soln2 - CS 151 Complexity Theory Spring 2011 Solution Set 2...

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Unformatted text preview: CS 151 Complexity Theory Spring 2011 Solution Set 2 Posted: April 18 Chris Umans 1. Suppose L NP coNP . Then there exist languages R 1 and R 2 in P for which L = { x : y, | y | | x | k 1 , ( x,y ) R 1 } L = { x : z, | z | | x | k 2 , ( x,z ) R 2 } On input x , our strong nondeterministic Turing Machine M will guess y and guess z . If ( x,y ) R 1 then we accept; if ( x,z ) R 2 then we reject; otherwise we output ?. The above equations imply that if x L then some path leads to accept and if x L then some path leads to reject, as required. Moreover, when x L , no computation path rejects (because that would imply that there exists a z for which ( x,z ) R 2 ) and when x L no computation path accepts (because that would imply that there exists a y for which ( x,y ) R 1 ). In the other direction, suppose we have strong nondeterministic Turing Machine M that decides L in time n k . We can modify it so that whenever it would have output ? it instead rejects. This gives an ordinary non-deterministic Turing Machine that decides L , and so L NP . We can also modify it so that whenever it would have rejected it instead accepts and vice versa, and whenever it would have output ? it instead rejects. This gives an ordinary non-deterministic Turing Machine that decides L , and so L coNP . We conclude that L NP coNP . 2. (a) Recall that R is the reduction from SAT to a unary languages U 1 . Consider x = R ( ), where is a formula in the self-reduction tree. If x 1 , then we can easily detect this, and we used this observation critically to assign a single color to such strings. Since every other color was identified with a string in 1 of length at most p ( n ) (where p ( n ) is a bound on the length of the string output by R ) the total number of colors was p ( n ) + 1. If we know only that U is sparse, we have a similar polynomial bound on the number of satisfiable colors, but not on the number of unsatisfiable colors, as it is perfectly legal for R to map unsatisfiable formulas to strings outside U , and we have no ecient way of detecting these strings and grouping them all into a single color, as we did with unary languages. Since there may be exponentially many different colors, the tree-traversal procedure is no longer guaranteed to run in polynomial time....
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soln2 - CS 151 Complexity Theory Spring 2011 Solution Set 2...

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