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Unformatted text preview: CS 341: Foundations of Computer Science II Prof. Marvin Nakayama Homework 10 Solutions 1. If A ≤ m B and B is a regular language, does that imply that A is a regular language? Answer: No. For example, define the languages A = { n 1 n  n ≥ } and B = { 1 } , both over the alphabet Σ = { , 1 } . Define the function f : Σ ∗ → Σ ∗ as f ( w ) = braceleftBig 1 if w ∈ A, if w negationslash∈ A. Observe that A is a contextfree language, so it is also Turingdecidable. Thus, f is a computable function. Also, w ∈ A if and only if f ( w ) = 1 , which is true if and only if f ( w ) ∈ B . Hence, A ≤ m B . Language A is nonregular, but B is regular since it is finite. 2. Show that A TM is not mapping reducible to E TM . In other words, show that no computable function reduces A TM to E TM . (Hint: Use a proof by contradiction, and facts you already know about A TM and E TM .) Answer: Suppose for a contradiction that A TM ≤ m E TM via reduction f . This means that w ∈ A TM if and only if...
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This note was uploaded on 02/22/2010 for the course CS 881 taught by Professor H.f. during the Spring '10 term at Shahid Beheshti University.
 Spring '10
 H.F.
 Computer Science

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