Consider the following transformation f that maps each pair <M,w> consisting of a

Turing machine M and input w to M to another Turing machine M ,w N〈 〉 .

The TM M ,w N〈 〉 when given an input string x over its input alphabet Σ behaves as follows.

M ,w N〈 〉 simulates the computation of M on w for |x| steps (where |x| is the length of x). If

M does not accept within these steps, then M ,w N〈 〉 accepts its own input x and halts. If M

accepts during these steps, then M ,w N〈 〉 rejects its input x and halts.

1. Suppose that M accepts w. What is L( M ,w N〈 〉 )?

2. Suppose that M does not accept w. What is L( M ,w N〈 〉 )?

3. Use the transformation f to show that the language { <N> | N is a Turing machine

whose language L(N) is infinite } is not recursively enumerable.

4. If you carry out the reduction in the proof of Rice’s theorem for the special case of the

property P: “infinite language”, does this reduction also show that the language

P L = { <N> | N is a Turing machine whose language L(N) is infinite } is not recursively

enumerable? Explain why it does or it does not.

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