Data structures as topological spaces (2002) [pdf]

http://mgs.spatial-computing.org/PUBLICATIONS/umc02.pdf

by mbustamanter • 2 years ago

170 42 2 years ago

misja111• 2 years ago

How is this fundamentally different than considering data structures as graphs?

anon2915• 2 years ago

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.

dlahoda• 2 years ago

feels so true.

StevenXC• 2 years ago

Also relevant:

https://en.m.wikipedia.org/wiki/Topological_data_analysis

JensW• 2 years ago

Discrete spaces can also be topological spaces, see discrete topology

aleph_minus_one• 2 years ago

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

mathgradthrow4• 2 years ago

Topologies are necessarily continuous

Kalanos• 2 years ago

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

anon2915• 2 years ago

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

Verdex5• 2 years ago

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.

Syzygies• 2 years ago

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.

reuben364• 2 years ago

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.

HackerThemAll• 2 years ago

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?

Verdex5• 2 years ago

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.

Cacti• 2 years ago

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?

Verdex5• 2 years ago

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.

amelius• 2 years ago

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?

Verdex5• 2 years ago

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.

iamwil• 2 years ago

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?

Verdex5• 2 years ago

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.

hackandthink• 2 years ago

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...

tudorw• 2 years ago

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?

bibanez• 2 years ago

Considering the ruliad consists of all possible rules and their applications, you could say that. But it's oh so much more!

mjhay• 2 years ago

Related to this, AlgebraicJulia has been doing a lot with applying concepts from algebra and category theory to data analysis and modelling.

https://www.algebraicjulia.org/

There's some blog posts that are also interesting:

https://blog.algebraicjulia.org/

whosthatguy• 2 years ago

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).

andoando2• 2 years ago

I wish I understood what this paper was saying.

andoando2• 2 years ago

This is very interesting. Has anyone considered using paths/vectors in 2d/3d space as data structures?

runsfromfire• 2 years ago

Jurassic Park definitely did

Data structures as topological spaces (2002) [pdf]

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