CSC242: Intro to AI. Lecture 13. Thursday, February 28, 13

Transcription

1 CSC242: Intro to AI Lecture 13

2

3 Recap

4 Rooms adjacent to pits will have breezes Socrates is a person All people are mortal Anybody s grandmother is either their mother s or their father s mother

5 Elements of First- Objects (in the world) Relations among (tuples of) objects Mappings from (tuples of) objects to objects (i.e., objects identified in terms of other objects) Order Logic Connectives Variables and Quantifiers

6 Syntax of First-Order Logic Constants Functional expressions } Terms Atomic sentences Connectives Variables and Quantifiers

7 Atomic Sentences Predicate symbol + list of terms (arguments) Brother(R, J) Married(Father(Richard), Mother(John))

8 Syntax of First-Order Logic Constants Functional expressions } Terms Atomic sentences Connectives Variables and Quantifiers

9 Complex Sentences Connectives combine sentences Brother(LeftLeg(R), J) Brother(R, J) Brother(J, R) King(R) King(J)

10 Syntax of First-Order Logic Constants Functional expressions } Terms Atomic sentences Connectives Variables and Quantifiers

11 Quantifiers If α is a sentence, then so are x α x α

12 Syntax of First-Order Logic Constants Functional expressions } Terms Atomic sentences Connectives Variables and Quantifiers

13 Semantics of First- Order Logic

14 Possible Worlds for PL Assignment of truth values to propositional variables

15 Semantics of PL B1,1 P1,2 P2,1 P1,2 P2,1 B1,1 (P1,2 P2,1) true true true true true false true true true false true false true true true false false true true false true true false true true false true false true false true false false false false false false false false true

16 Possible Worlds for FOL Domain: Non-empty set of objects in the world Relations: Sets of tuples of objects Functions: Total mappings from tuples of objects to objects

17 Interpretation Mapping from elements a sentence to elements of a possible world Specifies exactly which objects, relations, and functions are referred to by the constant, predicate, and function symbols

18 Interpretation Domain of objects D Mapping from constant symbols to objects: I(c) 2 D Mapping from predicate symbols to relations: I(p) D n for p of arity n Mapping from function symbols to (total) functions: I(f) :D n! D for f of arity n

19 Semantics of Unquantified Sentences P (hx 1,...,x n i) is true in interpetation I if hi(x 1 ),...,I(x n )i2i(p ) Connectives: use truth tables (as for PL)

20 Semantics of Quantified Sentences

21 Extended Interpretation Adds mapping from variables to objects: I(v) 2 D

22 Semantics of Quantified Sentences 8x is true in interpetation I if I 0 ( ) is true in every extended interpetation I 9x is true in interpetation I if I 0 ( ) is true in some extended interpetation I

23 Semantics of FOL Atomic sentences: Use interpretation Complex sentences: Use semantics of connectives 8x : True in every extended interpretation 9x : True in at least one extended interp.

24 Entailment If α entails β: = Whenever α is true, so is β Every model of α is a model of β Models(α) Models(β) α is at least as strong an assertion as β Rules out no fewer possible worlds

25 crown person brother brother on head person king R $ J left leg left leg 137,506,194,466 models with 6 objects!

26 Computing Entailment Number of models (probably) unbounded And anyway hard to evaluate truth in a model Can t do model checking Look for inference rules, do theorem proving

27 First-Order Inference

28 x King(x) Evil(x) King(J)

29 x King(x) Evil(x) King(J) Modus Ponens Evil(J)

30 Universal Instantiation 8xP(x) P (a),p(b),p(c),p(f(a)),p(f(b)),p(g(a)),...

31 Propositionalization x King(x) Evil(x) King(R) Evil(R) King(J) Evil(J) King(Father(R)) Evil(Father(R))...

32 Propositionalization Convert FOL sentences to PL sentences and do PL inference on them Seems kind of indirect Technical problem enumerating all the ground terms (could be infinite; see book)

33 x King(x) Evil(x) King(J)

34 x King(x) Evil(x) King(J)

35 King(J) Evil(J) King(J)

36 King(J) Evil(J) King(J) Evil(J)

37 x King(x) Greedy(x) Evil(x) King(J) Greedy(J), Greedy(R)

38 x King(x) Greedy(x) Evil(x) King(J) Greedy(J), Greedy(R) Evil(J)

39 x King(x) Greedy(x) Evil(x) King(J) Greedy(J), Greedy(R) Evil(J)

40 x King(x) Greedy(x) Evil(x) King(J) Greedy(J), Greedy(R) Evil(J)

41 x King(x) Greedy(x) Evil(x) King(J) Greedy(J), Greedy(R) Evil(J)

42 Substitution Replacement (binding) of variables by terms { x/j } { x/r, y/father(j) }

43 x King(x) Greedy(x) Evil(x) King(J) { x/j } Greedy(J), Greedy(R) If there is some substitution that makes the premises true, then the conclusion with the same substitution is also true

44 x King(x) Greedy(x) Evil(x) King(J) y Greedy(y) { x/j, y/j }

45 Generalized Modus Ponens p 0 1,p 0 2,,p 0 n, (p 1 p 2 p n q) Subst(,q) p i,p 0 i,q are atomic sentences is a substitution such that: Subst(,p 0 i)=subst(,p i )

46 Lifted Inference Rule Inference rule lifted from ground (variablefree) propositional logic to first-order logic The key advantage of lifted inference rules over propositionalization is that they make only those substitutions that are required to allow particular inferences to succeed

47 Inference Rules ( ) ( ) And-elimination Double negation DeMorgan s Laws ),, ( ) ) ( ) ) ( ) ) ( ) ), Modus Ponens Definition of biconditional

48 Unit Resolution Clause l 1 l i l k, m l 1 l i 1 l i+1 l k Unit Clause l1,..., lk are literals li and m are complementary Positive literal: P Negative literal: P Complementary literals

49 Resolution l 1 l i l k, m 1 m j m n l 1 l i 1 l i+1 l k m 1 m j 1 m j+1 m n l1,..., lk, m1,..., mn are literals li and mj are complementary Technical note: Resulting clause must be factored to contain only one copy of each literal.

50 Resolution Sound Complete Can it be lifted to FOL?

51 CNF for FOL Every sentence of first-order logic can be converted into an inferentially equivalent sentence in conjunctive normal form (CNF)

52 CNF for FOL Eliminate implications Move negation inwards Standardize variables apart Skolemize existential variables Drop universal variables Distribute over

53 CNF for FOL Every sentence of first-order logic can be converted into an inferentially equivalent sentence in conjunctive normal form (CNF) Not logically equivalent Inferentially equivalent: (un)satisfiable iff original sentence is (un)satisfiable

54 Resolution l 1 l i l k, m 1 m j m n l 1 l i 1 l i+1 l k m 1 m j 1 m j+1 m n l1,..., lk, m1,..., mn are literals li and mj are complementary Technical note: Resulting clause must be factored to contain only one copy of each literal.

55 x King(x) Greedy(x) Evil(x) King(J) y Greedy(y) { x/j, y/j }

56 Unification Unify(p, q) = where Subst(,p)=Subst(,q)

57 Knows(John,x) and Knows(John,Jane) { x/jane } Knows(John,x) and Knows(y, Bill) {x/bill, y/john } Knows(John,x) and Knows(y, Mother(y)) { y/john, x/mother(john) } Knows(John,x) and Knows(x,Elizabeth) Fails

58 Unification Unify(p, q) = where Subst(,p)=Subst(,q) Variables need to be standardized apart Occurs check Most general unifier: places the fewest restrictions on the variables Is unique AIMA Fig 9.1

59 FOL Resolution l 1 l i l k, m 1 m j m n Subst(,l 1 l i 1 l i+1 l k m 1 m j 1 m j+1 m n ) = Unify(l i, m j )

60 Proof by Resolution Convert KB to CNF Convert α to CNF and add to KB Apply resolution rule to complementary clauses (with unification) Until you derive the empty clause

61 American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x) Criminal(West) American(West) American(West) Weapon(y) Sells(West,y,z) Hostile(z) Missile(x) Weapon(x) Weapon(y) Sells(West,y,z) Hostile(z) Missile(M 1 ) Missile(y) Sells(West,y,z) Hostile(z) Missile(x) Owns(Nono,x) Sells(West,x,Nono) Sells(West,M 1,z) Hostile(z) Missile(M 1 ) Missile(M 1 ) Owns(Nono,M 1 ) Hostile(Nono) Owns(Nono,M 1 ) Enemy(x,America) Hostile(x) Owns(Nono,M 1 ) Hostile(Nono) Hostile(Nono) Enemy(Nono, America) Enemy(Nono, America)

62 The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all its missiles were sold to it by Colonel West, who is American. To prove: West is a criminal

63 It is a crime for an American to sell weapons to hostile nations. 8x, y, z American(x) Weapon(y) Sells(x, y, z) Hostile(z) ) Criminal(x) Nono... has some missiles. 9x Owns(Nono,x) Missile(x) All Nono s missiles were sold to it by Colonel West. 8x Missile(x) Owns(Nono,x) ) Sells(West,x,Nono) Missiles are weapons. 8x Missile(x) ) Weapon(x) An enemy of America counts as hostile. 8x Enemy(x, America) ) Hostile(x) West is an American. American(West) Nono is an enemy of America. Enemy(Nono, America)

64 8x, y, z American(x) Weapon(y) Sells(x, y, z) Hostile(z) ) Criminal(x) 9x Owns(Nono,x) Missile(x) 8x Missile(x) Owns(Nono,x) ) Sells(West,x,Nono) 8x Missile(x) ) Weapon(x) 8x Enemy(x, America) ) Hostile(x) American(West) Enemy(Nono, America)

65 Proof by Resolution Convert KB to CNF Convert α to CNF and add to KB Apply resolution rule to complementary clauses (with unification) Until you derive the empty clause

66 Convert to CNF Eliminate implications Move negation inwards Standardize variables apart Skolemize existential variables Drop universal variables Distribute over

67 8x, y, z American(x) Weapon(y) Sells(x, y, z) Hostile(z) ) Criminal(x) Eliminate implications: x, y, z [American(x) Weapon(y) Sells(x, y, z) Hostile(z)] Criminal(x) Move negation inwards (DeMorgan s Laws) x, y, z [ American(x) Weapon(y) Sells(x, y, z) Hostile(z)] Criminal(x) Standardize variables apart Skolemize existential variables Drop universal quantifiers American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Distribute over

68 8x, y, z American(x) Weapon(y) Sells(x, y, z) Hostile(z) ) Criminal(x) American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x)

69 8x, y, z American(x) Weapon(y) Sells(x, y, z) Hostile(z) ) Criminal(x) American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) 9x Owns(Nono,x) Missile(x) Owns(Nono,M 1 ), Missile(M 1 ) 8x Missile(x) Owns(Nono,x) ) Sells(West,x,Nono) Missile(x) Owns(Nono,x) Sells(West,x,Nono) 8x Missile(x) ) Weapon(x) Missile(x) Weapon(x) 8x Enemy(x, America) ) Hostile(x) Enemy(x, America) American(West) American(West) Hostile(x) Enemy(Nono, America) Enemy(Nono, America)

70 American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ) Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missile(x) Weapon(x) Enemy(x, America) Hostile(x) American(West) Enemy(Nono, America)

71 Proof by Resolution Convert KB to CNF Convert α to CNF and add to KB Apply resolution rule to complementary clauses (with unification) Until you derive the empty clause

72 West is a criminal Criminal(West) Negate and convert to CNF Criminal(West)

73 American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ) Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missile(x) Weapon(x) Enemy(x, America) Hostile(x) American(West) Enemy(Nono, America) Criminal(West)

74 Proof by Resolution Convert KB to CNF Convert α to CNF and add to KB Apply resolution rule to complementary clauses (with unification) Until you derive the empty clause

75 American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ) Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missile(x) Weapon(x) Enemy(x, America) Hostile(x) American(West) Enemy(Nono, America) Criminal(West)

76 American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ) Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missile(x) Weapon(x) Enemy(x, America) Hostile(x) American(West) { x/west } Enemy(Nono, America) Criminal(West)

77 American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Criminal(West) { x/west } American(West) Weapon(y) Sells(West,y,z) Hostile(z)

78 American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x) Criminal(West) American(West) American(West) Weapon(y) Sells(West,y,z) Hostile(z) Missile(x) Weapon(x) Weapon(y) Sells(West,y,z) Hostile(z) Missile(M 1 ) Missile(y) Sells(West,y,z) Hostile(z) Missile(x) Owns(Nono,x) Sells(West,x,Nono) Sells(West,M 1,z) Hostile(z) Missile(M 1 ) Missile(M 1 ) Owns(Nono,M 1 ) Hostile(Nono) Owns(Nono,M 1 ) Enemy(x,America) Hostile(x) Owns(Nono,M 1 ) Hostile(Nono) Hostile(Nono) Enemy(Nono, America) Enemy(Nono, America)

79 American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x) Criminal(West) American(West) American(West) Weapon(y) Sells(West,y,z) Hostile(z) Missile(x) Weapon(x) Weapon(y) Sells(West,y,z) Hostile(z) Missile(M 1 ) Missile(y) Sells(West,y,z) Hostile(z) Missile(x) Owns(Nono,x) Sells(West,x,Nono) Sells(West,M 1,z) Hostile(z) Missile(M 1 ) Missile(M 1 ) Owns(Nono,M 1 ) Hostile(Nono) Owns(Nono,M 1 ) Enemy(x,America) Hostile(x) Owns(Nono,M 1 ) Hostile(Nono) Hostile(Nono) Enemy(Nono, America) Enemy(Nono, America)

80 Resolution Proof Since KB { Criminal(West)} is unsatisfiable, KB = Criminal(West)

81 Forward Chaining Knowledge base of definite clauses Starting from known facts, trigger rules whose premises are satisfied Using substitution to match Add their conclusions to the KB Until the query is answered or no new facts can be generated

82 Backward Chaining Work backward from the goal, chaining through rules to find known facts that support the proof Allow substitutions when matching facts and rules DFS => incomplete The basis of logic programming (Prolog)

83 FOL Inference Semantics of first-order sentences Entailment: same as PL! Lifted inference rules Resolution (first-order CNF, unification) Proof by contradiction Forward and backward chaining

84 For Next Time: Review for Midterm AIMA Chapters 1-9 (per syllabus) Exam issues: me ASAP