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Data structures as topological spaces (2002) [pdf]
- misja111 3y agoHow is this fundamentally different than considering data structures as graphs?
- anon291 3y agoGraphs 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 3y agofeels so true.
- StevenXC 3y agoAlso relevant: https://en.m.wikipedia.org/wiki/Topological_data_analysis https://en.m.wikipedia.org/wiki/Topological_data_analysis
- JensW 3y agoDiscrete spaces can also be topological spaces, see discrete topology
- HighFreqAsuka 3y agoDid you not read the word “potentially”? Topological spaces are a more general case of spaces that contain the discrete case as a subset.
- aleph_minus_one 3y 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/Grothendieck_topology > https://en.wikipedia.org/wiki/Topos https://en.wikipedia.org/wiki/Topos
- anon291 3y ago> 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.
- mathgradthrow 3y agoTopologies are necessarily continuous
- anon291 3y agoPerhaps 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.
- m3ndax 3y agoThere 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.
- nyrikki 3y agoThe 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.
- anon291 3y agoConnected or complete then.
- bmacho 3y agoThis is not even false.
- Kalanos 3y agoso 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
- anon291 3y agoI wonder if this article is related to homotopy type theory at all, since they propose similar ideas.
- Verdex 3y agoUltimately, 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 3y agohttps://homotopytypetheory.org/book/ 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 3y agoI'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.
- richrichie 3y ago[flagged]
- xanderlewis 3y agoIt would have been slightly weird terminology even back then; it seems to have always been much more common to use the word 'children'.
- penteract 3y agoI suspect that English isn't the first language of the authors and there has been a translation from a language where all nouns are gendered. The phrase "an hexagon" also suggests that they don't pronounce 'h'.
- 0x264 3y agoThat's correct. It's (and they are) French.
- xanderlewis 3y agoAh — that makes sense.
- HackerThemAll 3y 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?
- Verdex 3y agoIt feels like the following PDF better expresses what they're trying to accomplish. http://mgs.spatial-computing.org/PUBLICATIONS/lami-RR72--computational-models-for-dynamical-structure.pdf 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/ 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.
- hackandthink 3y agoStephen 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://content.wolfram.com/sites/43/2021/11/1110swimg46.png https://writings.stephenwolfram.com/2021/11/the-concept-of-the-ruliad/ https://writings.stephenwolfram.com/2021/11/the-concept-of-t...
- tudorw 3y agoFascinating, 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 3y agoConsidering the ruliad consists of all possible rules and their applications, you could say that. But it's oh so much more!
- mjhay 3y agoRelated 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/ https://www.algebraicjulia.org/ There's some blog posts that are also interesting: https://blog.algebraicjulia.org/ https://blog.algebraicjulia.org/
- whosthatguy 3y 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 https://arxiv.org/abs/1408.5809).
- andoando 3y agoI wish I understood what this paper was saying.
- andoando 3y agoThis is very interesting. Has anyone considered using paths/vectors in 2d/3d space as data structures?
- runsfromfire 3y agoJurassic Park definitely did