False choice.

I’ve spent the week mulling over this. Bluntly, it is neither pictorial nor diagrammatic. The elements of Turing Machines et. al. are information.

We can abstract the output of a CA by agreeing on a convention and call this a diagram. A similar scheme on a universal CA, Turing Machine, or GOL simulation would net a visual appearance nearly as verbose as the original, and therefore would not be terribly diagrammatic. There’s no “meta” message to be pulled out of these systems other than the rule sets.

Nor can I gain much traction in describing even Lindenmayer systems as pictorial. They speak about topological relationships of the plant’s branching, but rarely fool me for a second as to their origin. Additionally, there is some fancy, and I mean _FANCY_, math on the back end to get {{a->a},{b->a,b}} to look like my houseplant. As for CA, come on… It’s just not a picture. It’s not representing anything beyond the information itself.

I feel like the pictorial trap is precisely where so many have gone wrong. These systems provide a rich test bed for theories of morphogenesis, social systems and a dizzying array of other phenomena, but it is often forgotten that these are models of phenomena and not the phenomena itself. The substitution of a model for its phenomena is endemic, and is demonstrated with whip cream on top the philosophy of Nick Bostrom, who believes we are living in a simulation. WTF?! A group of mathematicians has placed the probability at 1 in 20. No joke.

Perhaps this can feed into the other question on the table:

Computation between Geometry and Topology

On the one hand, these systems excite me to a degree that I’m uncomfortable relegating them to a prepositional phrase, yet they’ve enjoyed a grammatical identity in the hands of Noam Chomsky in his research into linguistics. In the interest of completely dorking out this evening I re-checked his Wikipedia entry, and, lo, he’s got this automata theory of formal languages. I read about this a while back and quite honestly couldn’t make any sense of it, but I would guess that Peter’s four food groups would have a place on Chomsky’s table. Turing machines are mentioned by name.

In his “Computational Theory of Morphogenesis”, Przemyslaw Prusinkiewicz, draws the axes of his consideration along three lines.

1. One or many
2. Computing capability (a finite automaton or an automaton with counters)
3. Information exchange with the environment

The last of these certainly involves communication, or this kind of go-between again. In modern agent based programing, the geometry takes the role of the discrete unit, but it seems like a computational grammar serves the role of the information exchange. (Or was that topology). I'm not positive these mix and I need to think about this. More later.