Collaboration Policy

For this assignment, you must work individually in regard to the design and implementation of your project.

Please make sure you adhere to the policies on academic integrity in this regard.

Topic: Propositional Logic
Related Reading: Ch. 2
Due: Thursday, 19 September 2013, 2:15pm

Please make sure you adhere to the policies on academic integrity.

1. (15 points)

   Convert the following formula to an equivalent formula in Conjunctive Normal Form (CNF), without any redundant or trivial clauses.

   ¬( (A ∨ B ∨ ¬C) ∧ (¬B ∨ C) ∧ (¬A ∨ C) )

2. (15 points)

   Demonstrate how to use resolution to prove the goal (¬A ∧ ¬B) given the following premises:

   1. B ⇔ (A ∨ D)
   2. D ⇒ F
   3. E ∧ C ⇒ ¬A
   4. D ⇒ A
   5. A ⇒ C
   6. A ⇒ E
   You should start by converting the combination of all premises into CNF form, numbering each clause, and then you should explicitly show each resolution step that is applied, giving the numeric labels for the two previous clauses that are being resolved.

3. (20 points)

   A propositional 2-CNF formula is a conjunction of clauses, each containing exactly 2 literals, such as the following:

   (A ∨ B) ∧ (¬A ∨ C) ∧ (¬B ∨ D) ∧ (¬C ∨ G) ∧ (¬D ∨ G)

   1. Prove, using resolution, that the above formula entails G.
   2. Two clauses are semantically distinct if they are not logically equivalent. How many semantically distinct 2-CNF clauses can be constructed from n proposition symbols?
   3. Using your answer to (b), prove that propositional resolution always terminates in time polynomial in n given a 2-CNF sentence containing no more than n distinct symbols.
   4. Explain why your argument in (c) does not apply to 3-CNF formulae.

Submitting your assignment

You may choose to submit either a hardcopy in class, or an electronic copy via the course website (details on the submission process).