soln1 - CS 151 Complexity Theory Spring 2011 Solution Set 1...

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Unformatted text preview: CS 151 Complexity Theory Spring 2011 Solution Set 1 Posted: April 11 Chris Umans 1. Let A be a language that is downward self-reducible. Given an input x , we simulate the polynomial-time computation that (with queries) decides A , and recursively compute the answer to the each query as it is made. Since the recursive calls are all on strings shorter than | x | , we will eventually reach the base case in which we query strings of length 1. The program we are describing will simply have the answers to these (constant number of) length- 1 queries hard-coded. The depth of the recursion is at most | x | , and at each level of recursion, we need to remember the state which requires space at most poly( | x | ). This last point holds because the basic computation runs in polynomial time, and hence polynomial space. Thus the overall procedure runs in PSPACE . 2. Assume both inequalities fail to hold. Then we have L = P and P = PSPACE which together imply L = PSPACE . But we know these two classes are different, by the Space Hierarchy Theorem. Thus one of the inequalities must hold....
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soln1 - CS 151 Complexity Theory Spring 2011 Solution Set 1...

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