CS4811: Homework 1 --- Predicate Calculus


Due: Monday, January 23, 2006, beginning of class. (Assigned: Friday, January 13, 2006.)

Reminder:

This is an individual assignment. All the work should be the author's and in accordance with the university's academic integrity policies. You are allowed to use any source in preparing your answers, but if you use any other source than the textbook and the class notes, you should specify it on your assignment.

Question 1(10 points)
We have defined 4 binary logical connectives (∧, ∨, →, and ≡). Are there any others that might be useful? How many binary connectives can there be? Explain your answers.

Question 2(15 points)
We noted in class that the daily ussage of words such as "and", "or", and "if ... then" might not always directly translate to logic. For each sentence below, give both a translation into first-order logic that preserves the intended meaning in English, and a straightforward translation (as if the logical connectives had their regular first-order logic meaning). Show an unintuitive consequence of the latter translation.

a. One more outburst like that and you'll be in contempt of court.

b. The game is on TV tonight, if you're interested.

c. Maybe I'll come to the party and maybe I won't.

Question 3.(15 points)
For each of the following statements decide whether it is valid, satisfiable, or unsatisfiable.
b. storm → accident
b. sun ∨ storm ∨ ¬storm
c. (snow → ski) → (¬snow → ¬ski)

Question 4.(15 points)

Represent the following sentences in first-order logic using quantifiers. Remember to define a consistent vocabulary and write its semantics in English.
a. Some plants need a lot of water.
b. CS4811 has exactly one section.
c. Everybody who takes CS4811 needs to take three exams.








Question 5.(15 points)
Minesweeper is a well-known computer game which is played on a rectangular grid of N squares with M invisible mines scattered among them. Any square my be probed by the agent; instant death follows if a mine is probed. Minesweeper indicates the presence of mines by revealing, in each probed square, the number of mines that are directly or diagonally adjacent. The goal is to have probed every unmined square.

Let Xi,j be true if square [i,j] contains a mine. Write down the assertion that there are exactly two mines adjacent to [1,1] as a sentence involving some logical combination of Xi,j propositions. The complete sentence would be very long, simply show a part of it and describe how it would be completed.

Question 6.(10 points)
Consider a knowledge base (KB) that contains two facts: p(a) and p(b). Does ∀X p(X) logically follow from this KB? Does ∃ X p(X)?

Question 7.(20 points)
Say whether or not the following pairs of expressions are unifiable. If unifiable, show the mgu and a non-mgu, if not, explain why.
a. p(X,b,b) and p(a,Y,Z)
b. p(Y,Y,b) and p(Z,X,Z)
c. p(f(X,X),a) and p(f(Y,f(Y,a)),a)
d. q(X) and ¬q(a)