4.2.b.1

### ATM is undecidable

Theorem 4.11:ATM is undecidable.

Proof idea. Suppose, for a contradiction, that ATM is decidable.

That is, there is a TM H that decides ATM.

Thus, that machine H behaves as follows:

accept if M accepts w

reject if M does not accept w

H(<M,w>) =

Using H as a subroutine, we can construct the following TM D:

D = “On input <M>, where M is a TM:

1. Run H on input <M,<M>>.

2. Do the opposite of what H does.

That is, if H accepts, reject, and if H rejects, accept.”

accept if M does not accept <M>

reject if M accepts <M>

Thus, D(<M>) =