This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nondeterministic 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 nondeterministic 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 selfreduction 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 treetraversal procedure is no longer guaranteed to run in polynomial time....
View
Full
Document
 Fall '09

Click to edit the document details