[Aardwolf]

Does there exist any formal terminology for "something that is like a Turing machine if it would have infinite memory"?

I'm asking because since the true definition of a Turing machine requires infinite memory, in theory nothing can be a Turing machine in the observable universe, so this definition doesn't help describe anything that's used in practice.

On the other hand, there's an obvious difference in power of a language like C, versus non Turing-complete languages like regexp. So it's useful to talk about this concept, but we could never formally call any of those Turing complete (C pointers are limited to so many bits, for example).

The discussion about this detail comes up every time, so is there some proper formal term for this?

Something related to how much code is required to use the finite pool of memory you have in any way you want, or so (where a non turing complete language may require enumerating all possibilites and thus too large code size)

[roywiggins]

You can assume that, if the machine runs out of memory, that someone comes along and installs another gigabyte and sets it going again. Predicting whether programs will halt under those conditions is exactly the same as if they really had an infinite tape.

It's a bit tricky with real programs because eventually you might expect you'd run out of address space, so you might have to relax constraints and say that at least some integer types don't have a defined maximum value, or stuff like that.