Consider a Context$-$Free Grammar $($CFG$)$ G$,$ with a language L$($G$)$ that describes a certain pattern

of strings over an alphabet $\backslash$Sigma $.$ Suppose that you have an intuition that L$($G$)$ could also be

described as a regular language.

$1.$ Provide an informal proof to explain why not all context$-$free languages can be described

as regular languages. Use the pumping lemma for regular languages as a part of your

argument.

$2.$ Suppose you found a Regular Expression $($RE$)$ R that seemingly describes the same

language as our CFG G$.$ Sketch a Deterministic Finite Automaton $($DFA$)$ corresponding to

this Regular Expression R$.$

$3.$ In light of your response to the first part of the question, explain under which conditions

our CFG G could indeed be equivalent to the Regular Expression R$.$ What specific

properties would the language L$($G$)$ need to have?