First order logic (FOL) Chapters 8 & 9

Transcription

1 First order logic (FOL) Chapters 8 & 9

2 Pros and cons of propositional logic Propositional logic is declarative Propositional logic is compositional Meaning in propositional logic is context-independent Propositional logic has very limited expressive power 2

3 First- order logic Propositional logic assumes the world consists of atomic facts First-order logic assumes the world contains objects: people, john, Relations: sibling(john, sally), bigger than(john, sally), functions: mother of(john), sqrt(x) 3

4 Constants: Predicates: Functions: Variables: KingJohn,2,... Brother,>,... Syntax of FOL Sqrt, LeftLegOf,... x, y, a, b,... Connectives: Equality: = Quantifiers: Term: Constant or Variable or Function(Term1,..., Termn ) Atomic sentence: Predicate(Term1,..., Term n ) or Term 1 = Term 2 Complex sentence: made from atomic sentences using connectives and quantifiers 4

5 Example Atomic sentence Complex sentence 5

6 Semantic of FOL Sentences are true with respect to a model and an interpretation Model contains objects (domain elements) and relations among them Interpretation specifies referents for constant symbols objects predicate symbols relations function symbols functional relations An atomic sentence Predicate(Term1,..., Term n )is true iff the objects referred to by Term 1,..., Term n are in the relation referred to by predicate 6

7 Model of FOL: Example 7

8 Universal quantification <variables> <sentence> x P is true in a model m iff P is true with x being each possible object in the model. Example: Everyone at UNC is smart x At(x, UNC) Smart(x) Roughly speaking, equivalent to the conjunction of instantiations of P (At(KingJohn,UNC) Smart(KingJohn)) (At(Richard,UNC) Smart(Richard)) (At(Berkeley,UNC) Smart(Berkeley))... 8

9 A common mistake to avoid Typically is the main connective with Common mistake: using as the main connective with : x [At(x, UNC) Smart(x)] means Every one is at UNC and every one is smart 9

10 Existential quantification <variables> <sentence> Some one at Stanford is smart: x At(x,Stanford) Smart(x) x P is true in a model m iff P is true with x being some possible object in the model Roughly speaking, equivalent to the disjunction of instantiations of P (At(KingJohn,Stanford) Smart(KingJohn)) (At(Richard,Stanford) Smart(Richard)) (At(Stanford,Stanford) Smart(Stanford))... 10

11 Another common mistake to avoid Typically, is the main connective with Common mistake: using as the main connective with : x At(x, Stanford) Smart(x) is true if there is any one who is not at Stanford! 11

12 Properties of quantifiers x y is the same as y x x y is the same as y x x y is not the same as y x x y Loves(x,y) There is a person who loves everyone y x Loves(x,y) Everyone is loved by p at least one person Quantifier duality: each can be expressed using the other x Likes (x, IceCream) x Likes(x, IceCream) x Likes (x, Broccoli) x Likes(x, Broccoli) 12

13 Equality term1=term2 is true under a given interpretation if and only if term1 and term2 refer to the same object E.g., x (Sqrt (x ),Sqrt (x ))=x are satisfiable 2=2is valid E.g., definition of (full) Sibling in terms of Parent: x,y Sibling(x,y) [ (x= y) m, f (m= f) Parent(m, x ) Parent(f, x ) Parent(m, y) Parent(f, y)] 13

14 Example Brothers are siblings x,y Brother(x,y) Sibling (x,y). One s mother is one s female parent x,y Mother(x,y) (Female(x) Parent(x,y)). A first cousin is a child of a parent s sibling x,y FirstCousin(x,y) p,ps Parent(p,x) Sibling(ps,p) Parent(ps,y) 14

15 Universal instantiation(ui) 15

16 Existential instantiation(ei) 16

17 Existential instantiation contd. 17

18 Reduction to propositional logic 18

19 Problem with propositionalization 19

20 Unification 20

21 Generalized modus ponens (GMP) 21

22 Example knowledge base 22

23 Example knowledge base 23

24 Forward chaining proof 24

25 Forward chaining proof 25

26 Forward chaining proof 26

27 Backward chaining proof 27

28 Backward chaining proof 28

29 Backward chaining proof 29

30 Backward chaining proof 30

31 Backward chaining proof 31

32 Backward chaining proof 32

33 Backward chaining proof 33

34 Resolution 34

35 Conversion to CNF 35

36 Conversion to CNF 36

37 Resolution proof 37

38 End of chapters 8/9 38