Motivation
Consider the following languages, are they context-free?
- $\{x \# y: x \neq y\}$
- $\{x y: |x|=|y|, x \neq y\}$
- $\{x \# y: |x|=|y|, x \neq y\}$
- $\{x y: |x|=|y|,d(x,y)>1\}$
- $\{x x\}$
The first three are explained here, the fourth one is this question, the last one is well-known.
I'm wondering whether there is an algorithm to solve this kind of problem in general.
Question
Given two strings $x,y$, let $\operatorname{zip}(x,y)$ denote the string $(x_1,y_1)(x_2,y_2)\dots(x_n,y_n)$. Note that letters in $\operatorname{zip}(x,y)$ are pairs. If one string is shorter, we pad it with an extra blank symbol. For example, $\operatorname{zip}(aab,cd)=(a,c)(a,d)(b,blank)$.
Is the following problem decidable?
Given a regular language $L$, is $ \{x y: \operatorname{zip}(x,y) \in L\}$ context-free?
If the answer is positive, we can solve the five problems from the motivation section by picking a suitable $L$:
- $\{x x\}$ can be written using $L=((a_1,a_1)+\dots+(a_n,a_n))^{\ast}$ where $a_i$ are all letters of the alphabet except blanks.
- $\{x y: |x|=|y|, x \neq y\}$ can be written using $L$ which checks that in the pairs $(a,b)$ there is at least one mismatch $a \neq b$ and there are no blanks.
- $\{x y: |x|=|y|,d(x,y)>1\}$ is similar, the language $L$ checks if there are at least two mismatches and no blanks.
- $\{x \# y: |x|=|y|, x \neq y\}$ is checking for at least one mismatch, and then expects a single symbol $(\#, blank)$.
- $\{x \# y: x \neq y\}$ is checking that the left component of the last character is $\#$, that there is at least one mismatch, and unlike the previous examples blanks are allowed when checking for a mismatch.
