Now you are mixing things up.
> It is far from trivial to work out what problems they can and can't solve.
Yes, and I am exactly saying that if you want to talk about explicitly finite machines, you need more sophisticated concepts. Formally defined 'computability' (with the usual definition) is a silly notion for these finite machines, because it captures nothing of the subtlety. It's trivial for finite machines. Trivial concepts are seldom useful.
The standard formal definition of computability doesn't care about your finite human life. Similar for the standard formal definition of 'regular language' etc.
It seems like you are agreeing with me, but don't really know the terms very well.
> Formally defined 'computability' (with the usual definition) is a silly notion for these finite machines, because it captures nothing of the subtlety. It's trivial for finite machines.
Computability is relative to the definition of an abstract machine - e.g. there are problems which aren’t computable given a standard Turing machine, but which are for e.g. a Turing machine with an oracle, or a Turing machine that can execute supertasks - the halting problem for a standard Turing machine is computable by a Turing machine with an oracle for the first-order halting problem (but its own halting problem remains uncomputable for it)
And if “computability” can be meaningful for abstract machines stronger than a Turing machine, it can also be a meaningful concept for weaker machines
e.g. a problem for which the fastest possible algorithm is O(n^2) isn’t computable by a linear bounded automaton, but problems for which the fastest possible algorithm is O(n) or O(log n) are computable by an LBA
> but don't really know the terms very well.
I think the same about you
> e.g. a problem for which the fastest possible algorithm is O(n^2) isn’t computable by a linear bounded automaton, but problems for which the fastest possible algorithm is O(n) or O(log n) are computable by an LBA
You are right that there's lots of interesting mysteries to explore. But the tools we commonly use only work for (potentially) infinite machines.
Alas, Computability for all finite machines is trivial. And big-O notation is another silly one for finite machines: it's all O(1) for them. Big-O notation is too blunt a tool to deal with finite machines, at least by itself.
You seem to be using the word “computability” to mean “computable by an abstract machine of Turing degree 0”, whereas I am using it in the broader sense of “computable by an abstract machine of specified type”. As I said, in that broader sense, computability is non-trivial for finite machines.
And that broader sense is very standard, you’ll find heaps of papers/etc containing variations of the phrase “computable by a Turing machine with an oracle”… but if that oracle is e.g. for the standard (Turing degree 0) halting problem, then that halting problem is computable by such a machine.
You were earlier claiming I wasn’t familiar with the definitions, but you are giving me the impression you aren’t familiar with “computable” as a relative term