For example, you can ask whether a Java program, run with infinite memory, will eventually halt. For any particular Java program, there's obviously an algorithm that says whether it halts or not. The algorithm is a single statement, which says either "yes" or "no". Might be hard to figure out which is the correct algorithm, but the Java program is fixed so the algorithm is definitely one of the two.
However, there is no algorithm which can take an arbitrary Java program as input and determine whether it will halt. It's about the class of all possible programs.
[1] https://en.wikipedia.org/wiki/Undecidable_problem
[2] https://en.wikipedia.org/wiki/Independence_(mathematical_log...
This isn't true.
In general, if a program hasn't halted yet you don't know if it will.
In particular, consider the Collatz conjecture. You can't even tell if your Java implementation of it will halt for a particular input, until it does.
We don't _know_ which algorithm it is, but that's not relevant to the definition of undecidability, which only requires that the algorithm exist.
This is slightly bizarre now I think about it: the definition of decidability allows the algorithm selection to be undecidable!
This is very helpful though, thank you.
Feel free to flag this comment if I get an answer. I do want to know.
It is depressing though, writing feels like it's in part becoming a game of outpacing the latest LLM's idiosyncrasies so we can signal authenticity, which perversely, is achieved through using an LLM enough so that you can become familiar with its flavor of communication.
I actually laughed quite a lot to begin with, GPT models saying things like "...might look like P, but is NP wearing a hat and a lab coat..." and "...is a haunted house disguised as a git repository..."; but alas when you've heard them a million times everywhere it really starts to bite.
My apologies, and I do appreciate your reply.
What was the book in 1997? That's about the time of my first UAP sighting.
For your Turing machine example: even if we built such a machine, it would never truly be giving an answer to the halting problem, because any stray cosmic particle could excite the electron and cause it to cross whatever plane.
For a more realistic example: the ground state of an molecule is a physically relevant quantity, and in theory any molecule alone should lose energy and attain it's ground state, even if finding the ground state electronic configuration is undecidable. But in reality, no molecule is ever truly isolated and so would never actually be guaranteed to enter it's ground state (or if it were truly isolated, it would not be observed at all rendering the question moot)
No, since you cannot physically build a Turing Machine. A Turing machine requires infinite tape. Any physically realizable machine doesn’t have that, so has finite states, so is decidable: enumerate the states in finite time - it halts or repeats, so all programs on a finite state machine are decidable.
Your example is not an undecudable physical process.
Godel things also don’t apply: Godel theorems are about proof of this or that from within the same system. In logic one can prove such things from an outside system, then construct towers, avoiding Godel theorems. Godel theorems also require a model of integers including multiplication (without multiplication, such systems were proven complete and decidable). However the universe does not contain a model of integers, as the physical universe is not unbounded: relativity places a finite limit in spacetime on what can interact.
Mixing math as reality fails at these requirements.
sqrt(1-exp(-t/T))|1> + sqrt(exp(-t/T))|0>
If you want to claim that we could predict which specific atom decays next, I'd really like to see a lot more explaintion and exposition as it would upend current understanding.
But at least we agree there is nothing we can predict.
sqrt(1-exp(-t/T))|1> + sqrt(exp(-t/T))|0>
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.