Question

If the production rules are given as:
\( S \to XY | W \)
\( X \to aXb | \epsilon \)
\( Y \to cY | \epsilon \)
\( W \to aWc | bZ | \epsilon \)
\( Z \to bZ | \epsilon \)
Then the language generated by these rules is _______ .

• A. \( \{ a^i b^i \mid i \geq 0 \} \)
• B. \( \{ a^i b^i \mid i \geq 1 \} \)
• C. \( \{ a^i b^j c^k \mid i \geq 0, j \geq 0, k \geq 0 \} \)
• D. \( \{ a^i b^j c^k \mid i = j \text{ or } i = k \} \)

Hint:

When analyzing CFGs, focus on how the productions relate different symbols in the string to identify the structure of the language generated.

Answer 1

The Correct Option is D

The given production rules generate strings where the number of \( a's \), \( b's \), and \( c's \) are related in specific ways. The productions \( X \to aXb \) and \( Y \to cY \) allow for the formation of strings where the number of \( a's \) and \( b's \) are the same, or the number of \( a's \) and \( c's \) are the same, as per the final production rules. Thus, the language generated is \( \{ a^i b^j c^k \mid i = j \text{ or } i = k \} \).