:



3.1
3.1.1

3.1.2
1.

1
3.1
2

3
3.1

2.
3.1.3




3.1.4








3.2
3.2.1
PQR…“T”“F”
TF

3.2.2
1.
“”P Q“PQ”
3.1
3.1
P Q
P∨Q
P∧Q
P→Q
P Q
2.
A→BAB
3.1




3.3
3.3.1



3.3


P(x1,x2,…,xn )
.
3.3


P(x1,x2,…,xn )
.
3.3

P(x)P(x,y)P(x1,x2,…,xn )n
P(x1,x2,…,xn )xi(i=1,2,…,n)xi
“”“”
3.3

3.3.2
1.
3.

3.5
1
3ABA∧BA∨BA→BAB
4Ax(x)A(x)A
51—4
3.3

4.
3.3.3
1
3.1 D={1,2}A=(x)(P(x)→Q(f(x),b))DA
()
2
3.8 PDFPDPP

3
3.3

3.3.4



2TSS
3CPRSR→S
4P=>QP∧QFQPP∧Q

(P1∧P2∧…∧Pn)∧Q
3.3.5
1.

3.12 {t1/x1,t2/x2,…,tn/xn}xitixixi xjIjij=1,2,…,n
3.3

{a/x,b/y,f(x)/z}{f(z)/x,y/z}
3.3

{t1·/x1,t2·/x2,…,tn·/xn,u1/y1,u2/y2,…,um/ym}
yi{ x1,…,xn}ui/yi
·
2.
E1 θ = E2 θ =…=En θ
E1E2…En θ
3.15 E1E2…En σE1E2…En λθ= σ · λ σE1E2…En mgu
3.3


E1E2E1yE2f(a)
1W={ E1,E2 }
3Wkσkmgu
4WkDk
5Dkxktk xktkxktk σk+1=σk ·{tk/xk }
Wk+1=wk{tk/xk } k=k+1 3
6Wmgu
3.3

3.5 E1=P(a,v,f(g(y)))E2=P(z,f(a),f(u))E1 E2mgu

3.4
3.4.1

P→QQ→RP→R



P→QQP P→Q Q P
P→QPQP→QPQ

3.4
3.4.3
3.5

3.5.1



Skolem
GSkolem
1  G→ A∨BA→B(A∧B)∨(A∧B)AB
2 “”
3
4 Skolem
5
6
3.5

2
3
4PP1∧P2∧…Pn
PP1∧P2∧…PnPSPPiSiSPS1∪S2∪S3…∪SnS1∪S2∪S3…∪SnSPS1∪S2∪S3…∪Sn
SPS1∪S2∪S3…∪Sn
3.5

3.5.2 Herbrand
1H
1 H0SSaDH0={a}
2
3.5

3.10 S{T(x)∨Q(z),R(f(y))}H
af(y)
H0={a}
H1={a,f(a)}
H2={a,f(a),f(f(a))}
2
3.21 S
3.5

3.22
3.23 SCHC
H{a,f(a),f(f(a)),…}
3.5

3H
DIHI*DSHI*SS|I=TS|I*=T
3.5

3.3 ISDIHI* S|I=TS|I*=T
3.4 SSH
SHI*SDI0S3.3HI*0SSH
3.5

3.5 SS′
Herbrand
3.5

3.5

3.5.3
3.6 C12C1C2

S
3.5

2
C12 =( C1σ{L1σ})∪(C2 σ{L2σ})
C1 C2L1 L2
3.5


2
3.5


1   C1C2
2   C1C1σ1C2
3   C1C2C2σ2
4   C1C1σ1C2C2σ2
3.5


C1=P(a)∨Q(x)∨R(x)C2 =P(y)∨Q(b) L1=P(a)L2=P(y)L1L2mguσ={a/y}3.27C1C2
C12 =( C1σ-{L1σ})∪(C2 σ-{L2σ})
=({P(a),Q(x),R(x)}-{P(a)})∪({P(a),Q(b)})-{P(a)}
=({Q(x),R(x)})∪({Q(b)})
C12 =P(a)∨R(b)∨P(y)
3
3.5

3.5.4
(2)    GS
3.5


B: ( x)R(x)→( x)( y)(P(x,y)→Q(y))
BA

7 Q(b) 46σ={a/x,b/y}
8 NIL 57
BA
3.5.5
3ANSWERS1S
4SANSWER
5ANSWER ANSWER
JohnPeterJohnDavidPeter

1
2
F1
ANSWER
PeteruFather(u,Peter)
GS2,S1S2S

6Brother(John,Peter)∨ANSWER(David)
7ANSWER(David) 26
ANSWER(David)u=DavidPeterDavid
3.5

3.6
3.6.1
1
3.5


3.5

2
3.5

3.6.2
1
3.5



C1C2σC1σC2C1C2P(x)∨Q(a)P(f(a))∨Q(a)∨R(y) σ={f(a)/x}
T(x)∨S(y)T(b)∨S(v)∨R(x) σ={b/xv/y}

2
3
3.5

4