How can you convert a generalized regular expression to a standard regular expression?

Question

Formally, regular expressions only contain the union, concatenation, and star operations. However, regular languages are also closed under intersection and complementation. This means that a generalized regular expression, which allows the intersection and complementation operators, is equivalent to some standard regular expression without these explicit operations. How could I obtain such a regular expression from a generalized one? I would prefer a way that converts to it directly, instead of converting it to an NFA and then a standard regular expression.

Answer 1

I'm not aware of any simple algorithm. I think the smallest regular expression might be exponentially larger than the corresponding generalized regular expression. Some references to investigate further:

• Is intersection used in regular expression?
• Given regular expression construct regex for the complement language
• https://cs.stackexchange.com/a/57998/755
• Can we say anything about the complement of a regular language?