The book and class leave it at that and proceed onto the limits of computability, which is the real point of the material. But there’s a natural question that isn’t presented in the book and which I never thought to ask:

Finite State Automata Regular Expressions
Pushdown Automata Context Free Grammars
Turing Machines ?

While we know that there are many models that equal Turing Machines, we could also construct other models that equal FSAs or PDAs. Why are RegExs and CFGs used as the parallel models of computation? With the machine model, we were able to just add a stack to move up at each level – is there a natural connection between RegExs and CFGs that we extrapolate to find their next level that is Turing equivalent?