Due Thursday, May 20 at 12:00 NOON. You can turn this in via e-mail in PDF format (to BOTH vrastogi at u.northwestern.edu and ddowney at eecs.northwestern.edu) OR by hardcopy on Thursday to Ford 3-345 (slide your assignment under the door). The reason for the hardcopy option is that typing this particular assignment might be burdensome. It's also okay to e-mail a PDF scan of a hand-written copy. If you send e-mail, use EECS 348 Homework 5 as the e-mail subject line. LATE ASSIGNMENTS WILL NOT BE GRADED.



Assignment

This homework covers logic. Submit a write-up responding to the questions below.

  1. (3x1/2 points) Translate the following first order logic sentences to English. Predicates and functions carry the obvious meanings.
    1. brother(Jack, Jill)
    2. ∀ x ∀ y greater(sum(x, y), y)
    3. ∀ x ∀ y ∀ z [greater(x, y)∧greater(y, z)] ⇒ greater(x, z)
  2. (3x1/2 points) Translate the following English sentences to first order logic. Name predicates and functions such that their meanings are obvious.
    1. Max is father of Bill and Eddie.
    2. Every integer is greater than some integer.
    3. For any numbers x and y, the result of x added to y is the same as that of y added to x.
  3. (6 points) Consider the following Knowledge base.
    1. Jim is the brother of Dan.
    2. Mandy is Jim's mother.
    3. Tina is Dan's daughter.
    4. If A's mother is B then B is also mother of A's brother.
    5. If A is B's daughter B is A's father.
    6. If A is B's brother B is A's brother.
    7. A father's brother is an uncle.
    8. A grandmother is father's mother.
    Translate each of the above and below English sentences to first-order logic. Using the above knowledge base only, i.e. not assuming any other real world facts, prove by resolution that
    1. Jim is Tina's uncle.
    2. Mandy is Tina's grandmother.
  4. (6 points) Consider the following two sentences in first order logic.
    1. ∀ X ∃ Y (X ⊇ Y)
    2. ∃ X ∀ Y (X ⊇ Y)
    1. Assume the variables range over all possible sets of integers (for example, X might be {1, 2, 6}, or the set of all negative odd numbers, or the empty set, etc.). The "⊇" symbol denotes "is a superset or equal to." Under this interpretation, translate (i) and (ii) into English. For the below, it might be helpful to know that by definition the empty set is a subset of any set.
    2. Is (i) true under this interpretation?
    3. Is (ii) true under this interpretation?
    4. Does (i) logically entail (ii)?
    5. Does (ii) logically entail (i)?
    6. Using resolution try to prove that (i) follows from (ii). Do this even if you think that (ii) does not logically entail (i); continue until the proof breaks down and you cannot proceed (if it breaks down). Show the unifying assumption for each substitution step. If the proof fails, explain exactly where, how, and why it breaks down.