Bai baa cung cap mot sothu~t toan lien quan den viec lam min t%p luat va x5.y dung h~ luat chinh qui cila h~ chuy en gia.. Nhu cluing ta deu biet, tron n arn thanh p an chinh cua h~ chuy
Trang 1T -p chf Tin hqc viDi'eu khi€n hoc, T, 17, S,2 (2001), 20-26
LE HAl KHOI
Abstract. In this paper we give some algorithms for refining the rules set and buildin the regular rule-based system of the expert system,
Torn t~t. Bai baa cung cap mot sothu~t toan lien quan den viec lam min t%p luat va x5.y dung h~ luat chinh qui cila h~ chuy en gia 'I'Inh dung d[n cda thudt toan dtro'c clui-ng minh e ~t che dtro'i g6e di? toan hoc
Nhu cluing ta deu biet, tron n arn thanh p an chinh cua h~ chuyen gia thl co' so' tri thirc va
mo to' suy di~n dong vai to quan trong n at Vi the, ngtro i ta can noi r~ng "He chuyen gia = Co' s()'tri thirc + Mo to' su di~ ", CO' s6' tri thirc d troc bie'u di~n b n nhie u pluro'ng phap: phiro'ng
ph ap logic, phiro'ng phap m~ g ngi n hia, phuo'ng phap mo hlnh, ph tro'ng phap h~ lu%t, phiro'ng
ph ap thOng qua khung , phu'o'ng phap bi? ba OAV (doi tucng - thuoc tinh - gia tr~ ), v v Trong so cac phirong ph ap nay thl phiro'ng phap bie'u di~n blng h lu%t la tu'o g d i ph5 bien, nho mot so U'U die'm nhir: tn true quan, tinh mo ,kh anang kie'm ta va xti:ly mau thuan ciing nhu' du th ira,
Vi the, cac bai tcan lien quan den h~ lu%t d tro'c nhieu ng o'i quan tam, Di?c gii co the' tlm to g [1,2,4,5] n h irng kien t.htrc CO's6've h~ chuyen gia ciing nlur cac phtrong ph ap bie'u di~n tri thtrc
Cr mfit bai bao truo'c [3], khi de c%pc c van de lien quan den vie bie'u di~n tri th irc b~ng h lufit , cluing toi dil trlnh bay thufit toan tlm bao d n cua t%psir kien va cac thu%t toan ve loai bo lu%t dir thira cda t%p lu%t cling nlnr s\!.'dtr thira cii a h luat Mi?t cau hoi t\!.' nhien d~t ra la: co the' n6i gl ve ban than cac luat? Cu the' hori, li~u trong ni?itai ctia tirng lu~t trong t%p lu~t co su: dir t.hia nao kho g ,va neu co thl lam the nao de' loai bo du thira di? Trong bai bao nay cluing toi nghien cti u van de d o
Cau t uc cu a bai bao nlnr sau Trong mvc 2 cluing toi neu lai cac thu%t toan ve tlm bao dong
cu a t~p su' kien va lo ai bo lu%t du thira cu a t~p lu~t ma dil d iro'c trlnh bay trong bai [3]' bo-iVIchung can thiet cho viec xay du'ng thu~t toan t5ng hop sau nay, Ngoai ra, de' cac t.huat toan do thu'c su: c6 y nghia, cluing toi d u'a ra vi du rninh hoa ch c c thu%t toan Muc 3 trinh bay thuat toan vet can t~p lu~t khong dtr thira, " nhtr la h qui cu a thu%t toan loai b lu~t dir thira cua t~p luat; Muc
4 de c~p th uat toan min h6a m9t lu~t cling nhu min hoa t~p lu%t, Trong m\!.c 5 cluing toi d ua ra khai niern h~ lu%t chinh qui va tren co' so' t5ng h p cac thu%t toan truo'c do, trlnh bay thudt toan xay dtrng h~ lu~t chinh qui tir mot h lu%tbat ky cho trtro'c
~::;"f :~4\&;uo:~ t.adung h lu%t bao gom cac cau "NEU " THI " de' bie'u di~n tri thirc theo c u true
'1':: t:t'rSC ' iU - ~:}, :t , ' J;
~ ; ; : i:':I:f! = - '~1r r: ;1
.:.:i)·k , ···· ot, " ) {" ,' ' A ,'" 'A , ' " ' A
:.: : { < ;"!:tiJ ~3 ' j.: ', - ~; THI (ket luan 1 ), (ket luan 2 ) "", (ket luan n/'
' :: ~ _: _:;:; - ~ r:;A::;'::;:; ' !""
H~ ~)l~t nay can c6 ten goi la h~ lu%t dang 1 (khac vo'i h~ lu%t dang 2 la h~ lu%t m a 6'do trong phan
"THI" cac "ket luan" d u'o'c thay b~ng cac "thirc hien"] Tro g h lu%t tren ca dieu kien va ket
Trang 2TH U A.T T O A.N LAM MIN T AP L U AT v ) H~ L U AT CHiNH Q U I CVA H~ C HUY EN GIA 21
Chung taco the' hmh tlurc hoa eao hori de' the' hi~n toan be?tri th uc trong mot h~ luat Cu the' nhir sau
Dinh nghia 2.1 H~luat ,kihieu la L =( F , R ) , gom hai thanh phan F ={II," , fp} la t%p cac s\!-'
kien, R = {rl, " r q} la t%p cac luat;
Theo c ac qui tae bien d5i cua Virong Hao, luon co the' eoi r~ g h~ lu%t chi gom ca lu%t voi ve
traitoan ph ep "va (1\) va ve ph ai co dung mot s\!-'kien, tu'c la h lu%tchi bao gom cac lu%t dang
6· day PI, P 2 , , PH va Q 1 cac su:kien De' dan gian chung ta thay dau 1\trong ve tr ai b~ng dau
p ay (,), khi d6 lu%tdu'o'cviet diro'i dang
Doi v 'i lu%t r chung ta kf hieu L e ft(r) la t%pcac su kien 6·ve trai, Righ t( r la su: kien & ve
p ai c a lu%t
Gi<i sti: co h luat L = ( F , R to g d F = { il, , f p } la t%pcac S I) 'kien, R = { l, "" rq} 1
t%pcac lu Ki hieu F * la t%pcac su- kien f E F thoa man dong thai hai di'eu kien:
(i) f co m~t & ve tr ,
(ii) f khong co m~t &ve phai,
to g tat d cac lu%t thucc R T%pF * nay du 'c goi la t4 p cdc S'I.!- · ki 4 goc.
DU'6'iday neu lai thu%t toan tlm bao dong cua t%pstr kien va loai b6 lu%tduothira ct a t%p lu%t
Nhimg th uat toan nay se du'oc su· dung trong qua trinh xay dung nh img thuat to an khac 6' cac muc tiep theo Chung ta su' dung k hieu ( , , ) de' chi day (tu'e la co thir t\!-·)cac ph an tU'
Neu F' ~ F, thl ba o do ng cua F ' doi vo'i R , kf hieu (Fi ? ) +, dtroc dinh nghia la t%p thu du o'c
tu:F ' sau khi ap dung tat ca ca lu%t co the' co cua R. Chung ta luon gi<i thiet la cac phep suy dien khong bi l~p (t ue la khong co chu trlnh)
T'huat toan 2.2 (tinh (Fi ? ) + )
In ut L = ( F, R ) vo'i F = ( , , f p ) , R = (l,"" r q ) va F' ~ F.
Output ( F i ?
- BU'6'ei neu co lu%t r E R tho a man dieu ki~ L e ft(r) ~ Ki-1 va Ri gh t (r) ¢: Ki-1, thl d~t
K , = Ki - 1 U Rig h t (r) :
- Qua trlnh diro'c l~p Iai eho den khi K ; =Ki + l
Lu do d~t (Fi ? )+ = Ki.
Merrh d 2.3 Tl i iuit to an 2.2 c6de; phuc tap la d« th s ic theo lu : « lu(tng C ' da F va R
Vi d u 2.4 (minh hoa Thu%t toan 2.2)
Ap dung thu%t toan, cluing ta co:
- Biro c0: K o =F' = {A , B , D, H }.
- Buoc 1: lu%t rl cho them SI rkien C ¢:. K o = F' , nen ta co K ; = {A, B , C, D, H }
- Bucc 3: lu%t r 4 cho them s\!-·kien I ¢: K 2, neri ta co K 3 ={A, B , C, D, E, H, I}
- BU ' ae 4: do khong co lu%t nao nira ma eho them s\!-·kien mo ikhong thuoc K nen K 4 = K
Trang 3LE HAl KHOl
Vfiy, ( Fi ? ) + = K 3 = {A , B, C, D , E,H , I }.
Bay gio' ch ung ta chuye n sang van de lui).tduothira V &i F * 1 1t .pcac su kien g c cua h lui.t
L = ( F, R ) , neu co r E R sao cho ( F t = ( FR\{T)t, thi luat r diro'c coi 111.l uit ih u :«v ve n uyen
tic cluin ta co th loai b lui).t nay di
T'huat to an 2.5 [Ioai 16lui).t thira]
Input: L = ( F , R ) vo'i F = ( I, , II ') va R = (l, ,r 'J )
(F R , t·
- Bu·&c0: D~t Ko =R, tfnh (FR ) +
- Buxrc i (1:s ;i:s;q - 1)
s , = { tc \ { rd, n u (F ~i _ , \{~;}t = ( FR ) +,
K i - l, neu ng o'c lai
- Biro'c q:
N e u K , - chi co r« thl d~t K q = K , - l.
Neu K ,, - chtra kho g chi co r ' J ' thl d~t
K q - l, neu ngU"<!clai
- Buc q+1: D~t R' =K q •
Merih de 2.6 Thu4t totiti 2.5 c o d ph U: c tap L da thU:c theo lu : « luotiq cila F va R
Vi du 2.7 (minh hoa Thuat toan 2.5)
Xet h lui.t L = ( F , R ), to g do F = {A , B , C , D,E,F, G ,H,I , J,K}, R = h , .,rG), voi
rl = A - + C, r z = C D - + E , r3 = EF - + G, r4 = DH - + I, r5 = I J - + K, rG= C E - + I Khi d6
F * = {A, B , D, F, H , J }
Ap dung thui).t toan, chiing ta co:
- Biroc 0: D~t K o = R , khi do (FR) + = {A , B , C,D , E , F ,G,H , I , J , K}
- Buoc 1: do (F~o\(r,)t = { A, B , D , F , H , I , J , K} = f (F Rt, nen t c , = K o
- Bircc 2: d (F ;{ - ,\ h)) + = (F ;{ -o \ h)) + = {A, B, C, D , F, H , I, J, K} = ( FRt, neri K 2 = K l·
- Bu'o c 3: d (F ;{ - 2 \ h} ) + = (F ;{ - O \ { T J) ) += {A , B ,C, D , E , F , H , I , J , K} = f (F R ) +, n n K 3= K 2.
- Bu-oc 4: d ( F; {- J\{T4}t = ( F;{ o \ {r } t = {A , B ,C, D ,E, F ,G, H , I , J , K } = (F nt, nen
K4 = K 3 \ { 4} = (rl' r 2, r 3, r 5,rG)·
- Burrc 6: do ( F;{ - ,\h )t = {A,B, C ,D ,E ,F,G,H, J } = f (F n t , nen K G= K 5.
- Burrc 7: Ch n ta dic R' = K G= (rl' r 2, r 3, r 5, r 6) va lui).tr4 1 1.lui.t thira
3 TAP. LUAT. RHONG DU THtrA Trong m1,lcnay chung ta xem x t khanang vet c~n tat d cac lui).tkhong duothia trong ti.p lui).t
R
D~ lam dieu nay, ta xep ca lui).t th anh mi?t day R = (rl, ,r,, ) , roi su dung thu~t toan loai
bo lu$.t duothira 2.5 cho tat d cac hoan vi cua day R Noi each khac, vo i m6i hoan vi cu a day nay,
chung ta ap dung Thuat toan 2.5 d loai luat duothira di Do ti).p lu%t R ={rl, ,r,,} co q phan
Trang 4THUAT ToAN LAM MIN TAP LUAT v ) H:¢ LUAT cHINH QUI CUA HI; CHUYEN GIA 23
T'h'uat to an 3.1 (vet c,!-n t~p luat khorig dtr thira)
Input: L = ( F, R ) v i F = {f l , "" II ' } v a R = {T I, " Tq } , RI , R 2 "'" Rql.
- (F~) + = ( FR.t,
- 'ITE R~ : R? =R: \ {T} luon co (F n:, ) += 1= (F n ) + ,
- Buoc q !: thuc h ien Thuat toan 2.5 doiv6- Lq! = ( F,Rq!), dU'9'CL~ ! = (F,R:1!)
Chu'ng m in h Chung ta phai clui-ng minh r~ g moi day lu~t kho g duo thua cia R deu sinh r a tv:
day nao d6 trong so cac day RI, R 2 , , Rq ! qua thuat toan neu tren
That v ay, lay day lu%t kho g duothira hilt ky R' = h" Ti" , Ti J Day nay tiro'ng irng voi
- Trtro'ng ho p t'5.m th iro'ng khi s = q, vi hie d6 (il,i2, ,i.) la mot hoan vi cua (1,2,: ,q), va
ban than R la t%p lu%t kho g du thira
- Tru'o g h9-Ps <q Xet day chi so (fl, i z, , J ; - '" iI, i2, , • sac cho day nay la mi?t hoan vi
Nhfin x et 3.3 1) Theo cong thirc St ling v i nhirng n 1 'n
n! = j27rn nn e - n +i!?n (0 <(}n< 1) ,
2) C6 th xay ra tru'o g ho'p la cac day lu%t khOn duothira nhan dU'9'C tuy khac nhau, nhirng
neu xet tir g6c di? t%p h9'P thi co nh irng t%p ho'p trung nh au Do d6, chung taphai so sanh cac phan
trong d6 PI , P2, , Pt va Q = R ig ht(T) la cac su' kien Tu' lu%t Tnay sau k i bo disu' kien P i se
n han Rig h t «) =Q , tuy nhien to g viec loai bo str kien P i can can n ac den ngir nghia cua su' kien
b6 su kien Pi trong ve tr ai cua T di, xong y nghia cua su' kien nay trong lu%t T tren thic te can ph ai
d 'c xem xet rat than tro g Nhir vay, viec min h6a mi?t.lui),t cho chung ta'kha nang bot di cac str
ki~n d othira ve m~t suy di~n logic d lu%tdrrqc gon ho-n
Trang 5LE HAl KHOI
Vi~c 111mmin mot lu~t hi~u theo nghia sau: doi vo'i m6i lu~t r E R, r = l\ i Pi > R i ght(r) ,
ki~m tra xem li~u co th~ loai bo mi?t so S\!' ki~n nao d6 trong so cac S\!' ki~n PI, P2 , ' " ,Pt sac cho
bao d6ng cua F * trong t~p lu~t m6i khong thay d5i (di'eu d6 c6 nghia la viec loai bo mi?t S1:1ki~n'
nao d6 ciing khong d iro c lam thay d5i t~p F * ), Nh irng su' kien d o, neu c6, coi nhir la th ira N6i
each kh ac, chung ta goi lu~t r = l\iPi > Right(r) v6i P = (PI, P2" " , P t) la lu4t min, neu nhir ( F~\ {r} u {r ) + i (Fi?) + , Vi = 1,2, "t, Vi~c xay dirng lu~t min t.ir mot lu~t cho trtro-c d u'o'c goi
ia m i n. s « lu~t d6, Di!' cho g9n cluing ta viet I\pE? thay cho l\ iPi'
'I'h uat t o a n 4 ,1 [rnin h6a mi?t luat]
Output: r' = I\ pE > R i ght(r') thoa man P' ~ P , (F~ \ {r} u {r,}t = (Fi ? ) + va VpE P' : P" =
P ' \ {p} , r" =I\ E > Ri g ht(r"), luon c6 (F~ \ {r} u {r l l}) + i (Ff l J +
- Bu'cc 0: d~t Ko = P
- Buoc i (1 : ::::i::::t -:1):
tc. = { K i-l \ {Pd,
K' 1 - 1,
neu R \ {r} U {r' = l\ p E K i_ l { Pi } ~ Ri g ( r')} R '
neu ngu'qc Iai
Neu K t l chi con Pt> thl d~t K, =K t -l ={Pd.
Neu K t - l chu a khOng chi P t, thl d~t
tc; = { tc , \ {Pd,
tc : «.
neu R\{r}u{r' = l\p E Kt _ l\ { Pd ~ Right(r')} - R ,
neu ngrro'c Iai,
- Butrc t +1: D~t P' = K t v a r' = I pE?' > Right(r')
Dirih ly 4,2, ThuM totiti 4 , 1 I i i du : ng va cho ktt qud La Lu4t r' = I \p E > Right(r') m~n.
Chu' ng minh. Chung ta chirng minh bhg phiro ng ph ap pharr chimg
Luu y din d~ c6 d u'o'c K t l cluing ta da kii!'m tra tinh du' thira cu a t - 1 s1:1ki' ~n (r ve phai cua
lu t rIa PI " ' " Pt - l, d do, nlnr thu~t toan da chi ro, c6 th~ xay ra hai kha nang sau:
- Kh<l.nang thli' nh at: K t - l chi clnra mot phan tli', The thl phan tli' nay chinh la P; v a do d o K t- l
khong th€ giam di diro'c nira V~y thl K; ~ K t- l, t.u'c la P' =K t ={Pd v a r' = I\pE > Right(r')
la lufit kho g dtr thira
- Kh<l.nan thU: hai: K t - l c6 it nhat hai phfin tli·, The thl ngo ai P; ra, K t - l con chira them it
nhat mdt phan tu: nU·a Khi d o, theo thu~t to an chung ta c6:
x , = { « \ {Pd ,
K t- l,
neu R \ {r } u {r'= l \p E K t _ l \{ P d ~Ri ght ( r '}} R,
neu ngiro'c lai
Gi<l.sl' rbg K; chu'a phai la toi UU, tu:c la P' C K t nlnrng P' i K t Dieu d6 c6 nghia
la to g K; v&n con S1:1'kien thira, n i each khac, 3 E K t sao cho voi P" = K, \ { p} thl
( F~\ {r} U l =l\ l 'E PI I ~Righ t ( r l l) ) + = ( Fi ?
Xet timg tru'ong ho'p doi vo i K t:
thu~n v6i viec trong K t ngoai P t ra v&n con it nhat mdt S1:1"kien nao d6 chira ki~m tra
(2 ) K ; =K t - l: tro g trtrong h 'p nay, theo thuat toan thl
( F n\ {r' \{ P, } ~ Ri g ht ( r ' ) } ) + i- (Fi? ) +,
Trang 6THUAT TOAN LAM MIN TAP LU :r vA HE LUAT C HiNH Q I CUA HE CH YEN GIA 25
tii'c la.Pt khong phai la.S \1'ki~n thira va nlnr v~y tat d c c S~' kien thudc P da d iro'c kiEim tra Dieu
Nhu v%y dieu gii.thiet d.n P' c K; 11 sai Vay, r' = ApE P' > R ig ht(r') la lu~t min co diro'c
D~dang chirn minh ket qua sau day
M~nh de 4.3 Tliui i : totiii 4 1 c o aq phU : c iqp ld d a th u: c th e lu : « lu ot u ; c si « t q, p s u : ki~ n d·ve phdi cd« l uq , a o.
PIP 2 P 3 > QI, r 2 = P 2 > Q 2 , r 3 = P4P6 > P r4 = P I P4P6 > Q rs = P 3 P 4Ps > QI Khi do
F * = {PI, P 2 , P 3 , P4, P6 }
- BU'C1C2: Xet KI \ {P2 } , khi do lu%t r~ = A pEK, \ {P , } > R ig h t r ~) ; d tro g truo-ng hop nay
- Buo c 3: Vi t.rong P chi can su kie P 3, n n d~t K 3 =K 2 =( P 3 ).
Nhir vay, l~~t rl da dtro'c min hoa b 'i lu%tr~ = P 3 > QI
Output: R ' = ( "r l ,r 2, . , rq' ) t oah' man-ri , la' rnrn heoa cu a rv vo',r mcr r e- 1 2, , , q.
- Biroc 2: thuc hi~ Thu%t toan 4.1 ch lu~t r 2 d 'o'c r ~
.- BU'C1Cq: thuc hi~n Thuat to an 4.1 cho luSt r q dtro'c r ;J '
Cho h~ luat L = ( F , R ) Chung ta noi rhg h lu~t L l a c h in h qui, ne u L tho a man cac dieu
kien sau:
(i) k o g co su' kien duoth ira trong t~p c ac su' kien F ,
(ii) khorig co.lu~t duo thira trong t~p lu~t R ,
(iii) kho g co su' kien duothira tro g cac lu~t cii a R.
T6ng hop cac thu%t toan da trnh bay chung ta co thu%t toan sau day ve viec xay du'ng h~ lu%t chinh qui tir mot h lu%t bat ky cho truoc
T'hua t t.oan 5.1 [xay dung h~ lu%t chinh qui)
Input: L = ( F , R ) vo'i F = (1 , , Jp ) , R = h, ,r q )
Output: L' " = ( F ' R" ) la.h lu%t chinh qui co d u'o'c tir h~ lu~t L
Trang 72 ,LE HAl KHOl
- Thuc h ien thu~t toan loai b6 lu~t dtr thira de' lo ai cac lu~t khong din thiet trong R : tir
L = ( F , R ) c6 L' =( F , R' ) , trong d6 R' la t~p lu~t khong du' th ira
- Thtc hien thu~t tcan tinh bao dong de' loai bo cac su' kien du' thira trong F, d u'o'c h~ lu~t
- Thu'c hien thuat toan min hoa cua t~p Iuat de' loai cac su: kien khong can thiet trong tat ea
cac lu~t cua R' , d 'oc R" di.dlIQ'Cmin hoa
- H~ lu~t L"' = ( F' , R" ) la h~ lu~t chinh qui can xay dung
D~ dang chung minh ket qua sau day,
L<ri ~:irn on Tac gii xin chan th anh earn empes TS VUDire Thi dii.dong gop nh irng y kien qui
bau to g qua trinh hoan th anh bai bao nay Ta g , cling xin earn on ngirci ph an bien ve nh iing
rhan xet gop y cho b ai bao du 'c tot ho'n
[1] Bach Hirng Khang, Hoang Kiem, Tri tuif nluin: tao : cdc phsioru; pluip va u:ng dV-ng, NXB Khoa
h9Cva K y thuat , 1 89
[2] Durkin J Expert 3ystems, Prentice Hall, 1994
[3] Le Hai KhOi, Thuat toan tim bao dong cu a t~p S1).·kien va loai bo lu%t duothira cu a t%p lu%t trong h~ lu~t cii a h~ chuyen gia, Tap chi Tin hoc va Di'eu khie"'n ho c 16(4) (2000) 79-84
[4] Sundermeyer K Knowledge Based Systems, Wissenschafts Verlag, 1991
[5] Turban E Decisions Support € Expert Systems - Management Support Systems, Prentice Hall,
1998
Nh~n ba.i ngay 26 iluitiq 10 niim 2000 Ntuin ba.i sau khi sda ngay 25 tluinq 4 niim 2001 Viifn Gong nghif thong tin