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# CSCI 6610 - Formal Languages and Automata Final Exam Due at 3:00 PM on Tuesday, May 4th , 2010 All problems are worth 10 points (for a total of 100...

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8. Let f be the addition function that maps < x,y > to < x + y > , where x,y N and numbers are represented using binary notation. Prove f is a computable function by writing down a full description (including the states, alphabet, and transition function) of a 2-tape Turing machine that computes f . Upon completion, tape 1 should contain < x + y > and the tape head should be at the beginning of the tape (tape 2’s contents are ignored). 9. Let L 1 ,L 2 NP T coNP . Show that L 1 L L 2 NP T coNP , where L 1 L L 2 = { x | x is in exactly one of L 1 ,L 2 } . 10. Show that there exists a function that is not time-constructible. You may either give an example of such a function and prove it is not time-constructible, or you prove that some non-time-constructible language exists using methods discussed in class. Sharing your work, using a prohibited resource or using a permitted resource without citation are all considered to be cheating. I will report any such cases and will fail any student(s) found violating these policies. 2

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