$(10$ points$)$ In class we cover the proof of Theorem $5\mathrm{.}3$ that the language

$RE{G}_{TM}$:$=\{($:$M$:$)|MisaTM$ and $L(M)is$ regular $\}$

is undecidable. We showed that $RE{G}_{TM}$ is not co$-$recognizable.

$($a$)(2$ points$)$ These two arguments $($not decidable and not co$-$recognizable$)$ can be

captured by a single mapping reduction. What is the mapping reduction? $($Give a

description of the function $f.)$

$($b$)(6$ points$)$ Give a different mapping reduction from $A_{TM}$ to show that $RE{G}_{TM}$ is

not recognizable.

$($c$)(1$ point$)$ Describe these two mapping reductions using the ${?}_m$ notation.

$($d$)(1$ point$)$ What is your final classification of $RE{G}_{TM}$ in the Chomsky hierarchy?

$($Please please DON'T USE AI Thank you! I promise I will give you thumbs up if you answer this correctly$)$