Prompt Title: palindrome cfg

Here's a context-free grammar (CFG) for the language $L(G) = \{wcw^R : w \in \{a, b\}^*\}$, where $w^R$ denotes the reverse of $w$:

cssCopy code
S -> aSa | bSb | A | B | ε
A -> a | ε
B -> b | ε

Explanation:

• $S$ is the start symbol. It generates strings of the form $wcw^R$.
• $A$ and $B$ are non-terminals used to generate the prefix and suffix of $w$, respectively. They can generate any combination of $a$'s and the empty string $\varepsilon$ (when $w$ is empty).
• The productions $S \rightarrow aSa$ and $S \rightarrow bSb$ generate strings of the form $wcw^R$ by recursively adding the same character to both ends of $w$.
• The production $S \rightarrow A$ generates the prefix $w$ of the string, allowing for the possibility of an empty prefix.
• The production $S \rightarrow B$ generates the suffix $w^R$ of the string, allowing for the possibility of an empty suffix.
• The productions $A \rightarrow a$ and $B \rightarrow b$ generate $a$ and $b$, respectively, contributing to the prefix and suffix of $w$.
• The productions $A \rightarrow \varepsilon$ and $B \rightarrow \varepsilon$ allow for the possibility of an empty prefix or suffix, respectively, when $w$ is empty.

This CFG ensures that any string generated by it will have the form $wcw^R$, where $w$ can be any string of $a$'s and $b$'s, including the possibility of an empty string.