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Regular expressions (in the original computer science sense) can be matched by finite-state automata!

Yes, but filling a grid in a way that satisfies all of the regular expressions is a much harder problem than matching a single regex.

Semi-related: generating fill for regular crosswords is NP-complete: https://www.sciencedirect.com/science/article/abs/pii/S03043...

Well, at this scale, it would take longer to set up a SAT solver than it would for the SAT solver to list all valid solutions.