Problem $27.$ Consider the following algorithm which takes a grammar $G=(V,,P,S)$ as input and returns True or False. As usual, $V$ is the set of non$-$terminals, $$ is the alphabet, $P$ is the set production rules, and SinV is the initial non$-$terminal.

Tlarr$\frac{O}{.}$

while $P$ contains a production rule of the form $A,$ where $A!$inT and $()**$ do TlarrT$\{A\}.$

end while

If SinT then return True, else return False.

What interesting property of $G$ is this algorithm testing? Prove your answer.

$($Homework $3$ Problem $5)$ Make minimal changes to the above algorithm so that your modified algorithm returns True if and only if $L(G)\frac{O}{?}.$ Prove that your algorithm is correct.