How to determine if a regex is orthogonal to another regex?

Question

I guess my question is best explained with an (simplified) example.

Regex 1:

^\d+_[a-z]+$

Regex 2:

^\d*$

Regex 1 will never match a string where regex 2 matches. So let's say that regex 1 is orthogonal to regex 2.

As many people asked what I meant by orthogonal I'll try to clarify it:

Let S1 be the (infinite) set of strings where regex 1 matches. S2 is the set of strings where regex 2 matches. Regex 2 is orthogonal to regex 1 iff the intersection of S1 and S2 is empty. The regex ^\d_a$ would be not orthogonal as the string '2_a' is in the set S1 and S2.

How can it be programmatically determined, if two regexes are orthogonal to each other?

Best case would be some library that implements a method like:

/**
 * @return True if the regex is orthogonal (i.e. "intersection is empty"), False otherwise or Null if it can't be determined
 */
public Boolean isRegexOrthogonal(Pattern regex1, Pattern regex2);

Answer 1

By "Orthogonal" you mean "the intersection is the empty set" I take it?

I would construct the regular expression for the intersection, then convert to a regular grammar in normal form, and see if it's the empty language...

Then again, I'm a theorist...