1996, Mang tinh toan & ung dung

8/7/2019 1996, Mang tinh toan & ung dung

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la R(x) (ky hieu x dung e ch bo bien < x1,x2,...,xm >). Ta co thethay rang quan he R(x) co the c bieu dien bi mot anh xa fR,u,vvi u v = x, va ta viet : fR,u,v : u p v, hay van tat la f : u p v.

oi vi cac quan he dung cho viec tnh toan, cach ky hieu trenbao ham y ngha nh la mot ham: ta co the tnh c gia tr cua

cac bien thuoc v khi biet c gia tr cua cac bien thuoc u.Trong phan sau ta xet cac quan he xac nh bi cac ham codang f : u p v, trong o u v = (tap rong). ac biet la cac quan heoi xngco hang (rank) bang mot so nguyen dng k. o la cacquan he ma ta co the tnh c k bien bat ky t m-k bien kia (ay x la bo gom m bien < x1,x2,...,xm >). Ngoai ra, trong trng hpcan noi ro ta viet u(f) thay cho u, v(f) thay cho v. oi vi cac quan hekhong phai la oi xng co hang k, khong lam mat tnh tong quat,ta co the gia s quan he xac nh duy nhat mot ham f vi tapbien vao la u(f) va tap bien ra la v(f); ta goi loai quan he nay laquan he khong oi xng xac nh mot ham, hay goi van tat la

quan he khong oi xng.

v du: quan he f gia 3 goc A, B, C trong tam giac ABC cho bi hethc:

A+B+C = 180 (n v: o)

1.2. Mang tnh toan va cac ky hieu:

Nh a noi tren, ta se xem xet cac mang tnh toan bao gommot tap hp cac bien M va mot tap hp cac quan he (tnh toan) Ftren cac bien. Trong trng hp tong quat co the viet:

M = _x1,x2,...,xna,F = _f1,f2,...,fma.

oi vi moi f F, ta ky hieu M(f) la tap cac bien co lien he trong

quan he f. D nhien M(f) la mot tap con cua M: M(f) M. Neu viet fdi dang:

f : u(f) p v(f)th ta co M(f) = u(f) v(f).

II.- BAI TOAN TREN MANG TNH TOAN :


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Cho mot mang tnh toan (M,F), M la tap cac bien va F la tapcac quan he. Gia s co mot tap bien A M a c xac nh vaB la mot tap bien bat ky trong M.

Cac van e at ra la:

1. Co the xac nh c tap B t tap A nh cac quan he trong Fhay khong? Noi cach khac, ta co the tnh c gia tr cua cac bienthuoc B vi gia thiet a biet gia tr cua cac bien thuoc A haykhong?

2. Neu co the xac nh c B t A th qua trnh tnh toan gia trcua cac bien thuoc B nh the nao?

3. Trong trng hp khong the xac nh c B, th can cho themieu kien g e co the xac nh c B.

Bai toan xac nh B t A tren mang tnh toan (M,F) c viet di

dang:A p B,

trong o A c goi la gia thiet, B c goi la muc tieu tnh toancua bai toan.

nh ngha 2.1:Bai toan A p B c goi lagiai ckhi co the tnh toan

c gia tr cac bien thuoc B xuat phat t gia thiet A. Ta noi rangmot day cac quan he _f1, f2, ..., fka F la mot li giaicua bai toanA p B neu nh ta lan lt ap dung cac quan he fi (i=1,...,k) xuat phat

t gia thiet A th se tnh c cac bien thuoc B. Li giai _f1, f2, ..., fkac goi la li giai totneu khong the bo bt mot so bc tnhtoan trong qua trnh giai, tc la khong the bo bt mot so quan hetrong li giai.

Viec tm li giai cho bai toan la viec tm ra mot day quan hee co the ap dung suy ra c B t A. ieu nay cung co ngha latm ra c mot qua trnh tnh toan e giai bai toan.

nh ngha 2.2 :Cho D = _f1, f2, ..., fka la mot day quan he cua mang tnh toan

(M,F), A la mot tap con cua M. Ta noi day quan he D la ap dung

ctren tap A khi va ch khi ta co the lan lt ap dung c cacquan he f1, f2, ..., fk xuat phat t gia thiet A.Nhan xet : Trong nh ngha tren, neu at : A0 = A, A1 = A0 M(f1),

. . . , Ak = Ak-1 M(fk), va ky hieu Ak la D(A), th ta co D la mot li giaicua bai toan A p D(A). Trong trng hp D la mot day quan he batky (khong nhat thiet la ap dung c tren A), ta van ky hieu D(A)la tap bien at c khi lan lt ap dung cac quan he trong day D


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nhng bien thuoc M co the tnh c t A; cuoi cung se c baoong cua A.

Thuat toan 3.1. tm bao ong cua tap A M :Nhap : Mang tnh toan (M,F),

A M.Xuat : A Thuat toan :

1. B n A;2. Repeat

B1 n B;for f F do

if ( f oi xng and Card (M(f) \ B) e r(f) ) or( f khong oi xng and M(f) \ B v(f) ) then

beginBn

B M(f);F n F \ _fa; // loai f khoi lan xem xet sauend;

Until B = B1;3. A n B;

2. Li giai cua bai toan :

tren ta a neu len cach xac nh tnh giai c cua baitoan. Tiep theo, ta se trnh bay cach tm ra li giai cho bai toan A pB tren mang tnh toan (M,F).

Menh e 3.4 : day quan he D la mot li giai cua bai toan Ap B khi va ch khi D ap dung c tren A va D(A) B.

Do menh e tren, e tm mot li giai ta co the lam nh sau:Xuat phat t gia thiet A, ta th ap dung cac quan he e m rongdan tap cac bien co gia tr c xac nh; va qua trnh nay tao ramot s lan truyen tnh xac nh tren tap cac bien cho en khi aten tap bien B. Di ay la thuat toan tm mot li giai cho baitoan A p B tren mang tnh toan (M,F).

Thuat toan 3.2. tm mot li giai cho bai toan A p B :Nhap : Mang tnh toan (M,F), tap gia thiet A M, tap bien can

tnh B M.Xuat : li giai cho bai toan A p BThuat toan :

1. Solution n empty; // Solution la day cac quan he se apdung


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2. if B A thenbeginSolution_found n true; // bien Solution_found = true khi bai

toan la // giai cgoto 4;

endelse

Solution_found n false;3. Repeat

Aoldn A;Chon ra mot f F cha xem xet;while not Solution_found and (chon c f) do

beginif ( f oi xng and 0 < Card (M(f) \ A) e r(f) ) or

( f khong oi xng and { M(f) \ A v(f) ) then

beginA n A M(f);Solution n Solution _fa;end;

if B A thenSolution_found n true;

Chon ra mot f F cha xem xet;end; _ while a

Until Solution_found or (A = Aold);4. if not Solution_found then

Bai toan khong co li giai;

elseSolution la mot li giai;

Ghi chu :1. Ve sau, khi can trnh bay qua trnh giai (hay bai giai) ta co the xuat phat t

li giai tm c di dang mot day cac quan he e xay dng bai giai.2. Li giai (neu co) tm c trong thuat toan tren cha chac la mot li giai

tot. Ta co the bo sung them cho thuat toan tren thuat toan e tm motli giai tot t mot li giai a biet nhng cha chac la tot. Thuat toan seda tren nh ly c trnh bay tiep theo ay.

nh ly 3.3. Cho D=_f1, f2, ..., fma la mot li giai cua bai toan Ap B. ng vi moi i=1,...,m at Di = _f1, f2, ..., fia, D0 = . Ta xay dngmot ho cac day con Sm, Sm-1, ..., S2, S1 cua day D nh sau :

Sm = neu Dm-1 la mot li giai,Sm = _fma neu Dm-1 khong la mot li giai,Si = Si+1 neu Di-1 Si+1 la mot li giai,Si = _fia Si+1 neu Di-1 Si+1 khong la mot li giai,

vi moi i = m-1, m-2, ..., 2, 1.Khi o ta co :


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(1) Sm Sm-1 ... S2 S1.(2) Di-1 Si la mot li giai cua bai toan A p B vi moi i=m, ...,

2, 1.(3) Neu Si la mot day con that s cua Si th Di-1 Si khong

phai la mot li giai cua bai toan Ap B vi moi i.

(4) S1 la mot li giai tot cua bai toan A p B.

Thuat toan 3.3. tm mot li giai tot t mot li giai a biet.Nhap : Mang tnh toan (M,F),

li giai _f1, f2, ..., fma cua bai toan Ap B.Xuat : li giai tot cho bai toan A p BThuat toan :

1. D n_f1, f2, ..., fma;2. for i=m downto 1 do

if D \ _fia la mot li giai thenD n D \ _fia;

3. D la mot li giai tot.

Trong thuat toan 3.3 co s dung viec kiem tra mot day quan he cophai la li giai hay khong. Viec kiem tra nay co the c thchien nh thuat toan sau ay:

Thuat toan kiem tra li giai cho bai toan :Nhap : Mang tnh toan (M,F), bai toan Ap B, day cac quan he

_f1, f2, ..., fma.Xuat : thong tin cho biet _f1, f2, ..., fma co phai la li giai

cua bai toan Ap B hay khong.Thuat toan :1. for i=1 to m do

if ( fi oi xng and Card (M(fi) \ A) e r(fi) ) or( fi khong oi xng and M(fi) \ A v(fi) ) thenA n A M(fi);

2. if A B then _f1, f2, ..., fma la li giaielse _f1, f2, ..., fma khong la li giai;

3. nh ly ve s phan tch qua trnh giai :

Xet bai toan A p B tren mang tnh toan (M,F). Trong muc nay taneu len mot cach xay dng qua trnh giai t mot li giai a biet.oi vi mot li giai, rat co kha nang mot quan he nao o danti viec tnh toan mot so bien tha, tc la cac bien tnh ra makhong co s dung cho cac bc tnh pha sau. Do o, chung ta canxem xet qua trnh ap dung cac quan he trong li giai va ch tnhtoan cac bien that s can thiet cho qua trnh giai theo li giai. nh


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ly sau ay cho ta mot s phan tch tap cac bien c xac nh theoli giai va tren c s o co the xay dng qua trnh tnh toan cacbien e giai quyet bai toan.

nh ly 3.4. Cho _f1, f2, ..., fma la mot li giai tot cho bai toan A

p B tren mot mang tnh toan (M,F). at :A0 = A, Ai = _f1, f2, ..., fia(A), vi moi i=1,...,m.Khi o co mot day _B0, B1, ..., Bm-1, Bma, thoa cac ieu kien sau ay:

(1) Bm = B.(2) Bi Ai , vi moi i=0,1,...,m.(3) Vi moi i=1,...,m, _fia la li giai cua bai toan Bi-1p Bi nhng

khong phai la li giai cua bai toan G p Bi , trong o G la mot tapcon that s tuy y cua Bi-1.

Ghi chu :(1) T nh ly tren ta co qua trnh tnh toan cac bien e giai bai toan ApB

nh sau:bc 1: tnh cac bien trong tap B1 \ B0 (ap dung f1).bc 2: tnh cac bien trong tap B2 \ B1 (ap dung f2).v.v...bc m: tnh cac bien trong tap Bm \ Bm-1 (ap dung fm).

(2) T chng minh cua nh ly tren, ta co the ghi ra mot thuat toan e xaydng day cac tap bien _B1, ..., Bm-1, Bma ri nhau can lan lt tnh toan trong quatrnh giai bai toan (Bi = Bi \ Bi-1) gom cac bc chnh nh sau:

y xac nh cac tap A0, A1, ..., Am .y xac nh cac tap Bm, Bm-1, ..., B1, B0 .y xac nh cac tap B1, B2, ..., Bm .

PHU LUC:Trong phan nay chung ta neu len 2 v du ng dung cua mangtnh toan trong toan hoc va trong hoa hoc.

V du1: Cho tam giac ABC co canh a va 2 goc ke la F, K ccho trc.

Tnh dien tch S cua tam giac.

e tm ra li giai cho bai toan trc het ta xet mang tnh toancua tam giac. Mang tnh toan nay gom :

1. Tap bien M = _a, b, c, E, F, K, ha, hb, hc, S, p, R, r, ...a,trong o a,b,c la 3 canh; E, F, K la 3 goc tng ng vi 3 canh; ha, hb, hc

la 3 ng cao; S la dien tch tam giac; p la na chu vi; R la ban knhng tron ngoai tiep tam giac; r la ban knh ng tron noi tieptam giac, v.v...

2. Cac quan he:

f1 : E + F + K = 180 f2 :a

sin

b

sinE F!


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f3 :c

sin

b

sinK F! f4 :

a

sin

c

sinE K!

f5 : p = (a+b+c) /2 f6 : S = a.ha / 2f7 : S = b.hb / 2 f8 : S = c.hc / 2f9 : S = a.b.sinK / 2 v.v ...

3. Yeu cau tnh : S (dien tch cua tam giac).

Theo e bai ta co gia thiet la : A = _a, F, Ka, va tap bien can tnh laB = _Sa.Ap dung thuat toan tm li giai (thuat toan 3.2) ta co mot li giaicho bai tnh la day quan he sau: _f1, f2, f3, f5, f9a. Xuat phat t tapbien A, lan lt ap dung cac quan he trong li giai ta co tap cacbien c xac nh m rong dan en khi S c xac nh :

_a, F, Ka 1f p _a, F, K, Ea 2f p _a, F, K, E, ba 3f p _a, F, K, E, b, ca5f

p _a, F, K, E, b, c, pa 9f p _a, F, K, E, b, c, p, Sa.

Co the nhan thay rang li giai nay khong phai la li giai tot vco bc tnh toan tha, chang han la f5. Thuat toan 3.3 se loc ra tli giai tren mot li giai tot la _f1, f2, f9a:

_a, F, Ka 1f p _a, F, K, Ea 2f p _a, F, K, E, ba 9f p _a, F, K, E, b, Sa.

Theo li giai nay, ta co qua trnh tnh toan nh sau :bc 1: tnh E (ap dung f1).bc 2: tnh b (ap dung f2).bc 3: tnh S (ap dung f9).

Qua trnh tnh toan (gom 3 bc) nay co the c dien at motcach ro rang tren s o mang sau ay:

V du2: Chung ta biet rang trong hoa hoc, viec xem xet cacphan ng hoa hoc la mot trong nhng van e quan trong. Ve mattri thc ngi ta a biet c nhieu chat va cac phan ng hoa hoc


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[7] Judea Pearl. (1984). Heuristics.ADDISON-WESLEY PUBLISHING COMPANY.

[8] J.D. Ullman. (1988). Principles of Database and Knowledge-base Systems,Vol I. Computer Science Press.

[9] J.D. Ullman. (1989). Principles of Database and Knowledge-base Systems,Vol II. Computer Science Press.

[10] D. Kapur & J.L. Mundy. (1988). Wus Method and Its Application to Perspective Viewing.[11] D. Kapur. (1988). A Refutational Approach to Geometry Theorem Proving.[12] Adam Blum. (1992). Neural Networks in C++. John Wiley & Sons, Inc.

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1996, Mang tinh toan & ung dung