai niem mo Cac cau chira khai niem mo dircc goi la cac menh de mo Vi du nhir "Minh con tre", "Sinh vien A h9Crat cham" , la cac menh ae mo hay t6 g quat la c c vi tir mo Dirci dan the hien cua bien ngon ngir, chung co the viet thanh "Tu6i cua Minh con tre", "Viec hoc cua Sinh vien A la rat cham" Nhir vay, mot each hinh thirc moi menh ae me C( J sa la mot cap (p, u) v ip la mot vi tir n-ngoi va u la mot khai niem mo,chan han (Tu6i (Minh), tre , (Vi~c h c ISinh vien A), rat cham)
Voi moi p , t~ cac khai niern mo u cua no se diroc nhung vao mot DSGT kho g thuan nhat hiru han aai xirng
trong do phep - la phep toan mot ngoi lay phan tu- aoi xirng Chang han aai voi vi tir
p = Tu6i (ngtro i) thl Gp = {tre, gia}, LH = {rat, co the, kho g, tiro g aai, } Tap tat d cac khai niem mo irng voi vi tir p , kf hieu la T ERp, co the dinh nghia nhirtro g n on ngir hinh thirc cua logic vi tir [13] Tuy nhien, ae tien su-dung cac tinh chat cua DSGT, chung
ta co the dinh nghia mot each true tiep nhir sau
Dinh nghia 2.1 [ 12 ] T ER p la mot bo phan cua D p thoa man cac dieu kien:
(i) c ,c :;;; T ER p,
(ii) Neu u ET ERp thi hu ET ERp v imoi b « H ,
(iii) Neu u ETERp thi -u E TERp.
Trang 4358 LE XUAN V I N
Nhir vay T ER p chira Gp va dong doi vricac phep toan mot ngoi tro g ( Dp, Gp, LH, -).
Tir cac menh de C(J sa , barig cac phep toan logic nhir V,1\ , -', -t co the xay dung cac
menh de phirc tap ho'n Ket qua la thu diroc tap cac menh de mer hay tap cac cong thirc,
kf hieu la F P, diroc dinh nghia nhirsau
D!nh nghia 2.2
(i) Menh de C(J sa (p,u) E FP voi moi U ETER p Voi P = (p,u), b « H ta qui iroc viet
(ii) V&i moi P , Q E F P, P vQ , P 1\ Q , P - t Q, - i: thuoc F P.
Nhir vay F P la tap be nhat chira cac menh de C(J sa va don kin doi voi cac phep toan
logic V,I\, - ' , - t
Chu y r~ng, trong Dinh nghia 2.2 chiing ta han che h E H, tire la khi ch truce ffi9t
khai niem mer u E T ERp thl tir menh de C(J sa (p, u), d i voiqui t1'ic (i) chi co h(p , u) voi
h E H (chir khong phai LH ) la con thirc Day la dieu gici han cua bai nay
Chung ta biet rang tap gia tu n uyen thuy H = H+ +H-, ho'n nira voi I la toan tl'r
d n nhat thi H + +I, H- +I la cac dan modular va VI vay cluing diroc phan hoach boi
ham dQ cao height (xem [1]) G9i cac gia tu tro g cling mot lap phan hoach la dong mire,
thiet r~ g so lap phan hoach tron H- va H+ bang nhau va cac phan tu dai dien cho cac
lap duoc s1'ipco tlnr tir:
h q, h q+ l , ,h- , I, hI, hq- I, hq (1)
sao ch cac phan tu ben trai I deu thuoc H-, ben p ai I deu thuoc H + va phan tu du g cang xa I thi dQ cao can Ion
(1) co the trc thanh cac day nhir sau:
Nhir vay, v i gia thiet tren, moi gia tu ton tai mot gia tu doi xirng qua I va n i chung
la khong duy nhat Dieu nay goi y cho chung ta dira ra dinh nghia sau
Dinh n hia 2.3 Phep doi xirng gia tu, ki hieu bci -, la mot tuong irng da tri tir H +I
t&i chinh no thoa man cac dieu kien sau day:
(i) 1-= I.
(ii) Vo imoi t i « H, n: = k khi va chi khi h eight(h) = height(k) va h, k kho g cling thuoc
Vi du: L ess- =Very, M ore - =Approximately hoac M ore- =N otso, Possibly- =More.
D e thay voi moi u « H ta co the chon gia tu doi mot each thich ho de ( h-)- = h.
Tro' lai van de tren, tuorig tir nhir logic kinh dien, moi menh de diroc gan mot gia tri
chan ly "dung", "sai", moi menh de tro g logic mer theo n hia Zadeh se diroc gan mot gia tri chan ly ngon ngir de bieu dat mire dQ dung d1'incua no Vi du nhir "Minh con tre" la
Trang 5MQT PHUONG PHAp LAp LUAN NGON NGU DVA TREN £)AIs6 GIA n KHONG THUAN NHAT 359
'' (It dung" Nhir vay, cluing ta da nhung cac menh de mo C ( sa VaG mien gia tri cua bien
ngon ngir T r uth. M a rong phep gan nay cho tap cac cong thirc F P la yeu diu tir nhien va
no se tro thanh ca sa de xac dinh mire d9 dung, sai cho cac menh de ket luan tro g qua
trlnh lap luan xap xi
Dinh nghia 2 4. Ch T = (T, G, LH ,:S, U,n,=}, -) la DSGT khong thuan n at hiru han doi xirng cua bien ngon ngir Truth Anh X 0 v : F P t T diroc goi la mot ham dinh gia tren
T neu cac dieu kien sau day dUQ'Cthoa man:
(i) Neu P = ( p , u) la menh de C( Jsa thi v( P) luon xac dinh, han nira, v( - ,( p , u)) = v( p , -u).
(ii) Neu P = (p , ku) thl v(hP) = 8lT khi va chi khi v(P) = 8l * h * T voi moi h , k , l E H va
T E G Trang do
{h * =h -, l * =l h* = h , l* = t:
h * =h, l * =l
neu k =N,
neu k =I N va h =N ,
neu k = I N va h = I N
(2)
(iii) Voimoi cong thirc P, Q ma v(P) va v(Q) xac dinh thi
v (P v Q) =v (P) Uv (Q) ,
v( P 1\Q) = v( P ) nv( Q ),
v( -, P ) = -v (P ),
a day, tro g ve trai la cac phep toan logic va trong ve phai la ca phep toan cua T
Hai cong trnrc P va Q dUQ'Cgoi la tuang diro g, kf hieu la P ' " Q neu voi moi phep dinh gia v, khi v( P ) va v( Q ) xac dinh thi v( P ) = v( Q )
Tir dinh nghia ham din gia va Dinh ly 1.1, ta co
Dinh ly 2.1 V 6i moi cang t l u i c P ,Q ,R , m oi h E LH va m oi v j t it p ta co
1 ) -,( p , u) ' " (p, -u) va (p, h - u) '" - ,(p, h u),
2) P '" P v a -"p ' " P ,
4) P vQ '" Q v P va P 1\Q '" Q 1\ P ,
5) P V (Q vR) '" (P v Q) vR va P 1\(Q 1\ R) '" (P 1\Q) 1\ R ,
6) P 1\ (P v Q) '" P va P v ( P 1\Q) '" P ,
7 ) -, (P vQ) '" - , p 1\-,Q va - ,( P 1\Q) '" - , p' V, Q ,
8) P t Q '" -,pvQ
Tinh chat phan phdi giira phep hci va tuyen noi chung khong thoa man VI DSGT khong
thuan nhat hiru han doi xirng la mot dan khong phan phoi
3 CAC QUI TAC SUY DIEN
Trong [7,8] cac tac gia da xay dung mot so qui titc su dien cho lap luan ngon ngir nhir
cac qui titc chuyen gia ta, qui titc ti l~, Cac qui titc nay giai quyet kha hieu qua cho phan
Trang 6360 LE XUAN VINH
"MQt sin vien hoc cham thl ket qua tot" va do do "Neu Minh h9C khong cham Himthi ket
qua co the la tot" Dieu nay chap nhan dUQ'CVI ket qua co the tot dircc hieu la kho g tot
litm Su dung qui titc tll~ cho cau thir hai ta thu diroc "Neu Minh h9Crat khong cham thl
do sir xuat hien cua "k o g cham litm" chira gia t u "kho g" (No t so) va tinh kho g thuan
n at cua no voigia tu "co the" (Possib l y) trong thanh p an con lai Cling VI 11do nay ma
titc suy dien moi nhir thay the gia tu-d n mire, phan ti 1~,nharn giai quyet cac tinh huong
neu ten
Chung ta biet rang qui titc su dien la mot sa do ma dira vao do ngtro: ta co the suy ra
cac ket luan tir mot tap cac khan dinh cho truce, no co dan :
( PI , td , ,( P n, t n)
(QI,Sl), ,(Qm,sm) ,
tro g do (P i , ti) la cac tien ae va (Qi , s.) la cac ket luan vai cacgia tri t s ; > W.
MQt qui titc suy dien diroc goi la dung din neu khi v(P i = t i, Vi = 1, ,n thl v( Q j) = Sj ,
Vj = 1, , m voi v la ham dinh gia bat ki
3 1 1 C ec qui t i ie e hu ye n g ia ta c ho cec diu tion g i c in
Tro g qua trinh lap luan ngon ngir & nhieu biroc ta can chuyen mot cau mer san dan
cua cac cau thu diroc:
(( P , h ku) , b lT ) (( P , k u), 6 l * *T) ,
( P , k u), b lhT) ((P, h * ku) , 6l * T)
(RTl)
(RT2) trong do b la xau gia tu- bat ki, h , k, l E H , la khai niern sinh nguyen thuy cua bien ngon
n ir Truth va h *, l * diroc xac dinh q a qui titc (2) cua Dinh nghia 2.4
Merih de 3.1 Cti c qui tac (RT1), (RT2) l a dung dan.
Chung minh Theo (ii) cua dinh nghia ham dinh gia vai chu y r~ g co the chon gia t u doi
Nhan xet 3.1 (i) Kh i su d1Lng(RT1), c h Ung t a uu t ien c h s v co mif , cua h , t sic l a neu
hk u co d!;mg hI T t h i h = hI va kh Ong ca n xe i Mn vai tro c' ll k.
(ii) »s« k h On g co tiu it l trong gi d th ie t c ' lla (RT1) va (RT2) , d i 'e u nay k e o th e o b la xau rong,
th i l * ciitu; k h O ng co mif , t trong ke t l u~n c ' lla cac qu i t ac n a y
Vi du Du g qui titc (RT1), ta co the chuyen: "Ket qua cua Minh co the to t la rat dung"
thanh "Ket qua cua Minh tot la rat co the dung" "Ket qua cua Minh kho g to t litm la f<3:t
dung" thanh "Ket qua cua Minh tot la it kho g dung litm"
Trang 7M OT PH U NG PH A P L A p LUAN N G ON NGU D VA TR EN B I s6 G I A T U KH O N G T H UAN N AT 36 1
Theo cau true cua DSGT khong thuan nhat co thg cho rcing "co thg tot" co nghia trong
diro g voi"kho g tot l~m", tire la hai c u can chuyen tro g vi d tren tirong dircng nhau
ve m~t n ir nghia Khi 00 hai ket qua thu diroc la phu hop boi VI "rat co th dung" lai co
rmrc 0 tirong dircng v 'i "It khong dung l~m"
3 1 2 Qui ute e hu y n g ia t o - eho ce c m ~ nh d e de n h o t h eo
Cho v la mot ham dinh gia va cac c u P,Q saDch v( P ), v(Q) deu xac dinh Nhir truce
day, kf hieu hP chi cho h(p, u ) hoac (p , h ) neu P = (p , u) Bay gio, ch n ta gioi thieu
mot so ki hieu va khai niem
Ta se viet P = h -, (Q ,ku) neu nhir P = -, h (Q, k u) va v(-,h(Q,ku)) = OlT keo theo
v(-, Q , k u)) =o l * h * T , v 'i h*, l * xa dinh theo (2) cua Dinh ng ia 2.4
Viet P = h((Q , ku) 0 (Q' , k v)) neu P = h (Q, k U) 0h (Q', k ), 0-day 0 la phep toan lo ic
cua Dinh nghia 2.4
P va Q oUQ'Cg i la tirong thich c10ivoi mot dinh gia v neu v(P) > W dong thai
n iroc lai
B5 de 3.1 V e m o t h a m rljn qui v c h tr uo c, t a co:
(i) - h( P , k u) = h - P , k u),
(ii) h ( P , ku) V h(Q , kv ) = h((P , ku) V ( Q , k v)) neu ( P , k U) v ( Q, kv) khOng tuang thic l i rloi
(iii) h(P , ku) /\ h(Q ,kv ) =h( ( P , ku) /\( Q , k v)) n u ( P ,ku) va (Q,k ) k Ong tuang ihicli rloi veri v,
(iv) h ( P ,ku) - + h (Q,k )= h ( (P ,ku) - +(Q,k )) n u (P,ku) va (Q,k ) tuang thich rloi uo i u.
C hun g mi nh
(i) Gia su v(-, h ( P , ku)) = O lT Theo Dinh nghia 2.4 (iii) va tinh chat cua DSGT khOn
thuan nhat hiru han ooi xirng, ta co v( h ( P , ku)) = -O l T = ol - T. Tir dieu nay va Dinh
nghia 2.4(ii), ta su ra v (P , ku) = ol * h* - T = - o l * h * T , trong 00 l *,h* xac dinh boi
cong thirc (2) Lai su dung Dinh nghia 2.4 (iii), ta diroc v( -, ( P,ku)) = o l *h* T , suy ra
v(( h *)* - P , k ) = O(l *)* T VI k khong 0 i va chu yrcing co th ch n thich hop og (h - )- = h
voi moi h E H nen o ng thirc cu i cling tiro g diro g voi v(h-,(P, ku)) = OlT. V~y (i) oa
diroc chirng min
(ii) Gia su v ( h(P, k u) V h( Q, kV ) = O l T, v(h ( P, kU)) = 01iIT1 va v(h(Q, kv)) = 02l2T2, 0
-day O ,Oila cac xau gia tu, l , l E H va T , Ti E {True, False} voi = 1,2 VI (P, kU) va (Q , kv)
khong tiro g thich nen co th gia su rcing T1=False va T 2= True Khi 00 olll T1<02l2T2 ,
keo theo v(h( P , k u)) <v(h( Q , k )) v a v ( h ( P , ku)) Uv( h (Q, k ) =0 l 2T2. Ket hop voi Dinh
n hia 2.4 (iii), ta su ra v( h(P, k u) V h ( Q, k v)) = 02 l 2 T 2 V~y 02 = 0, l 2 = l , T2= T.
Theo Dinh nghia 2.4 (ii), tir v (h( P, ku) ) =01 h T1 v a v( h (Q, k )) =02 l 2T2ta su ra v(P, ku) =
01 l h * T 1 va v( Q , kv) =0 l 2h*T2' Do T 1=False, T2= True nen v(P, ku) <v(Q, kv) , keo theo
v( P , ku) U (Q, kv) = 02 l 2 * T 2. Cling vc i dinh nghia ham dinh gia ta su ra v((P , ku) V
(Q, k v)) = 02 l 2 *T2 va VI v~y v(( h *)*(( , ku) V (Q, kv))) = 0 (l )*T2' VI k kh ng 06i va
( h - = h voimoi h E H nen oiing thirc cuoi cling chinh la v (h ((P, kU) V(Q, kv))) =0 l 2T2.
Trang 8362 LE x A V I N
K~t hop v i ket qua th diroc a tren, ta co v(h((P, ku) v (Q , kV) = 8lT. V~y (ii) da duoc
chimg minh
(iii) Ket qua nay diroc suy ra tir (ii) bang n uyen 11doi ngau
(iv) Duoc su ra tir (ii)vl v(h(P, ku) + h( Q , kv)) = v( -,h(P, ku)) u v(h( Q, kv)).
Bay g i o ch ng ta trinh bay cac qui tiic su dien ch menh de keo theo:
(h( P , ku) + h ( Q , k v), 8lT r u e), (( , k u), 8'True)
(( , k u ) + ( Q , k v), St : h * Tru e ) (RTIl)
(( , kU) + (Q, k ) 8lhTru ) ((P, ku), 8'True)
tro g do 8,8' la cac xau gia tt-t u y y, h , k , l E H va h* , l* xac dinh boi (2)
Tru n ho p kh ng co mat l tro g cac gia thiet cua qui tic su dien nay, cluing ta v ~ n
dung Nha xet 3.1(ii)
Menh de 3.2 Cac q ui t iie (RTIl), (RTI2) l acl u ng cl ii n.
Chung m i n Chung minh khang dinh nay dira theo Bo de 3.1 •
(P + Q ,8True) ,(P ,True)
(Q,8True) (RMP)
(P + Q ,8True), ( -, Q, True)
M~nh de 3.3 cs ; qui tiie (RMP), (RMT) la clung cliin.
3.2 Cac q ui t~e khac eho menh de keo theo
Trong thirc te nhieu menh de keo theo co tinh ti l~ giira hai thanh phan cua no Chan
han "Neu sinh vien hoc cang cham thl ket qua can tot" hoac "Tro i can ning thi nhiet d9
cang cao" Doi voi cac menh de nay, chung ta cling co the noi rang "Troi niing thi nhiet
d9 cao" la "tiro g doi dung" dan den "Tro'i rat ning thl nhiet d9 rat cao" hay "Tro'i khong
n n liim thl nhiet d9 kho g cao liim" cling se la "tirong doi dung" Tuy nhien, khi x at
hien gia tt- k h Ong (No t so) a dung mot tro g hai thanh phan thi se kh6 g con t l~ nira
Chung ta se goi tfn chat nay la phan t l~ Vi d "Neu Minh h9C kh6 g cham thi ket qua
co the tot" la "tiro g doi dung" kho g the su ra "Neu Minh h9C rat kho g cham thi ket
Trang 9MOT P UONG PHAp LAp LUAN NGON NGD " DVA TREN 8 \1s6GIA TL T K ONG THUA N N AT 363
qui nit co the tot" la "tirong d i dung" ma phai la "Neu Minh h9Cnit kh ng cham thl ket
q i it co the tot" mo i la "tirong doi dung"
Cac menh de vira de cap tren day co dan P (X*, hI U) -+ Q(X*, h 2V) trong do x * co the
la bien hoac hang, u, v la cac khai niem mo va hI, h la cac gia trt- Chia cac menh de keo
theo nay thanh hai loai khac nhau:
a) Loai ti l~: khi hI v a h kh6 g la gia trt-Not so (hI'" N va h2 ' " N) hoac dong thai
la gia trt-nay (hI = h2 = N).
b) Loai pha t l~: khi co dung mot gia trt-hI hoac h la No t so tire la ((hI = N hoac
h =N) va hI '" h )
Chung ta se xet cac qui utc khac nhau ch hai loai menh de nay
Qui tde ti l~
Doi voi cac menh de thuoc loai ti l~,tirong tv trong [7, 8]ta co qui tiic sau
( P (x*, hI u: -+ Q( x*, h 2v), 6True)
( hP (x*, hI u) -+ h Q(x*, h 2v), 6True) , (RPI)
trong do 6 la cac xau gia trt-, x * co the la hKn hoac bien, cac con thirc P , Q thuoc lap co
the chuyeri gia trt-,hI, h 2 la cac gia trt-tu y thoa dieu kien menh de loai ti l~
Tir (RPI), (RMP) va (RMT) vci a la hKn , ta suy ra:
- Qui tiic t l~Modus ponens
(P( x * , hI u ) -+ Q( x*, h 2v), 6True) , ( hP (a, h Iu), True)
( h Q(a, h2v), 6True) (RPMP)
- Qui tiic ti l~ Modus tollens
(P(X* , hIU) -+ Q( x*, h v), 6 Tru e ) -, h Q(a, h 2v ) Tru e)
(- ,hP ( a,hI u) , 6True) (RPMT)
Qui tde phdn ti l~
Doi vo'i cac menh de thuoc loai phan ti l~ta co qui tiic
( P (x*, h IU) -+ Q (x*, h 2v), 6True)
( hP (x*, h Iu) -+ h -Q(x*,h2v),6True)' (RNPI)
trong do x* co the la hKng hoac bien, 6 la xau gia trt-va hI, h2 la gia trt-bat ky thoa dieu
kien menh d loai phan t l~va n : la gia ttr doi xirng cua h
Tir cac qui tiic (PNPI), (RMP), (RMT) ta su ra
(P( x*, hIu) -+ Q( x*, h2V), 6True) ,(hP( a, hI u) , True)
(h -Q ( a, h 2v), 6True) (RNPMP)
( P (x*, hI u ) -+ Q (x*, h 2v), 6True) , (-, h Q(a, h v), True)
( -, h- P( a, hIu) , 6True) (RNPMT)
Trang 10P == Q, (F(P) , OT)
Viec thay the cac gia tu dong rmrc h va k ch nhau er V! trf t'en t6 cua khai niern mo
kho g lam thay doi y nghia cua menh de VI v~y ta co qui tac thay the gia tu don mi re
sau day
P (x* , hu)
Ngoai ra, tirong tv nhu trong [7,8] ta ciing co cac qui tac thay the cong thirc tirong duang
v aqui tac thay the han a ch bien x*
P (x* , u)
P (a, u) (RSUB)
Lap luan xap xi la tirn kiem cac ket luan kho g chac chan ban phtrong phap su dien
theo nghia xap xi tir cac tien de kho g chac chan Gia su cho trirrrc tap cac khan dinh K
bao gom cac cau mo co mire dQkhan dinh la gia tri chan ly ngon ngir dan oTrue, bKng
cac qui tac su dien noi tren chung ta se su ra diroc cac ket luan gi tir K ?
Ttrorig tv trong logic kinh dien, mot dan x at tir K la mot day hiru han cac khan
dinh (PI , td , , ( P n , t n ) sao cho veri moi i= 1, , n, ( P i, t i) thuoc K hoac ( P i, t i ) diroc su
ra tir cac khang dinh (PI , td, , ( P I , ti-l) bKng mot tro g c c qui tac su dien (RT1),
(RT2), (RTIl) , (RTI2), (RMP), (RMT), (RPI), (RPMP), (RPMT), (RNPI), (RNPMP), (RNPMT), (REH), (REF) va (RSUB) Khi do (P n, t n ) diroc goi la mot dan dircc tir K , ki
hieu la K f ( Pn, t n ) Tap cac he qua logic cua K la C(K) = {(P, t) :K f (P n , t n )} , chinh la
tap cac dan duoc tir K
K diroc goi la phi mau thuan neu C(K) khong ton tai dong thai (P , t) va ( , p , t/ ) ma
t t' 2 W Chung ta thira nhan rKng gia tri chan ly ngon ngir cua moi cong thirc n i chung khong duy nhat, chung co the nhan nhieu gia tri khac nhau mien la cac gia tri do deu Ion
hon hay be hen gia tri trung hoa W Chang han ch P , Q la hai cong thirc ma P + Q
thuoc loai dl~ eo gia tri chan 11la oTrue:
( P + Q,O"True),
verituy y h E L H , theo (RPI)
( hP + hQ , 0"True )
VI v~y theo (RTIl) nhieu t ru o g h p tro thanh
( P + Q , O"hTru e)