Ky y^u HQi thto ICT.rda'06 Proceedings of ICT.rda'06. Hanoi May. 20-
LOGIC MOTA MOf NGON N G C A fuzzy linguistic description logic
Tran Dinh Khang, Dinh KhSc Dung
Tom tat
Ngi dung bdi bdo de cgp din mien gid tri chdn ly ngdn ngir thod mdn linh chdt ke thira ngii nghia vd cac tinh chdt dan di^u, khong dimg lam lap gid tri nen cho logic mo Id. Tir dd, dinh nghta logic mo td ma ngon ngir vcri do thugc cua cdc cd the vdo khdi ni$m Id cdc gid tri ngon ngir. Bdi bdo cSng dua ra thu tuc gidi quyil bdi todn khdng thod ciia logic mo Id ma ngdn ngu dua tren thugt loan bdng.
Tir khda: logic md id, thu tyic quyet dinh Abstract
This paper deals with the domain of linguistic values satisfying the meaning inheritance property and other properties to make the based domain of description logics. After that,
we introduce a fuzzy linguistic description logic including linguistic values as memberships of an individual belong to a concept The paper also proposes an algorithm to solve the satisfiability problem based on the tableau procedure.
Keywords: fiizzy description logic, decision procedure
1. DAT VAN Dfe ALC_L va thu tyc giai quyet bai toan \
thoa ciia ALC L.
Logic mo Xk la hinh thurc bieu diln va xu ly tri thuc c6 clu true d\ra tren cic khai ni^m va quan h | giiJa cac khai ni^m, the hi^n tinh chat ciia cac ca the . Khi muc dp thupc cua ca th^ vao mpt khai ni^m la khong rd rang, dupe bilu dien bing dp thupc trong khoang [0,1], ta CO logic mo ta mo [4], [8], [10].
Thoi gian gan day, co nhieu cong trinh nghien cuu huong toi mo rpng kha n&ng bieu dien va xu ly cua logic mo ta mo, nhu them toan tu bien doi khai ni^m [1], [9], ho|ic tiSp c^n phan tich t^p gia trj nen ciia logic [4].
Npi dung bai bao nay se sii dyng t|ip gia trj ngon ngu' thoa man tinh chat ciia d^i so gia tu va mpt s6 tinh chat khac [7], [2], [3], [5] Ak djnh nghTa logic mo ta mo ngon ngir (ALC_L) va xay dyng cac thu tyc quylt djnh cho logic do, trong do quan trpng nhSt la bai toan khong thoa. Bai bao, ngoai phan mo diu va ket luan, gom CO ba phan chinh de cap den miln gia trj chan ly ngon ngu, logic mo ta mo ngon ngu
2. MIEN GIA TRI CHAN LY NGON P Truoc het, chung ta se nhie lai cac nifm CO ban vl d^i so gia tu. Vi npi dunj bai nay tap trung vao miln gia trj chi ngon ngQr cho nen chiing ta se chi quan sa dai so gia tu voi tap phan tii sinh G = {j
False). Cho H la tap gia tii, ta c6 H=H*ul• h€H* la gia tu duang, nghTa la h tang ngu- nghTa ciia cac phAn tu si
• heH" la gia tii dm, nghTa la h giam ngu- nghTa ciia cac phan tu si
• I la gia tu dan vi, nghTa la x: Ix=x:
• h la duang (positive) so vai k, n\
la h lam tang them muc dp nh^n k, va
• h la am (negative) so vai k, nghTa
lam giam mire dp nhan ciia k.
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Tir cac mure dp nhan ciia cac gia tur, co cic quan h? thir ty tren H*u{I} va H u { I } , ta xac djnh quan h^ thu ty tren Hu{I} nhu sau:
Cho h, k e Hu{I} thi h^k khi \k chi khi:
(i) heH* va keH", ho$c
(ii) h,k€H*u{I} va h co muc dp nhan manh hon ho$c bing k, ho^c
(iii) h,k6H'u{I} va h co muc dp nhSn ylu hon ho^c bang k.
h>k khi va chi khi h>k va h^k.
Miln gia trj chan ly ngon ngu trong [7] da dupe gioi h^n thoa mpt so dieu kien sau:
Dinh nghia: Dai s6 gia tu (A,G,H, ) thoa man tinh chat dan di4u va khong dimg, neu:
1.. Vh,k6H*(H'), thi h la duong so voi k 2. VheH*, keH", thi h la am so voi k,
va k la am so voi h (1)
3. Vh,keHu{l}, V x e X : (h;tk) suyra (hx
*kx)
Vi du: Dai so gia tu (X, G, {very, more, possible}, <) thoa man tinh chat don di?u.
That v$y, ta co:
very la duong so voi very, more, la am so voi possible;
more la duong so voi very, more, la am so voi possible;
possible la duong so voi possible, la am so voi very, more.
Thii ty ciia cac gia tir la: very > more > I
> possible.
Diiu kien (1) thyc chat lam cho vifc nhan biet cac gia tii lam tang hay giam ngii nghTa ciia cac gia trj ngon ngQ: khong bj phy thupc vio gia til diJmg truorc no trong chudi gia tii. d vi dy tren very luon lam tSng ngd nghTa va possible ludn lam giam ngir nghTa cua bit cu gia trj ngon ngir nao. Truoc kia, dl nh|in bilt diiu nay, ngudi ta thudrng phai su dyng ham dau (sign). Tu tinh chat don difu va khong dung, trong [7] da chiing minh dupe cac tinh chat sau day:
Tinh chat: Cho dai so gia tii {X,{TrueJ^alse},U,<) thoa man tinh chat don di?u va khong dung, cho h,keH, va o, 0|, o;
la cac chuoi gia tu, ta co
h > k <:> haTrue > luaTrue, va
h > k <:> haTrwe>korrue,va (2) a\True>a2True o aihrrMe>a2h7>Me.
Trong truong hpp tong quat, phin tu sinh CO the la True hoac False, thi tinh chat trer van dupe bao toan:
. VheH: ajCi > 0202 <^ Oi hci > 02 he2, trong do Ci, C2 e {TrueJFalse}
Tham chi, khi 5 la mpt chuoi gia tu:
V5eH*: OiCi > CT2C2 o Oi 6ci > 02 5c:
(3)
Cac tinh chat tren deu da dupe chung minh trong [7], cho phep djnh nghTa anh x$
ngupe cua cac gia tu. Xuat phat tii cac lu^il chuyin gia tu RTl, RT2 trong [3],
RTl: f(P. hu). oTrue) ((P, u), ohTn/e) RT2: ((P. u). ahTrue) ((P, hu), cTrue)
ta thiy gia tii dupe "djch chuyen" tir gij trj ngon ngQ cua biln ngon ngu sang gia tri chan ly cua m?nh 6e va ngupe lai. Vi du nhu.
menh dl "John is possible young" voi gia tr;
chan ly "very true" co thi chuyen thanh "Johr, is very possible young" voi gia trj chan 1>
"true", va ngupe lai. Ta co thi nhin nhan vi?c chuyin tii "very true" sang "true" nhu la tac dpng very' la anh xa ngupe ciia very vac
"very true". Chung ta se phan tich sau vl tin!
chat chuyin gia tii dl xay dyng cac tien Ai cho anh xa ngupe gia tur. Co cac nhan xet sau:
(i) Cac gia tii dupe chuyin la nhCmg gie tur nim ngoai ciing ben trai cua chuoi gia tii dumg truoc gia trj cua bier ngon ngil, va nim trong cung ber phai ciia chu6i gia tu tac dpng vao giz trj chan ly, nghTa la dung ngay truoc True ho^c False.
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(ii) Khi chuyin gia tu cung c6 thi lam chuSi gia tii ngin l^i, nghTa la anh xa ngupe CO tinh chat "khii gia tu". Diiu nay xay ra khi tac dpng h' vao ShTrue, cho kit qua BTrue.
(iii) D^i sd gia tiJr ma chung ta dang khao sat CO tinh chat don difu va khong dimg: OiCi > 0202 o Oi hci > 02 hc2.
Ma theo nh|ui xet (ii) a tren mong muon h(aihci) = OiCj, h"(o2hc2) = a2C2. Do vly, ta co h'(CTihcO > h"
(CT2hc2) o Ol hci > 02 hc2. Nhu vay, anh x^ h' co tinh chat don di^u tai cac gia trj chan ly c6 gia tir h dung ngay truoc phin tu sinh. Tong quat hoa y tucmg nay, chiing ta cd thi dua them tinh chat don dif u cho toan bp cac gia trj chan ly: OiC| > 02C2 o h"
(a,Ci)>h(a2C2).
Xuat phat tu cic nh|ui xet tren, ta co djnh nghTa:
Dinh nghta: Cho d^i s6 gia tii (X,G,H,<) thoa m§n cic tinh chat don di^u va khong dung, dnh xg ngugc cua gia tir heH dupe djnh nghTa:
h":X->X, trong do h(ohc) = oc
vdri mpi chu§i gia tii oeH* vi ceG (4)
vi OiCi > O2C2 o h'(ci|Ci) > h"(o2C2), vdi ai,a2€H* vi C|,C2eG
Khi do, lu|t RT2 c6 thi tong quit hoi thanh
GRT2: (fP.uVSrnie) ((P,hu),h-(5rnie))
Trong d6 RT2 li mpt trudng hpp dac bift cuaGRT2khi ah = 5.
Nhu v§y, vdi mdi gia tii cd the cd cic inh X9 ngupe khic nhau, nhung cin thoi man dieu kifn (4). Sau day li mpt vi dy vl inh xa ngupe cua cic gia tu.
Vi dft: Cho d?i so gia tu (X, [TrueJFalse], {very, more, possible}, <), cd
thi dl xuat inh xa ngupe ciia cic g sau:
very (a True) = 5 True, nlu a = 5 a possible possib
neu o ^ 5 vei more' (o True) =
a more very True, neu a = 5 5 True, nlu a = 5 more,
o possible possible True, neu
a = b possible hoac a = possible' (a True) =
8 True, nlu o = b possible,
a very very True, neu a * 8 possible Dl ding kiim tra dupe cic inh xa ngu thoa man dieu kien (4). Tir do cd tl dupe bing gii trj cua anh xa ngupe c.
chuoi gia tu cd dp dai < 2. (vdi v viet 1 very, m viet tat cho more, p viet t possible)
oc v v c m v c v c p v c v m c m m c m c p m c c(a=E) v p c m p c p c p p e
very'(ac) v c m c c p c v m p p c m m p p c m p p c p m p p c p p c v p p p c m p p p c p p p e p p p p c
more"(ac) v v m v c m v m v c v m v c p v m v c v c m c c p c p p c v p p p c m p p p c p p p e p p p p c
possibli v v V V c m v V V c V V v c p V V VC v m v v c m m V v c v v c p m v v c V v c v c m c c p c
Ngoii ra, chiing ta cung cd thi md rpng c nghTa inh xa ngupe cua gia tir thanh anh ngupe cua mpt chuoi gia tii 5 = hp ...hi:
5': X - • X, trong dd
5-(ac) = (hp ...h,)-c = hp(...(hp(ac)...) ( De dang chiimg minh dupe:
Oic, > a2C2 o 6'(o,c,) > 5 (ojCj)
ffyfaW
li thto ICT.rda'06 Proceedings of ICT.rda'06. Hanoi May. 20-21.2006 Tren diy li m^t so tinh chat ciia inh x? ngupegia ttr, vdi cdng thurc (4), chung ta cd dupe nipt quan hf thur ty cua inh x? niy giiip cho vifc djnh nghTa inh x^ ngugc de ding hon.
Tiep theo, luit chuyin gia tur GRT2 cung vdi RTl cung cho phep hudng tdi giii quylt cic bai toin lip luan xip xi tryc tiep tren gii trj ngdn ngiir, sg dupe trinh biy d phan tilp theo.
3. LOGIC M 6 TA M d NGON NGU"
Chiing ta se xem xet logic md ti md ALC_t. vdi cic gii trj chin ly diupc vio miln gii trj chin ly ngdn ngu' cd cic tinh chit nhu d phin trfn. Diy la sy md rpng cua cic logic mo ti md nhu ALC_F, ALC_FH, ALC_FLH,
... (xem [1], [9], [10]), trong dd cic khing djnh md cd d?ing <a:C • x), vdi • e {>,<.>,<}
va T li mpt gii trj chan ly ngon ng&, vi dy, very true,... trong khi dd d cac logic mo ta md ti^yln thdng, t la mpt gii tri sd trong khoang [0,1]. Cung nhu cic logic md ti khic, phan ngdn ngu gom cd hai thinh phin chinh la cic khii nifm vi quan hf. Cic khii nifm dupe djnh nghTa bdi cac luit cu phip dudi day, tren CO sd cic khii nifm vi quan hf nguyen thuy cho trudc:
C, D ^ A I (primitive concept) T 1 (top concept) 1 I (bottom concept) -iC I (concept negation) CnD I (concept conjunction) CuD I (concept disjimction)
5A j (concept modification) VR.C I (universal quantification)
BR.C I (extential quantification) Sy khic bift cua ALC L so vdi cic logic md ti md khic nim d mien gii trj chin ly ngdn ngO' thoi man tinh chit cua d^i so gia tu vi cic tinh chit khic nhu d Phin 2, thay cho khoing [0,1]. Cho trudc d^i s6 gia tur AX = (X, {true, false), H, ^) thoi min tinh chit dan difu, khong dimg, thi mpt diin dich /chinh l i mOt cSp (A^, '), trong dd A^ li mpt tip khong rong vi •' li mpt inh x^ tu cic phin tur ho$c khii nifm nguyen thuy vio mpt gii trj, hay him thu$c ngdn ngi^. Sy md rpng vl ngu'
nghTa cho cic khii nifm phiire hpp dupe tinI theo bing dudi day:
A' : A ' - > X R' : A ' X A ' ^ X
T'(d) = I, VdeA' l'(d) = 0, VdeA' (^C)'(d) = ^C'(d) (CrJ))'(d) = C'(d)Aiy(d) (CuD)'(d) = C'(d)viy(d)
(8A)'(d) = 8(A'(d)) (VR.C)'(d) =
(3R.C)'(d) =
Ad-,A/ {-R'(d,d') V C(d)}
Vd.,^{R'(d,d')A C(d)}
Dang luu y la ngO nghTa ciia mpt bien ddi khai nifm (5A)\d) la 5(A'(d)), vdi S'la inh xa ngupe cua chuoi gia tii 5, nhu djnh nghTa o Phin 2, cic phep toin -i, A, V tuong iimg vdi cac toan tii phu djnh, hpi, tuyln cua d^i so gia tu, 1 = sup(/rue), li phin tur Idn nhit ciia X, 0 la phin tur nhd nhit cua X. Vl ngO' nghTa, hai khai nifm C vi D li tuong duong, neu C'=iy, vdi mpi /.
Mpt CO sd tri thiic trong logic md ti md bao gom cic khing djnh md (a • x), vdi a cd d?ng a:C hojc (a,b):R, • e {>,<.^S}, xeX, vi cic tien dl thuit ngft li bao hdm khdi ni^m ma A:cC hoic dinh nghia khdi ni^m ma A:«C.
Mpt dien djch / thoi man A:eC, nlu VdeA':
A'(d) S C'(d), thoi man A:aC, nlu VdeA':
A'(d) = C'(d). ^k xur l^ tri thirc, cic co sd tri thiire thudng dupe chudn Aod [1,10], yi dy, vl d?ing chuin phu djnh, nghTa li, cic diu -. chi dimg trudc cic khii nifm nguyen thuy A, hoic biln doi cua khii nifm nguyen thuy 5A.
Ngoii ra, ngudi ta ciing cd the khai trien co sd tri thirc bing cich thay thi cic bao him khii nifm A:cC bing djnh nghTa khii nifm A:«CnA*, trong dd A* li mpt khii nifm nguyen thuy mdi, chiira cic die trung ciia A.
Vi dy, CHA:c CHAME, dupe thay bing CHA :« CHAME n DAN_ONG.
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Vifc xiir ly tri thurc trong logic md ta md dugc thyc hifn dya tren cic thu tyc quyet dinh, trong dd quan trpng nhit la bdi todn keo theo
£ |» Hf, chiimg minh khing djnh v)/ fur mpt co sd tri thurc £ cho trudc, nghTa la, vdi mpi dien djch /, nlu cic bilu thurc trong I diu thoi thi v)/ cung thoi, hoic bdi todn bao hdm, chiimg minh C:cP, tir co sd tri thurc I cho trudc. Tit ci cic bai toin niy diu cd thi chuyin thinh Bdi todn khong thod, vi dy, dl chumg minh I h (a • x), ngudi ta chuyin thinh chiimg minh I u (a o x) li khdng thoi, trong dd o li toin tiir ngupe ciia • (ngupe ciia >,<.^,< lin lupt li
<,>,<,>). 6 phan tiep theo, chiing tdi se trinh biy thii tyc giii quyet bii toin khdng thoi.
4. BAI TOAN
K H 6 N GTHOA
Y tudng giii quylt bii toin khdng thoi trong logic md ti md ngdn ngu* cung gi6ng nhu trong cic logic md ti khic nhu ALC_F, ALC_FH, ALC_FLH, ... li xuit phit tit mpt tip S chira cic ring bupc md, timg bude ip dyng cac luit lan truyin dl md r^ng tip S, cho din khi, hoic li tit ci cic md rpng dd diu cd chiira mau thuan, chiing td S khdng thoi, hoic li tip dd li diy du, nghTa la khdng cdn luit nio cd thi ip dyng them dupe n&a, vi khdng cd mau thuin, chijmg td tip S thoi.
Mpt tip ring bupc md S dupe gpi la cd miu thuan, nlu nd chura cic ring bupc nhu: <1 t x), vdi x>0, <T < x), vdi x<l, (a < 0), (a >
1), (1 > x), <T < x), vdi 0=inf(/2z/5e) vi l=sup(/rMe), X la phin tir ciia d^i so gia tiir, hoic chijra cic cip rang bupc doi ngiu nhu d dudi diy:
<a>x)
<a>x>
<o<y>
x > y x > y
<a<y>
x > y x > y
Ve luit lan truyin, chung ta cin them luit khiir gia tiir gidng nhu RTl [2]:
(hSA)'(a) = oc (5A)'(a) = ohc
Tir dd cd luit:
(8^): <w: 8A > ac> ^ (w: A ^ cac luat (5>),(5<),(8s) tuong ty.
Ngoai ra la cic luit cho cic k phuc hpp tuomg irng vdi cac phep -i, 3, va cac quan hf so sanh >, < , > , < ( nhu da dugc trinh bay trong [8]:
(^^) : <w: ^C > ac) ^ <w: C < oc') vdi c' la phin tii sinh ngupe vdi c, tuc
cac luat
(-I>),(-K),(->S)-(n^): <w:CnD > ac) -> (w:C > ac), ac), tuong ty la ci(
(n>),(n<),(n^.
(Ua): (w:CuD > ac) -> <w:C > ac) I ac), tuong ty li cac
(U>),(U<),(Us).
(V^ <w,: VR.C > ac), V|/ -> (W2: C nlu VJ/ doi ngiu vdi ((wi,W2): R (Vs) (w: VR.C < ac) -^ ((w,x): R > a.
C < ac), nlu X la mpt bien n khdng cd w' nao ma ca hai khin{
<(w,w'): R > ac'), <w': C < ac) i t^i trong tap rang bupc
(3^) (w: 3R.C > ac) -^ <(w,x): R > ac),
> ac), neu x la mpt biln mdi va 1<
cd w' nao mi ci hai khing ((w,w'): R > ac), (w': C > ac) da t(
trong tip rang bupc
(3^) <w,: 3R.C < ac), \\f -^ {^2. C < ac), v|/ doi ngau vdi ((W|,W2): R<ac) Sau day li mpt vi dy vl bii toin kl thoi:
Vi </fi : Cho CO sd tri thiire I gom cd khing djnh sau:
(1) (p: (-TI attr. very int) u wants > very ti (2) (tech: int t more true)
(3) <(p,tech): attr ^ poss more true)
vdi int, wants la cic khii nifm, attr
quan hf, p, tech la cic ca the. Cin chii
Ky yh Hfti thto ICT.rda'06 Proceedings of ICT.rda'06. Hanoi May. 20-21,2006 minh £ |» <p: wants ^ very true), nghTa l i
chiing minh £ cimg vdi
(4) (p: wants < very true) la khong thoa.
Ap dyng luit ( u ^ ) cho (1), se cho hai trudng hpp sau:
(5) (p: wants > very true)
(5*) (p: -i£ attr. very int > very true) d trudng hpp diu tien, ta cd ngay (S) mau thuin vdi (4), vi viy, chi cin xet tiep trudng hpp thu hai. Ap dung luat (-i^) cho (5*):
(6) (p: £ attr. very int < very false) Vi (3) doi ngau vdi <(p,tech): attr < very false), cho nen cd thi ap dung (3^):
(T) (tech: very int < very false) Tilp theo, ap dung (5s):
(8') (tech: int < very very false) mau thuan vdi (2).
Nhu vay, £ u {(4)} khdng thoa, hay la,
£ |« (p: wants > vetj true) dupe chirng minh.
5. KtT LUAN
Bai bao nay tilp ndi cic ket qua trong [7], siir dyng tap gia trj chin ly ngdn ngu bieu dien bdi cau true d^i sd gia tiir lam tap gia trj nIn cho logic md t i md ngon ngir. Tir dd xay dyng cac thii tyc quylt djnh ciia logic nay, giai quylt cic bai toin lip luan xip xi. Trong Phin 4 da trinh biy chi tilt ve thu tyc giii quylt bai toan khdng thoa. Day la nhirng kit qui bude diu theo hudng xiy dyng cic logic bieu dien va xiir ly tryc tiep tren gii trj ngdn ngS, cho phep phat trien them nhung hudng nghien ciiu mdi. Bai bio niy nim trong khuon kho dl tai nghien cuu co bin ve logic tinh toin v i ung dyng, thupc Chuong trinh Khoa hpc co bin cip nha nude.
T i i lifu tham khao
[I] StefTen Hoelldobler, Nguyen Hoang Nga, Tran Dinh Khang, The fuzzy description logic
ALCFLH, Proceedings of Internationa Description Logic Workshop, DL2005 Edinburgh, July 2005
[2] Nguyen Cat Ho, A method in linguistic reasoning on a knowledge based representing by sentences with linguistic belief degree Fundamenta Informalicae, 28: 247-259, 1996 [3] Nguyen Cat Ho, Tran Dinh Khang, Huynh Vai Nam, Nguyen Hai Chau, Linguistic-valuec logic and their application to iiizzy reasoning International Journal of Uncertainty, Fuzzines:
and Knowledge-based Systems, 7(4):347-361 1999
[4] Umberto Straccia, Uncertainty in descriptior logic, a lattice-based approach. Proceedings Oj the Iff'' International Conference or.
Information Processing and Management Oj Uncertainty in Knowledge-based Syslenu (IPMU-04):25\-25S,2004
[5] Tran Dinh Khang, Dinh Khac Dung, Suy dilr voi tap ma loai hai dira tren dai so gia tir, Tap chi tin hgc vd diiu khien hgc, 19(1): 28-43.
2003
[6] Lofti A. Zadeh, The concept of a linguistic variable and its application in approximate reasoning. Information Sciences, 8: 199-249, 301-357, 1975
[7] Dinh Khic Dung, Trin Dinh Khang, Mpt tilp can bilu dien mien gii h-j chan ly ngon ng&, Bdo cdo tgi Hdi thdo khoa hgc qudc gia lan II, Nghien ciru vd irng dung cdng ngh4 thdng lin (FAIR). TPHd Chi Minh, thdng 9/2005 [8] Trin Dinh Khang, Logic mo ta md, Kyyiu Hdi
thdo FAIR khoa hgc qudc gia lan thit nhdt
"Nghien ciru ca bdn vd img dung cdng nghi thdng tin", Nhd xudt bdn Khoa hgc vd Ky thudt, Hd Ngi. 2004 trang 160-167
[9] Steffen Hoelldoble, Tran Dinh Khang, Hans- Peter Stoerr, A fiizzy description logic with hedges as concept modifiers. Proceedings of InTech/VJFuzzy 2002. pages 25-34, 2002 [10] Umberto Sti^accia, Reasoning within fuzzy
description logics. Journal of Artificial Intelligent Research, 14: 137-166,2001
Ky yfai H»i thto lCT.ixla'06 Proceedings ofICT.rda'06. Hanoi May.
Ve die t i c gia:
PGS.TS. Trin Dinh Khang, t6t nghifp D^i hpc nganh Toin diiu khiln va 1^ thuit dnh toan t^i Trudng D^i hpc T6ng hpp K^ diuit Dresden, CHLB Diic, bio vf hiin in tiln sT t^i Trudng D^i hpc I Bach Khoa Ha N9i. PGS.
' ' Tiin £)inh Khai% hifn dang cdng tac t^i Khoa C6ng nghf didng tin, Truemg D;u hpc Bach Khoa Ha NfL
vin dl quan tim chinh: Logic tfnh toin.
Email: khangtd(a).it-hutedu.vn
Th9C s^ Dinh Kh t6t nghifp trudng Bach khoa Ha N<
nganh Cong nghf ' nam 2003, bao vf tfa^c s5r vl Logic 1 tai Trudng D^i hi hpp Ky thuat Dresd Cic van dl quan tai tinh toan, bieu dien tri thiire khdng chic c F.-mail: dinhkhacdungfSjgnix.net
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