handout4 - Handout Transformation of a formula A to a...

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Handout Transformation of a formula A to a formula B in conjunctive normal form, so that A is satisifable iff B is satisfiable. Given a formula A , consider its parse tree T A . To each node a of T A associate a proposi- tional variable p a , so that: i) If a is terminal, then p a = p , where p is the propositional variable occurring at a ; ii) If a is a non-terminal node, p a is different from all the propositional variables occurring in A ; iii) To distinct nodes we assign distinct variables. For each node a which is not terminal, we have one of the following five possibilities b a B ¬ B b B a ( B C ) C c b B a ( B C ) C c b B a ( B C ) C c b B a ( B C ) C c In the first case, associate to a the formula: p a ⇔ ¬ p b (1) In the second, associate to a the formula: p a ( p b p c ) (2) In the third case, the formula: p a ( p b p c ) (3) In the fourth case, the formula p a ( p b p c ) (4) and in the fifth, the formula p a ( p b p c ) (5)
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