Basically I'm wondering if the concatenation of two equal length string is context free. I've seen multiple proofs of this online using PDAs but we aren't covering them in my automata course and my professor says they aren't needed for the proof. Any help would be greatly appreciated!
This is the full question:
For any two languages $A, B$ over $\Sigma$, define $A \diamond B := \{xy\mid x \in A, y \in B, |x| = |y|\}$. Show that if $A, B$ are regular then $A \diamond B$ is context-free
One way to show this is as follows. Make another copy of your alphabet, $\Sigma'$. Let $B'$ be the language $B$ over $\Sigma'$. The language $AB'$ is regular. By intersecting this with an appropriate context-free language, we can get the language of all words $xy'$, where $|x|=|y'|$, $x \in A$, and $y' \in B'$. Finally, we can remove the tags, replacing $\Sigma'$ with $\Sigma$.
Another way is using grammars. Construct an automaton $D(A)$ for $A$ and $D(B^R)$ for the reverse of $B$. Construct a grammar whose nonterminals encode a pair of states $(q_a,q_b)$, where $q_a$ is a state in $D(A)$ and $q_b$ is a state in $D(B^R)$. There will be productions $(q_a,q_b) \to \sigma_a (q'_a,q'_b) \sigma_b$ which correspond to transitions in both automata simultaneously, and productions $(q_a,q_b) \to \epsilon$ for all pairs of final states.
Yet another way is using NPDAs. The NPDA first simulates an automaton $D(A)$ for $A$, pushing a symbol $X$ into the stack whenever it reads a letter. It then nondeterministically transitions to simulating an automaton $D(B)$ for $B$ (only when reaching a final state of $D(A)$!), popping a symbol $X$ from the stack whenever it reads a letter. You should arrange matters so that the NPDA accepts only if it reaches a final state of $D(B)$, and moreover exhausts the stack. (You can do this using a special symbol found at the bottom of the stack.)
I'll let you work out the details.
