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CISG 5115 Theory of Computation, Fall 2008 October 09, 2008 

MIDTERM EXAM 

Name: Student ID#: 

Duration: 75 min. 

Total: 100 pts. 

This exam is closed book. No books, notes, calculators or any other type of help material is 

permitted. 

Show all your work on the problems to receive full credit. Explain all the assumptions that 

you make (if any), reference all the rules, theorems and other sources that are used. Use the 

back side of the sheets in case you need extra space for your answers. Do not take the sheets 

apart and submit the entire exam back to the instructor. Clearly identify your answers in the 

form required by the question(s). 

1. (10 pts.) Describe the method of proof by contradiction: 

1


2. (10 pts.) Prove by mathematical induction that n! > 2 n , ∀n ≥ 3. 

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3. (10 pts.) Give definition of extended transition function for DFA. 

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4. (10 pts.) Compute the ɛ-closures of the states in the given ɛ-NDFA: 

ɛ a b c 

→ p ∅ {p} {q} {r} 

q {p} {q} {r} ∅ 

∗r {q} {r} ∅ {p} 

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5. (10 pts.) Describe the methods for converting a finite automaton to a regular expression. 

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6. (10 pts.) Prove or disprove, for regular expressions R, S: 

(R + S) ∗ S = (R ∗ S) ∗ 

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7. (10 pts.) State (without proof) the Pumping Lemma for regular languages. 

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8. (10 pts.) Is the language L = {a 2 b 3n |n ∈ Z + } regular? Thoroughly explain your answer. 

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9. (10 pts.) List at least five of the closure properties of the regular languages, i.e. at least five operations 

that preserve regularity. 

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10. (10 pts.) Given a regular language L, define the language 

init(L) = {w|∃x, such that wx ∈ L} 

Is the language init(L) necessarily regular? Thoroughly explain your answer. 

Make sure your name and student ID number appear on the title page! 

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