Some news after the anouncement of kali, or anharmonic lambda calculus. I use again two pages, which are updated in tandem:
At the moment I write this post the github.io version is more recent. I think the termination rewrites are better than before.
There is still some work on the exact choice of rewrites, among the many possible which are compatible with the underlying geometric structure. But you can see by looking at kali24.js that there is some connection between the nodes and the anharmonic group.
All this will be explained when I’ll arrive to the most satisfying version.
I added to the lambda calculus examples some more, not lambda calculus ones. Among them the much discussed 10-node quine and also the most amazing molecule I discovered until now. It appears in the menu as “10 nodes sometimes becomes quine from [graph A-L-FI-FOE 540213]” and please do reload it often and let it unfold. For an archived video see this one. It is a graph which basically shatters the notion that a quine is enough, conceptually, to describe life. More simply put, it can evolve in so many ways, among them in a chemlambda quine way, but not uniquely. Amazing.
You can see it also on the menu of the find a quine page. There the graphs look more compact and you can see more of the overall structure, but less the detailed linking of nodes.
Coming back to kali24, the chunk which I need to explain first is what does it have to do with em-convex. That is an enriched lambda calculus which describes my work on emergent algebras. It ends with the proof that in the presence of an axiom called “convex”, we end up with usual vector spaces over terms of type N (in the paper) and that also the term of type N have an em calculus themselves, which is a way of saying that we recover on them a structure which is like the Riemann sphere.
What is not proved are two things: (1) is there is a graphical rewrite system which can reproduce the proofs of em-convex, under the algorithm of rewrites used with chemlambda? (2) can we dispense of the lambda calculus part (the abstraction and application operations) and redo everything only with the pure em calculus?
Back to earth now, just for the fun, don’t you think that a geometrical model of lambda calculus on the Riemann sphere would be nice?