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Data structures as topological spaces (2002) [pdf]

mgs.spatial-computing.org

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Re: Data structures as topological spaces (2002) [pdf]

#41

Post by Verdex »

iamwil wrote: post #39
Verdex wrote: post #15

It feels like the following PDF better expresses what they're trying to accomplish. http://mgs.spatial-computing.org/PUBLICATIONS/lami-RR72--com... As far as I can tell, they're trying to model things like chemical reactions (and other stuff) where given a bunch of "stuff" in some solution it will combine with other "stuff" if it's in the same topological neighborhood (which I think is basically the idea that ALL of…

Figure 5 kinda reminds me of...Datalog? Where you have rules and a data set. Given the rules, you iteratively compute on the data set until there are no more rules that match the given data--the fixed point. Is that right?

I think that's accurate except it looks like you need to manually continue the applications of your transform.

It looks like they've got a pure functional thing going on.

Re: Data structures as topological spaces (2002) [pdf]

#42

Post by Verdex »

amelius wrote: post #37
Verdex wrote: post #15

It feels like the following PDF better expresses what they're trying to accomplish. http://mgs.spatial-computing.org/PUBLICATIONS/lami-RR72--com... As far as I can tell, they're trying to model things like chemical reactions (and other stuff) where given a bunch of "stuff" in some solution it will combine with other "stuff" if it's in the same topological neighborhood (which I think is basically the idea that ALL of…

I'm puzzled by what's happening on page 9 of that PDF. Ok, I quickly browsed through it, but a chemical reaction where two identical molecules react into those same molecules plus another molecule? How can this be possible?

I'm not up to speed on all of the things that computational biology is up to. But IIRC there are some metabolic pathways that cascade into producing a lot of whatever it is the organism is trying to produce. Which sounds kind of like what you're describing.

Re: Data structures as topological spaces (2002) [pdf]

#43

Post by xanderlewis »

mathgradthrow wrote: post #19
anon291 wrote: post #3

Earlier quoted context omitted.

Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (l…

Topologies are necessarily continuous

No, they are not.

In the usual mathematical sense of the words you are using, topologies aren’t even the right type of object to admit a notion of continuity. Your statement doesn’t even make sense. It’s maps between them that can be continuous.

In fact, a topological space is sort of the minimal amount of structure a set needs to have to be able to talk about continuity of maps to/from it.

Re: Data structures as topological spaces (2002) [pdf]

#44

Post by Syzygies »

Verdex wrote: post #17
anon291 wrote: post #4

I wonder if this article is related to homotopy type theory at all, since they propose similar ideas.

Ultimately, I don't think so. All the HoTT stuff seems to really be focusing on constructing proof objects so that the computer can run them on mathematics which would otherwise not have any computer verification run on it. More or less advanced type checking for math. Meanwhile, the MGS language's topological collections and associated transforms seems to be about simulating things like chemical reactions. Not reall…

https://homotopytypetheory.org/book/

I disagree. There's a migration from Haskell to Lean 4, which is influenced by HoTT, and is a credible general purpose programming language.

Arguably, _the_ credible general purpose programming language, if one believes that programming should feel like doing mathematics. Languages are shaped by their target tasks, and writing tactics for proofs subsumes any other task one might consider. Programming is recognizing pattern, and pattern runs deep in Lean.

When we're young, but past existential BS, we start to think that ten years of training will yield productive years that outstrip decades of muddling in ad hoc languages if we hadn't made the commitment. But soon, few want to make the commitment.

If programming is ever going to become far more advanced, it will take the form of successors to Haskell and Lean.

Re: Data structures as topological spaces (2002) [pdf]

#45

Post by mathgradthrow »

xanderlewis wrote: post #43
mathgradthrow wrote: post #19

Earlier quoted context omitted.

Topologies are necessarily continuous

No, they are not. In the usual mathematical sense of the words you are using, topologies aren’t even the right type of object to admit a notion of continuity . Your statement doesn’t even make sense. It’s maps between them that can be continuous. In fact, a topological space is sort of the minimal amount of structure a set needs to have to be able to talk about continuity of maps to/from it.

It is not always done, but it is still correct, to replace the objects of a category with the identity morphisms. So in Top, it is totally correct to think of topological space as the identity homeomorphism, which is indeed continuous.

Re: Data structures as topological spaces (2002) [pdf]

#46

Post by mathgradthrow »

skhunted wrote: post #24
mathgradthrow wrote: post #19

Earlier quoted context omitted.

Topologies are necessarily continuous

You and OP are using the word continuous in two different contexts. Generally one would not say that the integers with the trivial topology is continuous. It’s a discrete space with a topology. But when someone says a space is continuous generally they mean not discrete.

If you use the word topology as if you know what you are talking about, you sacrifice this benefit of the doubt.

Re: Data structures as topological spaces (2002) [pdf]

#47

Post by xanderlewis »

mathgradthrow wrote: post #45
xanderlewis wrote: post #43

Earlier quoted context omitted.

No, they are not. In the usual mathematical sense of the words you are using, topologies aren’t even the right type of object to admit a notion of continuity . Your statement doesn’t even make sense. It’s maps between them that can be continuous. In fact, a topological space is sort of the minimal amount of structure a set needs to have to be able to talk about continuity of maps to/from it.

It is not always done, but it is still correct, to replace the objects of a category with the identity morphisms. So in Top, it is totally correct to think of topological space as the identity homeomorphism, which is indeed continuous.

I'm aware; in mathematics it's possible to replace almost anything with some other thing to make the statement you want to be true come true. But it's usually just gonna confuse everyone.

Re: Data structures as topological spaces (2002) [pdf]

#48

Post by mathgradthrow »

xanderlewis wrote: post #47
mathgradthrow wrote: post #45

Earlier quoted context omitted.

It is not always done, but it is still correct, to replace the objects of a category with the identity morphisms. So in Top, it is totally correct to think of topological space as the identity homeomorphism, which is indeed continuous.

I'm aware; in mathematics it's possible to replace almost anything with some other thing to make the statement you want to be true come true. But it's usually just gonna confuse everyone.

Well the thing I chose to replace the thing with is actually isomorphic (type equivalent) to the thing I replaced. So that's quite a bit more constrained than "replacing anything with anything". Not only are the arrows the only thing that matters, but its cleaner to suppose that they're the only thing there is.
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