# Computing with space in the internet of things

My primary interest is “computing with space”.  Some remarks by  Allen Francom  make me believe that it might be a way to give a “space” to the internet of things by using graphic lambda calculus. By that I mean the following: the internet of things is a huge collection of objects which are somewhere on Earth, each carrying a very small processor. So the objects are in the physical space, but they change their places all the time, and their processors are tiny, so one better use tiny programs and simple executions for syncronizing their relative places one with respect to the others.

That is a kind of “computing with space” and it fits well with the following problem with e-books raised by  by Mark Changizi, who says that  e-books lack space and gives as an example his (physical library). He says he knows that book A is the one close to that calculus book he uses when he need some formulae, and that’s the way he is going to find book A. While, in a virtual library there is no space, the relative positions of a e-book wrt to others is irrelevant, because when you click a link is like you teleport (and ignore space) from one place to another. My interpretation of this is: the way Changizi finds a book in his library is different from the way a robot Changizi would find the book, because the robot Changizi would need the coordinates of the book A with respect to  a corner of the bookshelf, or some information like this, then it would go and fetch the book. On the contrary, the human Changizi (‘s brain) acquired a geometric competence by accumulating these simple bits of knowledge about his library, namely tht book A is near the calculus book, and the calculus book is used when he needs formulae, etc.

Is this a problem which the internet of things will have? Surely. Is this solved? I have no idea, but I know how to solve it, by satisfying also the “tiny programs and executions” constraint. This involves exactly that part (called by me “emergent algebra sector”) which gives the kind of programs for using space in a very concentrated form, not needing “geometric expertise” to generate and manipulate them. Then, imagine that each object from the internet of things has a relative representation of some  of his neighbours, under the form of a small graph, like the graphs from graphic lambda calculus or the molecules from the chemical concrete machine. These graphs are generated from simple rules concerning proximity relations (like the keys thing is on the couch thing, or the socks thing is in the drawer thing), more on that later, and changes in proximity relations are like moves acting on such tiny graph molecules. Then, if you want to know where you lost the keys thing, instead of endowing the keys with a GPS, you just need to evaluate (i.e. decorate, like in the discussions about decorations of graphic lambda calculus graphs) these tiny graphs and find out that the key thing is i the drawer thing.
Here is a more mathematical explanation. Suppose you have a set $T$ of things, objects. You want to describe how they are in space without eventually using any “geometric expertise” (like using a GPS coordinate system, for example).

As you know, the set of things $T$ is the same as the trivial groupoid of things $T^{2}$, with arrows being pairs of things $(a,b)$ and with composition of arrows being $(a,b)(b,c) = (a,c)$.  Now, let us define first what is the “space” around one object from T, let’s call it “$e$“. (The same works for each object).

For defining this you take a commutative group (could be $(0,\infty)$ or could be the free commutative group over an alphabet (containing for example “NEAR” an “FAR”, such that $NEAR^{-1} = FAR$, etc). Call this group the “scale group”. Then to any pair of objects $(a,b)$ and any scale $\varepsilon$, associate a “virtual pair at scale epsilon”  $(a,b)_{\varepsilon} = (b(\varepsilon)a, b)$, where $b(\varepsilon)a$  is like a term in lambda calculus constructed from variable names “a” and “b”, only that it does not satisfy the lambda calculus definition, but the emergent algebra definition. Of course you can see such terms as graphs in graphic lambda calculus made by the epsilon gates, along with the emergent algebra moves from graphic lambda calculus, as in

Moreover, you can mix them with lambda calculus terms, as it is possible in graphic lambda calculus.

So we have (in our platonic universe, because in practice we can’t “have” an infinite set of all things possible) a collection of term-like objects made from a, b, etc and from scales epsilon, mu, etc. We may imagine them visually either as binary trees, like when we represent graphically compositions of operations,
or as graphs in graphic lambda calculus, if we apply to them the algorithm for eliminating variable names. This is a kind of collection of all programs for using a space (each one is a me for space). (Mind you that there is no constraint, like euclidean space, or other.) The space around one object, say e, is obtained from this construction applied to the groupoid of pairs of pairs with arrows $((a,e),(b,e))$  and the scale acts like this $((a,e),(b,e))_{\varepsilon} = ((a,e)_{\varepsilon}, (b,e)_{\varepsilon})$.  (Remark: in the epsilon deformed groupoid the composition of arrows is transported by the epsilon deformation, so the composition changed from trivial to more interesting.)
If you are still reading this and is too long, the shorter description is that the space around an object e is defined in terms of virtual geometric relations (mes for space)  between all the other things from the set $T$ (which is finite, of course), with respect to $latex e$, and that each such virtual geometric relation is a small graph in graphic lambda calculus. More than this, suppose you want to translate from a virtual geometric relation to another (or from a virtual place to another). Such a “translation” is itself a small graph, actually made by only 4 nodes.
And here comes the final part: in order to know where the things from $T$ are, one with respect to the others, it is enough to give,  to attribute to each pair of objects $(a,b)$ a “virtual geometric relation” in the space around $b$, i.e. a graph. If the things move one with respect  to the other, they can update their relative geometric relations by moves in graphic lambda calculus.
If you want to know where a thing is in the physical space then you have to start from somewhere (some object) and decorate the virtual space relations,as they appear as graphs. This is like evaluations of  lambda calculus terms.

In conclusion, I am very interested in practical applications of this, and I can at least formulate a list of needs, some of them with an IT flavour, others applied math. As for the pure math needs, they are pure fun and play. All this together would lead to really interesting things. One of the things to keep in mind is that would also endow the internet of things with a synthetic chemical connectome, as an extension of The chemical connectome of the internet .

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