Automated Reasoning

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Automated Reasoning
General Game Playing Lecture 3
Michael Genesereth Spring 2005
2
Unification
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Substititions
A substitution is a finite set of pairs of
variables and terms, called replacements. ?
((?x . A) ((?y . (F B)) (?V . ?W)) The result of
applying a substitution ? to an expression ? is
the expression ?? obtained from ? by replacing
every occurrence of every variable in the
substitution by its replacement. (P ?x ?x ?y
?Z)? (P A A (F B) ?Z)
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Unification
A substitution ? is a unifier for an expression ?
and an expression ? if and only if ????. ?
((?x . A) ((?y . (F B)) (?V . ?W)) ? ((?x . A)
((?y . (F B)) (?V . ?W)) (P ?x ?y)? (P A B) (P
?x ?y)? (P A B) If two expressions have a
unifier, they are said to be unifiable.
Otherwise, they are nonunifiable. (P ?x ?y) (P A
B)
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Non-Uniqueness of Unification
Unifier 1 p(x,y)x?a,y?b,v?bp(a,b) p(a,v)x?a,y
?b,v?bp(a,b) Unifier 2 p(x,y)x?a,y?f(w),v?f(w
)p(a,f(w)) p(a,v)x?a,y?f(w),v?f(w)p(a,f(w))
Unifier 3 p(x,y)x?a,y?vp(a,v) p(a,v)x?a,y?v
p(a,v)
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Most General Unifier
A substitution ? is a most general unifier (mgu)
of two expressions if and only if it is as
general as or more general than any other
unifier. Theorem If two expressions are
unifiable, then they have an mgu that is unique
up to variable permutation. p(x,y)x?a,y?vp(a,v
) p(a,v)x?a,y?vp(a,v) p(x,y)x?a,v?yp(a,y) p
(a,v)x?a,v?yp(a,y)
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Most General Unification
(defun mgu (x y al) (cond ((eq x y) al)
((varp x) (mguvar x y al)) ((atom x)
(cond ((varp y) (mguvar y x al))
((atom y) (if (equalp x y) al)))) (t (cond
((varp y) (mguvar y x al)) ((atom y)
nil) ((setq al (mgu (car x) (car y) al))
(mgu (cdr x) (cdr y) al))))))
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Most General Unification (continued)
(defun mguvar (x y al) (let (dum) (cond
((setq dum (assoc x al)) (mgu (cdr
dum) y al)) ((eq x (setq y (mguval y
al))) al) ((mguchkp x y al)) nil)
(t (acons x y al))))) (defun mguval (x al)
(let (dum) (cond ((and (varp x) (setq dum
(assoc x al))) (mguval (cdr dum) al))
(t x))))
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Problem
hates(X,X) hates(Y,f(Y))
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Solution
Before assigning a variable to an expression,
first check that the variable does not occur
within that expression. This is called, oddly
enough, the occur check test. Prolog does not do
the occur check (and is proud of it).
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Most General Unification (concluded)
(defun mguchkp (p q al) (cond ((eq p q))
((varp q) (mguchkp p (cdr (assoc q al)) al))
((atom q) nil) (t (some '(lambda (x)
(mguchkp p x al)) q))))
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Example
(mgu (p ?x b) (p a ?y)) Calling (MGU (P ?x B)
(P A ?y) ((t . t))) Calling (MGU P P ((t .
t))) MGUEXP returned ((t . t)) Calling
(MGU ?x A ((t . t))) MGUEXP returned ((?x .
A) (t . t)) Calling (MGU B ?y ((?x . A) (t .
t))) MGUEXP returned ((?y . B) (?x . A) (t .
t)) MGUEXP returned ((?y . B) (?x . A) (t .
t)) ((?y . B) (?x . A) (t . t))
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Example
Call (mgu (p ?x ?x) (p a b) ((t . t)))
Call (mgu p p ((t . t))) Exit ((t . t))
Call (mgu ?x a ((t . t))) Exit ((?x .
a) (t . t)) Call (mgu ?x b ((?x . a) (t
. t))) Call (mgu a b ((?x . a) (t . t)))
Exit nil Exit ((?y . b) (?x . a) (t .
t)) Exit ((?y . b) (?x . a) (t . t))
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Example
Call (mgu (p (f ?x) (f ?x)) (p ?y (f a))
((t.t))) Call (mgu p p ((t . t))) Exit
((t . t)) Call (mgu (f ?x) ?y ((t . t))
Exit ((?y . (f ?x)) (t . t))) Call (mgu (f
?x) (f a) ((?y . (f ?x)) (t . t))) Call (mgu
f f ((?y . (f ?x)) (t . t))) Exit ((?y . (f
?x)) (t . t))) Call (mgu ?x a ((?y . (f ?x))
(t . t))) Exit ((?x . a) (?y . (f ?x)) (t .
t) Exit ((?x . a) (?y . (f ?x)) (t .
t)) Exit ((?x . a) (?y . (f ?x)) (t . t))
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Example
(mgu (p (f ?x) (f ?x)) (p ?y (f ?y))) Call
(MGU (P ?x ?x) (P ?y (F ?y)) ((t . t))) Call
(MGU P P ((t . t))) Exit ((t . t))
Call (MGU (F ?x) ?y ((t . t)) Exit ((?y .
(F ?x)) (t . t))) Call (MGU (F ?x) (F ?y)
((t . t)) Call (MGU F F ((?y . (F ?x)) (t
. t))) Exit ((?y . (F ?x)) (t . t)))
Call (MGU ?x ?y ((?y . (F ?x)) (t . t)))
Exit NIL Exit NIL Exit NIL NIL
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Evaluate
(defun myevaluate (thing p theory) (let
(answers) (myeval p nil truth)
(nreverse (remove-duplicates answers)))) (defun
eval (p pl al) (cond ((atom p) (evalrs p pl
al)) ((eq 'not (car p)) (evalunprovable p
pl al)) ((eq 'and (car p)) (eval
(cadr p) (append (cddr p) pl) al)) (t
(evalrs p pl al)))) (defun evalexit (pl al)
(cond (pl (eval (car pl) (cdr pl) al)) (t
(setq answers (cons (plug thing
al) answers)))))
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Evaluate (continued)
(defun evalunprovable (p pl al) (unless (eval
(cadr p) (cdr p) al) (evalexit pl
al))) (defun evalrs (p pl al) (do ((l theory
(cdr l)) (rule) (bl)) ((null l))
(setq rule (stdize (car l))) (when (setq bl
(mguexp p rule al)) (evalexit pl bl))))
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Unification
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Substititions
A substitution is a finite set of pairs of
variables and terms, called replacements. X?a,
Y?f(b), V?W The result of applying a
substitution ? to an expression ? is the
expression ?? obtained from ? by replacing every
occurrence of every variable in the substitution
by its replacement. p(X,X,Y,Z)X?a,Y?f(b),V?Wp(
a,a,f(b),Z)
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Unification
A substitution ? is a unifier for an expression ?
and an expression ? if and only if
????. p(X,Y)X?a,Y?b,V?bp(a,b) p(a,V)X?a,Y?b,
V?bp(a,b) If two expressions have a unifier,
they are said to be unifiable. Otherwise, they
are nonunifiable. p(X,X) p(a,b)
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Non-Uniqueness of Unification
Unifier 1 p(X,Y)X?a,Y?b,V?bp(a,b) p(a,V)X?a,Y
?b,V?bp(a,b) Unifier 2 p(X,Y)X?a,Y?f(W),V?f(W
)p(a,f(W)) p(a,V)X?a,Y?f(W),V?f(W)p(a,f(W))
Unifier 3 p(X,Y)X?a,Y?Vp(a,V) p(a,V)X?a,Y?V
p(a,V)
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Most General Unifier
A substitution ? is a most general unifier (mgu)
of two expressions if and only if it is as
general as or more general than any other
unifier. Theorem If two expressions are
unifiable, then they have an mgu that is unique
up to variable permutation. p(X,Y)X?a,Y?Vp(a,V
) p(a,V)X?a,Y?Vp(a,V) p(X,Y)X?a,V?Yp(a,Y) p
(a,V)X?a,V?Yp(a,Y)
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Most General Unification
(defun mgu (x y al) (cond ((eq x y) al)
((varp x) (mguvar x y al)) ((atom x)
(cond ((varp y) (mguvar y x al))
((atom y) (if (equalp x y) al)))) (t (cond
((varp y) (mguvar y x al)) ((atom y)
nil) ((setq al (mgu (car x) (car y) al))
(mgu (cdr x) (cdr y) al))))))
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Most General Unification (continued)
(defun mguvar (x y al) (let (dum) (cond
((setq dum (assoc x al)) (mgu (cdr
dum) y al)) ((eq x (setq y (mguval y
al))) al) ((mguchkp x y al)) nil)
(t (acons x y al))))) (defun mguval (x al)
(let (dum) (cond ((and (varp x) (setq dum
(assoc x al))) (mguval (cdr dum) al))
(t x))))
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Problem
hates(X,X) hates(Y,f(Y))
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Solution
Before assigning a variable to an expression,
first check that the variable does not occur
within that expression. This is called, oddly
enough, the occur check test. Prolog does not do
the occur check (and is proud of it).
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Most General Unification (concluded)
function mguchkp (p,q,al) cond(pq, al
varp(p), mguchkp(p,cdr(assoc(q,al)),al)
atom(q), nil t, some(lambda(x).mguchkp(p,
x,al),q))
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Most General Unification (concluded)
function mguchkp (p,q,al) if (pq) al else
if varp(p) mguchkp(p,cdr(assoc(q,al)),al)
else if atom(q) nil else some(lambda(x).mguchk
p(p,x,al),q)
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Example
Call mgu(p(X,b),p(a,Y),) Call
mgu(p,p,) Exit Call
mgu(X,a,) Exit X?a Call
mgu(b,Y,X?a) Exit Y?b,X?a Exit
Y?b,X?a
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Example
Call mgu(p(X,X),p(a,b),) Call
mgu(p,p,) Exit Call
mgu(X,a,) Exit X lt- a Call
mgu(X,b, X lt- a) Call mgu(a,b,X lt- a)
Exit nil Exit nil Exit nil
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Example
Call mgu(p(f(X),f(X)),p(Y,f(a)),)
Call mgu(p,p,) Exit Call
mgu(f(X),Y,) Exit Y lt- f(X)
Call mgu(f(X),f(a),Y lt- f(X)) Call
mgu(f,f,Y lt- f(X)) Exit Y lt- f(X)
Call mgu(X,a,Y lt- f(X) Exit X lt- a,Y
lt- f(X) Exit X lt- a,Y lt- f(X) Exit
X lt- a,Y lt- f(X)
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Example
Call mgu(p(X,X),p(Y,f(Y)),) Call
mgu(p,p,) Exit Call
mgu(f(X),Y,) Exit Y lt- f(X)
Call mgu(f(X),f(Y),Y lt- f(X)) Call
mgu(f,f,Y lt- f(X)) Exit Y lt- f(X)
Call mgu(X,Y,Y lt- f(X)) Exit nil
Exit nil Exit nil
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Evaluation
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Reasoning Subroutines
Database Call theory Exit m(a,b), m(b,c),
p(X,Y)ltm(X,Y) Subroutines Call
findp(p(X,Y),theory) Exit true Call
findx(X,p(X,Y),theory) Exit A Call
finds(X,p(X,Y),theory) Exit A,B
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Evaluate
(defun myevaluate (thing p theory) (let
(answers) (myeval p nil truth)
(nreverse (remove-duplicates answers)))) (defun
eval (p pl al) (cond ((atom p) (evalrs p pl
al)) ((eq 'not (car p)) (evalunprovable p
pl al)) ((eq 'and (car p)) (eval
(cadr p) (append (cddr p) pl) al)) (t
(evalrs p pl al)))) (defun evalexit (pl al)
(cond (pl (eval (car pl) (cdr pl) al)) (t
(setq answers (cons (plug thing
al) answers)))))
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Evaluate (continued)
(defun evalunprovable (p pl al) (unless (eval
(cadr p) (cdr p) al) (evalexit pl
al))) (defun evalrs (p pl al) (do ((l theory
(cdr l)) (rule) (bl)) ((null l))
(setq rule (stdize (car l))) (when (setq bl
(mguexp p rule al)) (evalexit pl bl))))
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Deduction
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Reasoning Subroutines
Database Call theory Exit m(a,b),
m(b,c),p(X,Y)ltm(X,Y) Subroutines Call
findp(p(X,Y),theory) Exit true Call
findx(X,p(X,Y),theory) Exit A Call
finds(X,p(X,Y),theory) Exit A,B
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Backward
(defun myevaluate (thing p theory) (let
(answers) (myeval p nil truth)
(nreverse (remove-duplicates answers)))) (defun
eval (p pl al) (cond ((atom p) (evalrs p pl
al)) ((eq 'not (car p)) (evalunprovable p
pl al)) ((eq 'and (car p)) (eval
(cadr p) (append (cddr p) pl) al)) (t
(evalrs p pl al)))) (defun evalexit (pl al)
(cond (pl (eval (car pl) (cdr pl) al)) (t
(setq answers (cons (plug thing
al) answers)))))
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Backward (continued)
(defun evalunprovable (p pl al) (unless (eval
(cadr p) (cdr p) al) (evalexit pl
al))) (defun evalrs (p pl al) (do ((l theory
(cdr l)) (rule) (bl)) ((null l))
(setq rule (stdize (car l))) (when (setq bl
(mguexp p rule al)) (evalexit pl bl))))
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(No Transcript)
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Backward Chaining
Backward Chaining is the same as reduction except
that it works on rule form rather than clausal
form.
Reduced literals need be retained only for
non-Horn premises. Cancellation and Dropping
are analogous.
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Example
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Example

• Given q(x) ? p(x) and p(a), find a term ? such
  that q(?) is
• true.
• q(x) ? p(x) Premise
• p(a) ? Premise
• goal(z) ? q(z) Goal
• goal(z) ? p(z) 1, 3
• goal(a) ? 2, 4

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Example

• Given q(x) ? p(x) and p(a) and p(b), find a term
  ? such that
• q(?) is true.
• q(x) ? p(x) Premise
• p(a) ? Premise
• p(b) ? Premise
• goal(z) ? q(z) Goal
• goal(z) ? p(z) 1, 4
• goal(a) ? 2, 5
• goal(b) ? 3, 5

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Example

• Given q(x) ? p(x) and p(a) ? p(b), find a term ?
  such that
• q(?) is true.
• q(x) ? p(x) Premise
• p(a) ? p(b) ? Premise
• goal(z) ? q(z) Goal
• goal(z) ? p(z) 1, 3
• goal(a) ? p(b) ? 2, 4
• goal(a) ? goal(b) ? 4, 5