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# hw2 - M x accepts the correct result f x is written on the...

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University of Maryland CMSC652 — Complexity Theory Professor Jonathan Katz Homework 2 Due at the beginning of class on Sept. 28 I suggest to use L A T E Xwhen typing up your solutions. 1. An undirected graph G can be k -colored if each of its vertices can be assigned a “color” in { 1 ,...,k } such that no vertices that share an edge have the same color. Let 3COL = { G : G can be 3-colored } . Prove that 3COL is NP -complete. 2. Let L be an NP -complete language. Prove that if L co NP then NP = co NP . 3. Prove that Deﬁnition 4.19 (the certiﬁcate-based deﬁnition) yields the same class NL as the deﬁnition of NL based on non-deterministic Turing machines. 4. A non-deterministic machine M computes a function f : { 0 , 1 } * → { 0 , 1 } * if the following holds: For every x , there exists a computation path such that M ( x ) accepts. In any computation path on which
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Unformatted text preview: M ( x ) accepts, the correct result f ( x ) is written on the output tape when M halts. (On computation paths where M ( x ) does not accept, anything may be written on the output tape.) Answer the following questions: (a) Let f ( G,s,t ) be the function that outputs a path from s to t in directed graph G (or ⊥ if there is no path). Show that f can be computed by a non-deterministic log-space machine. (b) Show that f can be computed by a deterministic machine in space O (log 2 n ). (c) Show that any function that can be computed by a non-deterministic machine in space s ( n ) can be computed by a deterministic machine in space O ( s ( n ) 2 ). 5. Barak-Arora, Exercise 4.5. ( Hint : reduce 2SAT to conn .) 1...
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