Question 51

A language is context-free if and only if it can be recognized by a pushdown automaton (PDA), which uses a single LIFO stack and can verify nested (last-in, first-out) dependencies.

Analyze each language:

• Option A: $\{w x w^R x^R \mid w, x \in \{0,1\}^*\}$.
  Here $w$ is matched with $w^R$ and $x$ is matched with $x^R$, but they overlap in cross-serial fashion: $w$ starts first, then $x$, then $w^R$ finishes before $x^R$. This requires matching two independent pairs in an interleaved order, which cannot be handled by a single stack. This language is not context-free.

• Option B: $\{w w^R x x^R \mid w, x \in \{0,1\}^*\}$.
  This is the concatenation of two even-length palindrome languages: $L_{\text{pal}} \cdot L_{\text{pal}}$, where $L_{\text{pal}} = \{u u^R \mid u \in \{0,1\}^*\}$. Since $L_{\text{pal}}$ is context-free and CFLs are closed under concatenation, this language is context-free. Option B is correct.

• Option C: $\{w x w^R \mid w, x \in \{0,1\}^*\}$.
  This is the set of all odd-length or general palindromes with an arbitrary center $x$. A PDA can push $w$ nondeterministically, read $x$ without touching the stack, and then pop and verify against $w^R$. This language is context-free. Option C is correct.

• Option D: $\{w x x^R w^R \mid w, x \in \{0,1\}^*\}$.
  Notice that $w x x^R w^R = (w x)(w x)^R$. Letting $u = w x$, this is simply $\{u u^R \mid u \in \{0,1\}^*\}$, which is the standard language of even-length palindromes. Hence it is context-free. Option D is correct.