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q2-solutions

# q2-solutions - 6.045J/18.400J Automata Computability and...

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6.045J/18.400J: Automata, Computability and Complexity Prof. Nancy Lynch Quiz 2: Solutions April 4, 2007 Elena Grigorescu Please write your name in the upper corner of each page. Problem Score 1 2 3 4 5 Total SQ2-1

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Name: Problem 1 : True or False (20 points). Full credit will be given for correct answers. If you include justification for your answers, you may obtain partial credit for incorrect answers. 1. True or False: If , , and are all Turing-recognizable, then must be Turing- recognizable. True. Turing-recognizable languages are closed under union and intersection. 2. True or False: If , , and are all Turing-recognizable, then must be Turing- recognizable. False. Turing recognizable languages are not closed under complementation. Take for example and . 3. True or False: The Post Correspondence Problem for a 3-letter alphabet is decidable. False. See recitations notes for the binary alphabet case. 4. True or False: If and is Turing-recognizable, then must also be Turing-recognizable. False.
Name: 5. True or False: Rice’s Theorem implies that is a Turing machine and accepts the strings and and does not accept is undecidable. True. The property above is a non-trivial property of the language of M 6. True or False: Rice’s Theorem implies that is recognized by a Turing machine with at most states and at most tape symbols is undecidable.

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