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hwsol2.4 - Ted Kaminski CSci 5403 Computational Complexity...

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Ted Kaminski CSci 5403 - Computational Complexity Homework 2 Problem 4 Solution Corrected problem statement Prove that if NP S c N dtime ( n c log n ) = dtime ( n O (log n ) ), then for each i there exists a constant k such that Σ i dtime ( n O ((log n ) k ) ). Conclude that under this hypothesis, PH S k N dtime ( n (log n ) k ) = dtime ( n log O (1) n ). Proof of the second part We show that PH S k N dtime ( n (log n ) k ) = dtime ( n log O (1) n ) under the above hypothesis. Recall the definition of PH : PH = [ k N Σ k Our hypothesis is that Σ i dtime ( n O ((log n ) k ) ), from which we can conclude: [ k N Σ k [ k N dtime ( n O ((log n ) k ) ) And to be absolutely precise, we can show that S k N dtime ( n O ((log n ) k ) ) = S k N dtime ( n (log n ) k ) because dtime ( n O ((log n ) k ) ) dtime ( n (log n ) k 0 ) for some k 0 > k . Thus demonstrating that PH [ k N dtime ( n (log n ) k ) Proof of the first part We show by induction that assuming NP S c N dtime ( n c log n ) = dtime ( n O (log n ) ), we can conclude that for every i , there is a constant k
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