hwsoln10 - CS 341 Foundations of Computer Science II Prof...

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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 = { 0 n 1 n | n 0 } and B = { 1 } , both over the alphabet Σ = { 0 , 1 } . Define the function f : Σ Σ as f ( w ) = braceleftBig 1 if w A, 0 if w negationslash∈ A. Observe that A is a context-free language, so it is also Turing-decidable. 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 f ( w ) E TM , which is equivalent to saying w negationslash∈ A TM
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