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29 PS3 Solutions

# 29 PS3 Solutions - 1 Handout#29 CS103 Robert Plummer...

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1 Handout #29 CS103 April 26, 2010 Robert Plummer Problem Set #3 Solutions 1. Construct a proof by resolution that shows that the following CNF (conjunctive normal form) sentence is not satisfiable. Do so by drawing a resolution diagram as shown in class. If you can derive the empty clause (regardless of whether you still have other clauses left), then the sentence is not satisfiable. (A ¬ C B) ¬ A (C B A) (A ¬ B) A good strategy is to deal with unit clauses first. That is, pick a unit clause (if any), do all the resolutions you can with it, then pick another unit clause, etc. (A ¬ C B) ¬ A (C B A) (A ¬ B) ( ¬ C B) (C B) ( ¬ B) ( ¬ C ) (C ) ± Since we arrive at an empty clause, the formula is not satisfiable.

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2 2. Construct a resolution proof that shows that the following sentence is not satisfiable. Since the sentence is not in CNF, you will have to convert it to CNF first. ¬¬ A ( ¬ A (( ¬ B C) B)) ¬ C Note that in creating the CNF you arrive at (B ¬ B) (B C) for part of the expression. This should be simplified to F (B C), which simplifies to (B C). The final CNF is A ( ¬ A B) ( ¬ A C) ¬ C C ± 3. Suppose A , B , and C are sets. Prove that [(A C) (B C)] [(A B) C] Suppose that (i) (A C) and (ii) (B C). Now suppose that x (A B). There are two cases to consider: if x A, then x C by (i), and if x B, then x C by (ii). So x (A B) means that x C, i.e., (A B) C.
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