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assign4ans_2 - CS5371 Theory of Computation Homework...

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CS5371 Theory of Computation Homework 4 (Suggested Solution) 1. Ans. Suppose on the contrary that T is decidable. Let R be its decider. Then, the following TM Q is a decider for A TM : Q = “On input h M, w i , 1. Construct a TM M 0 as follows: M 0 = “On input x , 1. If x 6 = 011 , accept . 2. Run M on w . 3. If M accepts w , accept .” 2. Run R to decide if h M 0 i is in T . 3. If yes (i.e., R accepts), accept . 4. Else, reject .” It is easy to check that Q runs in finite steps. Also, in Step 1, M 0 has the property that: (i) If M accepts w , L ( M 0 ) = Σ * , so that h M 0 i ∈ T . (ii) Else, L ( M 0 ) = Σ * - { 011 } , so that h M 0 i / T . So, if Q accepts h M, w i , it must mean that R accepts h M 0 i , which implies that h M 0 i ∈ T , which implies M accepts w . On the other hand, if Q rejects h M, w i , R rejects h M 0 i , which in turn implies that M does not accept w . Thus, Q is a decider for A TM , and a contradiction occurs. So, we conclude that T is undecidable. 2. In the silly Post Correspondence Problem , we see that if a set of dominoes S is in SPCP if and only if S contains a piece whose top string matches exactly the bottom string. Thus,
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