Introduction to Artificial Intelligence 2 nd semester 2016/2017. Chapter 9: Inference in First-Order Logic

Transcription

1 Introduction to Artificial Intelligence 2 nd semester 2016/2017 Chapter 9: Inference in First-Order Logic Mohamed B. Abubaker Palestine Technical College Deir El-Balah 1

2 Outlines Reducing first-order inference to propositional inference Unification Generalized Modus Ponens Forward chaining Backward chaining Resolution 2

3 Inference in First Order Logic The idea is that first-order inference can be done by converting the knowledge base to propositional logic and using propositional inference. This technique is known as propositionalization. Given: x King(x) Greedy(x) Evil(x) One can infer: King(John) Greedy(John) Evil(John) King(Richard) Greedy(Richard) Evil(Richard) King(Father (John)) Greedy(Father (John)) Evil(Father (John)) 3

4 Inference in First Order Logic Universal Instantiation (in a rule, substitute all symbols) Existential Instantiation ( in a rule, substitute one symbol, use the rule and discard) Skolem constancts is a new variable that represents a new inference. 4

5 Universal Instantiation (UI) we can infer any sentence obtained by substituting a ground term (a term without variables) for the variable. Let SUBST(θ,α) denote the result of applying the substitution θ to the sentence α. Then the rule is written: for any variable v and ground term g. For example, the three sentences given earlier are obtained with the substitutions {x/john}, {x/richard}, and{x/father (John)}. 5

6 Existential instantiation (EI) For any sentence α, variable v, and constant symbol k (that does not appear elsewhere in the knowledge base): For example, from the sentence x Crown(x) OnHead(x,John) we can infer the sentence Crown(C1) OnHead(C1,John) where C 1 is a new constant symbol, called a Skolem constant Existential and universal instantiation allows to propositionalize any FOL sentence or KB 6

7 Reduction to propositional inference Suppose KB: x King(x) Greedy(x) Evil(x) King(John) Greedy(John) Brother(Richard, John) Apply UI using {x/john} and {x/richard} King(John) Greedy(John) Evil(John) King(Richard) Greedy(Richard) Evil(Richard) And discard the Universally quantified sentence. We can get the KB to be propositions. 7

8 Problems with Propositionalization Propositionalization generates lots of irrelevant sentences So inference may be very inefficient For example: x King(x) Greedy(x) Evil(x) King(John) y Greedy(y) Brother(Richard,John) it seems obvious that Evil(John) is entailed, but propositionalization produces lots of facts such as Greedy(Richard) that are irrelevant 8

9 Unification Subst(θ, p) = result of substituting θ into sentence p Unify algorithm: takes 2 sentences p and q and returns a unifier if one exists Unify(p,q) = θ where Subst(θ, p) = Subst(θ, q) Example: p = Knows(John,x) q = Knows(John, Jane) Unify(p,q) = {x/jane} 9

10 Unification examples simple example: query = Knows(John,x), i.e., who does John know? p q θ Knows(John,x) Knows(John,Jane) {x/jane} Knows(John,x) Knows(y,Bill) {x/bill,y/john} Knows(John,x) Knows(y,Mother(y)) {y/john,x/mother(john)} Knows(John,x) Knows(x,Elizabeth) {fail} Last unification fails: only because x can t take values John and Elizabeth at the same time But we know that if John knows x, and everyone (x) knows Elizabeth, we should be able to infer that John knows Elizabeth Problem is due to use of same variable x in both sentences Simple solution: Standardizing apart eliminates overlap of variables, e.g., Knows(x 17, Elizabeth) 10

11 Unification examples p q θ Knows(John,x) Knows(x 17,Elizabeth) {x 17 /John, x/elizabeth} Knows(John,x) Knows(y,z) {y/john, x/z} OR {y/john, x/john, z/john} For the last one: The first unifier is more general -most general unifier (MGU) - than the second. 11

12 Example x King(x) Greedy(x) Evil(x) King(John) y Greedy(y) Brother(Richard,John) And we would like to infer Evil(John) without propositionalization 12

13 Generalized Modus Ponens (GMP) For atomic sentences p i, p i and q, where there is a substitution θ such that SUBST(θ, p i )=SUBST(θ, p i ), for all i, 13

14 Completeness and Soundness of GMP GMP is sound Only derives sentences that are logically entailed GMP is complete for a KB consisting of definite clauses Complete: derives all sentences that are entailed OR answers every query whose answers are entailed by such a KB 14

15 Inference approaches in FOL Forward-chaining Uses GMP to add new atomic sentences Useful for systems that make inferences as information streams in Requires KB to be in form of first-order definite clauses Backward-chaining Works backwards from a query to try to construct a proof Can suffer from repeated states and incompleteness Useful for query-driven inference Resolution-based inference (FOL) Refutation-complete for general KB Can be used to confirm or refute a sentence p (but not to generate all entailed sentences) Requires FOL KB to be reduced to CNF Uses generalized version of propositional inference rule Note that all of these methods are generalizations of their propositional equivalents 15

16 Horn Clauses and Definite Clauses Horn Form KB conjunction of Horn clauses. Horn clause (at most one literal is Positive not negated ) For example: ( P Q V) is a Horn clause So is ( P W). But, ( P Q V) is not. Definite Clause: exactly one literal is positive. Horn clauses can be re-written as implications Proposition symbols (fact) or Conjunction of symbols (body or premise) symbol (head) Example: ( C B A) becomes (C ^ B A) Modus Ponens for Horn KB: 16

17 Forward and Backward Chaining 17

18 Forward chaining example OR Gate AND gate 18

19 Forward chaining example 19

20 Forward chaining example 20

21 Forward chaining example 21

22 Forward chaining example 22

23 Forward chaining example 23

24 Forward chaining example 24

25 Backward chaining Idea: work backwards from the query q check if q is known already, or prove by BC all premises of some rule concluding q Hence BC maintains a stack of sub-goals that need to be proved to get to q. Avoid loops: check if new sub-goal is already on the goal stack Avoid repeated work: check if new sub-goal 1. has already been proved true, or 2. has already failed 25

26 Backward chaining example 26

27 Backward chaining example 27

28 Backward chaining example 28

29 Backward chaining example we need P to prove L and L to prove P. 29

30 Backward chaining example 30

31 Backward chaining example 31

32 Backward chaining example 32

33 Backward chaining example 33

34 Backward chaining example 34

35 Backward chaining example 35

36 Forward vs. backward chaining FC is data-driven May do lots of work that is irrelevant to the goal BC is goal-driven Complexity of BC can be much less than linear in size of KB 36

37 Knowledge Base in FOL 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 of its missiles were sold to it by Colonel West, who is American. Prove that Colonel West is a Criminal? 37

38 Knowledge Base in FOL 38

39 Forward chaining proof 39

40 Forward chaining proof 40

41 Forward chaining proof 41

42 Forward chaining proof 42

43 Backward chaining example 43

44 Backward chaining example 44

45 Backward chaining example 45

46 Backward chaining example 46

47 Backward chaining example 47

48 Backward chaining example 48

49 Backward chaining example 49

50 Resolution in FOL Full first-order version: l 1 l k, m 1 m n Subst(θ, l 1 l i-1 l i+1 l k m 1 m j-1 m j+1 m n ) where Unify(l i, m j ) = θ. The two clauses are assumed to be standardized apart so that they share no variables. For example, Rich(x) Unhappy(x), Rich(Ken) Unhappy(Ken) with θ = {x/ken} Apply resolution steps to CNF (KB α); complete for FOL 50

51 Converting FOL sentences to CNF Original sentence: Everyone who loves all animals is loved by someone: x [ y Animal(y) Loves(x,y)] [ y Loves(y,x)] 1. Eliminate implications x [ y Animal(y) Loves(x,y)] [ y Loves(y,x)] 2. Move inwards: Recall: x p x p, x p x p x [ y ( Animal(y) Loves(x,y))] [ y Loves(y,x)] x [ y Animal(y) Loves(x,y)] [ y Loves(y,x)] x [ y Animal(y) Loves(x,y)] [ y Loves(y,x)] 51

52 Conversion to CNF contd. 3. Standardize variables: each quantifier should use a different one x [ y Animal(y) Loves(x,y)] [ z Loves(z,x)] 4. Skolemize: a more general form of existential instantiation. Each existential variable is replaced by a Skolem function of the enclosing universally quantified variables: x [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x) (reason: animal y could be a different animal for each x.) 52

53 Conversion to CNF contd. 5. Drop universal quantifiers: [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x) (all remaining variables assumed to be universally quantified) 6. Distribute over : [Animal(F(x)) Loves(G(x),x)] [ Loves(x,F(x)) Loves(G(x),x)] Original sentence is now in CNF form can apply same ideas to all sentences in KB to convert into CNF Also need to include negated query Then use resolution to attempt to derive the empty clause which show that the query is entailed by the KB 53

54 Example KB: Everyone who loves all animals is loved by someone Anyone who kills animals is loved by no-one Jack loves all animals Either Curiosity or Jack killed the cat, who is named Tuna Query: Did Curiosity kill the cat? Inference Procedure: Express sentences in FOL Convert to CNF form and negated query 54

55 Resolution-based Inference 55

56 Resolution-based Inference H Animal (Tuna) E,F {x/tuna} I Kills(Jack,Tuna) D,G J Animal(F(Jack)) Loves(G(Jack), Jack) A2,C {x/jack, F(x)/x} K Loves(G(Jack), Jack) J,A1 {F(x)/F(Jack), x/jack} L Loves(y,x) Kills(x, Tuna) H,B {z/tuna} M Loves(y, Jack) I,L {x/jack} N. M,K {y/g(jack)} 56

57 Resolution-based Inference Confusing because the sentences Have not been standardized apart 57

58 Recall: Example Knowledge Base in FOL... it is a crime for an American to sell weapons to hostile nations: American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x) Nono has some missiles, i.e., x Owns(Nono,x) Missile(x): Owns(Nono,M 1 ) and Missile(M 1 ) all of its missiles were sold to it by Colonel West Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missiles are weapons: Missile(x) Weapon(x) An enemy of America counts as "hostile : Enemy(x,America) Hostile(x) West, who is American American(West) The country Nono, an enemy of America Enemy(Nono,America) Convert to CNF Q: Criminal(West)? 58

59 Resolution proof 59

60 Resolution proof 60 Adapted from Lecture Notes on Inference Methods, Dr. Mustafa Jarrar, University of Birzeit

61 END 61