1. Knowledge Representation III First-Order Logic
CSE 473

2. Why Not Propositional Logic?
“All monkeys are fuzzy”
Propositional logic requires:
fuzzy1, fuzzy2, …, fuzzyN
Why? Lack of structure in/between atoms.
What we’d like is a way to talk about objects
and groups of objects, and to define
relationships between them.
First-order logic (aka “predicate logic”)
© D. Weld, D. Fox

3. FOL Definitions Constants: a,b, dog33. Variables: X, Y.
Name a specific object.
Variables: X, Y.
Refer to an object without naming it.
Functions: dad-of
Mapping from objects to objects.
Terms: dad-of(dog33)
Refer to objects
Relations: in, hungry.
State relationships between objects.
Atomic Sentences: in(dad-of(dog33), food6)
Can be true or false
Correspond to propositional symbols P, Q
© D. Weld, D. Fox

4. More Definitions Logical connectives: and, or, not, => Quantifiers:
 Forall
 There exists
Examples
George is brown
Monkeys are curious
There is a curious monkey
Brown(George)
m: Monkey(m)  Curious(m)
m: Monkey(m) ^ Curious(m)
© D. Weld, D. Fox

5. Quantifier / Connective Interaction
M(x) == “x is a monkey”
C(x) == “x is curious”
x: M(x)  C(x)
x: M(x) C(x)
x: M(x)  C(x)
x: M(x) C(x)
“Everything is a curious monkey”
“All monkeys are curious”
“There exists a curious monkey”
“There exists an object that is either a curious
monkey, or not a monkey at all”
© Daniel S. Weld

6. Nested Quantifiers: Order matters!
x y P(x,y)  y x P(x,y)
Examples
Every dog has a tail
Every dog shares a tail! ?
d t has(d,t)
t d has(d,t)
Someone is loved by everyone
x y loves(y, x)
© D. Weld, D. Fox

7. Semantics
Semantics: what the arrangement of symbols means in the world
Propositional logic
Basic elements are variables
(references to facts)
Possible worlds: mappings from variables to T/F
First-order logic
Basic elements are terms
(references to objects)
Possible worlds: mappings from terms to real-
world elements.
© Daniel S. Weld

8. Models
Depiction of one possible model
© D. Weld, D. Fox

9. Interpretations=Mappings syntactic tokens  model elements
Depiction of one possible interpretation, assuming
Constants: Functions: Relations:
Richard John
LegOf(p)
On(x,y) King(p)
© D. Weld, D. Fox

10. Interpretations=Mappings syntactic tokens  model elements
Another interpretation, same assumptions
Constants: Functions: Relations:
Richard John
LegOf(p)
On(x,y) King(p)
© D. Weld, D. Fox

11. Satisfiability, Validity, & Entailment
S is valid if it is true in all interpretations
S is satisfiable if it is true in some interp
S is unsatisfiable if it is false all interps
S1 entails S2 if
forall interps where S1 is true,
S2 is also true |=
© D. Weld, D. Fox

12. First-Order Wumpus World
Objects
Squares, wumpuses, agents,
gold, pits, stinkiness, breezes
Relations
Square topology (adjacency),
Pits/breezes,
Wumpus/stinkiness
© D. Weld, D. Fox

13. Wumpus World: Squares Square topology relations?
Each square as an object:
Square1,1, Square1,2, …,
Square3,4, Square4,4
Square topology relations?
Adjacent(Square1,1, Square2,1) …
Adjacent(Square3,4, Square4,4)
Better: Squares as lists:
[1, 1], [1,2], …, [4, 4]
Square topology relations:
x, y, a, b: Adjacent([x, y], [a, b]) 
[a, b] Є {[x+1, y], [x-1, y], [x, y+1], [x, y-1]}
© D. Weld, D. Fox

14. Wumpus World: Pits Each pit as an object: Problem?
Pit1,1, Pit1,2, …,
Pit3,4, Pit4,4
Problem?
Not all squares have pits
List only the pits we have:
Pit3,1, Pit3,3, Pit4,4
Problem?
No reason to distinguish pits. Complicates inference.
Better: pit as unary predicate
Pit(x)
Pit([3,1]); Pit([3,3]); Pit([4,4]) will be true
© D. Weld, D. Fox

15. Wumpus World: Breezes Represent breezes like pits,
as unary predicates:
Breezy(x)
“Squares next to pits are
breezy”:
x, y, a, b:
(Pit([x, y]) ^ Adjacent([x, y], [a, b]))  Breezy([a, b])
© D. Weld, D. Fox