What has speed brought ?

Many good things, to be sure, but more has been omitted.

Perhaps Kent Pitman expressed it the best:

My only problem with this is that things are ALREADY sped up. What’s the point of running a zillion times faster than the machines of yesteryear, yet still not be willing to sacrifice a dime of it to anything other than doing the same kinds of boring computations that you did before? I want speedups not just to make my same old boring life faster, but to buy me the flexibility to do something I wasn’t willing to do at slower speeds.

But then I had a whimsical thought; the sort of thing that seems at once not-impossible and yet such a long shot that one can just relax and enjoy exploring it without feeling under pressure to produce a result in any particular timeframe (and yet, I have moved my thinking forward on this over the years, which keeps it interesting).

What if we could find a way to take advantage of the fact that our logic is embedded in a computational system, by somehow bleeding off the paradoxes into mere nontermination? So that they produce the anticlimax of functions that don’t terminate instead of the existential angst of inconsistent mathematical foundations ?

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?