Notably, this was exactly the sort of belief I was trying to show is false
Please point out if there is a specific claim I made in my comment that you believe to be false. I said that “I don’t think a TC computer can ever be built in our universe.”, which you don’t seem to argue with? (If we assume that we can only ever get access to a finite number of atoms. If you dispute this I won’t argue with that, neither of us has a Theory of Everything to say for certain.)
Just to make precise why I was making that claim and what it was trying to argue against, take this quote from the post:
yes, you can consider a finite computer in the real world to be Turing-complete
I dropped the “finite computer” constraint and interpreted the phrase “real world” to mean that “it can be built in our universe”, this is how I arrived at the “a TC computer can be built in our universe” statement, which I claimed was false.
I actually agree that if we assume that there’s a finite maximum of atoms, we could in principle reformulate the universal computer as a finite state automaton, and if we were willing to accept the non-scalability of a finite state automaton, this could actually work.
The fundamental problem is that now we would have software that only works up to a specified memory limit, because we essentially burned the software into the hardware of the finite automaton and if you are ever uncertain of how much memory or time a problem requires, or more worryingly if we were ever uncertain about how much resources we could actually use, then our “software” for the finite automaton is no longer usable and we’d have to throw it away and recreate a new computer for every input length.
Turing Machine models automatically handle arbitrarily large inputs without having to throw away expensive work on developing the software.
So in essence, if you want to handle the most general case, or believe unbounded atoms are possible, like me, then you really want the universal computer architecture of modern computers.
The key property of real computers that makes them Turing Complete in theory is that they can scale with more memory and time arbitrarily without changing the system descriptior/code.
More below:
(If we assume that we can only ever get access to a finite number of atoms. If you dispute this I won’t argue with that, neither of us has a Theory of Everything to say for certain.)
Please point out if there is a specific claim I made in my comment that you believe to be false. I said that “I don’t think a TC computer can ever be built in our universe.”, which you don’t seem to argue with? (If we assume that we can only ever get access to a finite number of atoms. If you dispute this I won’t argue with that, neither of us has a Theory of Everything to say for certain.)
Just to make precise why I was making that claim and what it was trying to argue against, take this quote from the post:
I dropped the “finite computer” constraint and interpreted the phrase “real world” to mean that “it can be built in our universe”, this is how I arrived at the “a TC computer can be built in our universe” statement, which I claimed was false.
I think I understand the question now.
I actually agree that if we assume that there’s a finite maximum of atoms, we could in principle reformulate the universal computer as a finite state automaton, and if we were willing to accept the non-scalability of a finite state automaton, this could actually work.
The fundamental problem is that now we would have software that only works up to a specified memory limit, because we essentially burned the software into the hardware of the finite automaton and if you are ever uncertain of how much memory or time a problem requires, or more worryingly if we were ever uncertain about how much resources we could actually use, then our “software” for the finite automaton is no longer usable and we’d have to throw it away and recreate a new computer for every input length.
Turing Machine models automatically handle arbitrarily large inputs without having to throw away expensive work on developing the software.
So in essence, if you want to handle the most general case, or believe unbounded atoms are possible, like me, then you really want the universal computer architecture of modern computers.
The key property of real computers that makes them Turing Complete in theory is that they can scale with more memory and time arbitrarily without changing the system descriptior/code.
More below:
https://www.dwarkesh.com/p/adam-brown