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TR TU NHN T OLogic v t

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N i dung trnh byLogic b c nh t (First Order Logic)C php v ng nghaCc l ng t

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Php thThu t gi i ng nh t

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T i sao s d ng logic b c nh t?Logic m nh ch x l trn ccs ki n, l cckh ng nh c gi tr ho c ng ho c sai.Logic v t (logic b c nh t) cho php chng tani v cc i t ng, tnh ch t c a chng, quan

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h gi a chng, pht bi u v t t c cc it ng hay m t i t ng no m khng c nli t k t ng minh.

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Logic B c nh tCc cu khng th bi u di n b ng logic m nh nhng c th b ng logic b c nh t

Socrates l ng i nn socrates ch tKhi sn m t h p b ng mu xanh, n s tr thnh h p

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xanhM t ng i c cho php truy c p trang web n u h c c p quy n chnh th c hay quen bi t v i ai cphp truy c p

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C php c a FOLTerm (tham chi u i t ng)

K hi u h ng: Lan, Tuan, DHKHTN,Bi n: x, y, a,K hi u hm p d ng cho m t hay nhi u term: f(x), tuoi(Lan),anh-cua(Tuan)

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Cu (pht bi u v i t ng)M t k hi u v t (predicate) p d ng cho m t hay nhi uterm:Thuoc(Lan, DHKHTN), La-anh-em(Tuan, Lan), La-ban-be(anh-cua(Tuan), Lan),t1= t2N u x l m t bi n v l m t cu thx. v x. l m t cuCc cu t o t cc cu khc v i ton t n i cu:

Tn v s l ng i s c a hm hay v t c g i l

functor v arity.

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Ng ngha c a FOLC m t t p ng m nhU cc i t ng m m t cu

FOL pht bi u trn ,U g i l t p v tr hay mi npht bi u.Term tham chi u n i t ng trongU :

H n tham chi u n m t i t n c th .

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Bi n tham chi u n m t i t ng no (c n l ng tha).Hm tham chi u n m t i t ng thng qua cc it ng khc.

Cu l pht bi u trn cc i t ng:V t th hi n tnh ch t ho c quan h gi a cc i t ng.L ng t gip pht bi u trn ton b cc i t ng (l ngt v i m i) hay m t i t ng no (l ng t t n t i) mkhng c n li t k hay nu ra t ng minh.

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L ng t v i m i

Sinh vin CNTT th thng minh:x Sinh-vin(x,CNTT) Thng-minh(x)

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ng v i m ix thu c U .Ngha l, tng ng v i phpn i li nc a cc it ng trongU

Sinh-vin(Lan,CNTT) Thng-minh(Lan) Sinh-vin(Minh,CNTT) Thng-minh(Minh) Sinh-vin(Tu n,CNTT) Thng-minh(Tu n)

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L i th ng g p c n trnhThng th ng, l php n i th ng i v i

L i th ng g p: dng lm php n i chnh i v i :

x Sinh-vin(x,CNTT)Thng-minh(x)

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ngha l M i ng i u l sinh vin CNTT v m i ng i u thng minh

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L ng t T n t i

C sinh vin CNTT thng minh:x Sinh-vin(x,CNTT)Thng-minh(x)

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.ng v i m tx no thu c U .Ngha l, tng ng v i phpn i r ic a cc it ng trong U

Sinh-vin(Lan,CNTT)Thng-minh(Lan) Sinh-vin(Minh,CNTT)Thng-minh(Minh) Sinh-vin(Tu n,CNTT)Thng-minh(Tu n)

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L i th ng g p khc c n trnhThng th ng, l php n i chnh v i

L i th ng g p: dng lm php n i chnh v i:

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x n -v n x, ng-m n xng n u c b t k ai khng l sinh vin CNTT!

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Vi t FOL nh th noMo l ng v t c v [Mo1, ng-v t-c-v1]

x.Mo(x) ng-v t-c-v(x)Lan l sinh vin h c gi i [Sinh-vin1, H c-gi i1,Lan]

Sinh-vin(Lan)H c-gi i(Lan)

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Chu l con c a anh em [Chu2, Anh-em2, Con2]x,y.Chu(x,y) z.(Anh-em(z,y)Con(x,z))

B ngo i l m c a m [cc hm: b-ngo i, m ]

xy. x= b-ngo i(y) z.(x= m (z) z= m (y))M i ng i u yu ai [Yu2]

x, y.Yu(x, y)

x, y.Yu(x, y)

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Vi t FOL nh th no (tt)Khng ai yu Lan

x. Yu(x, Lan) x. Yu(x,Lan)

Ai cng c m t cha

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x, y. Cha(y,x)Ai cng c m t cha v m t m

x, yz. Cha(y,x)M (z,x)

B t k ai c m t cha cng c m t mx. [[y.Cha(y,x)] [y.M (y,x)]]

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Suy d n trong FOLKB suy d n S: v i m i th hi n I, n u KB tho trong I thS cng tho trong I

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Ni chung tnh ton suy d n l khng kh thi v c v scc mi n pht bi u c thNgay c vi c tnh ton tnh tho cng khng kh thi iv i cc mi n pht bi u v h n

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Ch ng minh v Suy d nSuy d n xu t pht t khi ni m t ng qut c a php ko

theoKhng th tnh ton tr c ti p b ng cch li t kDo , ta s lm theo cch ch ng minh

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Trong FOL, n u KB suy d n c S th c m t t p h uh n cc ch ng minh c a S t KBT n t i h ch ng minh FOL v ng

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H p gi i B c nh tx, P(x) Q(x)

P(A)Q(A)

Tam o n lu n:M i ng i u ch t

Socrates l ng iSocrates ch t

Hai v n m i: bi n i FOL thnhd ng m nh (clausalform)

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x, x xP(A)Q(A)

P(A)Q(A)P(A)Q(A)

Tng ng theo nh ngha c a phpSuy ra

Thay A vo x, v nng

khi

H p gi i M nh

h p gi i v i bi n

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D ng m nh (Clausal Form)c u trc ngoi gi ng CNF

khng c l ng t

x. y. P(x) R(x, y)

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P(x)R(x, F(x))

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Bi n i thnh d ng m nh 1. Lo i b cc d u mi tn

( ) ( )

2. Phn ph i ph nh

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( ) ( ) x. x.x. x.

3. i tn cc bi n thnh ph nx.y.( P(x) x. Q(x,y)) x1.y.( P(x1) x2. Q(x2,y))

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Skolem ho4. Skolem ho

thay tn m i cho t t c l ng t t n t ix. P(x) P(Lan)

x,y.R(x,y) R(Thing1, Thing2)

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.x. P(x) x. Q(x) P(Frog) Q(Grog)

y, x. Loves(x,y) x.Loves(x, Englebert)thay hm m i cho t t c cc l ng t t n t i t m v c ngoi

x y. Loves(x,y) x.Loves(x, beloved(x))x y z w. P(x,y,z)R(y,z,w) x z. P(x,F(x),z)

R(F(x),z,G(x,z))

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Bi n i thnh d ng m nh 5. B cc l ng t v i m i

x. y Loves(x,y) Loves(x, beloved(x))6. Ph n ph ior vo and; tr v cc m nh

P(z) (Q(z,w)R(w,z))

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{P(z)Q(z,w), P(z)Q(w,z)}7. i tn cc bi n trong t ng m nh

{P(z)Q(z,w), P(z)Q(w,z)}

{P(z1) Q(z1,w1), P(z2) Q(w2,z2)}

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H p gi i B c nh tx, P(x) Q(x)

P(A)Q(A)

Tam o n lu n:M i ng i u ch t

Socrates l ng iSocrates ch t

i u ch y u l tm cc

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x, x xP(A)Q(A)

P(A)Q(A)P(A)Q(A)

Tng ng theo nh ngha c a phpSuy ra

Thay A vo x, v nng

khi

H p gi i M nh

p p t ng n c occ bi n

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Php thP(x, f(y), B): m t cu FOL

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Cc th hi n Php th{v1/t1, v2/t2 }

Ghi ch

P(z, f(w), B) {x/z, y/w} i tn bi n

P(x, f(A), B) {y/A}

P(g(z), f(A), B) {x/g(z), y/A}P(C, f(A), B) {x/C, y/A} Php th c s

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Php thP(x, f(y), B): m t cu FOL

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Cc th hi n Php th{v1/t1, v2/t2 }

Ghi ch

P(z, f(w), B) {x/z, y/w} i tn bi n

P(x, f(A), B) {y/A}

p d ng m t php thP(x, f(y), B) {y/A} = P(x, f(A), B)P(x, f(y), B) {y/A, x/y} = P(A, f(A), B)

P(g(z), f(A), B) {x/g(z), y/A}P(C, f(A), B) {x/C, y/A} Php th c s

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ng nh tHai bi u th c 1 v 2 l ng nh t c(unifiable) khi

vo ch khi t n t i php th s sao cho 1s = 2sG i 1 = x v 2 = y, d i y l ccphp ng nh t:

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{y/x} x x

{x/y} y y

{x/f(f(A)), y/f(f(A))} f(f(A)) f(f(A))

{x/A, y/A} A A

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ng nh t t ng qut nh t ng nh tKnows(John,x)v Knows(y,z), ta c cc

php th = {y/John, x/z } hay = {y/John, x/John, z/John}

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Php ng nh t u tin t ng qut hn ci th hai.

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ng nh t t ng qut nh tg l php ng nh t t ng qut nh t (most general unifier

- MGU)c a 1 v 2 khi v ch khi v i m i php ngnh t s, t n t i s sao cho 1.s = (1.g)s v 2.s =(2.g)s

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1

2

P(x) P(A) {x/A}

P(f(x), y, g(x)) P(f(x), x, g(x)) {y/x} hay {x/y}

P(f(x), y, g(y)) P(f(x), z, g(x)) {y/x, z/x}

P(x, B, B) P(A, y, z) {x/A, y/B, z/B}

P(g(f(v)), g(u)) P(x, x) {x/g(f(v)), u/f(v)}

P(x, f(x)) P(x, x) Khng c MGU!

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Qui t c ng nh t26E1 ng nh t v i E2 khi:

N u E1, E2 l h ng: ng nh t n u E1 gi ng E2.E1 l bi n, E2 b t k: lun ng nh t v i E2 thayth cho E1.

E2 l bi n, E1 b t k: lun ng nh t v i E1 thayth cho E2.N u E1, E2 l hm ho c v t : ng nh t khi E1,

E2:cng functor v aritycc i s ng nh t v i nhau theo ng th tcc php th dng trong ng nh t cc i s ph itng thch nhau.

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M t s v d v ng nh t

1 2 MGUA(B,C) A(x,y) {x/B, y/C}

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, , , , ,

A(x, y) A(f(C, y), z) {x/f(C,y), y/z}P(A, x, f(g(y))) P(y, f(z), f(z)) {y/A, x/f(z), z/g(y)}P(x, g(f(A)), f(x)) P(f(y), z, y) Khng cP(x, f(y)) P(z, g(w)) Khng c

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i u c n n m ngha c a logic b c nh t

C php c a logic b c nh tC th bi u di n m t v n d i d ng logic

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Chuy n i KB thnh d ng m nh Hi u c php th v php ng nh t

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Th c m c29