GATE CS 2019 Q.29 (CS Q.19): Parsing and Syntax Analysis (LL and LR)

Consider the grammar given below:

\begin{aligned} S &\to Aa \\ A &\to BD \\ B &\to b \mid \epsilon \\ D &\to d \mid \epsilon \end{aligned}

Let $a, b, d,$ and $\$$ be indexed as follows:

Symbol	$a$	$b$	$d$	$\$$
Index	3	2	1	0

Compute the $\text{FOLLOW}$ set of the non-terminal $B$ and write the index values for the symbols in the $\text{FOLLOW}$ set in descending order. (For example, if the $\text{FOLLOW}$ set is $\{a, b, d, \$\}$ , then the answer should be 3210)

Answer: ________.

Your answer

Official answer and solution

Official answer: 31
From: Official GATE 2019 answer key

Solution

Calculate FIRST and FOLLOW sets for the grammar symbols according to standard compiler parsing rules.

The given grammar is:

S \to Aa

A \to BD

B \to b \mid \epsilon

D \to d \mid \epsilon

Step 1: Compute FIRST sets:

$\text{FIRST}(D) = \{d, \epsilon\}$
$\text{FIRST}(B) = \{b, \epsilon\}$
$\text{FIRST}(A)$ : Since $A \to BD$ , $\text{FIRST}(A)$ includes $\text{FIRST}(B) \setminus \{\epsilon\} = \{b\}$ . Because $B \Rightarrow^* \epsilon$ , $\text{FIRST}(A)$ also includes $\text{FIRST}(D) = \{d, \epsilon\}$ . Thus $\text{FIRST}(A) = \{b, d, \epsilon\}$ .

Step 2: Compute FOLLOW sets:

For start symbol $S$ : $\text{FOLLOW}(S) = \{\$\}$ .
For non-terminal $A$ : In $S \to Aa$ , $A$ is followed by terminal $a$ . Hence $\text{FOLLOW}(A) = \{a\}$ .
For non-terminal $B$ $B$ : In $A \to BD$ $A \to B D$ , $B$ $B$ is followed by $D$ $D$ .
- $\text{FOLLOW}(B)$ contains $(\text{FIRST}(D) \setminus \{\epsilon\}) = \{d\}$ .
- Since $D \Rightarrow^* \epsilon$ , $\text{FOLLOW}(B)$ also contains everything in $\text{FOLLOW}(A) = \{a\}$ .

Therefore:

\text{FOLLOW}(B) = \{a, d\}

The given symbol indices are:

a: 3, \quad b: 2, \quad d: 1, \quad \$: 0

For the symbols in $\text{FOLLOW}(B) = \{a, d\}$ , the indices are $3$ and $1$ .
Arranging them in descending order gives 31.

Final answer

Common trap

Forgetting that D can derive epsilon, which causes symbol 'a' to be missed from FOLLOW(B).

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

Source

Paper: GATE 2019 Computer Science and Information Technology (CS)
Question: 29 of 65, printed as CS Q.19
Section: Computer Science and Information Technology

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