Download Sri Lanka Institute of Information Technology MSc in IT/IM Problem

Sri Lanka Institute of Information Technology
MSc in IT/IM
573 Intelligent Systems
Problem Set 2
1. Prove each of the following assertions:
a. α is is valid if and only if T rue |= α.
b. For any α, F alse |= α.
c. α |= β if and only if the sentence (α =⇒ β) is valid.
d. α |= β if and only if the sentence (α ∧ ¬β) is unsatisfiable.
2. Given the following, can you prove that the unicorn is mythical? How about magical? Horned?
If the unicorn in mythical, then it is immortal, but if it not mythical, then it
is a mortal animal. If the unicorn is either immortal or a mammal, then it is
horned. The unicorn is magical if it is horned.
3. Write axioms describing the predicates GrandChild, GreatGrandP arents, Brther, Sister, Daughter,
Son, Aunt, U ncle, BrotherInLaw, SisterInLaw, and F irstCousine. Find out the proper definition of mth cousin n times removed, and write the definition in first-order logic.
Now write down the basic facts depicted in the family tree in Figure 1. Using a suitable logical
reasoning system, Tell it all the sentences you have written down, and Ask who are Elizabeth’s
grandchildren, Diana’s brothers-in-laws, and Zara’s great-grandparents.
George
Spencer
Kydd Elizabeth
Diana
Charles
William Harry
Mum
Philip
Anne
Peter
Margaret
Mark
Andrew
Sarah
Edward
Zara Beatrice Eugenie
Figure 1: A typical family tree. The symbol o
n connects spouses and arrows point to children
4. From “Horses are animal” it follows that “Head of an horse is the head of an animal.” Demonstrate that this inference is valid by carrying out the following steps.
a. Translate the premise and the conclusion into the language of first-order logic. Use three
predicates: HeadOf (h, x) (meaning h is the head of x), Horse(x), and Animal(x).
b. Negate the conclusion, and convert the premise and the negated conclusion into conjunctive
normal form.
c. Use resolution to show that the conclusion follows from the premise.
5. Consider the following sentences:
– John likes all kinds of foods.
– Apples are food.
– Chicken is food.
– Anything anyone eats and isn’t killed by is food.
– Bill eats peanuts and is still alive.
– Sue eats everything Bill eats.
a. Translate these sentences into formulas in predicate logic.
b. Prove that John likes peanuts using backward chaining.
c. Convert the formulas of part (a) into clause form.
d. Prove that John likes peanuts using resolution.
e. Use resolution to answer the question, “What food does Sue eat?”
6. Consider the following information:
– Animals can outrun any animals that they eat.
– Carnivores eat other animals.
– Outrunning is transitive: If x can outrun y and y can outrun z, then x can outrun z.
– Lions eat Zebras.
– Zebras can outrun dogs.
– Dogs are carnivores.
Use resolution to find three animals that Lions can outrun.