Not really, formal mathematics in complexity theory is involved when talking about asymptotics, constants like “for N < 10 trillion, a desktop computer is good enough” isn’t very interesting from a mathematical perspective.
That said, some simple intuition is the following: PSPACE is a subset of EXP is a subset of EXPSPACE
(We think these are all strict separations but technically that’s not fully proven)
If you use the shorthand intuition that we can handle polynomial scaling but can’t handle exponential scaling, this means that we hit a time barrier (EXP-complete problems) before we hit a space barrier (EXPSPACE-complete problems)
Another bit of intuition: you can store a very big number in very few bits in binary because binary holds an exponentially large number in linear bits. But you can’t loop over that number in a humans’ lifespan.
Edit:
> they can be treated as Turing-complete for a subset of programs that are well-behaving - i.e. don't end up hitting the memory limit
Just to be clear, it’s a matter of input size and not the programs themselves. Technically you could say we haven’t “solved” sorting on real hardware because nobody can sort the first 2^1000 digits of pi. But realistically we don’t care to do so.
That said, some simple intuition is the following: PSPACE is a subset of EXP is a subset of EXPSPACE
(We think these are all strict separations but technically that’s not fully proven)
If you use the shorthand intuition that we can handle polynomial scaling but can’t handle exponential scaling, this means that we hit a time barrier (EXP-complete problems) before we hit a space barrier (EXPSPACE-complete problems)
Another bit of intuition: you can store a very big number in very few bits in binary because binary holds an exponentially large number in linear bits. But you can’t loop over that number in a humans’ lifespan.
Edit:
> they can be treated as Turing-complete for a subset of programs that are well-behaving - i.e. don't end up hitting the memory limit
Just to be clear, it’s a matter of input size and not the programs themselves. Technically you could say we haven’t “solved” sorting on real hardware because nobody can sort the first 2^1000 digits of pi. But realistically we don’t care to do so.