GATE CS 2015 Set 2 Q.45 (CS Q.35): Regular Expressions, Finite Automata and Regular Languages

Consider the alphabet $\Sigma = \{0, 1\}$ , the null/empty string $\lambda$ and the sets of strings $X_0, X_1$ , and $X_2$ generated by the corresponding non-terminals of a regular grammar. $X_0, X_1$ , and $X_2$ are related as follows:

X_0 = 1 X_1

X_1 = 0 X_1 + 1 X_2

X_2 = 0 X_1 + \{\lambda\}

Which one of the following choices precisely represents the strings in $X_0$ ?

$10(0^* + (10)^*)1$

$10(0^* + (10)^*)^*1$

$1(0 + 10)^*1$

$10(0 + 10)^*1 + 110(0 + 10)^*1$

Official answer and solution

Official answer: C
From: Official GATE 2015 answer key

Solution

Solve the system of regular grammar equations using Arden's Theorem.

The system of equations is:

$X_0 = 1 X_1$
$X_1 = 0 X_1 + 1 X_2$
$X_2 = 0 X_1 + \{\lambda\}$

Substitute equation (3) into equation (2):

X_1 = 0 X_1 + 1(0 X_1 + \lambda)

X_1 = 0 X_1 + 10 X_1 + 1

X_1 = (0 + 10) X_1 + 1.

Applying Arden's Theorem ( $X = AX + B \implies X = A^*B$ since $\lambda \notin A$ ):

X_1 = (0 + 10)^* 1.

Now substitute $X_1$ into equation (1):

X_0 = 1 X_1 = 1(0 + 10)^* 1.

This matches option C.

Final answer

Common trap

Incorrectly distributing $(0 + 10)^*$ into $0^* + (10)^*$ , which is not equivalent to $(0 + 10)^*.$

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

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 2
Question: 45 of 65, printed as CS Q.35
Section: Computer Science and Information Technology

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