1. First-Order Logic Prof. Dr. Widodo Budiharto 2018
Course : Artificial Intelligence
First-Order Logic Prof. Dr. Widodo Budiharto 2018

2. Outline First-Order Logic Quantifiers
Inference using First-order Logic
Exercise

3. Pros and Cons of Propositional Logic
Propositional logic is declarative: pieces of syntax correspond to facts
Propositional logic is compositional:
meaning of B1,1 ^ P1,2 is derived from meaning of B1,1 and of P1,2
Meaning in propositional logic is context-independent
Propositional logic has very limited expressive power
E.g., cannot say “pits cause breezes in adjacent squares” except by writing one sentence for each square

4. Introduction to 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, …

5. Introduction to First-Order Logic
English
Squares adjacent to pits are breezy
Propositional Logic
B1,1  (P1,2  P2,1), B2,1  (P1,1  P2,2  P3,1 ), etc
We should define all possible facts that satisfy the “English”
First-Order Logic
 s Breezy(s)   r Adjacent(r,s)  Pit(r)
We can satisfy the “English” with only a sentence

6. Introduction to First-Order Logic
Standard Logic Symbols
 = For all ; [e.g : every one, every body, any time, etc]
 = There exists ; [e.g : some one, some time, etc]
 = Implication ; [ if … then ….]
 = Equivalent ; biconditional [if … and … only … if …]
 = Not ; negation
 = OR ; disjunction
 = AND ; conjunction

7. Introduction to First-Order Logic
Some kind of logic
Epistemological commitment: the truth of what can be said about a sentence
Language
Ontological
Epistemological
Propositional logic
facts
true/false/unknown
First-order logic
facts, objects, relations
Temporal logic
facts, objects, relations, times
Probability theory
Degree of belief є [0, 1]
Fuzzy logic
Degree of truth є [0, 1]
known interval value

8. Syntax and Semantic of First-Order Logic
What are the objects? Relations? Functions?

9. Syntax and Semantic of First-Order Logic
Basic elements
Constants : KingJohn, 2, Binus, … (Objects)
Predicates : Brother, >, loves, … (Relations)
Functions : Sqrt, LeftLegOf, … (Functions)
Variables : x, y, a, b,...
Connectives : , , , , 
Equality : =
Quantifiers : , 

10. Syntax and Semantic of First-Order Logic
Term = function(term1,...,termn) or constant or variable
A logical expression that refers to an object
“Richard the Lionheart is the brother of King John”.
e.g., Brother(Richard, John)  A term
“King John’s left leg”
e.g., LeftLeg(John)  A term

11. Syntax and Semantic of First-Order Logic
Atomic sentence
formed from a predicate symbol optionally followed by parenthesized list of terms
Brother(Richard, John)
Complex terms as arguments in atomic sentences is :
married (Father(Richard), Mother(John))
An atomic sentence is true in a given model if
The relation referred to by the predicate symbol holds among the objects referred to by the arguments.

12. Syntax and Semantic of First-Order Logic
Complex sentence
Are made from atomic sentences using connectives
Examples:
 Brother(LeftLeg(Richard), John)
Brother(Richard, John)  Brother(John, Richard)
King(Richard)  King(John)
 King(Richard)  King(John)

13. Universal quantifiers
Sentence: All Kings are persons
Variable x = {Richard, King John, the crown}
FOL: x King(x)  Person(x)
Richard is a King  Richard is a person
King John is a King  King John is a person
The crown is a King  the crown is a person

14. Universal quantifiers
Sentence: All Kings are persons
Variable x = {Richard, King John, the crown}
Attention! Don’t use  for 
FOL: x King(x)  Person(x)
Richard is a King  Richard is a person
King John is a King  King John is a person
The crown is a King  the crown is a person

15. Universal quantifiers

16. Existential quantifiers
Sentence: The King John has a crown on his head
Variable x = {Richard, King John, the crown}
FOL: x Crown(x)  OnHead(x, John)
x is pronounced “There exists an x such that …” or “For some x …”
x P says that P is true in at least one extended interpretation that assigns x to a domain element

17. Existential quantifiers
Sentence: The King John has a crown on his head
Variable x = {Richard, King John, the crown}
FOL: x Crown(x)  OnHead(x, John)
Richard is a Crown  Richard is on John’s head
King John is a Crown  King John is on John’s head
The crown is a Crown  the crown is on John’s head
(True) then the sentence is true, at least one

18. Existential quantifiers

19. Nested quantifiers Sentence: Brothers are siblings
x y Brother(x, y)  Sibling(x, y)
Consecutive quantifiers of the same type can be written as one quantifier with several variables
To say that siblinghood is a symmetric relationship:
x,y Sibling(x,y)  Sibling(y,x)

20. Nested quantifiers “Everybody loves somebody”: x y Loves(x, y)
A mixture:
“Everybody loves somebody”:
x y Loves(x, y)
“There is someone who is loved by everyone” :
y x Loves(x, y)

21. Nested quantifiers

22. Using First-Order Logic
Assertions and queries in first-order logic
Sentences are added to a knowledge base using TELL, exactly as in propositional logic (Assertion)
We can assert that John is a king, Richard is a person, and all kings are persons:
TELL(KB, King(John)) .
TELL(KB, Person(Richard)) .
TELL(KB, ∀ x King(x) ⇒ Person(x)) .

23. Using First-Order Logic
Assertions and queries in first-order logic
We can ask questions of the knowledge base using ASK.
ASK(KB, King(John))
Answer: True
We can ask what value of x makes the sentence true, using ASKVARS
ASKVARS(KB, Person(x))
Answer: {x/John} and {x/Richard}

24. Inference in First-Order Logic
We begin with some simple inference rules (Instantiation) that can be applied to sentences with quantifiers to obtain sentences without quantifiers
These rules lead naturally to the idea that first-order inference can be done by converting the knowledge base to propositional logic and using propositional inference

25. Universal Instantiation
Universal Instantiation (UI)
for any variable v and ground term g,
Axiom that all greedy king are evil :
e.g., x King(x)  Greedy(x)  Evil(x) yields:
King(John)  Greedy(John)  Evil(John)
King(Richard)  Greedy(Richard)  Evil(Richard)
King(Father(John))  Greedy(Father(John))  Evil(Father(John))

26. Existential Instantiation
Existential Instantiation (EI)
For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base:
e.g., x:Crown (x)  OnHead (x, John) yields:
Crown (C1)  OnHead (C1, John)
provided C1 is a new constant symbol, called a Skolem constant

27. Unification

28. Unification

29. Unification

30. Unification

31. Unification

32. Unification To unify Knows (John, x) and Knows (y, z), could return:
θ = {y/John, x/z} or θ = {y/John, x/John, z/John}
The first unifier is more general than the second.
There is a single most general unifier (MGU) that is unique up to renaming of variables.
MGU = { y/John, x/z }

33. Generalized Modus Ponens (GMP)

34. Example: Knowledge Base
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

35. Example: Knowledge Base
... 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,M1)
Missile(M1)
… all of its missiles were sold to it by Colonel West
Missile(x)  Owns(Nono, x)  Sells(West,x,Nono)

36. Example: Knowledge Base
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)

37. Resolution

38. Conversion to CNF
Sentence : “Everyone who loves all animals is loved by someone”:
x [y Animal(y)  Loves(x, y)]  [y Loves(y, x)]
Eliminate implications
x [y Animal(y)  Loves(x, y)]  [y Loves(y, x)]
Move  inwards:
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)]

39. Conversion to CNF
Standardize variables: each quantifier should use a different one
x [y Animal(y)  Loves(x, y)]  [z Loves(z, x)]
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)
Drop universal quantifiers:
[Animal(F(x))  Loves(x,F(x))]  Loves(G(x),x)
Distribute  over  :
[Animal(F(x))  Loves(G(x),x)]  [Loves(x, F(x))  Loves(G(x), x)]

40. Conversion to CNF
American(x)  Weapon(y)  Sells(x, y, z)  Hostile(z)  Criminal(x)
American(x)  Weapon(y)  Sells(x, y, z)  Hostile(z)  Criminal(x)
Owns(Nono,M1) and Missile(M1)
Owns (Nono, M1)
Missile(M1)
Missile(x)  Owns(Nono, x)  Sells(West, x, Nono)
Missile(x)  Owns(Nono, x)  Sells(West, x, Nono)

41. Conversion to CNF Missile(x)  Weapon(x) Missile(x)  Weapons(x)
Enemy(x, America)  Hostile(x)
Enemy(x, America)  Hostile(x)
American(West)
Enemy(Nono, America)

42. Resolution Knowledge Base in CNF
American(x)  Weapon(y)  Sells(x, y, z)  Hostile(z)  Criminal(x)
Missile(x)  Owns(Nono, x)  Sells(West, x, Nono)
Enemy(x, America)  Hostile(x)
Missile(x)  Weapons(x)
Owns (Nono, M1)
Missile(M1)
American(West)
Enemy(Nono, America)

43. Resolution

44. References
Widodo Budiharto and Derwin Suhartono. (2014). Artificial Intelligence, Andi Offset Publisher
Stuart Russell, Peter Norvig Artificial Intelligence : A Modern Approach. Pearson Education. New Jersey. ISBN:

45. Exercise Given sentences as premise: John is a student
John is in the Informatics department
Each Informatics’ student must be an engineering student
Mathematic is a difficult lesson
Each engineering student would definitely like Mathematic or hate it
Each student would definitely like a lesson
Students who have never attended difficult lesson certainly do not like the lesson
Peter has never attended the Mathematic lesson
Based on given premises above, please create:
FOL
Convert FOL in part a) to CNF
Proof by Resolution, that Peter hate Mathematic.