This question is about semi$-$decidability, decidability, and reduction.

$($a$)$ Which of the following problems are semi$-$decidable? No explanation needed.

i$.$ there does not exist a number $n$ such that $d=$ulcornernurcorner

ii$.\{\begin{array}{cc}\text{u}\text{l}\text{c}\text{o}\text{r}\text{n}\text{e}\text{r}\text{p}\text{u}\text{r}\text{c}\text{o}\text{r}\text{n}\text{e}\text{r}& |\end{array}$ there is a dinD such that $\{$:$([[)p]{]}^{\text{W}\text{H}\text{H}\text{L}\text{E}}(d)=\}$

iii. $\{dinD|din\{ulcorner1$urcorner$\}\}$

iv$.$ Halting Problem

$($b$)$ For the following sets Asube WHILE$-$data, state whether they are WHILE$-$decidable

and explain your answer. In cases where $A$ is decidable this explanation should

consist of a description of the decision procedure. Recall that ulcornerpurcorner denotes the

encoding of WHILE$-$program $p$ in $D.$

i$.A=\{ulcornerpurcorner|[p{]}^{\text{W}\text{H}\text{I}\text{L}\text{E}}(d)=c\}$ where $c,$dinD are fixed.

ii$.$ with input nil terminates in $k$ steps or less