TITLE: How to prove two regular expressions are identical in mathematical way?
QUESTION [8 upvotes]: I'm currently working on "regular expression" exercises in the textbook ("An Introduction to Formal Languages and Automata"), and the problem that I'm facing is, most of the time, my solution is different than the book's ones. However, I realize that they could both express the same language, but I couldn't find a way to prove it. I wonder if is there a mathematical way (like induction) to prove that the two regular expression are actually identical? Any suggestion?
Thank you,

REPLY [6 votes]: Salomaa (1966) gave two complete proof systems for proving that two regular expressions are equivalent ("identical"). If you can't access the link above, there is a description in Kozen's lecture notes. Kozen himself gave another complete proof system.
As mentioned above, it is PSPACE-complete whether a regular expression generates all possible strings (this is a particular case of regular expression equivalence known as universality). This doesn't necessarily mean that the proofs are necessarily long, only that they are hard to find.