GATE CS 2016 Set 1 Q16 — Context-free Grammars and Push-down Automata

The given grammar is:

Let's analyze the language generated by this grammar:

• The production $S \rightarrow \varepsilon$ allows generating the empty string ($\varepsilon$).
• The productions $S \rightarrow aS$ and $S \rightarrow bS$ allow us to prepend either '$a$' or '$b$' to any string generated by $S$.

By repeatedly applying these productions, we can generate any arbitrary sequence of '$a$'s and '$b$'s of any length. For example, to generate the string $bba$:

Thus, the language generated by the grammar is the set of all possible strings over the alphabet $\{a, b\}$, which is represented by $\{a, b\}^*$.

Why other options are incorrect:

• Option A: $\{ a^nb^m \mid n,m \geq 0\}$ represents strings where all $a$'s must precede all $b$'s. However, the grammar can generate strings like $ba$, which do not belong to this language.
• Option B: This language only contains strings with an equal number of $a$'s and $b$'s. The grammar can easily generate strings like $a$ or $aab$, which have unequal counts.
• Option C: This language does not contain strings like $aba$ or $bab$, which are easily generated by the grammar.