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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 Σ (that is, not equal to ∅ or Σ ∗ ). 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 Σ ∗ ), 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 NPcomplete, A ∈ NP and B ∈ P , then A must be NPcomplete. 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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This note was uploaded on 08/12/2009 for the course CS 420 taught by Professor Nancylynch during the Spring '07 term at New Mexico Junior College.
 Spring '07
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