To determine which of the given languages are context-free, we need to understand the structure of each language and analyze whether they can be represented using a context-free grammar (CFG) or generated by a pushdown automaton (PDA).
This language requires recognition of a nested pattern: a string \(w\) followed by a string \(x\), then the reverses of these strings in the same order. This requires checking for two palindromes back-to-back, which is known to be beyond context-free capabilities as it demands more than one stack operation for matching pairs. Hence, this language is not context-free.
This language consists of a string \(w\) followed by its reverse, followed by a string \(x\) followed by its reverse. Each pairing is a palindrome, which is trivial for a CFG to generate individually, as it can independently handle finite nesting levels. Therefore, this language is context-free.
This language consists of a string \(w\), followed by any string \(x\), and then the reverse of \(w\). This pattern is similar to the structure that a single-stack PDA can manage, as it requires recognizing \(w\) and then comparing it after potentially ignoring \(x\). Thus, this language is context-free.
This language consists of a string \(w\), followed by string \(x\), then the reverse of \(x\) and the reverse of \(w\). As this corresponds to a sequence where two strings and their reverses are paired, it can be handled with a context-free grammar by appropriately managing nesting using non-terminals. Hence, this language is context-free.
The correct context-free languages from the options are:
Thus, the correct answer is the combination of the second, third, and fourth options listed.