GATE CS 2017 Set 2 Q.39 (CS Q.39): Regular Expressions, Finite Automata and Regular Languages

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

$\delta$	$\epsilon$	$a$	$b$
$\to q_0$	$\{q_2\}$	$\{q_1\}$	$\{q_0\}$
$q_1$	$\{q_2\}$	$\{q_2\}$	$\{q_3\}$
$q_2$	$\{q_0\}$	$\emptyset$	$\emptyset$
$q_3$	$\emptyset$	$\emptyset$	$\{q_2\}$

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

$\emptyset$

$\{q_0, q_1, q_3\}$

$\{q_0, q_1, q_2\}$

$\{q_0, q_2, q_3\}$

Official answer and solution

Official answer: C
From: Official GATE 2017 answer key

Solution

Compute the extended transition function $\hat{\delta}(q_2, aba)$ by computing $\epsilon$ -closures at each step.

First, compute $\epsilon\text{-closure}$ of the starting state $q_2$ :

$q_2 \xrightarrow{\epsilon} q_0$
$q_0 \xrightarrow{\epsilon} q_2$
Thus, $\epsilon\text{-closure}(q_2) = \{q_0, q_2\}$ .

Now trace the string $aba$ symbol by symbol:

Read first symbol ' $a$ ':
- $\delta(q_0, a) = \{q_1\}$
- $\delta(q_2, a) = \emptyset$
  Union of transitions: $\{q_1\}$ .
  Now compute $\epsilon\text{-closure}(\{q_1\})$ :
- $q_1 \xrightarrow{\epsilon} q_2 \xrightarrow{\epsilon} q_0$
  So $\epsilon\text{-closure}(\{q_1\}) = \{q_0, q_1, q_2\}$ .
Read second symbol ' $b$ ' from $\{q_0, q_1, q_2\}$ ${q_{0}, q_{1}, q_{2}}$ :
- $\delta(q_0, b) = \{q_0\}$
- $\delta(q_1, b) = \{q_3\}$
- $\delta(q_2, b) = \emptyset$
  Union of transitions: $\{q_0, q_3\}$ .
  Now compute $\epsilon\text{-closure}(\{q_0, q_3\})$ :
- $\epsilon\text{-closure}(q_0) = \{q_0, q_2\}$
- $\epsilon\text{-closure}(q_3) = \{q_3\}$
  So current states after ' $ab$ ' are $\{q_0, q_2, q_3\}$ .
Read third symbol ' $a$ ' from $\{q_0, q_2, q_3\}$ ${q_{0}, q_{2}, q_{3}}$ :
- $\delta(q_0, a) = \{q_1\}$
- $\delta(q_2, a) = \emptyset$
- $\delta(q_3, a) = \emptyset$
  Union of transitions: $\{q_1\}$ .
  Now compute $\epsilon\text{-closure}(\{q_1\}) = \{q_0, q_1, q_2\}$ .

Therefore, $\hat{\delta}(q_2, aba) = \{q_0, q_1, q_2\}$ , matching option C.

Final answer

Common trap

Forgetting the initial $\epsilon$ -closure of state $q_2$ before reading the first input symbol ' $a$ '.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2017 Computer Science and Information Technology (CS) Session 2
Question: 39 of 65, printed as CS Q.39
Section: Computer Science and Information Technology

Report an error in this question

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.

Question

Solution

Source