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practice-final-solutions

# practice-final-solutions - 6.045J/18.400J Automata...

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6.045J/18.400J: Automata, Computability and Complexity Prof. Nancy Lynch 6.045 Final Exam May 20, 2005 Vinod Vaikuntanathan Name: Please write your name on each page. This exam is open book, open notes. There are two sheets of scratch paper at the end of this exam. Questions vary substantially in difficulty. Use your time accordingly. If you cannot produce a full proof, clearly state partial results for partial credit. Good luck! Part Problem Points Grade Part I 1–10 50 1 20 2 15 3 25 Part II 4 15 5 15 6 10 Total 150 final-1

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Name: Part I Multiple Choice Questions. (50 points, 5 points for each question) For each question, any number of the listed answers may be correct. Clearly place an “X” in the box next to each of the answers that you are selecting. Problem 1 : Which of the following are true statements about regular and nonregular languages? (All lan- guages are over the alphabet { 0 , 1 } ) If L 1 L 2 and L 2 is regular, then L 1 must be regular. If L 1 and L 2 are nonregular, then L 1 L 2 must be nonregular. X If L 1 is nonregular, then the complement of L 1 must also be nonregular. If L 1 is regular, L 2 is nonregular, and L 1 L 2 is nonregular, then L 1 L 2 must be nonregular. X If L 1 is regular, L 2 is nonregular, and L 1 L 2 is regular, then L 1 L 2 must be nonregular. Problem 2 : Which of the following are guaranteed to be regular languages ? L 2 = { ww : w ∈ { 0 , 1 } } . L 2 = { ww : w L 1 } , where L 1 is a regular language. X L 2 = { w : ww L 1 } , where L 1 is a regular language. X L 2 = { w : for some x , | w | = | x | and wx L 1 } , where L 1 is a regular language. X L 2 = { w : w L 1 and no proper prefix of w is in L 1 } , where L 1 is a regular language. Problem 3 : Which of the following are known to be true? X If a language L is recognized by an NFA, then L is also recognized by some DFA. X If a language L is recognized by a nondeterministic Turing machine, then L is also recognized by some deterministic Turing machine. X If a language L is decided by a nondeterministic Turing machine, then L is also decided by some deterministic Turing machine. If a language L is decided in polynomial time by a nondeterministic Turing machine then L is also decided in polynomial time by some deterministic Turing machine. If a language L is decided in log space by a nondeterministic Turing machine then L is also decided in log space by some deterministic Turing machine.
Name: Problem 4 : Which of the following languages are undecidable ? CLIQUE X {( M ) : M is a Turing machine and L ( M ) = CLIQUE } X {( M ) : M is a Turing machine that recognizes a nonempty language } . X {( M ) : M is a Turing machine that recognizes an NP-complete language } {( M ) : M is a Turing machine that recognizes a language that is also recognized by some other Turing machine M with an even number of states } .

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