## AI: Problem Set 3

Assigned: Feb. 16
Due: Mar. 2

Let L be the following first-order language, over a universe of people (that is, all terms refer to people):

c(X,Y) --- X is a child of Y.
g(X,Y) --- X is a grandchild of Y.
s(X,Y) --- X is a sibling of Y. (This includes half-siblings; also, a person is considered his own sibling.)
r(X) --- X is in room 400.
e -- Constant: Ed.

### Problem 1

Represent the following statements in L.
• A. X is a sibling of Y if and only if there exists Z such that both X and Y are children of Z.
• B. X is a grandchild of Y if and only if there exists Z such that X is a child of Z and Z is a child of Y.
• C. Everyone is the child of someone.
• D. If X is a sibling of Y, then there exists Z such that both X and Y are grandchildren of Z.
• E. Everyone in room 400 is a child of Ed.
• F. Any two people in room 400 are siblings.

### Problem 2

Give a resolution proof of D from A, B, and C. Please note:
• You need not show the intermediate stages of Skolemization. You must show all the clauses generated. You must show all the resolutions.
• No credit will be given for a proof that is not a resolution proof.

### Problem 3.

Give a resolution proof of F from A and E.