final - CS 151 Complexity Theory Spring 2011 Final Out: May...

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Unformatted text preview: CS 151 Complexity Theory Spring 2011 Final Out: May 26 Due: 1pm June 2 This is a final exam. You may consult any of the course materials and the text (Papadimitriou), but not any other source or person. There are 4 problems on three pages. Please attempt all problems, and write clear, concise solutions. Good luck! To facilitate grading, please turn in each problem on a separate sheet of paper and put your name on each sheet. Do not staple the separate sheets. Instructions for turning in the exam: Please bring your exam to Diane Goodfellow in Annen- berg 246 any time before the deadline. 1. Define L i to be the class of languages decidable by a deterministic Turing Machine using at most O (log i n ) space, and NL i to be the class of languages decidable by a non-deterministic Turing Machine using at most O (log i n ) space. The classes L 1 and NL 1 should be familiar they are just deterministic logspace and nondeterministic logspace, respectively. (a) Show that for all i , NC i L i . (b) Show that for all i , NL i has O (log 2 i n ) depth, fan-in 2, Boolean circuits. Your circuits do not need to be uniform. (c) It is tempting to try to show that for all i , NL i NC 2 i (since this holds for i = 1). Show that this would solve a major open problem. Try to give the strongest implication you can, i.e., if the containment implies A , and A implies B , you should pick...
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final - CS 151 Complexity Theory Spring 2011 Final Out: May...

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