> Even given infinite space, there still wouldn't exist a Regular Expression (the CS kind) you could write down in finite time that would match only primes.

The point is, neither would there exist a non-(Regular Expression) method that could do such, if you're running it on a finite state machine (your computer).

But that argument is not interesting. "We can't count arbitrarily high because the universe is finite." Yeah, duh. (Observable universe, for the pedants.)

It's more interesting to ask: suppose you did have a magical stick of RAM that had infinite storage, and suppose you augmented your computer to store and query that memory. Then -- yes -- your computer could determine whether any number, however large, is prime. An FSM could not.

If you have an infinite stick of RAM, regular expressions wouldn't be FSMs, either, they could have infinite state space. They would terminate in finite time that is exponential in the size of the prime but nevertheless would still halt.

The point being made is that “regular expressions”, in the original CS meaning, do not have stack space or anything analogous. “Regular expressions” in modern programming practice are often different and more powerful, of course.

A finite state machine can’t have infinite state space.. also, you can write a finite Turing program that does exactly that, but you can’t write a finite description of an “infinite” state space, which is the whole point.

Ad absurdum everything would be computable (recognizable) by FSMs, simply by listing all the infinite number of possibilities. listOfAllPrimes.concat(“|”), here you are..

No, you could create arbitrarily large FSMs, but there are no infinite finite state machines.

The important point here was given infinite space.

There is, the property of being a Turing machine would only be thrown out when you actually hit the limit. Besides computers being able to modify state not only in their memory but e.g. outside memory, I can also look at my computer as a lazy-taped Turing machine that chugs along writing only to its memory, and when it were to go off it pauses and waits for me to add additional memory.

This is just useless pedantism that diminishes the very imporrant categorization of Chomsky.

> At least IIRC, it's been a 20+ years since I last took a Discrete Math course :)

Your memory is correct and there is very simple intuition for this. Regular languages are those that can be matched by a machine with bounded memory. So, let's say that bound is "n" bits. Then your machine will not be able to read in unary strings larger than 2^n.