## Artificial Intelligence: Problem Set 3

Assigned: Feb. 22

Due: Mar. 7

Consider a domain whose entities are people.
Let L be the first-order language containing the following predicates:

e(X) --- X is male.

f(X) --- X is female.

p(X,Y) -- X is a parent of Y

m(X,Y) -- X is a mother of Y

f(X,Y) -- X is a father of Y

g(X,Y) -- X is a grandparent of Y

t(X) --- X is tall.

### Problem 1

Express the following sentences in L.
- 1. Everyone is either male or female.
- 2. No one is both male and female.
- 3. A mother is a female parent.
- 4. A father is a male parent.
- 5. Everyone has both a father and a mother.
- 6. X is a grandparent of Y if and only if for some Z, X is a parent
of Z and Z is a parent of Y.
- 7. No one is both a father of one person and a mother of another person.
- 8. If X is a grandparent of Z and all X's children are tall, then
Z has a tall parent.
- 9. Everyone has a grandparent.

### Problem 2

Convert sentences 1-6 above to clausal form. You need not show the
intermediate steps of Skolemization.
### Problem 3

Using resolution, prove 7 and 8 from axioms 1-6. You must show the
Skolemized form of the negated goals and each step of
the resolution proof. You need not show the intermediate steps of
Skolemization.
### Problem 4

Statement 9 above can be proven from the Horn clauses that appear in the
Skolemization of sentences 1-6. Give a backwards chaining proof of 9 and
a forward chaining proof of 9.
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