Saturday, June 18, 2011

Coecke's Diagramatic Calculus: from factories to autocatalysis

Just today, I was thinking about using the diagramatic calculus of Coecke and Joyal and Penrose to think about different aspects of day to day life. A typical diagram in the calculus is like this:

Specifically, I wanted to use these diagrams to think about various things like :
  1. Factories
  2. Economies
  3. Chemical reaction networks
In the case of a factory, there are always many sub-assemblies that are made before they are assembled into the final product. Each sub-assembly is a process and all these processes hook together and probably make a diagram like those that Coecke uses to talk about Quantum protocols. In the case of an economy, we simply take each factory as a single process which has a bunch of material inputs (plus energy input) and spits out something and then all these factories link together in a network which is probably like the ones Coecke uses.
Then I started thinking about chemical processes and I wondered if you can bend a wired around back and attach it to the input of a box. That is something you definitely do in the diagrammatic calculus.

I then realized by looking at the diagram that what I had invented was the catalyst, in an abstract sense. This means that catalysts, when seen in networks of chemical processes are actually traces in the category. This also kind of indicates that chemical process networks should be modeled as these kinds of diagrams. But then this only makes sense since chemical reactions are exactly maps in the category of Hilbert spaces and the axioms of this category are equivalent to the rewrite axioms of these diagrams.
Looking around online for anything written about this, I didn't find anything so I wrote about it here. What I did find was the concept of autocatalysis. This is where the product of a process is its own catalyst and that diagram is demonstrated here:

We see that there is no output except the catalyst. This means that the process itself is simply there to encourage more of itself. Some say that this has something to do with life itself. The metaphor would be something like: life is a process that simply makes stuff that increases the likelihood of that process happening again.
I am wondering if diagrams like this exist in Coecke's diagrammatic calculus and I think they must.

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