Presentations text content in Chapter 4: Prolog (Substitution,
Slide1
Chapter 4: Prolog (Substitution, Unification and Resolution)
Dr Youcef Djenouridjenouri@imada.sdu.dk
DM552: Part 2 Programing Logic
2017-2018
Slide2Substitution: Definition (1/2)Substitution θ is an operation allowing to replace some variables occurring in a formula with terms.
The goal of applying a substitution is to make a certain formula more specific so that it matches another formula. Substitutions allow for unification of formulae (or
terms).
Slide3Substitution: Definition (2/2)A substitution is any finite mapping of variables into terms of the form: θ
: V ⇒ TERAny (finite) substitution θ can be presented asθ={t
1/X1, t2/X2… tn/Xn}ti is the term to be substituted for a variable Xi, i= 1…n. No two elements in the set have the same variable after the / symbol,
Slide4Substitution: Example{ y/X, f(y)/Z } is a substitution. { a/Y, b/X}
is a substitution.{a/X, b/X} is not a substitution.{X/f(Y)}
is not a substitution.
Slide5Instance (1/2) Let θ ={t1/X1, t2/X2…
tn/Xn} be a substitution and let E be an expression. Then E
θ is an expression obtained by replacing all occurrences of every vi in E with the corresponding term ti, Eθ is an instance of E.
Slide6Instance (2/2) θ ={ a/X, f(b)/Y, c/Z} , E=P(X, Y, Z)
Eθ= P(a, f(b), c) is an instance of E. θ ={f(f(a))/X, X/Y} , E
=(P(X ) ∨ Q(Y)) Eθ= (P(f(f(a))) ∨ Q(X)) is an instance of E. θ ={ Y/X, a/Y} , E=(P(X ) ∨ Q(Y)) Eθ= (P(Y) ∨ Q(a)) is an instance of E.
Slide7Composition of substitutionsLet θ= {t1/X1, t2/X2…
tn/Xn} and Γ={
u1/Y1, u2/Y2… um/Ym} be two substitutions. The composition of θ and Γ is denoted by θ o Γ, and it is obtained by building the set {t1 Γ
/
X
1
,
t
2
Γ
/
X
2
…
t
n
Γ
/
X
n
,
u
1
/Y
1
,
u
2
/Y
2
…
u
m
/
Y
m
}
R
emove
the
following
elements
:
t
j
Γ
/
X
j
such that
t
j
Γ
=
X
j
u
i
/
Y
j
such that
Y
j
is in {X
1
,
X
2
…
X
n
}
Slide8Exampleθ= {t1/X1, t2/X2}= {
f(Y)/X, Z/Y} Γ={u1
/Y1, u2/Y2,u3/Y3}= {a/X, b/Y, Y/Z}{t1 Γ /X1, t2 Γ /X2, u1/Y1, u2/Y2,u3/Y3
}={f(b
)/X,
Y
/Y
,
a/X
,
b/Y
,
Y
/Z}
θ
o
Γ
={f(b
)/X,
Y
/Z}
Slide9ClassworkConsider θ = {f(Y)/X} and µ = {
f(Z)/Y} Compute θ o µ and µ o
θ ?
Slide10UnificationA substitution θ is called a unifier for a set {E1…Ek} if and
only if: E1θ =E2θ = E
3θ =…= EkθThe set {E1…Ek} is unifiable if and only if there exists a unifier for it. A unifier θ is a most general unifier for a set {E1…Ek} if and only if for each
unifier
Γ
there
exisits
a substitution
μ
such
that
Γ
= θ o
μ
.
Slide11ExampleE1=P(X), E2= P(f(Y)) are unifiable by
Γ={f(f(Z))/X, f(Z)/Y}E
1 Γ = E2 Γ = P(f(f(Z))). θ= {f(Y)/X} is a most general unifier, we can find μ ={f(Z)/Y} such that θ o μ= {f(f(Z))/X, f(Z)/
Y
}=
Γ
Slide12ClassworkProve that the following expressions are all unifiers by θ = {f(b)/X, b/Y, u/Z}
E = f(X, b, g(Z)).
F = f(f(Y), Y, g(u)).
Slide13Resolution: Used to prove a consequence from a set of logical formulaeStates: Sets of clauses L1
… LmInitially: The consequence we want to prove is negated and added to the set of clauses representing the initial state
Goal: Derive the empty clause ()
Slide14Example (1/6)¬P(X) P(f(X))
¬Q(a, Y) ¬R(Y, X
) P(X)R(b, g(a, Z))Q(a, b)Goal: P(f(g(a, c)))
Slide15Example (2/6)Negate goal, add to formulas1. ¬P(X)
P(f(X))2. ¬Q(a, Y) ¬R(Y, X)
P(X)3. R(b, g(a, Z))4. Q(a, b)5. ¬P(f(g(a, c)))
Slide16Example (3/6)Resolve (4) and (2) by applying {b/Y}
1. ¬P(X) P(f(X))2. ¬Q(a, Y)
¬R(Y, X) P(X)3. R(b, g(a, Z))4. Q(a, b)5. ¬P(f(g(a, c)))¬R(b, X) P(X)
Slide17Example (4/6)Resolve (6) and (3) by applying {g(a,Z
)/X}1. ¬P(X)
P(f(X))2. ¬Q(a, Y) ¬R(Y, X) P(X)3. R(b, g(a, Z))4. Q(a, b)5. ¬P(f(g(a, c)))6. ¬R(b, X)
P(X)
P(X)
Slide18Example (5/6)Resolve (7) and (1) by applying {g(a,Z
)/X}1. ¬
P(X) P(f(X))2. ¬Q(a, Y) ¬R(Y, X) P(X) 3. R(b, g(a, Z))4. Q(a, b) 5. ¬P(f(g(a, c)))6. ¬R(b, X) P(X)
7.
P(g(a,
Z))
P(f(X))
Slide19Example (6/6)Resolve (8) and (5) by applying {g(a,c)/
X}1. ¬P(X) P(f(X))
2. ¬Q(a, Y) ¬R(Y, X) P(X)3. R(b, g(a, Z))4. Q(a, b)5. ¬P(f(g(a, c)))6. ¬R(b, X) P(X)7. P(g(a, Z))
8.
P(f(X))
□
Slide20SLDTreeFacts: grand_father(jacob, X)
father(john, merry).father(jacob, jones).father(jones, sylia
). father(jacob, Z), father(Z, X ) father(jacob, Z), mother(Z, X)mother(sylia, anes). mother(merry, relly).Rules: father(jacob, jones), father(jones, X)
Exit
grand_father
(X, Y)
:-
father
(X
, Z),
father
(Z
, Y).
grand_father
(X
, Y)
:-
father
(X
, Z
),
mother(Z
, Y
).
Question:
grand_father
(
jacob
,
X
)
father
(
jacob
,
jones
),
father
(
jones
, sylia)
Chapter 4: Prolog (Substitution,
Download Presentation - The PPT/PDF document "Chapter 4: Prolog (Substitution," is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.