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Computation as a universal and fundamental concept

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Re: Computation as a universal and fundamental concept

#21

Post by jojogeo »

tromp wrote: post #17
jojogeo wrote: post #9

Something has always nagged me about the halting problem, might be my mis-understanding of the problem space but; - You have a piece of software - That software does in memory compute only - The software does not touch any peripherals, networking, or any other external source which introduce unpredictability (x) I'm convinced that somehow this can be solved/proven whether the execution will halt or not. (x) The secon…

It can be solved if the memory is bounded. But unbounded memory comes with undecidable problems.

This truly leads into "computation"; when we're dealing with known quantities, yes, we can "solve" the halting problem. The second you move into "we don't know the answer yet", the can of worms opens. Thank you.

Re: Computation as a universal and fundamental concept

#23

Post by jojogeo »

docfort wrote: post #20
jojogeo wrote: post #9

Something has always nagged me about the halting problem, might be my mis-understanding of the problem space but; - You have a piece of software - That software does in memory compute only - The software does not touch any peripherals, networking, or any other external source which introduce unpredictability (x) I'm convinced that somehow this can be solved/proven whether the execution will halt or not. (x) The secon…

Related: the Busy Beaver problem https://news.ycombinator.com/item?id=40857041

Thank you internet stranger, for introducing me to hard-maths drugs; am hooked!! \o/

I love the idea of this. So the BB problems are individual iterations of the halting problem right? To truly solve the problem one would have to come up with a program which would operate on all possible BB numbers?

Re: Computation as a universal and fundamental concept

#24

Post by anon291 »

jojogeo wrote: post #9

Something has always nagged me about the halting problem, might be my mis-understanding of the problem space but; - You have a piece of software - That software does in memory compute only - The software does not touch any peripherals, networking, or any other external source which introduce unpredictability (x) I'm convinced that somehow this can be solved/proven whether the execution will halt or not. (x) The secon…

If the memory is bounded then your software is a simple finite automaton, and can be decided in finite time. The issue is with unbounded memory. The issue with the halting problem is a simple characteristic of infinity. This is actually what people are noticing when they say that computation is a fundamental part of the universe. They are correct! The universe deals with infinitisemals all the time. As humans, we have only discovered ways of dealing with certain classes of infinitesemals (calculus). The others remain beyond our ability to characterize. Indeed, some have been proven to be uncharacterizable.

Re: Computation as a universal and fundamental concept

#25

Post by anon291 »

jojogeo wrote: post #21
tromp wrote: post #17

Earlier quoted context omitted.

It can be solved if the memory is bounded. But unbounded memory comes with undecidable problems.

This truly leads into "computation"; when we're dealing with known quantities, yes, we can "solve" the halting problem. The second you move into "we don't know the answer yet", the can of worms opens. Thank you.

Computation is the study of infinity. That is how I like to think about it. It doesn't seem that way when you're building a website (well, in some ways because it's not at that point), but every algorithm, data structure, etc is an investigation into a certain part of infinity. Think of the way in which we generally categorize algorithms (Big-O notation)... that's just characterizing infinity.

Re: Computation as a universal and fundamental concept

#26

Post by jerf »

plastic-enjoyer wrote: post #8
sgt101 wrote: post #2

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 rea…

> up to the point that many computer scientists now seem to equate computation with the functioning of the universe. Do you think that's a kind of tunnel vision? If the only thing you focus on is computation, you'll probably end up seeing computation everywhere - it became a way of seeing the world.

It is a common accusation. There's a somewhat famous quote I've seen a few times:

"It's interesting to look back through history on this one. Each age has its pinnacle of technology, and each age uses that technology as a metaphor for nature, for the universe. In ancient Greece, the technological marvels were musical instruments and the ruler and compass. The Greek philosophers tried to build an entire cosmology from number, harmony, proportion, form, and so on — from mathematics, basically. Remember the music of the spheres? The Pythagoreans believed that nature was a manifestation of rational mathematics. Later on the pinnacle of technology was the clockwork. Newton wanted a clockwork universe, the entire universe as a gigantic clockwork mechanism, with all the parts interlocking and ticking over with infinite precision. Then in the 19th century along came steam power, and the universe was then depicted as an enormous heat engine, or thermodynamic machine, running down toward its heat death. Today the computer is the pinnacle of technology, so it's now fashionable to talk about nature as a computational process."

Which seems to source from https://www.edge.org/conversation/paul_davies-time-loops .

While "computer" may give us impressions of something with "a CPU" and "RAM" and "a disk drive", it does at least seem plausible that the universe as computation is a plausible base level, though. Unlike "the music of the spheres", which to the extent that it made predictions of the world, it got them wrong in the most basic way, viewing it through a lens of computation allows us to put some quite subtle and interesting limits on things. "Computation" is a pretty flexible substrate; it is difficult to imagine how the proposition "the universe is a computation and subject to the limitations thereto" could be falsified, and if it could, it is difficult to imagine how we would be able to know it was so falsified. Nevertheless the math of computation allows us to say non-trivial things about the universe as a result; it is not a vacuous generalization, though it is certainly a loose one... being able to say yet more concrete things about the nature of the computation, such as "this is exactly how gravity works", has quite a bit more utility.

Re: Computation as a universal and fundamental concept

#27

Post by jojogeo »

anon291 wrote: post #24
jojogeo wrote: post #9

Something has always nagged me about the halting problem, might be my mis-understanding of the problem space but; - You have a piece of software - That software does in memory compute only - The software does not touch any peripherals, networking, or any other external source which introduce unpredictability (x) I'm convinced that somehow this can be solved/proven whether the execution will halt or not. (x) The secon…

If the memory is bounded then your software is a simple finite automaton, and can be decided in finite time. The issue is with unbounded memory. The issue with the halting problem is a simple characteristic of infinity. This is actually what people are noticing when they say that computation is a fundamental part of the universe. They are correct! The universe deals with infinitisemals all the time. As humans, we hav…

Ahhh thank you it's effectively the known-vs-unknown space;

- How long does it take to get from A to B? => Easy if you know where A and B are, and what mode of transport you're taking to get there.

- How long does it take to get from A to _somewhere_ => As long as it takes!!

Re: Computation as a universal and fundamental concept

#28

Post by ctoa »

jojogeo wrote: post #21
tromp wrote: post #17

Earlier quoted context omitted.

It can be solved if the memory is bounded. But unbounded memory comes with undecidable problems.

This truly leads into "computation"; when we're dealing with known quantities, yes, we can "solve" the halting problem. The second you move into "we don't know the answer yet", the can of worms opens. Thank you.

There remain undecidable problems even with finite memory/state space.

Linear bounded automata (LBA) the halting problem is decidable. But many properties of LBA are undecidable:

Emptiness: Does an LBA reject all possible inputs? Universality: Does an LBA accept all possible inputs over its alphabet? Equivalent: Do two LBA accept the same language? Finiteness: Does an LBA accept a finite number of strings.

Re: Computation as a universal and fundamental concept

#29

Post by jerf »

sgt101 wrote: post #13
gradys wrote: post #4

Earlier quoted context omitted.

It can be the case that both: - The physics of the universe can be completely modeled as computation, and - 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".

Of course, if our universe is undecidable it must be the case that computable processes can be executed within it, and it might be the case that all of the processes that are ever executed within it are computable... but it might be that some of the processes that are executed are not computable... because the machine may.. or may not?

I think there's an equivocation of "computable" going on here. Mathematicians talk about a lot of things like "uncomputable sequences" but that is usually making a statement about the sequence, not necessarily any individual member. The Busy Beaver sequence is uncomputable. You can, however, quite trivially compute BB(2), even in your head if you're a bit careful. You can set up individual elements of an uncomputable sequence in our universe, and you may be unable to state in advance what the system would do with anything less than simply letting it run and see what happens due to the complexity of the system, but being a member of an uncomputable sequence doesn't mean that you can't in fact set those things up and watch them run. The Universe doesn't throw an "UncomputableCircumstance" exception or anything. It just keeps advancing to the next state. Your inability to make certain statements about that next state or some future state is not its problem.

Re: Computation as a universal and fundamental concept

#30

Post by joelshep »

Maxatar wrote: post #5
sgt101 wrote: post #2

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 rea…

>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 purel…

I may be misremembering Godel's proof or misunderstanding your last paragraph, but I thought Godel's proof actually presented a specific undecidable statement. The hope then was that somehow undecidable statements could be cordoned off from decidable statements, and Turing's result showed that that wasn't possible. Perhaps that's what you mean by "the nonexistence of a single algorithm that correctly answers every instance in that family"?
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