Chapter 8 First Order Logic

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1 1 Chapter 8 First Order Logic BBM 405 Artificial Intelligence Pinar Duygulu Slides are mostly adapted from AIMA and MIT Open Courseware CS461 Artificial Intelligence Pinar Spring

2 Pros and cons of propositional logic 2 Propositional logic is declarative Propositional logic allows partial/disjunctive/negated information (unlike most data structures and databases) Propositional logic is compositional: meaning of B 1,1 P 1,2 is derived from meaning of B 1,1 and of P 1,2 Meaning in propositional logic is context-independent (unlike natural language, where meaning depends on context) Propositional logic has very limited expressive power (unlike natural language) E.g., cannot say "pits cause breezes in adjacent squares except by writing one sentence for each square

3 First Order Logic 3

4 FOL Motivation 4

5 First-order logic Whereas propositional logic assumes the world contains facts, first-order logic (like natural language) assumes the world contains Objects: people, houses, numbers, colors, baseball games, wars, Relations: red, round, prime, brother of, bigger than, part of, comes between, Functions: father of, best friend, one more than, plus, (relations in which there is only one value for a given input) 5

6 Syntax of FOL: Basic elements 6 Constants : KingJohn, 2,... Predicates: Brother, >,... Functions : Sqrt, LeftLegOf,... Variables x, y, a, b,... Connectives,,,, Equality = Quantifiers,

7 FOL Syntax 7

8 Atomic sentences 8 Atomic sentence = predicate (term 1,...,term n ) or term 1 = term 2 Term = function (term 1,...,term n ) or constant or variable E.g., Brother(KingJohn,RichardTheLionheart) > (Length(LeftLegOf(Richard)), Length(LeftLegOf(KingJohn)))

9 Complex sentences 9 Complex sentences are made from atomic sentences using connectives S, S 1 S 2, S 1 S 2, S 1 S 2, S 1 S 2, E.g. Sibling(KingJohn,Richard) Sibling(Richard,KingJohn) >(1,2) (1,2) >(1,2) >(1,2)

10 Truth in first-order logic 10 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(term 1,...,term n ) is true iff the objects referred to by term 1,...,term n are in the relation referred to by predicate

11 Models for FOL: Example 11

12 FOL Interpretations 12

13 Holds 13

14 Semantic of Quantifiers 14

15 Example Domain 15

16 Example Domain 16

17 Example Domain 17

18 Example Domain 18

19 Writing FOL 19 There is somebody who is loved by everybody

20 Writing FOL 20

21 Universal quantification 21 <variables> <sentence> All Kings are persons: x King(x) Person(x) x P is true in a model m iff P is true with x being each possible object in the model Roughly speaking, equivalent to the conjunction of instantiations of P Richard the Lionheart is a king Richard the Lionheart is a person King John is a king King John is a person Richard s left leg is a king Richard s left leg is a person John s left leg is a king John s left leg is a person The crown is a king The crown is a person

22 A common mistake to avoid 22 Typically, is the main connective with Common mistake: using as the main connective with : x King(x) Person(x) means Everyone is a king and everyone is a person

23 Existential quantification <variables> <sentence> 23 x Crown(x) OnHead(x,John) 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 The crown is a crown the crown is on John s head Richard the Lionheart is a crown Richard the Lionheart is on John s head King John is a crown King John is on John s head...

24 Another common mistake to avoid 24 Typically, is the main connective with Common mistake: using as the main connective with : x Crown(x) OnHead(x,John) is true even if there is anything which is not a crown

25 Properties of quantifiers 25 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 in the world y x Loves(x,y) Everyone in the world is loved by 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)

26 Equality 26 term 1 = term 2 is true under a given interpretation if and only if term 1 and term 2 refer to the same object E.g., definition of 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)]

27 Using FOL 27 The kinship domain: Brothers are siblings x,y Brother(x,y) Sibling(x,y) One's mother is one's female parent m,c Mother(c) = m (Female(m) Parent(m,c)) Sibling is symmetric x,y Sibling(x,y) Sibling(y,x)

28 Using FOL 28 The set domain: s Set(s) (s = {} ) ( x,s 2 Set(s 2 ) s = {x s 2 }) x,s {x s} = {} x,s x s s = {x s} x,s x s [ y,s 2 } (s = {y s 2 } (x = y x s 2 ))] s 1,s 2 s 1 s 2 ( x x s 1 x s 2 ) s 1,s 2 (s 1 = s 2 ) (s 1 s 2 s 2 s 1 ) x,s 1,s 2 x (s 1 s 2 ) (x s 1 x s 2 ) x,s 1,s 2 x (s 1 s 2 ) (x s 1 x s 2 )

First-Order Logic. Michael Rovatsos. University of Edinburgh R&N: February Informatics 2D

First-Order Logic. Michael Rovatsos. University of Edinburgh R&N: February Informatics 2D First-Order Logic R&N: 8.1-8.3 Michael Rovatsos University of Edinburgh 4 February 2016 Outline Why FOL? Syntax and semantics of FOL Using FOL Wumpus world in FOL Pros and cons of propositional logic Propositional