[HN Gopher] Categories of Nets
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Categories of Nets
Author : chmaynard
Score : 51 points
Date : 2021-01-18 11:42 UTC (11 hours ago)
(HTM) web link (johncarlosbaez.wordpress.com)
(TXT) w3m dump (johncarlosbaez.wordpress.com)
| FabGenovese wrote:
| I'm one of the authors of the paper. Ask me anything!
| turnersr wrote:
| 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.
| FabGenovese wrote:
| 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!
| rocqua wrote:
| 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.
| FabGenovese wrote:
| 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
| carterschonwald wrote:
| Have you sat down with any epidemiology researchers to relate
| this to the compartment models they usually use?
| FabGenovese wrote:
| 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. :)
| bovermyer wrote:
| 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?
| FabGenovese wrote:
| 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-...
| spekcular wrote:
| 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.
| wires wrote:
| 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
| ulrikrasmussen wrote:
| 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.
| ampdepolymerase wrote:
| 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.
| FabGenovese wrote:
| 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. :)
| ampdepolymerase wrote:
| We may disagree on this point but thanks for taking the
| time to reply nonetheless :) I will take a look at your
| paper.
| wires wrote:
| 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.
| akimball wrote:
| 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.
| crawfordcomeaux wrote:
| 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.
| spekcular wrote:
| "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
| FabGenovese wrote:
| 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.
| spekcular wrote:
| 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.
| crawfordcomeaux wrote:
| 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.
| spekcular wrote:
| 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.
| FabGenovese wrote:
| 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.
| spekcular wrote:
| This mischaracterizes my position. I'm not against pure
| mathematics research, but I am against claiming
| applications exist when they don't.
| crawfordcomeaux wrote:
| 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."
| timvdalen wrote:
| 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.
| FabGenovese wrote:
| I've got a paper about colored nets as well :)
| https://arxiv.org/abs/2002.02762 Timed nets is in the works
| ^_^
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