Which of the following grammars generates the language L? L = {WWR | W ∈ {a,…

Concept

A context-free grammar for even-length palindromes can build strings from the centre outward.

Starting with ε, each recursive step must place the same terminal on both ends; this preserves reversal symmetry and increases the length by two.

Application

1. Let W = x1x2…xn. Then WWR = x1x2…xnxn…x2x1, which is an even-length palindrome.

2. The base derivation S ⇒ ε represents W = ε.

3. The productions S → aSa and S → bSb add a matching pair around the current string, so every derived terminal string remains an even-length palindrome.

4. Conversely, remove the equal outer symbols from any nonempty even-length palindrome. Repeating this process reaches ε, so the grammar derives every string of the required form.

Cross-check and contrast

• Using only aSb and bSa places different symbols at the two ends at every step.

• Allowing all four wrappers aSb, bSa, aSa, and bSb permits arbitrary pairs of outer symbols and therefore generates strings beyond the target language.

• A grammar with no ε or other terminal-only base production cannot finish a derivation to a terminal string.

Result

Therefore, the required grammar is S → aSa | bSb | ε.