Question

Show that A is Turing-recognizable if and only if A is m-reducible to ATM (the Acceptance problem for Turing machines).

A) A is m-reducible to ATM implies A is Turing-recognizable
B) A is Turing-recognizable implies A is m-reducible to ATM
C) Both A is m-reducible to ATM and A is Turing-recognizable
D) Neither A is m-reducible to ATM nor A is Turing-recognizable

Answer 1

A is Turing-recognizable if and only if A is m-reducible to ATM.

Step-by-step explanation:

The statement “A is Turing-recognizable if and only if A is m-reducible to ATM” is true. Both parts of the statement hold:

A) If A is m-reducible to ATM, then A is Turing-recognizable. This means that if we can solve ATM, we can also solve problem A. Since ATM is Turing-recognizable, A must also be Turing-recognizable.

B) If A is Turing-recognizable, then A is m-reducible to ATM. This means that if we can solve problem A, we can reduce it to ATM. Since ATM is Turing-recognizable, we can solve A by reducing it to ATM and using the solution to ATM.

Therefore, option C) Both A is m-reducible to ATM and A is Turing-recognizable is the correct answer.