I don’t know yet what exactly is “computing with space”, but I almost know. Further is a description of the road to this goal, along with an invitation to join.
Before starting this description, maybe is better to write what this explanation is NOT about. I have arrived to the idea of “computing with space” by branching from a beautiful geometry subject: sub-riemannian geometry. The main interest I have in this subject consists in giving an intrinsic (i.e. by using a minimal bag of tools) description of the differential structure of a sub-riemannian space. The fascinating part is that these spaces, although being locally topologically trivial, have a differential calculus which is not amenable to the usual differential calculus on manifolds, in the same way as the hyperbolic space, say, is not a kind of euclidean space, geometrically. I consider very important and yet not well known the discovery of the fact that there are spaces where we can define an intrinsic differential calculus fundamentally different than the usual one (locally, not globally, as it is the case with manifolds which admits different GLOBAL differential structures, although at the local level they are just pieces of an euclidean space). But in this post I shall NOT go to explain this. The road to computing with space branches from this one, however there are paths represented by mathematical hints which are criss-crossing both these roads.
1. In “Dilatation structures I. Fundamentals” I propose, in section 4 “Binary decorated trees and dilatations” a formalism for making easy various calculations with dilation structures (or “dilatation structures”, as I called them at the moment; notice that dilation vs dilatation is a battle won by dilations in math, but by dilatation in other fields, although the correct word historically is dilatations).
This formalism works with moves acting on binary decorated trees, with the leaves decorated with elements of a metric space. It was extremely puzzling that in fact the formalism worked without needing to know which metric space I use. It was also amazing to me that reasoning with moves acting on binary trees gave proofs of generalizations of results involving elaborate calculations with pseudo-differential operators and alike. At a close inspection it looked like somewhere in the background there is an abstract nonsense machine which is just applied to this particular case of metric spaces.
Here is an example of the formalism. The moves are (I use the names from graphic lambda calculus):
Define the following tree (and think about it as being the graphical representation of an operation):
Think that it represents , with respect to the base point . Then we can prove that
which is a kind of associativity relation. The proof by binary trees has nothing to do with sub-riemannian geometry, right? An indirect confirmation is that the same formalism works very well on the ultrametric space given by the boundary of the infinite dyadic tree, see Self-similar dilatation structures and automata.
As a conclusion for this part, it seemed that in order to unravel the abstract nonsense machine from the background, I needed to:
- find a way to get rid of mentioning metric spaces, so in particular to get rid of decorations of the leaves of binary trees by points in in some space, (or maybe use these decorations as a kind of names)
- express this proof based on moves applied to binary trees as a computation, (i.e. as something like a reduction procedure).
Otherwise said, there was a need for a kind of logic, but which one?