UPDATE (09.01.2013): There is now a web tutorial about graphic lambda calculus on this blog. At some point a continuation of “Computing with space …” will follow, with explicit use of this calculus, as well as applications which were mentioned only briefly, like why the diagram explaining the “emergence” of the Reidemeister III move gives a discretized notion of scalar curvature for a metric space with dilations.
Explanations. In the “Computing with space…” paper I claimed that:
1. – there is a “computing” part hidden behind the idea of emergent algebras
2. – which is analogous with the hypothetical computation taking place in the front-end visual system.
The 1. part is done, essentially. The graphic version of -Scale is in fact very powerful, because it contains as sectors:
– lambda calculus
– (discrete, abstract) differential calculus
– the formalism of tangle diagrams.
These “sectors” appear as subsets of graphs in (see the preview paper for definitions), for which the condition $G \in S$ is global, together with respective selections of local or global graphic moves (from those available on ) which transform elements of into elements of .
For example, for lambda calculus the relevant set is -GRAPH and the moves are (ASSOC) and the graphic move (actually, in this way we obtain a formalism a bit nicer than lambda calculus; in order to obtain exactly lambda calculus we have to add the stupid global FAN-OUT and global pruning moves).
For differential calculus we need to restrict to graphs like those in -GRAPH, but also admitting dilation gates. We may directly go to -Scale, which contains lambda calculus (made weaker by adding the (ext) rules, corresponding to -conversion) and differential calculus (via emergent algebras). The moves are (ASSOC), graphic move, (R1), (R2), (ext1), (ext2) and, if we want a dumber version, some global FAN-OUT and pruning moves.
For tangle diagrams see the post Four symbols and wait for the final version of the graphic calculus paper.
SO now, I declare part 1. CLOSED. It amounts to patiently writing all details, which is an interesting activity by itself.
Part 2. is open, albeit now I have much more hope to give a graph model for the front-end of the visual system, which is not relying on assumptions about the geometric structure of the space, linear algebra, tasks and other niceties of the existing models.
UPDATE 02.07.2012. I put on arxiv the graphical formalism paper, it should appear on 03.07.2012. I left outside of the paper a big chunk of very intriguing facts about various possible crossing definitions, for another paper.