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Categories of Nets

johncarlosbaez.wordpress.com

58 pointschmaynard37 comments

The neural field literature has also been exploiting (attempting to exploit?) connections between quantum mechanics and statistical mechanics (at least in the loose sense as used on this blog).

Readers may be interested in the works of Buice and colleages¹²³⁴.

You might also enjoy The Complex Chemical Langevin Equation⁵, which uses complex-valued particle counts to handle negative particle counts in a stochastic model.

1 https://journals.plos.org/ploscompbiol/article?rev=1&id=10.1...

2 https://journals.aps.org/pre/abstract/10.1103/PhysRevE.75.05...

3 https://www.mitpressjournals.org/doi/abs/10.1162/neco.2009.0...

4 https://www.sciencedirect.com/science/article/pii/S007961070...

5 https://aip.scitation.org/doi/full/10.1063/1.4885345?casa_to...

I'm one of the authors of the paper. Ask me anything!

Have you sat down with any epidemiology researchers to relate this to the compartment models they usually use?

No. I don't know why everyone is getting so much fixated with the epidemiological aspect, that in our paper is barely mentioned. Ours is a technical contribution that uses results from groupoid theory and homotopy theory to provide a framework where different flavors of nets given in the last 30 years or so can be nicely interconnected.

Applications are not the central focus of this paper. There are about a ton of applied papers out there employing Petri nets in computing, chemistry, epidemiology etc. This paper is not one of them. We just mentioned, in passing, how different flavors of nets have been applied in the last decades. We deem this to be an interesting paper for people working in Petri nets theory, because it systematizes decades of research. I'm pretty sure it will be of little interest for anyone not directly involved in Petri net research. :)

Ok. I misread the intro as making the epi application a selling point.

:D I know especially John is very interested in finding an application of this stuff to solve modelling problems for climate change etc, but yes, this is not the content of this work ^_^

What is the difference between a net and a Graph.

I made it more than halfway through the article, but without this simple intuition it was really hard to grasp the ideas.

The main difference is that a Petri net is basically an hypegraph, where you have directed edges connecting multiple vertexes both in the source and in the target.

Graphs give you finite state machines in the obvious way: You mark the vertex you are in and walk the arrows.

Hypergraphs give you Petri nets: You mark each vertex as many times as you want and walk the arrows to move marks around. This tells us two things: 1. Petri nets are a calculus of resources. The marking is not telling anymore "what state you are in". A state is an allocation of resources to each vertex in the net. 2. Petri nets are concurrent: you don't have to move stuff around by walking one edge at a time: Two different hyperedges in two different places of the hypergraph can "act at the same time", since the "what state you are in" thing makes no sense anymore.

Anyway, this paper is pretty complicated and for sure there are waaaaay easier places to start. Such as this one: https://arxiv.org/abs/1906.07629

When evaluating the dynamics of a net, do all tokens move each discrete step or are there other choices that can be made?

Are any forms where the edges are weighted, or does each edge necessarily have the same weight?

Related to the previous question, if you have a finite number of tokens at a vertex with multiple outgoing edges, how do you choose which edges they follow? I suppose that for any given allocation there may be multiple succeeding allocations.

Finally, the structure seems very similar to neural nets. Are they actually similar, or very different?

There are countless different flavors of Petri nets. The edges can be weighted, meaning that a transition can get more than one token from a given input place to fire, and can put more than one token in a output place when it fires.

About the choosing which edges they follow: You don't. In standard Petri nets firing is concurrent: If tokens can be used by more than one transition at the same time, they will non-deterministically go one way or another. You can actually refine this situation by extending your formalism, e.g. to timed nets.

I am not an expert of neural nets, but I'd guess they are more similar to signal flow graphs. These are related to Petri nets tho, but in a very deep and complicated way that I have no chance of explaining here right now. Check out the work of Sobocinski, Piedeleu and Zanasi about additive relations if you are interested in this!

Do you know of any interesting natural ismorphism between the categories you define in your paper and the category of finite-dimensional hilbert spaces? Curious if you have thought about applications to categorical quantum mechanics.

Natual isomorphisms with FDHilb are very difficult to get here: Different flavors of Petri nets present different flavors of FREE monoidal categories. "Free" here means that we have categories that satisfy exactly the equations that are needed to be (symmetric, commutative) monoidal, nothing more. Instead, FDHilb is compact closed, and even more, hypergraph. This means that it has a lot more structure beyond monoidality: It has products (that are actually biproducts), cups and caps (because it is compact closed), etc. So there is no way to generate this kind of stuff from one of our nets: FDHilb has waaay more equations than just a monoidal cat. What you can get, tho, is functors from our categories to FDHilb. This is what "freeness" means. :) In https://arxiv.org/abs/1805.05988 we were able to tweak the definition of Petri net a bit to let it generate free compact closed categories, and I feel this is the best we can do.

The kind of graphical gadget that generates FDHilb (in the sense that the graphical calculus is sound and complete wrt FDHilb) is called ZX calculus (or one of its equivalent variants, such as ZW). It took roughly 10 years to prove that ZX is complete wrt FDHilb! In any case, a string diagram in ZX calculus looks like a hypegraph with extra properties and equations. But you lose the dynamic interpretation of tokens moving in the net, there are no tokens in ZX!

Suppose there exists consciousness and there exist human selves through which consciousness flows. Suppose a new human self is created during every session of sleep and the previous one is archived. Suppose it's also possible to arbitrarily construct additional selves, which some people do out of trauma.

What kind of object is the self in this case? I've been contemplating this stuff and I'm not clear on how to model things that evolve over time in this way.

Do you want to be friends with someone looking to use CT in their daily life, including to change their language to improve alignment with reality (is that commutation with reality?) and to see how such changes impact the development of a child? Because the kid's 2 now and I could really use some sanity checking on my principles cause they're learning things way fast.

How would you recommend I go about making friends in the CT community? I'm a full-time parent without a degree and see CT as something worth trying to teach my child now (though without all the jargon).

Please don't, a 2yr old kid should basically play and have fun imho, there's so much time to experience the absolute misery of mathematical frustration.

I think you misunderstand what I'm suggesting. I'm looking to help them (the child, Uni) understand the things they're/we're playing with and having fun doing. I'm also experimenting with how I talk to them and I'm wondering if Modal Homotopy Type Theory suggests ways to form sentences to more clearly state things.

Also, Uni chooses what they do and when they do it. This includes diaper changes and baths. We seek to, at most, influence through what we say/do and configure the environment. We also moderate their food intake when it comes to things like sugar.

So with that in mind, does that change your answer at all?

So Petri nets are directed graphs, but with divergent paths describing chemical reactions along the way?

I'm not a chemist and my math is not amazing. Could someone explain how this would be used for, say, population dynamics, as mentioned in the article?

The "applications" are speculative at best. Chemists and biologists don't seem to care. This is really just category theory for the sake of category theory.

Petri nets are a quite general model of concurrency that has also found use in many places in computer science. They are used in model checking, and also as the foundation for giving a semantics to BPMN, for example.

I think it's not unreasonable to consider Petri nets as a generalization of automata (finite or infinite) with concurrency. Just like automata, they find applications in many places depending on how you further extend or restrict them.

I honestly doubt it will have any practical applications in biology. Computation chemistry and synthetic biology run on differential equations, anything discrete is generally useless. Perhaps it will have some use in a couple of decades when synthetic biology progresses beyond brute force simulations but at the current level it is the wrong abstraction.

You can turn Petri nets into stochastic nets using some formal procedures. Stochastic nets have a "master equation" which spits out a system of PDEs representing reactions where many many things happen at the same time, and so it makes more sense to think in terms of concentrations etc.

We are working to capture this categorically as well. We are perfectly conscious of how a discrete system won't help you if you have an Avogadro number of things going around. Just give us time. :)

We may disagree on this point but thanks for taking the time to reply nonetheless :) I will take a look at your paper.

you have to understand that the real power behind this work is not the Petri nets but that fact that they are described so abstractly that they can take on different shapes, such as stochastic nets, or coloured nets, or...

It puts the different models on the same footing (and hence allowing you to unify tools and do more work)

Also, I'd even argue that biology/chemistry needs to embrace categorical methods and we would see some deep discoveries.

The differential equations are themselves in many cases an abstraction over statistical numbers of discrete events. Arguably, in those cases, they are the wrong abstraction - a thesis which can only be proven by demonstrating the superiority of a different abstraction. Has one been demonstrated? Not yet, for most purposes. Is this a promising avenue of study? IMO, yes.

There's no speculation about the usefulness of having a mathematical foundation for understanding.

Chemists and biologists not caring can have a lot to do with hyperspecialization and fear of math. Category theory is still relatively esoteric.

"Mathematical foundations" are not always useful to practitioners – or even at all. Algebraic quantum field theory is a good example of the latter, to a large degree [0].

The fact that workers in the field with the presumed "applications" don't seem to care is pretty telling. My guess is that working at a level of generality where subject-specific insights are suppressed automatically limits the usefulness of the work.

[0]: http://philsci-archive.pitt.edu/8890/1/critique_sep10.pdf

It's funny, this is exactly what people have been saying of Categorical Quantum Mechanics and ZX calculus for 14 years, until they didn't, and now ZX calculus is used to draft quantum protocols and businesses like Cambridge Quantum Computing are investing heavily in it.

The fact that mathematical foundations don't have applications now does not mean they won't have applications in the future. Gregorio Ricci-Curbastro's work was also considered pretty useless until Einstein decided to build General Relativity over it. If it were for opinions like this one, progress in many field would be incremental at best.

Your characterization of the development of coordinate-free differential geometry is incorrect. Intrinsic geometry was not a seemingly "useless" generalization; it was motivated by concrete and specific problems about surfaces and mathematical physics. For example, one of Riemann's papers was literally titled "A mathematical work that seeks to answer the question posed by the most distinguished academy of Paris." (Perhaps not literally, given that's a translation, but you get the point.)

I don't know anything about ZX calculus, but if people are using it to solve real-world problems, that sounds good to me. What I object to is pure mathematicians giving "applications" of their work that aren't useful or real. And when the authors allude to applications in chemistry and biology, and no chemists or biologists are doing anything with category-theoretical analyses of Petri nets, I think it's reasonable to point this out.

Do you expect something that's useful to be immediately acknowledged as such in the field, much less adopted?

Human cultures are more complex than that. Capitalistic pressures to produce results have had a noted impact on research and development. There will likely need to be a practitioner willing to do the field work to bring about the big "Aha!" for that field. Based on the silo'd history of so many fields, it's a safe bet to say that may be necessary in every field where CT can be applied.

Yes, I expect practitioners to care about something useful, assuming the result has been properly written up and explained in a way they can understand. Scientists want good results and will adopt new tools if you can make a case that they will help them publish impactful papers.

Again, it's pretty telling specific applications aren't mentioned, and instead there's just vague gesturing. How can we expect scientists to "do the field work" necessary to bring the applications to fruition if we can't even tell them what those applications are?

Also, regarding your first sentence, the vast majority of influential mathematics was indeed "immediately acknowledged," because it made progress on some important problem. Perfectoid spaces are a recent example.

Nice, if everyone thought like you do, number theory wouldn't have existed, elliptic curves along with it, and hence much of public key cryptography as well. Indeed, number theory has been exquisitely useless for a couple of millennia, give or take.

I feel you are one of those people that just hate category theory because they consider it too abstract and useless for any purpose. We get that a lot, from a lot of people. Still, since Grothendieck, categorical methods are absolutely central in modern algebraic geometry and topology, and this centrality is only destined to grow, because CT is the best theory we have to manage emerging complexity in describing systems. In my opinion, the approach of "if it's not immediately useful then it's useless" really will lead you farther and farther away to understand modern developments in applied mathematics.

This mischaracterizes my position. I'm not against pure mathematics research, but I am against claiming applications exist when they don't.

I am a practitioner of life sciences. I use category theory to understand what I'm looking at around me in the world. I don't know if I'm even applying it properly and it's still been incredibly useful to me. I've developed basic principles for myself I use in daily life, to handle my emotions, and for raising my child.

Practitioner apathy and willingness to dismiss things as not useful can reflect a lack of intellectual humility in their cultures and lives.

The generality is useful if one practices seeking subject-specific insights from the generality. It sometimes leads to new options I haven't even thought of or leads me to reconsider decisions.

"Useful" is a judgment and most judgments like this are made on too short of a timeline. What's actually happening is someone hasn't found a use for the thing labeled as "useless" and is blaming it on the thing, themselves, or where the thing came from. Instead, a growth mindset focused on learning, accountability, and responsibility might be "I did this thing and the outcome I wanted didn't occur. I haven't yet learned how to use it for said outcome."

Not at all. We have been applying category theory for the past few years to software and systems design and this paper in particular came as a result of those implementations.

Soon these methods will be usable by many people for all kinds of purposes. You can think of cryptographic contracts, business process execution, game theory, functional reactive programming, quantum protocols and digital or analogue electronics.

The only difference is what category the diagrams live in, or put another way, which semantics are you assigning to thee boxes and wires.

You can stay tuned on statebox.org and process.io

Petri nets (and variant colored petri nets) are just ways of modeling states. We used them in my CS program to model application states for program verification.

I've got a paper about colored nets as well :) https://arxiv.org/abs/2002.02762 Timed nets is in the works ^_^

Petri nets can be used for populations dynamics, and indeed the structured cospans approach to Petri net is being used to simulate epidemic diffusion right now: https://www.algebraicjulia.org/blog/post/2020/10/structured-...

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