Data structures as topological spaces (2002) [pdf](mgs.spatial-computing.org)
mgs.spatial-computing.org
Data structures as topological spaces (2002) [pdf]
http://mgs.spatial-computing.org/PUBLICATIONS/umc02.pdf
47 comments
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 the "stuff" doesn't have to be next to each other b/c if you let a solution just sit there eventually reactions are going to react).
It's kind of neat, but their website seems to indicate that it is not being actively supported. Or at the very least they don't seem to have any reason to make publicly available documentation after since around 2010.
EDIT: So for example, you could use their "trans" transform primitive to implement conway's game of life by defining the birth rule as a pattern of an empty cell with three alive cells and a transform that results in the empty cell being alive and the alive cells being the same. The death rules in similar fashion. Neighbor here would be defined as being physically next to something (but the point is that because this is about topologies, then neighbor doesn't have to mean physical proximity ... although I'm not sure where that's defined ... in the collection maybe?).
And then you just run the transform on a collection with some initial state.
EDIT EDIT: Yeah, the notion of neighbor is defined on the collection. This allows you to use the same transform on different collection types and get the appropriate result.
ALSO checkout figure 5 in the PDF I linked because it's an incredibly concise description of exactly what they're doing.
EDIT EDIT EDIT:
This also feels vaguely similar to what the Egison language is doing with their pattern matching. Documentation for Egison feels better at least to me.
https://www.egison.org/
However, I don't think that egison allows you to define arbitrary notions of neighbors in a collection like MGS does. But I haven't exactly tried to use it very much.
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 the "stuff" doesn't have to be next to each other b/c if you let a solution just sit there eventually reactions are going to react).
It's kind of neat, but their website seems to indicate that it is not being actively supported. Or at the very least they don't seem to have any reason to make publicly available documentation after since around 2010.
EDIT: So for example, you could use their "trans" transform primitive to implement conway's game of life by defining the birth rule as a pattern of an empty cell with three alive cells and a transform that results in the empty cell being alive and the alive cells being the same. The death rules in similar fashion. Neighbor here would be defined as being physically next to something (but the point is that because this is about topologies, then neighbor doesn't have to mean physical proximity ... although I'm not sure where that's defined ... in the collection maybe?).
And then you just run the transform on a collection with some initial state.
EDIT EDIT: Yeah, the notion of neighbor is defined on the collection. This allows you to use the same transform on different collection types and get the appropriate result.
ALSO checkout figure 5 in the PDF I linked because it's an incredibly concise description of exactly what they're doing.
EDIT EDIT EDIT:
This also feels vaguely similar to what the Egison language is doing with their pattern matching. Documentation for Egison feels better at least to me.
https://www.egison.org/
However, I don't think that egison allows you to define arbitrary notions of neighbors in a collection like MGS does. But I haven't exactly tried to use it very much.
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?
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.
It looks like they've got a pure functional thing going on.
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?
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.
Is this for modeling a single reaction or chain of reactions? ie you don’t need neighborhoods if you are modeling 7 trillion of one thing combined with 14 trillion of another?
It looks like chains of reactions etc. As you say for something simple you don't need something complex.
It's a DSL for computational biology where very little documentation has apparently made it to a public forum. I can decode the programming language part of it but I'm not up to scratch on computational biology so I'm not sure exactly what they want it for.
It's a DSL for computational biology where very little documentation has apparently made it to a public forum. I can decode the programming language part of it but I'm not up to scratch on computational biology so I'm not sure exactly what they want it for.
Stephen Wolfram's ruliad seems to be some sort of concurrent topological computations.
There are some beautiful pictures.
https://content.wolfram.com/sites/43/2021/11/1110swimg46.png
https://writings.stephenwolfram.com/2021/11/the-concept-of-t...
There are some beautiful pictures.
https://content.wolfram.com/sites/43/2021/11/1110swimg46.png
https://writings.stephenwolfram.com/2021/11/the-concept-of-t...
Fascinating, so would this mean an LLM is an approximator of a certain proportion of a 'ruliad'? Apologies for using GPT here, but it's a bit beyond my math to make a statement like that without reaching for my sidekick..."in a broad sense, one could conceptualize a Large Language Model (LLM) as an approximator of a specific, limited subsection of the ruliad" so I guess so, ish?
Considering the ruliad consists of all possible rules and their applications, you could say that. But it's oh so much more!
How is this fundamentally different than considering data structures as graphs?
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 (linear programming for example)
Given the recent success with vectors as a general model for data (as witnessed by the continued success with deep neural networks), it's an interesting discussion to have.
Given the recent success with vectors as a general model for data (as witnessed by the continued success with deep neural networks), it's an interesting discussion to have.
> 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.
If you argue with more generality: why not consider sites (and, relatedly, topoi) instead of topological spaces then:
> https://en.wikipedia.org/wiki/Grothendieck_topology
> https://en.wikipedia.org/wiki/Topos
If you argue with more generality: why not consider sites (and, relatedly, topoi) instead of topological spaces then:
> https://en.wikipedia.org/wiki/Grothendieck_topology
> https://en.wikipedia.org/wiki/Topos
> If you argue with more generality: why not consider sites (and, relatedly, topoi) instead of topological spaces then:
I thought linear programming would be something everyone knows, and I am not the original author so I can't speak for why they chose topological spaces instead of anything listed here. I think their e-mails are on the paper. Perhaps e-mailing them will help elucidate their choice.
I thought linear programming would be something everyone knows, and I am not the original author so I can't speak for why they chose topological spaces instead of anything listed here. I think their e-mails are on the paper. Perhaps e-mailing them will help elucidate their choice.
so topologies are grids (e.g. coordinates) in this case? it's not a great naming choice as "topology" is frequently used to describe network/graph
feels so true.
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.
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.
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.
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.
Perhaps the confusion is that I should have said topological spaces can be continuous. There are discrete topological spaces. Topologies (which I believe is typically used to refer to the collection of open sets in a topological space) are not functions or relations themselves, so I'm not sure a useful notion of continuity applies there, but if I'm wrong, please inform.
There isn't really such a thing as a 'continuous topological space'. Technically speaking, continuity is a property of functions between topological spaces. I think you're being tempted to use the terms continuous and discrete in a more colloquial sense mapping more to uncountable vs countable/countable and finite perhaps. But yeah, you really wouldn't use the term continuous to describe a topological space or a topology.
The classic middle-thirds Cantor Set being a topologically set is one of the easiest counter examples to the above misconception that the sets need to be continuous themselves.
Being able to define a neighborhood or a concept of closeness is required, but the concept of distance is not required.
If you can define a distance a topological space is a metric space
If it is locally euclidean it may be a manifold.
Really the union and finite intersection of subsets is the formal way of showing something is a topological space. Too har do describe here but that is where the concept of continuity arises.
Being able to define a neighborhood or a concept of closeness is required, but the concept of distance is not required.
If you can define a distance a topological space is a metric space
If it is locally euclidean it may be a manifold.
Really the union and finite intersection of subsets is the formal way of showing something is a topological space. Too har do describe here but that is where the concept of continuity arises.
Connected or complete then.
This is not even false.
You're going to tell me the discrete topology is continuous!?
Discrete spaces can also be topological spaces, see discrete topology
Did you not read the word “potentially”? Topological spaces are a more general case of spaces that contain the discrete case as a subset.
> 1. selects a sub-collection B of A whose elements match the path pattern β,
> 2. computes a new collection C as a function f of B and its neighbors,
> 3. and specifies the insertion of C in place of B into A.
This sounds a lot like the presentation of comonads as directed containers (e.g. https://arxiv.org/abs/1408.5809).
> 2. computes a new collection C as a function f of B and its neighbors,
> 3. and specifies the insertion of C in place of B into A.
This sounds a lot like the presentation of comonads as directed containers (e.g. https://arxiv.org/abs/1408.5809).
I wish I understood what this paper was saying.
I'm left confused as to what the gluing in the rule replacement is. Must the boundary of a rule match on both sides? Also what examples there would be of what an example would of having topology that is not induced from a graph if it is at all possible.
This is very interesting. Has anyone considered using paths/vectors in 2d/3d space as data structures?
Jurassic Park definitely did
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 really verification so much as exploration.
Although, to be sure, I think both are examples of how mathematics (even highly abstract mathematics like topology) can be useful to other disciplines.
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 really verification so much as exploration.
Although, to be sure, I think both are examples of how mathematics (even highly abstract mathematics like topology) can be useful to other disciplines.
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.
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.
richrichie(1)
> So, the expression
> 1, 1+1, 2+1, ():set
> builds the set with the three elements 1, 2 and 3
Regarding the "():set" part, and the "():something" idiom repeating in the article, is that from the same competition for the most absurd syntax where Golang got most of its awkwardness?
> 1, 1+1, 2+1, ():set
> builds the set with the three elements 1, 2 and 3
Regarding the "():set" part, and the "():something" idiom repeating in the article, is that from the same competition for the most absurd syntax where Golang got most of its awkwardness?
https://www.algebraicjulia.org/
There's some blog posts that are also interesting:
https://blog.algebraicjulia.org/