Question

Let 𝛿 denotes the transition function and $\hat{\delta }$ denotes the extended transition function of the ε - NFA whose transition table is given below :

𝛿	ε	a	b
$\to q_0$	$\left\{q_2\right\}$	$\left\{q_1\right\}$	$\left\{q_0\right\}$
$q_1$	$\left\{q_2\right\}$	$\left\{q_2\right\}$	$\left\{q_3\right\}$
$q_2$	$\left\{q_0\right\}$	$\varphi$	$\varphi$
$q_3$	$\varphi$	$\varphi$	$\left\{q_2\right\}$

The $\hat{\delta }$ $(q_2,aba)is$

Answer 1

Converting the table to a state diagram, we get,

$\hat{\delta }(q_2,aba)$ = All states reachable from $q_2$ by aba

If aba is broken as ε$\cdot$a ε$\cdot$b ε$\cdot$c Then from $q_2$ we can reach $q_1$ and from there by null transition we can reach state $q_2$ as well as $q_0$.

$\hat{\delta }(q_2,aba)=\{q_0,q_1,q_2\}$