Exercises: Artificial Intelligence. Automated Reasoning: Good to walk

Transcription

1 Exercises: Artificial Intelligence Automated Reasoning: Good to walk

2 Automated Reasoning: Resolution INTRODUCTION

3 Introduction ProveQfromT TranslateQandTtologic LetT bet({ÿq} Transform all formula in T to implicative normal form: A 1 / /A n B 1 - -B m false B 1 - -B m A 1 / /A n true Derive contradiction: false true

4 Introduction Resolution propositional logic: Given two formula: /p/ - - / / -p- Derive new formula: ( /p/ )/( / / ) ( - - )-( -p- )

5 Introduction Dealing with variables: Substitution is assignment of values to variables: E.g.replacexbyA: q={x/a} (p(x,y))q=p(a,y) GivenatomsA,A Most general unifying substitution(mgu) is SubstitutionqsuchthatAq=A q q does not replace variables unless necessary Example: A=p(x,y), A =p(a,z) ï mgu(a,a )={x/a,y/z}

6 Introduction Resolution predicate logic: Given two formula: /A/ - - / / -A - Suchthatthereexistsanmgu(A,A )=q Derive new formula: (( /A/ )/( / / ) ( - - )-( -A - ))q

7 Introduction Another inference rule: Factoring Givenformula:( /A/A / - - ) Suchthatthereexistsanmgu(A,A )=q Derive new formula: ( /A/A / - - )q Givenformula:( / / -A-A - ) Suchthatthereexistsanmgu(A,A )=q Derive new formula: ( / / -A-A - )q

8 Automated Reasoning: Good to walk PROBLEM

9 Problem Convert to logic: Itisraining,itissnowingoritisdry. Itiswarm. Itisnotraining. Itisnotsnowing. Iftheweatherisnice,thenitisgoodtowalk. Iftheweatherisdryandwarm, theweatherisnice.

10 Automated Reasoning: Good to walk SOLUTION

11 Itisraining,itissnowingoritisdry. raining/snowing/dry( true) Itiswarm. Itisnotraining. Itisnotsnowing. Iftheweatherisnice,thenitisgoodtowalk. If the weather is dry and warm, the weather is nice.

12 raining/snowing/dry( true) Itiswarm. warm( true) Itisnotraining. Itisnotsnowing. Iftheweatherisnice,thenitisgoodtowalk. If the weather is dry and warm, the weather is nice.

13 raining/snowing/dry( true) warm( true) Itisnotraining. false raining OR Ÿraining Itisnotsnowing. Iftheweatherisnice,thenitisgoodtowalk. If the weather is dry and warm, the weather is nice.

14 raining/snowing/dry( true) warm( true) false raining OR Ÿraining Itisnotsnowing. false snowing OR Ÿsnowing Iftheweatherisnice,thenitisgoodtowalk. If the weather is dry and warm, the weather is nice.

15 raining/snowing/dry( true) warm( true) false raining OR Ÿraining false snowing snowing OR Ÿsnowing Iftheweatherisnice,thenitisgoodtowalk. walk nice If the weather is dry and warm, the weather is nice.

16 raining/snowing/dry( true) warm( true) false raining OR Ÿraining false snowing snowing OR Ÿsnowing walk nice If the weather is dry and warm, the weather is nice. nice dry-warm

17 raining/snowing/dry( true) warm( true) false raining OR Ÿraining false snowing OR Ÿsnowing walk nice nice dry-warm

18 Convert sentences to implicative normal form: raining/snowing/dry( true) warm( true) false raining false snowing walk nice nice dry-warm

19 Automated Reasoning: Good to walk PROBLEM

20 Problem Provebyresolution: Itisgoodtowalk

21 Automated Reasoning: Good to walk SOLUTION

22 Provebyresolution: Itisgoodtowalk Weassumethatitisnotgoodtowalk: false walk

23 Weassumethatitisnotgoodtowalk: Given: false walk raining/snowing/dry( true) warm( true) false raining false snowing walk nice nice dry-warm

24 false walk

25 walk nice false walk false nice

26 walk nice nice dry -warm false walk false nice false dry -warm

27 walk nice nice dry -warm warm true false walk false nice false dry -warm false dry

28 walk nice nice dry -warm warm true raining /snowing /dry true false walk false nice false dry -warm false dry raining /snowing true

29 walk nice nice dry -warm warm true raining /snowing /dry true false raining false walk false nice false dry -warm false dry raining /snowing true snowing true

30 walk nice nice dry -warm warm true raining /snowing /dry true false raining false snowing false walk false nice false dry -warm false dry raining /snowing true snowing true false true

31 Provebyresolution: Itisgoodtowalk Weassumethatitisnotgoodtowalk: false walk This leads to a contradiction: false true Thus, Itisgoodtowalk

32 Exercises: Artificial Intelligence Automated Reasoning: MGU

33 Automated Reasoning: MGU INTRODUCTION: UNIFICATION

34 Procedure Unify(a,b): mgu:={a=b};stop:=false; WHILE(not(stop) AND mgu contains s=t) Case1:tisavariable,sisnotavariable: Replaces=tbyt=sinmgu Case2:sisavariable,tistheSAMEvariable: Deletes=tfrommgu mgu Case3:sisavariable,tisnotavariableandcontainss: stop:=true Case4:sisavariable,tisnotidenticaltonorcontainss: Replacealloccurrencesofsinmgubyt Case5:sisoftheformf(s 1,,s n ),tofg(t 1,,t m ): Iffnotequaltogormnotequaltonthenstop:=true Elsereplaces=tinmgubys 1 =t 1,,s n =t n

35 Automated Reasoning: MGU PROBLEM

36 Problem Whatisthem.g.u.of: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) p(a,x,f(g(y)))=p(z,f(z),f(a)) q(x,x)=q(y,f(y)) = q(y,f(y)) f(x,g(f(a),u))=f(g(u,v),x)

37 Automated Reasoning: MGU SOLUTION

38 What is the m.g.u. of: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Init: p(f(y),w,g(z,y)) = p(x,x,g(z,a))

39 What is the m.g.u. of: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Init: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Case5:f(y)=x,w=x,g(z,y)=g(z,A)

40 What is the m.g.u. of: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Init: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Case5:f(y)=x,w=x,g(z,y)=g(z,A) Case1:x=f(y),w=x,g(z,y)=g(z,A)

41 What is the m.g.u. of: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Init: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Case5:f(y)=x,w=x,g(z,y)=g(z,A) Case1:x=f(y),w=x,g(z,y)=g(z,A) Case4:x=f(y),w=f(y),g(z,y)=g(z,A)

42 What is the m.g.u. of: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Init: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Case5:f(y)=x,w=x,g(z,y)=g(z,A) Case1:x=f(y),w=x,g(z,y)=g(z,A) Case4:x=f(y),w=f(y),g(z,y)=g(z,A) Case5:x=f(y),w=f(y),z=z,y=A

43 What is the m.g.u. of: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Init: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Case5:f(y)=x,w=x,g(z,y)=g(z,A) Case1:x=f(y),w=x,g(z,y)=g(z,A) Case4:x=f(y),w=f(y),g(z,y)=g(z,A) Case5:x=f(y),w=f(y),z=z,y=A Case2:x=f(y),w=f(y),y=A

44 What is the m.g.u. of: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Init: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) Case5:f(y)=x,w=x,g(z,y)=g(z,A) Case1:x=f(y),w=x,g(z,y)=g(z,A) Case4:x=f(y),w=f(y),g(z,y)=g(z,A) Case5:x=f(y),w=f(y),z=z,y=A Case2:x=f(y),w=f(y),y=A Case4:x=f(A),w=f(A),y=A

45 What is the m.g.u. of: p(f(y),w,g(z,y)) = p(x,x,g(z,a)) MGU: x/f(a),w/f(a),y/a Result: p(f(a),f(a),g(z,a))

46 Automated Reasoning: MGU SOLUTION

47 What is the m.g.u. of: p(a,x,f(g(y))) = p(z,f(z),f(a)) Init:p(A,x,f(g(y)))=p(z,f(z),f(A))

48 What is the m.g.u. of: p(a,x,f(g(y))) = p(z,f(z),f(a)) Init:p(A,x,f(g(y)))=p(z,f(z),f(A)) Case5:A=z,x=f(z),f(g(y))=f(A)

49 What is the m.g.u. of: p(a,x,f(g(y))) = p(z,f(z),f(a)) Init:p(A,x,f(g(y)))=p(z,f(z),f(A)) Case5:A=z,x=f(z),f(g(y))=f(A) Case1:z=A,x=f(z),f(g(y))=f(A)

50 What is the m.g.u. of: p(a,x,f(g(y))) = p(z,f(z),f(a)) Init:p(A,x,f(g(y)))=p(z,f(z),f(A)) Case5:A=z,x=f(z),f(g(y))=f(A) Case1:z=A,x=f(z),f(g(y))=f(A) Case4:z=A,x=f(A),f(g(y))=f(A)

51 What is the m.g.u. of: p(a,x,f(g(y))) = p(z,f(z),f(a)) Init:p(A,x,f(g(y)))=p(z,f(z),f(A)) Case5:A=z,x=f(z),f(g(y))=f(A) Case1:z=A,x=f(z),f(g(y))=f(A) Case4:z=A,x=f(A),f(g(y))=f(A) Case5:z=A,x=f(A),g(y)=A

52 What is the m.g.u. of: p(a,x,f(g(y))) = p(z,f(z),f(a)) Init:p(A,x,f(g(y)))=p(z,f(z),f(A)) Case5:A=z,x=f(z),f(g(y))=f(A) Case1:z=A,x=f(z),f(g(y))=f(A) Case4:z=A,x=f(A),f(g(y))=f(A) Case5:z=A,x=f(A),g(y)=A Case5:stop:=true

53 Automated Reasoning: MGU SOLUTION

54 Whatisthem.g.u.of:q(x,x)=q(y,f(y)) Init: q(x,x) = q(y,f(y))

55 Whatisthem.g.u.of:q(x,x)=q(y,f(y)) Init: q(x,x) = q(y,f(y)) Case5:x=y,x=f(y)

56 Whatisthem.g.u.of:q(x,x)=q(y,f(y)) Init: q(x,x) = q(y,f(y)) Case5:x=y,x=f(y) Case4:x=y,y=f(y)

57 Whatisthem.g.u.of:q(x,x)=q(y,f(y)) Init: q(x,x) = q(y,f(y)) Case5:x=y,x=f(y) Case4:x=y,y=f(y) Case3:stop:=true

58 Automated Reasoning: MGU SOLUTION

59 What is the m.g.u. of: f(x,g(f(a),u)) = f(g(u,v),x) Init: f(x,g(f(a),u)) = f(g(u,v),x)

60 What is the m.g.u. of: f(x,g(f(a),u)) = f(g(u,v),x) Init: f(x,g(f(a),u)) = f(g(u,v),x) Case5:x=g(u,v),g(f(a),u)=x

61 What is the m.g.u. of: f(x,g(f(a),u)) = f(g(u,v),x) Init: f(x,g(f(a),u)) = f(g(u,v),x) Case5:x=g(u,v),g(f(a),u)=x Case4:x=g(u,v),g(f(a),u)=g(u,v)

62 What is the m.g.u. of: f(x,g(f(a),u)) = f(g(u,v),x) Init: f(x,g(f(a),u)) = f(g(u,v),x) Case5:x=g(u,v),g(f(a),u)=x Case4:x=g(u,v),g(f(a),u)=g(u,v) Case5:x=g(u,v),f(a)=u,u=v

63 What is the m.g.u. of: f(x,g(f(a),u)) = f(g(u,v),x) Init: f(x,g(f(a),u)) = f(g(u,v),x) Case5:x=g(u,v),g(f(a),u)=x Case4:x=g(u,v),g(f(a),u)=g(u,v) Case5:x=g(u,v),f(a)=u,u=v Case1:x=g(u,v),u=f(a),u=v

64 What is the m.g.u. of: f(x,g(f(a),u)) = f(g(u,v),x) Init: f(x,g(f(a),u)) = f(g(u,v),x) Case5:x=g(u,v),g(f(a),u)=x Case4:x=g(u,v),g(f(a),u)=g(u,v) Case5:x=g(u,v),f(a)=u,u=v Case1:x=g(u,v),u=f(a),u=v Case4:x=g(f(a),v),u=f(a),f(a)=v

65 What is the m.g.u. of: f(x,g(f(a),u)) = f(g(u,v),x) Init: f(x,g(f(a),u)) = f(g(u,v),x) Case5:x=g(u,v),g(f(a),u)=x Case4:x=g(u,v),g(f(a),u)=g(u,v) Case5:x=g(u,v),f(a)=u,u=v Case1:x=g(u,v),u=f(a),u=v Case4:x=g(f(a),v),u=f(a),f(a)=v Case1:x=g(f(a),v),u=f(a),v=f(a)

66 What is the m.g.u. of: f(x,g(f(a),u)) = f(g(u,v),x) Init: f(x,g(f(a),u)) = f(g(u,v),x) Case5:x=g(u,v),g(f(a),u)=x Case4:x=g(u,v),g(f(a),u)=g(u,v) Case5:x=g(u,v),f(a)=u,u=v Case1:x=g(u,v),u=f(a),u=v Case4:x=g(f(a),v),u=f(a),f(a)=v Case1:x=g(f(a),v),u=f(a),v=f(a) Case4:x=g(f(a),f(a)),u=f(a),v=f(a)

67 What is the m.g.u. of: f(x,g(f(a),u)) = f(g(u,v),x) MGU: x/g(f(a),f(a)),u/f(a),v/f(a) Result: f(g(f(a),f(a)),g(f(a),f(a)))

68 Exercises: Artificial Intelligence Automated Reasoning: Resolution

69 Automated Reasoning: Resolution PROBLEM

70 Problem Isthereanyonewhoisamother-in-lawofPeter? mother-in-law(x,y) mother(x,z)- married(z,y) mother(x,y) female(x)- parent(x,y) female(an)( true) parent(an,maria)( true) married(maria,peter)( true)

71 Automated Reasoning: Resolution SOLUTION

72 Assumption: Peter has no mother-in-law false mother-in-law(x,peter) Given: mother-in-law(x,y) mother(x,z)- married(z,y) mother(x,y) female(x)- parent(x,y) female(an)( true) parent(an,maria)( true) married(maria,peter)( true)

73 false mother-in-law(x,peter)

74 false mother-in-law(x,peter) mother-in-law(x,y ) mother(x,z )-married(z,y ) {x /x,y /Peter} false mother(x,z )- married(z,peter)

75 false mother-in-law(x,peter) false mother(x,z )- married(z,peter) mother(x,y ) female(x )-parent(x,y ) {x /x,y /z } false female(x)- parent(x,z )- married(z,peter)

76 false mother-in-law(x,peter) false mother(x,z )- married(z,peter) false female(x)- parent(x,z )- married(z,peter) female(an) {x/an} false parent(an,z )- married(z,peter)

77 false mother-in-law(x,peter) false mother(x,z )- married(z,peter) false female(x)- parent(x,z )- married(z,peter) false parent(an,z )- married(z,peter) parent(an,maria) {z /Maria} false married(maria,peter)

78 false mother-in-law(x,peter) false mother(x,z )- married(z,peter) {x/an} false female(x)- parent(x,z )- married(z,peter) false parent(an,z )- married(z,peter) false married(maria,peter) married(maria,peter) false true( )

79 Automated Reasoning: Resolution PROBLEM

80 Problem Is there a valid colouring of a map of Belgium, the Netherlands, and Germany? color(red)( true) color(green)( true) color(blue)( true) neighbour(x,y) color(x), color(y), diff(x,y) diff/2 succeeds when arguments cannot be unified

81 Automated Reasoning: Resolution SOLUTION

82 Assumption: There is no valid colouring false nb(b,g),nb(g,n),nb(n,b) Given: c(r)( true) c(g)( true) c(b)( true) nb(x,y) c(x), c(y), diff(x,y) diff/2 succeeds when arguments cannot be unified

83 false nb(b,g)-nb(g,n)-nb(n,b)

84 false nb(b,g)-nb(g,n)-nb(n,b) nb(x,y ) c(x ),c(y ),diff(x,y ) {x /b,y /g} false c(b)-c(g)-diff(b,g)-nb(g,n)-nb(n,b)

85 false nb(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-nb(g,n)-nb(n,b) nb(x,y ) c(x ),c(y ),diff(x,y ) {x /g,y /n} false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-nb(n,b)

86 false nb(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-nb(n,b) nb(x,y ) c(x ),c(y ),diff(x,y ) {x /n,y /b} false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-diff(n,b)

87 false nb(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-diff(n,b) c(r) {b/r} false c(g)-diff(r,g)-c(n)-diff(g,n)-diff(n,r)

88 false nb(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-diff(n,b) false c(g)-diff(r,g)-c(n)-diff(g,n)-diff(n,r) c(g) {g/g} false diff(r,g)- c(n)- diff(g,n)- diff(n,r)

89 false nb(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-diff(n,b) false c(g)-diff(r,g)-c(n)-diff(g,n)-diff(n,r) false diff(r,g)- c(n)- diff(g,n)- diff(n,r) c(b) {n/b} false diff(r,g)- diff(g,b)- diff(b,r)

90 false nb(b,g)-nb(g,n)-nb(n,b) {b/r,g/g,n/b} false c(b)-c(g)-diff(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-diff(n,b) false c(g)-diff(r,g)-c(n)-diff(g,n)-diff(n,r) false diff(r,g)- c(n)- diff(g,n)- diff(n,r) false diff(r,g)- diff(g,b)- diff(b,r) Built-in diff/2: succeeds for different arguments false true( )

91 Alternative solution false nb(b,g)-nb(g,n)-nb(n,b) {b/b,g/g,n/r} false c(b)-c(g)-diff(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-diff(n,b) false c(g)-diff(b,g)-c(n)-diff(g,n)-diff(n,b) false diff(b,g)- c(n)- diff(g,n)- diff(n,b) false diff(b,g)- diff(g,r)- diff(r,b) Built-in diff/2: succeeds for different arguments false true( )

92 Or consistency = Continue search false nb(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-nb(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-nb(n,b) false c(b)-c(g)-diff(b,g)-c(n)-diff(g,n)-diff(n,b) false c(g)-diff(r,g)-c(n)-diff(g,n)-diff(n,r) false diff(r,r)- c(n)- diff(r,n)- diff(n,r) false diff(r,r)- diff(r,b)- diff(b,r) diff(r,r) is false false false

93 Exercises: Artificial Intelligence Automated Reasoning: Predicate Resolution

94 Automated Reasoning: Predicate Resolution PROBLEM

95 Problem Resolution in predicate logic: Given, the formula in first order predicate logic: "xp(x)/ÿr(f(x)) "x"yr(f(x))/r(f(f(y))) Here,xandyarevariables. Give an explicit resolution proof(graphical) for: "x$yp(f(x))-r(y)entailedbythegivenformula

96 Automated Reasoning: Predicate Resolution SOLUTION

97 Formula in implicative normal form: "xp(x)/ÿr(f(x)) p(x) r(f(x)) "x"yr(f(x))/r(f(f(y))) r(f(x))/r(f(f(y)))( true) Assumption Ÿ["x$yp(f(x))-r(y)]ñ$x"yŸ[p(f(x))-r(y)]ñ "yÿ[p(f(a))-r(y)]ñfalse p(f(a))-r(y)

98 false p(f(a))-r(y) Solution

99 false p(f(a))-r(y) p(x ) r(f(x )) {x /f(a)} false r(f(f(a)))-r(y)

100 false p(f(a))-r(y) false r(f(f(a)))-r(y) Factoring:mgu(r(f(f(A)))=r(y))={y/f(f(A))} false r(f(f(a)))-r(f(f(a)))

101 false p(f(a))-r(y) false r(f(f(a)))-r(y) false r(f(f(a)))-r(f(f(a))) r(f(x ))/r(f(f(y )))( true) Factoring:mgu(r(f(x ))=r(f(f(y ))))={x /f(y )} r(f(f(y )))( true) {y /A} false true( ) {y/f(f(a))}