[toastking]

I always saw it as his way of teaching of grammars. But I think it is kind of bad that the grammar isn't actually creating a solvable system.

[ineedasername]

it is kind of bad that the grammar isn't actually creating a solvable system

But that was precisely the point. That given a system with a set of rules, the space of all possible problems may be larger than the space of all solutions, i.e., not all problems have a solution.

To say it's useful to be able to determine if a problem is solvable is a vast understatement.