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# hw1 - “(where M above is a deterministic Turing machine...

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University of Maryland CMSC652 — Complexity Theory Professor Jonathan Katz Homework 1 Due at the beginning of class on Sept. 14 I suggest to use L A T E Xwhen typing up your solutions. It will yield more readable solu- tions, and will provide you with a skill that you will use your entire (graduate) career. 1. Prove that if time ( n 2 ) = ntime ( n 2 ) then time ( n 5 ) = ntime ( n 5 ). (Hint: given a language L ntime ( n 5 ), consider the language L 0 = { x 0 | x | 2 . 5 : x L } .) 2. Assume L 1 ,L 2 ∈ NP and S 1 ,S 2 co NP . For the purposes of this problem, you should also assume that NP 6 = co NP . Answer each of the following with proof or with a counterexample: (a) Is L 1 L 2 necessarily in NP ? Is L 1 S 1 necessarily in co NP ? (b) Is L 1 L 2 necessarily in NP ? Is L 1 L 2 necessarily in co NP ? 3. Prove that P = co P . 4. Prove that the following language L = ' ( M,x, 1 t ) : w ∈ { 0 , 1 } t s.t. M ( x,w ) halts within t steps with output 1
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Unformatted text preview: . “ . (where M , above, is a deterministic Turing machine) is NP-complete. 5. A language L is co NP-hard if for every L ∈ co NP it holds that L ≤ p L ; it is co NP-complete if furthermore L ∈ co NP . Prove or disprove: L is NP-complete iﬀ ¯ L is co NP-complete. 6. Challenge question – optional. Show a universal non-deterministic Turing ma-chine U such that (1) U ( M,x ) = M ( x ) for any non-deterministic Turing machine M for which M ( x ) is deﬁned, and (2) for every M there exists a constant c such that if M ( x ) runs in time T then U ( M,x ) runs in time O ( T ) (once again, the constant in the big-O notation may depend on M ). 1...
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