Call a language $ L$ Hamming connected iff, for every pair of strings $ x, y \in L^2$ , where $ |x|=|y|$ , $ x$ may be transformed into $ y$ by a sequence of single symbol in-place replacements, so that after every replacement, the resulting string is in $ L$ .
Is Hamming connectivity a decidable property for regular languages? Note that the brute-force algorithm of checking all string lengths fails, as $ L$ may contain infinitely many strings.
Example:
Define $ L$ on the binary alphabet to be the set of strings that do not contain $ 101$ or $ 010$ as a substring. $ L$ is regular. We now show that $ L$ is Hamming connected.
Informal Proof: We can grow internal “runs” out to the ends to transform any string in $ L$ into the all 1’s or all 0’s string without creating a forbidden substring, e.g., 001100 -> 011100 -> 111100 -> 111110 -> 111111. This transformation can also be done in reverse to reach any string. Thus, $ L$ is Hamming connected.
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