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?
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 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!
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!
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. :)
: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 ^_^
