ps1 - PSPACE . 2. Show that one of the following...

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CS 151 Complexity Theory Spring 2011 Problem Set 1 Out: March 31 Due: April 7 Reminder: you are encouraged to work in groups of two or three; however you must turn in your own write-up and note with whom you worked. You may consult the course notes and the text (Papadimitriou). Please attempt all problems. 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. 1. Downward self-reducibility. For a language A , define A <n = { x A : | x | < n } . Language A is said to be downward self-reducible if it is possible to determine in polynomial time if x A using the results of queries of the form “ y A < | x | ?” The queries may be adaptive , meaning that the polynomial time procedure may choose later queries depending on the results of earlier ones. Show that every downward self-reducible language is in
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Unformatted text preview: PSPACE . 2. Show that one of the following inequalities must hold: L = P or P = PSPACE . Note that both are believed to be true, and no one knows how to prove either one is true. 3. Show that logspace reductions are closed under composition. Then use the same ideas to prove that if language A is P-complete, then A L implies L = P . 4. Use a padding argument to show that if L = P then PSPACE = EXP . 5. Prove that SPACE( O ( n ) ) = P . (Note that while this is an interesting result, it doesnt seem to shed any light on the major open questions L ? = P and P ? = PSPACE ). Hint: consider a language A and a padded version of A . How are the two languages related with respect to space? How are they related with respect to time? 1-1...
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