Consider the following languages over the alphabet $\Sigma = \{0, 1, c\}$: $$\begin{aligned} L_1 &= \{0^n 1^n \mid n \ge 0\} L_2 &= \{w c w^r \mid w \in \{0, 1\}^*\} L_3 &= \{w w^r \mid w \in \{0, 1\}^*\} \end{aligned}$$ Here, $w^r$ is the reverse of the string $w$. Which of these languages are deterministic Context-free languages?

A language is a Deterministic Context-Free Language (DCFL) if it can be accepted by a Deterministic Pushdown Automaton (DPDA).

1. $L_1 = \{0^n 1^n \mid n \ge 0\}$: A DPDA can push all 0's onto the stack and then pop a 0 for each 1 seen. Since there is no ambiguity about where the 0s end and 1s begin, $L_1$ is a DCFL.

2. **$L_2 = \{w c w^r \mid w \in \{0, 1\}^*\}$: The marker symbol $c$ clearly indicates the midpoint of the string. The DPDA can push symbols of $w$ onto the stack, transition deterministically upon reading $c$, and then compare subsequent symbols with the stack to match $w^r$. Thus, $L_2$ is a DCFL**.

3. **$L_3 = \{w w^r \mid w \in \{0, 1\}^*\}$: $L_3$ consists of even-length palindromes without a middle marker. A pushdown automaton cannot deterministically guess where $w$ ends and $w^r$ begins; it requires non-deterministic guessing. Thus, $L_3$ is a non-deterministic CFL and not** a DCFL.

Therefore, only $L_1$ and $L_2$ are deterministic context-free languages, which corresponds to option (C).