I continue from Packing and unpacking arrows in graphic lambda calculus and I want now to describe another sector of the graphic lambda calculus, called the strand networks sector.
Strand networks are a close relative to spin networks (in fact they may be used as primitives, together with some algebraic tricks in order to define spin networks). See the beautiful article by Roger Penrose, Angular momentum: an approach to combinatorial space-time, in Quantum Theory and Beyond, ed. T. Bastin, Cambridge University Press, Cambridge, 1971. where strand networks appear in and around figure 17.
However the strand networks which I shall write about here are oriented. Moreover, a more apt name for those would be “states of strand networks”. The main thing to retain is that the algebraic glue can be put over them (for example, as concerns free algebras generated by those, counting states, grouping them into equivalence classes which are the real strand networks and so on).
In graphic lambda calculus we may consider graphs in which are obtained by the following procedure. We start with a finite collection of loops and we clip together, as we please, strands from these loops. How can we do this? Is simple, we choose our strands we want to clip together and the we cross them with an arrow. Here is an example for three strands:
We want to make this clipping permanent, so to say, therefore we apply as many graphic beta moves as are needed, three in this example:
The beautiful thing about this procedure is that in the middle of the graph from the right hand side we have now three arrows with the same orientation. So, let’s pack them into one arrow. The packing procedure for three arrows, with the same orientation, is the following:
So, we pack the arrows, we connect them and we unpack afterwards.
With the previous trick, we may define now decorated arrows of strand networks, but mind that we have bundles, packs of oriented strands, so the decoration has to be more precise than just indicating the number of strands.
That is about all, only we have to be careful to discern among clipped strands which become decorated arrows and clipped strands which become nodes of the strand network. But this is rather obvious, if you think about.
Conclusion. Let’s think about the purpose of the algebraic glue which is needed in order to define a spin network and then an evolving spin network. As you can see all comes down to another way of deciding which is the sequence of applications of (graphic) beta moves, which is this time probabilistic. It is another type of computation than the one from the lambda calculus sector, where we use the combinators and zippers (as well as some strategy of evaluation) in order to make a classical computation. So at the level of graphic lambda calculus, it becomes transparent that both are computations, but with slightly different strategies. (This is a kind of a theorem, but to put it in a precise form one has to make ornate notations and unnecessary orderings, so it would be just a manifestation of the cartesian disease, which we don’t need.) The last point to mention is that, compared to the lambda calculus strategy (based on combinators plus choice of evaluation), this computation which comes from physics, is more cartesian disease free.