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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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CSCI 6610 – Formal Languages and Automata Final Exam Due at 3:00 PM on Tuesday, May 4 th , 2010 All problems are worth 10 points (for a total of 100 points). This exam is open book and open notes. You should not discuss this exam with your classmates – not even in loose terms! The only person you should discuss this test with before handing it in is me (Shelby). You may use resources other than Sipser, but you must write up your solutions without looking at the resource. Do not simply copy results found in other resources! You should feel confident that you could reproduce your work without consulting any other resources when you hand your test in. If you use any resources other than your notes or Sipser while solving a problem, please indicate which source you used. If the source is a book or an article, state the author(s) and title. If the source is a website, state the url. Please do this for each problem. Thanks and good luck! 1. [Sipser 8.29] Show that A NFA is NL -complete. 2. A 2-tape read-only Turing machine is a 2-tape Turing machine that never changes the contents of either tape. For a given 2-tape read-only Turing machine M 2 RO , we say ( w 1 ,w 2 ) L ( M 2 RO ) if M 2 RO reaches the accept state when started with w 1 on tape 1 and w 2 on tape 2 (of course, both tape heads start at the beginning of their respective strings). Let 2 RO = { < M,w 1 ,w 2 > | M is a 2-tape read-only TM that accepts input ( w 1 ,w 2 ) } . Show that 2 RO is undecidable. 3. [Sipser 9.20] Show there exists a language C such that NP C 6 = coNP C . 4. Let Σ be any alphabet. For any w Σ, let S w = { < M > | M is a Turing machine and L ( M ) = { w }} . Prove that S and S are both unrecognizable. 5. Recall that NP can be defined in 2 ways – either using non-deterministic polynomial time Turing machines or using deterministic polynomial time verifiers. (a) Show that Σ 2 P also has a verifier-style definition. Specifically, show that L Σ 2 P if and only if there exist polynomials p 1 ( n ) ,p 2 ( n ) ,p 3 ( n ) and Turing machine M TIME ( p 1 ( n )) such that for every string w w L ⇔ ∃ u ∈ { 0 , 1 } p 2 ( | w | ) v ∈ { 0 , 1 } p 3 ( | w | ) M accepts ( w,u,v ) . ( Hint: If L satisfies the property above then a fairly straightforward application of the defini- tion presented in class and in the book can show that L Σ 2 P . To show that L Σ 2 P implies the above property holds, consider the strings u and v as paths in the non-deterministic tree of the ATM that decides L .) (b) Show that 2 P = NP SAT ( Hint: Recall ϕ / SAT iff ϕ TAUTOLOGY ). 6. Let L = { < R > | R is a regular expression and L ( R ) = Σ * } . Prove L is PSPACE -complete. 7. We say ϕ EXACTLY-ONE-SAT if ϕ is a 3CNF formula and if there exists a satisfying assignment for φ in which each clause has exactly one true literal. Prove 3 SAT P EXACTLY-ONE-SAT. 1
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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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