Prove $\{xy: x \in A \land y \in B \land |x| = |y|\}$ is context-free

Question

This is problem 2.44 from Introduction to the theory of computation by Michael Sipser.

If $A$ and $B$ are languages, define $A \diamond B = \{xy: x \in A \land y \in B \land |x| = |y|\}$

Show that if $A$ and $B$ are regular languages, then $A \diamond B$ is a $CFL$.

My try:

Let us define the following languages:

$$L_1 = \{ x\#y : |x| = |y| \}$$

$$L_2 = \{ x\#y : x \in A \land x\in B \}$$ $L_1$ is context-free, can be proven in a similar way to as done here

$L_2$ is concatenation of regular languages, and hence regular.

Context-free languages are closed under intersection with regular languages, and hence $L_1 \cap L_2 = \{x\#y: x \in A \land y \in B \land |x| = |y|\}$ is context free.

Let us define the homomorphism $h$ such that $h(\#)=\epsilon$ and as the identity homomorphism for all other symbols.

$h(L_1 \cap L_2)=A \diamond B$, and since Context-free languages are closed under homomorphism, we conclude the requested result.

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Does my proof make sense?

Answer 1

There are many ways to prove this. Here is another one, which proves a stronger result: $A \diamond B$ is linear.

Suppose that both $A,B$ are languages over a common alphabet $\Sigma$. Let $\langle Q_A,q_{0A},\delta_A,F_A \rangle$ and $\langle Q_B,q_{0B},\delta_B,F_B \rangle$ be DFAs for $A^R$ (the reverse of $A$) and for $B$. We construct a linear grammar with a nonterminals $\{S\} \cup Q_A \times Q_B$, with the following rules:

• For every $f_A \in Q_A$ and $f_B \in Q_B$, the rule $S \to \langle f_A, f_B \rangle$.
• For every $q_A \in Q_A$, $q_B \in Q_B$, and $\sigma_A,\sigma_B \in \Sigma$, the rule $\langle \delta_A(q_A,\sigma_A), \delta_B(q_B,\sigma_B) \rangle \to \sigma_A \langle q_A, q_B \rangle \sigma_B$.
• The rule $\langle q_{0A},q_{0B} \rangle \to \epsilon$.

In fact, your proof also shows that $A \diamond B$ is linear, since $\{a^n \# b^n : n \geq 0 \}$ is easily seen to be linear, and all the closure properties you use also hold for linear languages.

Answer 2

$A$ and $B$ are languages, when $A$ and $B$ both are regular

• Now, take $A=a^{n}|n>=0$
• take $B=b^{n}|n>=0$

So, $A.B=a^{n}b^{n}|n>=0$ Here, $A.B$ is CFL

But it may not be case. Say, $A=(a+b)^{n}$ and $B=(a+b)^{n}$ $A.B=(a+b)^{2n}$

-Here it will be regular.Hence we can say , $A.B$ will be regular or CFL.