4) Convert the following set of sentences to CNF form. S1: A ? (B ? E). S2: E ? D. S3: C ? F ? ¬ B. S4: E ? B. S5: B ? F. S6: B ? C
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Use the Tseitin encoding to convert the following formulas into 3CNF (CNF in which every clause has at most three literals): (a) (A ↔ (A ∧ B)) → (A → B) (b) (A ⊕ B) → (A ∨ B) (c) ((A ∧ B) → C) → (A → C) (d) ((A ∨ B) ∧ (¬A ∨ C)) → (B ∨ C)
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(a) Prove using resolution that the following sentence entails G: (A ∨ B) ∧ (¬ A ∨ C) ∧ (¬ B ∨ D) ∧ (¬ C ∨ G) ∧ (¬ D ∨ G) (b) 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? (c) 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. (d) Explain why your argument in (c) does not apply to 3-CNF.
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Translate each English sentence into a logical expression using the propositional variables defined below. Then negate the entire logical expression using parentheses and the negation operation. Apply De Morgan's law to the resulting expression and translate the final logical expression back into English. p: the applicant has written permission from his parents e: the applicant is at least 18 years old s: the applicant is at least 16 years old (a) The applicant has written permission from his parents and is at least 16 years old. (b) The applicant has written permission from his parents or is at least 18 years old.
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