010_2008_midterm

# 010_2008_midterm - CS3234 Logic and Formal Systems Midterm...

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Unformatted text preview: CS3234 Logic and Formal Systems Midterm Examination Questions 18/09/2008 This examination question booklet has 9 pages, including this cover page, and contains 16 questions. Answer all questions. All questions have equal weight. You have 50 minutes to complete the examination. Use a B2 pencil to fill up the provided MCQ form. Leave Section A blank. Fill up Sections B and C. After finishing, place the MCQ sheet on top of the question sheet and leave both on the table, when you exit the room. 1 Question 1 : Consider the following two formulas in propositional logic: φ = p ∧ q, ψ = r → ( p ∧ q ) Which of the following statement is true? 1 A If a truth assignment makes φ true, it also makes ψ true. 1 B If a truth assignment makes ψ true, it also makes φ true. 1 C If a truth assignment makes φ false, it also makes ψ false. 1 D None of the above. Answer 1 : 1 A If the conclusion of an implication is true, the implication is true, regardless of the premise. The other alternatives are false by similar simple arguments. Question 2 : Which formula captures the following statement most accurately? When the next large bank gets into trouble ( t ), the financial system collapses ( c ) unless the Fed buys the bank ( b ). 2 A ( ¬ c → b ) → t 2 B ( c ∧ ¬ b ) → ¬ t 2 C ( c ∧ ¬ b ) → t 2 D t → ( ¬ c → b ) 2 E t → ( ¬ b → c ) Answer 2 : 2 D “ b unless c ” in English means “if not c then b ”. Of course, t is the premise of an implication, where the unless-clause is the conclusion. Note that t → ( ¬ c → b ) and t → ( ¬ b → c ) are equivalent. Thus both answers D and E are valid. 2 Question 3 : Let us say you managed to prove that ( p ∧ q ) ∧ r → ( r ∧ q ) ∧ p is a tautology. Which known fact about propositional logic allows you to conclude that there is a natural deduction proof for ` ( p ∧ q ) ∧ r → ( r ∧ q ) ∧ p ? 3 A Law of Excluded Middle (LEM) 3 B Modus Tollens (MT) 3 C soundness of natural deduction 3 D completeness of natural deduction 3 E compactness of natural deduction Answer 3 : 3 D The fact that a tautology is provable using natural deduction follows from completeness of natural deduction, see Section 1.4.4. Question 4 : There is no polynomial algorithm known that can test the satisfiability of arbitrary formulas in propositional logic. However, for clauses of a formula in conjunctive normal form (CNF), Lemma 1.43 (page 56) claims: A clause L 1 ∨ L 2 ∨ ··· ∨ L m is valid iff there are indices i,j with 1 ≤ i,j ≤ m such that L i is ¬ L j . This suggests that we can test the validity of a formula by translating it to CNF, and then check each clause for validity. Which of the following statements is most accurate?...
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010_2008_midterm - CS3234 Logic and Formal Systems Midterm...

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