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Automated Reasoning

This lecture details the fundamental concepts of substitutions and unification in automated reasoning. A substitution is defined as a finite set of pairs of variables and their replacements. This lecture covers how to apply substitutions to expressions and introduces the notion of unifiability, which indicates whether two expressions can be made identical through substitutions. Additionally, it discusses the concept of most general unifiers (MGUs) and the non-uniqueness of unification, illustrated with various examples. The session provides essential insights for understanding logical expressions in computational contexts.

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Automated Reasoning

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  1. General Game Playing Lecture 3 Automated Reasoning 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){xa,yb,vb}=p(a,b) p(a,v){xa,yb,vb}=p(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v}=p(a,v) p(a,v){xa,yv}=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){xa,yv}=p(a,v) p(a,v){xa,yv}=p(a,v) p(x,y){xa,vy}=p(a,y) p(a,v){xa,vy}=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))

  13. 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))

  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))

  15. 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

  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. {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W}=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){Xa,Yb,Vb}=p(a,b) p(a,V){Xa,Yb,Vb}=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){Xa,Yb,Vb}=p(a,b) p(a,V){Xa,Yb,Vb}=p(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V}=p(a,V) p(a,V){Xa,YV}=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){Xa,YV}=p(a,V) p(a,V){Xa,YV}=p(a,V) p(X,Y){Xa,VY}=p(a,Y) p(a,V){Xa,VY}=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. Most General Unification (concluded) 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. Most General Unification (concluded) 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: {Xa} | | Call: mgu(b,Y,{Xa}) | Exit: {Yb,Xa} | Exit: {Yb,Xa}

  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 (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*)))))

  36. 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))))

  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 (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*)))))

  40. 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))))

  41. 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.

  42. Example

  43. 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

  44. 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

  45. 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

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