regular grammar

Question

What will be regular grammar for given regular expression
$r=(ab^*+b)^*ab$

Comments

$S_{1}\rightarrow aS_{2}|bS_{2}|ab$

$S_{2}\rightarrow bS_{2}|S_{1}$

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or
$S_{1}\rightarrow aS_{2}|S_{2}$

$S_{2}\rightarrow bS_{2}|S_{1}|ab$

Answer 1

Regex can also be written as $\left (a+b \right )^{*}ab$

Corresponding DFA is

and grammar will be

$S \rightarrow bS|aA$
$A \rightarrow aA|bB$
$B\rightarrow λ |aA|bS$