Let P be the language of palindromes over the alphabet Σ = {0, 1}. and let P‘ be the subset of the palindromes with different numbers of 0s and 1s

Question

Let P be the language of palindromes over the alphabet Σ = {0, 1}. and let P‘ be the subset of the palindromes with different numbers of 0s and 1s. Is P' context-free? I know that for the language of the set of palindromes with the same numbers of 0s and 1s is not context-free.

Answer 1

Intuitively you cannot check palindromicity and (un)equality of numbers using a single pushdown, so also $P'$ must be non-context-free.

Formally proving unequal-repeating-numbers to be non-context-free is quite a challenge, see How to prove L := { a^n b^n c^m | n,m >= 0 & n != m } is not context-free? for the language $\{ a^nb^nc^m \mid n\neq m \}$. For that language one needs a special $p!$ trick, as well as Ogden, a stronger form of the classical pumping lemma.

If one intersects your language $P'$ of strings that are palindromes and have unequal numbers of $0,1$ with $0^*(11)^*0^*$ one obtains the language $\{ 0^m 1^{2n} 0^m \mid m\neq n \}$. It seems reasonable to attack that language with similar methods. Then it would follow that $P'$ itself is not context-free as context-free languages are closed under intersection with regular languages.