Problem $2$: $L=\{0^n{1}^m,|nm2n+1\}$ is given.

$($a$)$ Construct a grammar that generates all and only strings in $L.$

$($b$)$ Check if your grammar is ambiguous or not. Explain why.

$($c$)$ Generate a PDA which is equivalent to CFG that you generated in $($a$).($gen$-$

erated PDA should accept $L$ by empty stack.$)$

$($d$)$ Is it possible to generate a PDA which accepts the language $L$ by final state?

Explain why.