In the $\epsilon$-NFA, $M=({q_0,q_1,q_2,q_3},{a},\delta,q_0,{q_3})$ where $\delta$ is given in the transition table below, what is the minimum length of string to reach to the final state?

	$\epsilon$	$a$
$q_0$	{$q_1$}	{ }
$q_1$	{$q_2$}	{ }
$q_2$	{ }	{$q_2 , $ $q_3$ }
$q_3$	{}	{}

Initial state is:

$q_0$

From $q_0$, using $\epsilon$ move:

$q_0 \xrightarrow{\epsilon} q_1$

From $q_1$, again using $\epsilon$ move:

$q_1 \xrightarrow{\epsilon} q_2$

So, without reading any input symbol, we can reach:

$q_2$

But final state is:

$q_3$

Now from $q_2$, on input symbol $a$:

$q_2 \xrightarrow{a} {q_2,q_3}$

So, after reading one $a$, we can reach final state $q_3$.

Therefore, the minimum string is:

$a$

Length of $a$ is:

$1$