CS 4700: Artificial Intelligence

Transcription

1 CS 4700: Foundations of Artificial Intelligence Fall 2017 Instructor: Prof. Haym Hirsh Lecture 16

2 Cornell Cinema Thursday, April 13 7:00pm Friday, April 14 7:00pm Sunday, April 16 4:30pm

3 Cornell Cinema Thursday, April 13 7:00pm Co-Sponsored by CIS Friday, April 14 7:00pm Sunday, April 16 4:30pm

4 Today First Order Logic (R&N Ch 8-9) Tuesday, March 28 First Order Logic (R&N Ch 8-9) Machine Learning (R&N Ch 18)

5 Mother(Alice,Charlie) Father(Bob,Charlie) First-Order Logic: Examples

6 First-Order Logic: Examples Mother(Alice,Charlie) Father(Bob,Charlie) x,y Mother(x,y) Parent(x,y) x,y Father(x,y) Parent(x,y)

7 First-Order Logic: Examples Mother(Alice,Charlie) Father(Bob,Charlie) x,y Mother(x,y) Parent(x,y) x,y Father(x,y) Parent(x,y) x,y Parent(x,y) Ancestor(x,y) x,y,z Ancestor(x,y) Ancestor(y,z) Ancestor(x,z)

8 First-Order Logic: Examples Mother(Alice,Charlie) Father(Bob,Charlie) x,y Mother(x,y) Parent(x,y) x,y Father(x,y) Parent(x,y) x,y Parent(x,y) Ancestor(x,y) x,y,z Ancestor(x,y) Ancestor(y,z) Ancestor(x,z) x y Mother(y,x) x y Father(y,x)

9 First-Order Logic: Examples Mother(Alice,Charlie) Father(Bob,Charlie) x,y Mother(x,y) Parent(x,y) x,y Father(x,y) Parent(x,y) x,y Parent(x,y) Ancestor(x,y) x,y,z Ancestor(x,y) Ancestor(y,z) Ancestor(x,z) x y Mother(y,x) x y Father(y,x) x y Mother(x,y)

10 First Order Logic Sentence AtomicSentence ComplexSentence AtomicSentence Predicate Predicate(Arguments) Term = Term Arguments Term Term,Arguments Term Function(Arguments) Constant Variable ComplexSentence (Sentence) [Sentence] Sentence Sentence Sentence Sentence Sentence Sentence Sentence Quantifiers Sentence Quantifiers Quantifier Variables Quantifier Variables Quantifiers Quantifier Constant/Predicate/Function <strings starting with upper case letters> Variables Variable Variable,Variables Variable <strings comprised of lower case letters>

11 First Order Logic Sentence AtomicSentence ComplexSentence AtomicSentence Predicate Predicate(Arguments) Term = Term Arguments Term Term,Arguments Term Function(Arguments) Constant Variable ComplexSentence (Sentence) [Sentence] Sentence Sentence Sentence Sentence Sentence Sentence Sentence Quantifiers Sentence Quantifiers Quantifier Variables Quantifier Variables Quantifiers Quantifier Constant/Predicate/Function <strings starting with upper case letters> Variables Variable Variable,Variables Variable <strings comprised of lower case letters>

12 Inference in First-Order Logic: Modus Ponens 1. x Man(x) Mortal(x) 2. Man(Socrates)

13 Inference in First-Order Logic: Modus Ponens 1. x Man(x) Mortal(x) 2. Man(Socrates) Mortal(Socrates)

14 Inference in First-Order Logic: Modus Ponens 1. x Man(x) Mortal(x) 2. Man(Socrates) Mortal(Socrates) 1,2 [x/socrates]

15 Inference in First-Order Logic: Modus Ponens 1. x Man(x) Mortal(x) 2. Man(Socrates) Mortal(Socrates) 1,2 [x/socrates] 1. x,y Mother(x,y) Parent(x,y) 2. Mother(Alice,Charlie)

16 Inference in First-Order Logic: Modus Ponens 1. x Man(x) Mortal(x) 2. Man(Socrates) Mortal(Socrates) 1,2 [x/socrates] 1. x,y Mother(x,y) Parent(x,y) 2. Mother(Alice,Charlie) Parent(Alice,Charlie)

17 Inference in First-Order Logic: Modus Ponens 1. x Man(x) Mortal(x) 2. Man(Socrates) Mortal(Socrates) 1,2 [x/socrates] 1. x,y Mother(x,y) Parent(x,y) 2. Mother(Alice,Charlie) Parent(Alice,Charlie) 1,2 [x/alice, y/charlie]

18 Inference in First-Order Logic: Modus Ponens 1. x Man(x) Mortal(x) 2. Man(Socrates) Mortal(Socrates) 1,2 [x/socrates] 1. x,y Mother(x,y) Parent(x,y) Substitutions 2. Mother(Alice,Charlie) Parent(Alice,Charlie) 1,2 [x/alice, y/charlie]

19 Inference in First-Order Logic: Modus Ponens 1. x P(x,B) Q(x) 2. y P(A,y)

20 Inference in First-Order Logic: Modus Ponens 1. x P(x,B) Q(x) 2. y P(A,y) Q(A) 1,2 [x/a, y/b]

21 Inference in First-Order Logic: Modus Ponens 1. x,y,z Ancestor(x,y) Ancestor(y,z) Ancestor(x,z) 2. Ancestor(David,Ed) 3. Ancestor(Ed,Fran)

22 Inference in First-Order Logic: Modus Ponens 1. x,y,z Ancestor(x,y) Ancestor(y,z) Ancestor(x,z) 2. Ancestor(David,Ed) 3. Ancestor(Ed,Fran) Ancestor(David,Fran) 1,2,3 [x/david, y/ed, z/fran]

23 Inference in First-Order Logic: Modus Ponens 1. x,y Ancestor(x,y) Ancestor(x,Child(y)) 2. Ancestor(Alice,Bob)

24 Inference in First-Order Logic: Modus Ponens 1. x,y Ancestor(x,y) Ancestor(x,Child(y)) 2. Ancestor(Alice,Bob) 3. Ancestor(Alice,Child(Bob)) 1,2 [x/a, y/bob]

25 Inference in First-Order Logic: Modus Ponens 1. x,y Ancestor(x,y) Ancestor(x,Child(y)) 2. Ancestor(Alice,Bob) 3. Ancestor(Alice,Child(Bob)) 1,2 [x/a, y/bob] 4. Ancestor(Alice,Child(Child(Bob))) 1,3 [x/a, y/child(bob)]

26 Inference in First-Order Logic: Modus Ponens 1. x Relative(x,Bob) Happy(x) 2. y Relative(Child(y),y)

27 Inference in First-Order Logic: Modus Ponens 1. x Relative(x,Bob) Happy(x) 2. y Relative(Child(y),y) Happy(Child(Bob)) 1,2 [x/child(y), y/bob]

28 Inference in First-Order Logic: Modus Ponens 1. x Relative(x,Friend(x)) Happy(x) 2. y Relative(Child(y),y)

29 Inference in First-Order Logic: Modus Ponens 1. x Relative(x,Friend(x)) Happy(x) 2. y Relative(Child(y),y)???? 1,2 [x/child(y), y/friend(x)] CYCLE!

30 Unification Unify(Man(x),Man(Socrates)) = [x/socrates] Unify(Mother(x,y),Mother(Alice,Charlie)) = [x/alice, y/charlie] Unify(P(x,B),P(A,y)) = [x/a, y/b] Unify(Ancestor(x,y),Ancestor(Alice,Child(Bob))) = [x/alice, y/child(bob)] Unify(Relative(x,Bob),Relative(Child(y),y)) = [x/child(y), y/bob] Unify(Relative(x,Friend(x)),Relative(Child(y),y)) = ERROR Occurs Check

31 Unification Unify(x,y,answer): {wrong!} If answer= fail then Return( fail ) Else if x=y then Return(answer) Else if Variable(x) then Return(answer+x/y) Else if Variable(y) then Return(answer+y/x) Else if Head(x)=Head(y) then For i=1 to #args(head(x)) answer answer+unify(arg(x,i),arg(y,i)) Return(answer) Else Return(Fail)

32 Unification Unify(x,y,answer): If answer= fail then Return( fail ) Else if x=y then Return(answer) Else if Variable(x) then Return(UnifyVar(x,y,answer)) Else if Variable(y) then Return(UnifyVar(y,x,answer)) Else if Head(x)=Head(y) then For i=1 to #args(head(x)) answer answer+unify(arg(x,i),arg(y,i)) Return(answer) Else Return(Fail)

33 Unification UnifyVar(x,y,answer) If x/val answer then Return(Unify(val,y,answer)) If y/val answer then Return(Unify(x,val,answer)) Else if OccursCheck(x,y) then Return( fail ) Else Return(answer+x/y)

34 Inference in First Order Logic First-order Horn clauses are an analogous subset of first-order logic They have analogs of forward chaining and backward chaining Uses unification to match rules and facts Prolog is a programming language based on backward chaining on first-order Horn clauses There is a conjunctive normal form for First-Order Logic There is a resolution inference rule for First-Order Logic

35 CNF Replace α β with α β everywhere [no more ] Replace (α β) with α β [push inward until Replace (α β) with α β you can t any more] Replace α with α [ only before prop symbols] Distribute over : Rewrite α (β γ) as (α β) (α γ)

36 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Distribute over : Rewrite α (β γ) as (α β) (α γ)

37 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Distribute over : Rewrite α (β γ) as (α β) (α γ)

38 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Distribute over : Rewrite α (β γ) as (α β) (α γ)

39 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Distribute over : Rewrite α (β γ) as (α β) (α γ)

40 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Distribute over : Rewrite α (β γ) as (α β) (α γ)

41 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Distribute over : Rewrite α (β γ) as (α β) (α γ)

42 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Skolemization Distribute over : Rewrite α (β γ) as (α β) (α γ)

43 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Skolemization Distribute over : Rewrite α (β γ) as (α β) (α γ)

44 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Skolemization Drop universal quantifiers Distribute over : Rewrite α (β γ) as (α β) (α γ)

45 Skolemization x P(x) becomes P(A) where A is a constant that is new

46 Skolemization x P(x) becomes P(A) where A is a new constant x,y z P(z) becomes P(F(x,y)) where F is a new function

47 Skolemization x P(x) becomes P(A) where A is a new constant x,y z P(z) becomes P(F(x,y)) where F is a new function Replace existentially quantified variables with Skolem functions new functions whose arguments are all universally quantified variables of higher scope

48 Resolution in First Order Logic Recall: (l α 1 α k ) where each α i and β j are propositional ( l β 1 β n ) symbols or their negations and l is a (α 1 α k β 1 β n ) propositional symbol

49 Resolution in First Order Logic Here: (l α 1 α k ) [l and l can be made identical by ( l β 1 β n ) substituting variables with terms] Subst(ѳ, α 1 α k β 1 β n ) [the necessary substitution] where ѳ = Unify(l, l )