Computation has turned out to be a far more general concept than I think was imagined, up to the point that many computer scientists now seem to equate computation with the functioning of the universe. Recently it's been shown that there are real, physical processes which are undecidable (we cannot know if a latice of atoms has a spectral gap or not, we cannot determine if a specific particle in a fluid flow will reach a specific place or not, we cannot determine if a ray of light will reach a specific target in certain configurations of reflection).
Our world appeared computable, but it isn't, even if P=NP.
- The physics of the universe can be completely modeled as computation
- It's possible to pose undecidable problems about the way the universe unfolds
This is intrinsic to the idea of undecidability even for Turing machines, e.g. "we equate computation with the functioning of Turing machines, but there are real processes executable in Turing machines that are undecidable".
>Recently it's been shown that there are real, physical processes which are undecidable
I want to push back a bit on this claim along two dimensions.
Imagine a physical Turing machine built out of atoms, gears, levers, and an electron parked on the read/write head and ask whether that electron ever crosses some fixed plane in space, which it does only when the machine enters its halt configuration. That's now a purely physical question about a trajectory (does this electron ever reach a certain target), yet answering it for the whole family of such machines is literally the halting problem, so there's a physical process that's undecidable.
Your examples about physical processes being undecidable are all basically just this... there examples of using reflections of light, or the flow of liquid, etc... and demonstrating that these physical processes in principle are sufficient to model a universal Turing machine.
And while it's fascinating that certain things you may not have expected can be used to model computation, it's misleading, or rather it's too strong of a claim to believe that there exist actual/real physical processes whose outcomes are undecidable. That's a subtle but very common misinterpretation of what undecidability is.
Undecidability, whether in physics or computer science, only applies to the infinitely broad class of a problem as a whole, it never applies to a specific instance of a problem. So it can never be the case that there's a certain configuration of reflections for which it's undecidable whether a ray of light reaches a target. Nor can it be the case that for a specific lattice of atoms, it's undecidable whether it has a spectral gap or not. It can only be the case that for the problem as a whole where the parameter space is entirely unbounded, there is no single algorithm that can decide if a ray of light reaches a specific target for all possible arbitrary (and infinitely many) configurations. Once you fix a specific system, then the undecidability goes away.
Not claiming that you are necessarily making this misconception, but I often see people misinterpret undecidability to mean that there exists a specific problem, like with specific inputs, where it's somehow impossible to know what the answer will be. Undecidability always requires an infinite family of instances, and it's a statement about the nonexistence of a single algorithm that correctly answers every instance in that family. It says nothing about any particular instance being unknowable/undecidable.
If I am wrong, please pardon. I suspect I am. But was this comment edited by Claude? I ask specifically because it is well written, substantive, all which is expected here, but the "push back" part, to me, must be a) an artifact of Claude, either by osmotic assimilation (Which is happening to many innocent users) or b) Claude itself.
Feel free to flag this comment if I get an answer. I do want to know.
No Claude was not involved in any way in me writing it, and honestly it's kind of getting depressing how many comments are constantly questioning peoples use of LLMs.
What even is computation? State-based inference. But intelligence itself does not rely on computation, only its biological counterweight seems to and only in certain situations. If Computation is a "Universal Concept" then there are at least 4 or 5 more "Universal Concepts" analogous to intuition and spontaneity.
Computation has turned out to be a far more general concept than I think was imagined, up to the point that many computer scientists now seem to equate computation with the functioning of the universe. Recently it's been shown that there are real, physical processes which are undecidable (we cannot know if a latice of atoms has a spectral gap or not, we cannot determine if a specific particle in a fluid flow will reach a specific place or not, we cannot determine if a ray of light will reach a specific target in certain configurations of reflection).
Our world appeared computable, but it isn't, even if P=NP.
It can be the case that both:
- The physics of the universe can be completely modeled as computation
- It's possible to pose undecidable problems about the way the universe unfolds
This is intrinsic to the idea of undecidability even for Turing machines, e.g. "we equate computation with the functioning of Turing machines, but there are real processes executable in Turing machines that are undecidable".
>Recently it's been shown that there are real, physical processes which are undecidable
I want to push back a bit on this claim along two dimensions.
Imagine a physical Turing machine built out of atoms, gears, levers, and an electron parked on the read/write head and ask whether that electron ever crosses some fixed plane in space, which it does only when the machine enters its halt configuration. That's now a purely physical question about a trajectory (does this electron ever reach a certain target), yet answering it for the whole family of such machines is literally the halting problem, so there's a physical process that's undecidable.
Your examples about physical processes being undecidable are all basically just this... there examples of using reflections of light, or the flow of liquid, etc... and demonstrating that these physical processes in principle are sufficient to model a universal Turing machine.
And while it's fascinating that certain things you may not have expected can be used to model computation, it's misleading, or rather it's too strong of a claim to believe that there exist actual/real physical processes whose outcomes are undecidable. That's a subtle but very common misinterpretation of what undecidability is.
Undecidability, whether in physics or computer science, only applies to the infinitely broad class of a problem as a whole, it never applies to a specific instance of a problem. So it can never be the case that there's a certain configuration of reflections for which it's undecidable whether a ray of light reaches a target. Nor can it be the case that for a specific lattice of atoms, it's undecidable whether it has a spectral gap or not. It can only be the case that for the problem as a whole where the parameter space is entirely unbounded, there is no single algorithm that can decide if a ray of light reaches a specific target for all possible arbitrary (and infinitely many) configurations. Once you fix a specific system, then the undecidability goes away.
Not claiming that you are necessarily making this misconception, but I often see people misinterpret undecidability to mean that there exists a specific problem, like with specific inputs, where it's somehow impossible to know what the answer will be. Undecidability always requires an infinite family of instances, and it's a statement about the nonexistence of a single algorithm that correctly answers every instance in that family. It says nothing about any particular instance being unknowable/undecidable.
If I am wrong, please pardon. I suspect I am. But was this comment edited by Claude? I ask specifically because it is well written, substantive, all which is expected here, but the "push back" part, to me, must be a) an artifact of Claude, either by osmotic assimilation (Which is happening to many innocent users) or b) Claude itself.
Feel free to flag this comment if I get an answer. I do want to know.
No Claude was not involved in any way in me writing it, and honestly it's kind of getting depressing how many comments are constantly questioning peoples use of LLMs.
What even is computation? State-based inference. But intelligence itself does not rely on computation, only its biological counterweight seems to and only in certain situations. If Computation is a "Universal Concept" then there are at least 4 or 5 more "Universal Concepts" analogous to intuition and spontaneity.