[FRIAM] comonadic-style: WAS: stygmergy, CA's, and [biological] development

[Jon Zingale]

(reposted for readability)

Comonads, like trees or stacks, are data structures which guide a
computation along a chosen line and imply a mode of reasoning. That they
first arose in the context of category theory should not make them lesser
citizens.

What the comonad captures is the essence of locality[◊], like it's dual
captures the essence of encapsulation. The comonadic cellular automata,
assigns to each cell a world with that cell at its center (subjectivity?).
That this assignment can be done *naturally* (via the natural lifting
δ: W -> W²) and progressed (via the composition W_β ○ δ) means that one
can handle many of the implementation details in the coalgebra, which a
language's compiler can be tuned to exploit. Two points of exploitation
that come immediately to mind being:

1. parallelization (especially wrt adaptive chunking)
2. algebrao-functional reduction (I'm not sure what to technically call
this)

Beyond this, I rather like not having to build in the topological
constraints in the low level details. Instead, as in my implementation[!],
one can allow computation over doubly unbounded lists and lists of lists
while only making the judicious choice of board size and it's accompanying
details later. In practice, I like this style of implementation because
it generalizes nicely while remaining structured (maintainability) and
suggests a clearer perspective from which to understand cellular automata,
that of a naturally localized computation.

The reason I mention this is that the comonadic-style walls-off each cell,
in a sense, much like David Lewis' counterparts. To write a shuffle in
this style is easy, but I suspect that to purely write a sort this way is
impossible. I would love to know whether this is, in fact, true.

[□] Yes, I know that his theory is less popular than Kripke's, but hey.
https://en.wikipedia.org/wiki/Counterpart_theory

[◊] Arguably, this "essence of locality" is an expression of *possible
worlds*, via the adjunction: X -> □Y iff X -> Y, or ◊ ⊣ □. More on this
can be read here:
https://ncatlab.org/nlab/show/necessity+and+possibility#:~:text=3.-,Possible%20worlds%20via%20first-order%20logic%20and%20type%20theory,-One%20common%20philosophical

[!] But the original idea here isn't mine, I followed up on ideas written
here: http://blog.sigfpe.com/2006/12/evaluating-cellular-automata-is.html
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