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hw04 - CS 154 Intro to Automata and Complexity Theory...

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CS 154 Intro. to Automata and Complexity Theory Handout 23 Autumn 2009 David Dill October 20, 2009 Problem Set 4 Due: October 27, 2009 Homework: (Total 100 points) Do the following exercises. Problem 1. [10 points] In this problem you will establish some closure properties for languages of Turing machines. (a). Are recursive languages closed under intersection ? Justify your answer. (b). Are recursively enumerable languages closed under intersection ? Jus- tify your answer. Problem 2. [20 points] Consider the following problem concerning Turing machines with a tape alphabet Γ. Given a Turing machine M , input string w , and a symbol X Γ, decide whether M , when running on input w , will ever write the symbol X on its tape. Show that this problem is undecidable. Is this problem recursively enumer- able? ( Hint: Can you give a reduction from the universal language L u ? It may help to look at the solution to Exercise 9.3.7(a), although Exercise 9.2.1 is more directly relevant.) Problem 3. [20 points] (Exercise 9.3.6(c) in the textbook, on page 400 of

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