In this post I want to emphasize a strange similarity between the beta rule in lambda calculus and the chora construction (i.e. encircling a tangle diagram).
Motivation? Even if now clearer, I am still not completely satisfied by the degree of interaction between lambda calculus and emergent algebras, in the proposed lambda-Scale calculus. I am not sure if this is because lambda-Scale is yet not explored, or because there exist a more streamlined version of lambda calculus as a macro over the emergent algebras.
Also, I am working again on the paper put on preview (version 05.06.2012) about planar trivalent graphs ans lambda calculus, after finishing the course notes on intrinsic sub-riemannian geometry.
So, I let my mind hovering over …
As explained in the draft paper, the beta rule in lambda calculus is a LOCAL rule, described in this picture (advertised here):
It is made by two halves: the left half contains the lambda abstraction, the right half contains the application operation. In between there is a wire. The rule says that these two halves annihilate somehow and the wire is replaced by a dumb crossing with no information about who’s on top.
Let us contemplate an elementary chora, made also by two halves:
We can associate to this figure a move, which consists in the annihilation of the left (difference gate) and right (sum gate) halves, followed by the replacement of the “wire” by an equivalent crossing