Show that if Q is both semi-decidable and co-semi-decidable, then it is decidable.

Question

Question

engineering Computer-Science

Answer 1

Answer #1

Proof:

Q is a language.

(=>) We already know decidable implies semi-decidable(CE).

->If Q is decidable, then compliment of M is decidable by flipping accept/reject.

->so Q is co-semi-decidable(co-CE).

Now the other way,

(<=) We have Turing machine M that recognizes Q and Turing machine M' that recognizes complement of Q.

->on input x, simulate M, M' in parallel

->if M accepts, then accept; if M' accepts, then reject.

Answer 2

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