The minimum DFA does not correspond in any particular sense to a minimal regular expression.

Of course not - my bad!

Yes they do. For reference, check Sipser.

Could you give a page number? I skimmed the regular expression section and couldn't find it. Also, while it is self-evident to me that there is a minimal form for each regex, the construction for finding one isn't immediately obvious to me (I can surely enumerate several ways of eliminating redundancy). Beyond that, I'm not sure a minimal form would minimize character length.

Of (a \star b \star ) \star and ( a ∪ b ) \star, which is the canonical form? The transformation from the former to the latter is nontrivial as a \star b \star is not equivalent with a ∪ b.

Of course, there is a shortest equivalent regular expression, but in general it's PSPACE-complete to find it: https://en.wikipedia.org/wiki/PSPACE-complete#Regular_expres...

It remains PSPACE-complete if you're given a corresponding DFA: "Jiang and Ravikumar [7] show moreover that the minimization problem for ... regular expressions remains PSPACE-complete, even when specifying the regular language by a dfa" from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.60....

In that sense, the grandparent is incorrect. It is unknown, in both practice and theory, how to find the shortest equivalent regex.

It's not unknown, it just takes a long time.

I think you should think deeper on the idea of "unknown"

I don't understand what you mean. Regex minimization is in PSPACE, so take any PSPACE complete problem, eg Q-Sat, do the reduction and apply the algorithm. Q-Sat can be solved by recursively enumerating all possibilities. So there is a method for minimizing regular expressions. It just takes a long time to run. If you have time you can probably think of an algorithm that works directly on the regex problem without doing the Q-Sat thing. It might even be faster.

Heck, there might be an algorithm that works really well in practice. People who do model checking and verification laugh PSPACE completeness in the face and are not even really concerned with undecidability.

I would be very interested in finding an approach that works well in practice.

Okay, fine, but in this case N=7. What are the constant factors?