4. (15 MARKS) For each of the following operations, give a high-level explanation of why the...

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4. (15 MARKS) For each of the following operations, give a high-level explanation of why the decidable languages are closed u

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Answer 1

Answer #1

(a) The proof for decidable languages is closed under Concatenation.

Let L1 and L2 be two Turing Decidable languages. Given an input w, use non determinism

and guess a partition w(say w = xy). Now run the respective Turing machines of L1 and L2

on x and y respectively. If both accepts then accept else reject.

(b) The proof for decidable languages is closed under Intersection.

Run the TMs of both the languages on the given input. accept if and only if both the machines

accept. In the case of intersection we can run the TMs of L1 and L2 one after the other.

(c) The proof for decidable languages is closed under Complementation.

Decidable languages are closed under complementation. To design a machine for the

complement of a language L, we can simulate the machine for L on an input. If it accepts

then reject and vice versa.

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