For a string \(w\), we define \(w^R\) to be the reverse of \(w\). For example,…

Summary: All four given languages are context-free. Below are short justifications or grammars for each.

• { w x w^R x^R | w,x ∈ {0,1}* }: This equals { y y^R | y ∈ {0,1}* } (take y = w x), the set of even-length palindromes. A simple CFG is S -> ε | 0 S 0 | 1 S 1, so it is context-free.

• { w w^R x x^R | w,x ∈ {0,1}* }: This is the concatenation of two languages of the form {z z^R}. Each {z z^R} is context-free (same CFG as above), and CFLs are closed under concatenation, so this language is context-free.

• { w x w^R | w,x ∈ {0,1}* }: Generate the arbitrary middle x and then wrap matching symbols for w. For example, grammar: S -> T | 0 S 0 | 1 S 1 and T -> ε | 0 T | 1 T. This produces exactly strings of the form w x w^R, hence context-free.

• { w x x^R w^R | w,x ∈ {0,1}* }: The middle x x^R is an even palindrome; generate it first and then symmetrically wrap w and w^R. For example, let M -> ε | 0 M 0 | 1 M 1 (generates x x^R), and S -> M | 0 S 0 | 1 S 1. This CFG yields all strings of the given form, so the language is context-free.

Conclusion: Each language can be described by a context-free grammar (or obtained by CFL closure operations), so all four languages are context-free.