Matching regular expressions using regular expressions
Even if we require that the expressions are "human-readable", and the conversion between the standard notation and the new notation is fairly easy to compute, the answer is "yes".

For example, we can modify the standard notation as follows to obtain "compressed" regular expressions:


You are allowed to remove any prefix that consists of a sequence of ('s
You are allowed to remove any suffix that consists of a sequence of )'s


That is, ((a|b)*c)de(f|g) can be expressed in the "compressed" notation using, for example, any of the following forms: a|b)*c)de(f|g or ((a|b)*c)de(f|g or (a|b)*c)de(f|g).

The compressed notation is unambiguous in the sense that you can recover a semantically equivalent expression by simply adding a sufficient number of leading and trailing parentheses to make everything balanced. Of course, a|b)*c)de(f|g might represent ((a|b)*c)de(f|g) or (((a|b)*c)de(f|g)), but adding extra parentheses does not change the language defined by the expression.

The "compressed" notation is a regular language.

And yes, this is cheating.