Question $6$:

Accept, Halt, Loop on input $.$ In this problem, we consider the languages

$ACCEP{T}_{}=\{($:$M$:$)|M$ accepts $\}$

$HAL{T}_{}=\{($:$M$:$)|M$ halts $on$ input $\}$

$\frac{L}{O}O{P}_{}=\{($:$M$:$)|M$ loops $on$ input $\}$

It is known from class $($slide $155,$ lecture $22$b$)$ that language $ACCEP{T}_{}$ is undecidable.

$($a$)$ Prove that $HAL{T}_{}$ is undecidable by constructing a mapping reduction $ACCEP{T}_{}{?}_{m}HAL{T}_{}.$

$($b$)$ Prove that $HAL{T}_{}$ is recognizable by describing a recognizer.

$($c$)$ Prove that $\frac{L}{O}O{P}_{}$ is not recognizable.