There’s no reason to say that “ca(t:d)og” is a “more correct” parsing than “(cat):(dog)”. You did hit the nail on the head insofar as that you realized that we as programmers have built strong habits and make assumptions on the basis of those habits. But you didn’t take it to the logical conclusion and didn’t realize that having a text-based syntax to represent regexes is also such a habit/assumption.
In pure theoretical computer science, regular expressions exist as an abstract concept independent from syntax or parsers. They are an “algebra”, which means they are composed of elements connected with operators, but they are not inherently tied to a syntax. In the most fundamental formulation of regular expressions (the one in the Chomsky hierarchy), the only operators are alteration (which modern syntaxes express as “|”), the Kleene star (“*”) and — notably — concatenation, which modern syntaxes simply omit, in a way comparable to how modern mathematics notation omits the multiplication operator when you write “2x”.
In the same way that maths needs rules to define whether “2x²” means “(2x)²” or “2(x²)”, regex syntax needs such rules too. This is called operator precedence. I’m sure you’ve heard that before, but you just might not have realized that the regular expression “ab” has an operator in it because it is typically not written.
Now I’m not going to argue that the operator precedence in maths notation is haphazard or without reason — but it is arbirary. It was arbitrarily chosen to be the most useful to mathematicians using the notation. And it turns out that giving exponentiation higher precedence than (invisible) multiplication (meaning: “2x²” means “2(x²)” rather than “(2x)²”) is more useful.
So coming back to the original example, whether “cat:dog” means “ca(t:d)og” or “(cat):(dog)” is simply a matter of defining the precedence of the “:” operator relative to the concatenation operator. You can argue (and I would agree with you) that one is more useful than the other, and therefore preferable (in the same way that “(cat)|(dog)” is more useful than “ca(t|d)og”), but neither of them is more fundamentally correct or primal or, as you put it, “supposed to formally expand to”.
I agree with the point that precedence is arbitrary. The current version looks like this:
1 Escaped characters
2 []
3 ()
4 * + ? {m,n}
5 :
6 . (implicit concatenation)
7 |
I have some reasons to put it that way. I want : to be somewhat 'atomic'. If you think about '*' or '+' they can be lower in the table as well. Anyway, I will try to put : lower in the next version and see how it goes.
In pure theoretical computer science, regular expressions exist as an abstract concept independent from syntax or parsers. They are an “algebra”, which means they are composed of elements connected with operators, but they are not inherently tied to a syntax. In the most fundamental formulation of regular expressions (the one in the Chomsky hierarchy), the only operators are alteration (which modern syntaxes express as “|”), the Kleene star (“*”) and — notably — concatenation, which modern syntaxes simply omit, in a way comparable to how modern mathematics notation omits the multiplication operator when you write “2x”.
In the same way that maths needs rules to define whether “2x²” means “(2x)²” or “2(x²)”, regex syntax needs such rules too. This is called operator precedence. I’m sure you’ve heard that before, but you just might not have realized that the regular expression “ab” has an operator in it because it is typically not written.
Now I’m not going to argue that the operator precedence in maths notation is haphazard or without reason — but it is arbirary. It was arbitrarily chosen to be the most useful to mathematicians using the notation. And it turns out that giving exponentiation higher precedence than (invisible) multiplication (meaning: “2x²” means “2(x²)” rather than “(2x)²”) is more useful.
So coming back to the original example, whether “cat:dog” means “ca(t:d)og” or “(cat):(dog)” is simply a matter of defining the precedence of the “:” operator relative to the concatenation operator. You can argue (and I would agree with you) that one is more useful than the other, and therefore preferable (in the same way that “(cat)|(dog)” is more useful than “ca(t|d)og”), but neither of them is more fundamentally correct or primal or, as you put it, “supposed to formally expand to”.