automated reasoning

Automated Reasoning

General Game Playing Lecture 3

Michael Genesereth Spring 2005

2

Unification

3

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)

4

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)

5

Non-Uniqueness of Unification

Unifier 1:p(x,y){xa,yb,vb}=p(a,b)p(a,v){xa,yb,vb}=p(a,b)

Unifier 2:p(x,y){xa,yf(w),vf(w)}=p(a,f(w))p(a,v){xa,yf(w),vf(w)}=p(a,f(w))

Unifier 3:p(x,y){xa,yv}=p(a,v)p(a,v){xa,yv}=p(a,v)

6

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){xa,yv}=p(a,v)p(a,v){xa,yv}=p(a,v)

p(x,y){xa,vy}=p(a,y)p(a,v){xa,vy}=p(a,y)

7

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))))))

8

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))))

9

Problem

~hates(X,X) hates(Y,f(Y))

10

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).

11

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))))

12

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))

14

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))

16

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*)))))

17

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))))

18

Unification

19

Substititions

A substitution is a finite set of pairs of variables and terms, called replacements.

{Xa, Yf(b), VW}

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){Xa,Yf(b),VW}=p(a,a,f(b),Z)

20

Unification

A substitution is a unifier for an expression and an expression if and only if =.

p(X,Y){Xa,Yb,Vb}=p(a,b)p(a,V){Xa,Yb,Vb}=p(a,b)

If two expressions have a unifier, they are said to be unifiable. Otherwise, they are nonunifiable.

p(X,X)p(a,b)

21

Non-Uniqueness of Unification

Unifier 1:p(X,Y){Xa,Yb,Vb}=p(a,b)p(a,V){Xa,Yb,Vb}=p(a,b)

Unifier 2:p(X,Y){Xa,Yf(W),Vf(W)}=p(a,f(W))p(a,V){Xa,Yf(W),Vf(W)}=p(a,f(W))

Unifier 3:p(X,Y){Xa,YV}=p(a,V)p(a,V){Xa,YV}=p(a,V)

22

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){Xa,YV}=p(a,V)p(a,V){Xa,YV}=p(a,V)

p(X,Y){Xa,VY}=p(a,Y)p(a,V){Xa,VY}=p(a,Y)

23

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))))))

24

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))))

25

Problem

~hates(X,X) hates(Y,f(Y))

26

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).

27


function mguchkp (p,q,al) {cond(p=q, al; varp(p), mguchkp(p,cdr(assoc(q,al)),al); atom(q), nil; t, some(lambda(x).mguchkp(p,x,al),q))}

28


function mguchkp (p,q,al) {if (p=q) al; else if varp(p) mguchkp(p,cdr(assoc(q,al)),al); else if atom(q) nil; else some(lambda(x).mguchkp(p,x,al),q)}

29

Example

Call: mgu(p(X,b),p(a,Y),{}) | | Call: mgu(p,p,{}) | Exit: {} | | Call: mgu(X,a,{}) | Exit: {Xa} | | Call: mgu(b,Y,{Xa}) | Exit: {Yb,Xa} | Exit: {Yb,Xa}

30

Example

Call: mgu(p(X,X),p(a,b),{}) | | Call: mgu(p,p,{}) | Exit: {} | | Call: mgu(X,a,{}) | Exit: {X <- a} | | Call: mgu(X,b, {X <- a}) | Call: mgu(a,b,{X <- a}) | Exit: nil | Exit: nil | Exit: nil

31

Example

Call: mgu(p(f(X),f(X)),p(Y,f(a)),{}) | | Call: mgu(p,p,{}) | Exit: {} | | Call: mgu(f(X),Y,{}) | Exit: {Y <- f(X)} | | Call: mgu(f(X),f(a),{Y <- f(X)}) | Call: mgu(f,f,{Y <- f(X)}) | Exit: {Y <- f(X)} | Call: mgu(X,a,{Y <- f(X)} | Exit: {X <- a,Y <- f(X)} | Exit: {X <- a,Y <- f(X)} | Exit: {X <- a,Y <- f(X)}

32

Example

Call: mgu(p(X,X),p(Y,f(Y)),{}) | | Call: mgu(p,p,{}) | Exit: {} | | Call: mgu(f(X),Y,{}) | Exit: {Y <- f(X)} | | Call: mgu(f(X),f(Y),{Y <- f(X)}) | Call: mgu(f,f,{Y <- f(X)}) | Exit: {Y <- f(X)} | Call: mgu(X,Y,{Y <- f(X)}) | Exit: nil | Exit: nil | Exit: nil

33

Evaluation

34

Reasoning Subroutines

Database: Call: theory Exit: {m(a,b), m(b,c), p(X,Y)<=m(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}

35

Evaluate




36

Evaluate (continued)



37

Deduction

38

Reasoning Subroutines

Database: Call: theory Exit: {m(a,b), m(b,c),p(X,Y)<=m(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}

39

Backward




40

Backward (continued)



42

Backward Chaining

Backward Chaining is the same as reduction except that it works on rule form rather than clausal form.

ϕ⇐ ϕ1 ∧...∧ϕm

ψ ∧ψ1 ∧...∧ψ n?

ϕ1 ∧...∧ϕm∧[ψ ]∧ψ1 ∧...∧ψ n?σ

where σ =mgu(ϕ,ψ )

Reduced literals need be retained only for non-Horn premises.

Cancellation and Dropping are analogous.

43

Example

1. m⇐ Premise

2. p⇐ m Premise

3. q⇐ m Premise

4. r⇐ p∧q Premise

5. r? Goal

6. p∧q? 4,5

7. m∧q? 2,6

8. q? 1,7

9. m? 3,8

10. ? 1,9

44

Example

Given q(x) p(x) and p(a), find a term such that q() is true.

1. q(x) p(x) Premise2. p(a) Premise3. goal(z) q(z) Goal4. goal(z) p(z) 1, 35. goal(a) 2, 4

45

Example

Given q(x) p(x) and p(a) and p(b), find a term such that q() is true.

1. q(x) p(x) Premise2. p(a) Premise3. p(b) Premise4. goal(z) q(z) Goal5. goal(z) p(z) 1, 46. goal(a) 2, 57. goal(b) 3, 5

46

Example

Given q(x) p(x) and p(a) p(b), find a term such that q() is true.

1. q(x) p(x) Premise2. p(a) p(b) Premise3. goal(z) q(z) Goal4. goal(z) p(z) 1, 35. goal(a) p(b) 2, 46. goal(a) goal(b) 4, 5

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