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Unformatted text preview: 6.045J/18.400J: Automata, Computability and Complexity Nancy Lynch Homework 10 Due: April 23, 2007 Elena Grigorescu Readings: Sections 7.4, 7.5 Problem 1 : Let A and B be nontrivial languages over an alphabet (that is, not equal to or ). State whether each of the following is KNOWN TO BE TRUE, KNOWN TO BE FALSE, or UNKNOWN. Explain carefully why. For example if you claim that a reduction exists, then you should actually define the reduction. 1. If A P B , then A P B . Answer: Yes. 2. If B P and A is nontrivial (not equal to or ), then A B P A . Answer: Yes, f ( x ) = x if x in B, f ( x ) = c , if x not in B, for c a fixed element of A if x not in B 3. If B P and A is nontrivial, then A B P A . Answer: f ( x ) = x if x negationslash B and f ( x ) = a , if x B for some fixed element a A . 4. If A B is NP-complete, A NP and B P , then A must be NP-complete. Answer: True. Since B P , it follows that A B P A . To see this, consider an element x negationslash A . We can map an instance y of A B into an instance z of A as follows. Decide if y is in B . If so, let z =...
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- Spring '07