## AI: Problem Set 4

Assigned: March 4
Due: March 25

### Problem 1

Let D be a domain containing solid objects. Let L be the first-order language over D with the following non-logical symbols:
i(O1,O2) --- Object O1 is inside O2.
h(O) --- O is hollow.
r(O) --- O is red.
s(O) --- O is small.
m(O) --- O is medium-sized.
b(O) --- O is big.
ox --- Constant symbol.

A. Represent the following statements as Horn clauses in L.

• a. If O1 is inside O2, then O2 is hollow.
• b. If O1 is inside O2 and O2 is inside O3, then O1 is inside O3.
• c. Every small object is inside some medium-sized object.
• d. Every medium-sized object is inside some big object.
• e. Every small object is red.
• e. If O1 is inside O2, O1 is red, and O2 is big, then O2 is red.
• f. ox is small.

B. Prove the statement ``There exists a hollow red object'' using backward chaining.

C. Prove the statement ``There exists a hollow red object'' using forward chaining.

### Problem 2

Let A, B, C, D be Boolean random variables.

Given that:

Prob(A=T) = 0.8.
Prob(B=T | A=T) = 0.2
Prob(B=T | A=F) = 0.6
Prob(C=T | A=T) = 0.7
Prob(C=T | A=F) = 0.4
Prob(D=T | B=T) = 0.1
Prob(D=T | B=F) = 0.2
B and C are conditionally independent given A.
A and D are conditionally independent given B.

Evaluate the following expressions.

• Prob(B=T).
• Prob(D=T).
• Prob(B=T,C=T | A=F).
• Prob(B=T,D=T | A=F)
• Prob(A=T | B=T)
• Prob(A=T | B=F, C=T).
• Prob(A=T | D=T).
• Prob(C=T | B=F).