Comment by taliesinb | Hacker News Reader

Comment by taliesinb | Hacker News Reader

Some of the ideas in this post might pan out, some might not. Regardless, I do think token event graphs will turn out to be important. Of course, I'm biased: I coined the name TEG -- although the underlying idea originated with Max Piskunov and his "local multiway systems" [0]

What's promising about TEGs (and their incidence hypergraph, the rewrite hypergraph) is that they offer a clean methodology to decompose the behavior of a non-deterministic automaton into its causally independent parts. We're still trying to understand how to think about them, but the most promising approach seems to use the lens of (modular) representation theory, which gives us a rich mathematical toolkit to work with.

If this methodology works, there will be possibility to represent many kinds of systems in disparate fields, ranging from distributed computation to physics to biology to machine learning, in the common language of TEGs and their representations. Of course it may turn out to be merely a recasting of older ideas. In particular the Krohn-Rhodes theorem [1], categorical Petri nets [2], and the GNS construction [3] seem like they might be describing the same or an analogous procedure.

I hope to soon be describing this approach in full detail using quiver geometry [4].

[0]: https://github.com/maxitg/SetReplace/blob/master/Research/Lo...

[1]: https://www.wikiwand.com/en/Krohn–Rhodes_theory

[2]: https://arxiv.org/abs/2101.04238

[3]: https://www.youtube.com/watch?v=OmaSAG4J6nw

[4]: https://quivergeometry.net

Do you know, whether anyone of the approaches (yours or one of that you mention in the links) has had any kind of practical success, or maybe has been adopted by some community? I mean, it's nice to generalize, but did it help anyone yet? like practically (e. g., did epidemiologists or biologists or math. modellers adopted any of the approaches?)

I'm not an expert in Petri nets, but they are definitely used for modeling in various fields. Krohn-Rhodes theory also claims some successes if you read that article.

As for the stuff I'm actively working on: even if it is successful, and uniquely suited to solve some particular problem, I'd imagine it would take at least a few years to be actively applied in the right places -- and it might end up being me helping to apply it!

If you look back at for example graph theory, it was explored on the pure mathematics and computer science side for many decades before it spawned e.g. network science and got applied in sociology, economics, etc.

So, yeah, don't hold your breath! If you'd prefer not to read any blog posts heralding XYZ as the next big thing before XYZ has already led to a concrete breakthrough, I think that's a totally fair. I myself am quite happy to work quietly on things until there is a satisfying and complete application, but Stephen isn't like that. Both stances have pros and cons.

Petri nets are interesting, and used e.g. in biology to model some cell regulatory processes.

However, it's really hard to infer properties from them in a static manner using model checking or abstract interpretation.

How are we going to infer properties about petri nets or the models Wolfram is proposing? Geometry or differential equations have been so successful because they are amenable to algebraic and analytic approaches.

I agree, it's a high priority to build discrete versions of differential geometry, calculus, PDEs and so on. That's what I'm trying to do

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