Symmetries of conflicts

Can you find hidden symmetries in the conflict diagrams of chemlambda? There are now two pages to play with, if you want to help (and have some fun maybe): [1] and [2].

What’s that? Each rewrite/chemical reaction in the artificial chemistry chemlambda has a pattern which triggers the reaction, and then the pattern is replaced by another one by an invisible “enzyme”. Now, the pattern which triggers the reaction is called LP (left pattern).

[3] is the list of rewrites, to be clear!

In chemlambda all LP consist of a pair of nodes, with some ports connected by an external bond (the internal bonds being those which connect the node with its ports). It happens that there exist two pairs of nodes, say LP1 and LP2, which overlap: they have a node in common.

In such a case it means that there are two possible reaction which may happen, but which one will happen? They can’t both happen at the same time, because the patterns overlap and so the results would be incompatible. That’s a conflict.


In a conflict diagram, as the one which is represented here, you see the nodes of chemlambda which enter in he composition of possible LPs, here you see the FI and L (red, but with different colors and radii of ports) FO and A (green, with different ports etc) and FOE (yellow). For each possible LP (i.e. for each rewrite which is triggered by an LP) there is a bond (thin white line) which connects the relevant ports of the pair of nodes of the LP.

What we get is not a chemlambda molecule, because there may be many bonds which connect a port with others. Here for example you see the FOE port middle_in which is connected with 4 other ports, indeed, this is because we find FOE in all DIST rewrites L-FOE, A-FOE, FO-FOE and in the rewrite FI-FOE.

But such a diagram allows to quickly read the possible conflicts.

Now, to symmetries: the conflict diagram shows the role of the nodes not absolutely, but as they enter in relative configurations. Therefore any symmetry of the nodes, ports and bonds (i.e. of rewrites) tells us something about chemlambda which is not obvious in isolation.

For example, I used the conflicts diagram [2] to group the nodes and bonds in order to show that in a sense the rewrites which involve L (the lambda abstraction) and the FI (the fanin node) are “the same”. THere are hints towards an overall symmetry of ports and rewrites in the diagram.

You can find others, in [2] or [3] which has also the node T (here the color of the node is red) and the PRUNING rewrites added.

For this you can move the nodes and fix their position by click and drag, or release them by double click.

Tell me if something seems weird, hidden there.




Do triangulations of oriented surfaces compute?

In a precise sense, which I shall explain, they do. But the way they do it is hidden behind the fact that the rewrites seem non local.

  1. They compute, because ribbon graphs with colored, trivalent nodes and directed edges do compute, via the encoding of untyped lambda terms into this family of graphs, provided by chemlambda. Indeed, a chemlambda molecule is a ribbon graph with these properties. If you want to encode a lambda term into chemlambda then there is a simple procedure: start from the lambda term on a form which eliminates the need of any alpha conversion. Then build the syntactic tree and replace the nodes by A nodes for application and L nodes for lambda abstraction (don’t forget that L nodes have one in and 2 out ports, differently from the syntactic tree node for lambda abstraction). Then eliminate the variables which are at the leaves by grafting trees of FO (green fanout) nodes from the lambda abstraction node to the places where the variables occur, or by grafting T (terminal) nodes to the lambda node which issues a variable which does not occur later, or simply by just erasing the variable label for those variables which are not issued from an abstraction. That’s it, you get a ribbon graph which is open (it has at least the root half-edge and maybe the half-edges for the variables which don’t come from an abstraction), but then you may add FRIN (free in) and FROUT (free out) nodes and think about them as tadpoles and you get a trivalent ribbon graph. The dual of this graph is (equivalent to) a triangulated, oriented surface, which has faces colored (corresponding to the nodes of the graph), directed edges, such that there are no faces with the 3 edges directed in a cyclic way.
  2. How they compute? Chemlambda uses a set of graph rewrites which has some classic ones, like the Wadsworth-Lamping graphical version of the beta move, but it has two types of fanouts (FO and FOE), one FANIN, and different than usual rules for distributivity. Look at the moves page to see them. All these rewrites are local, in the sense that there is a small number, fixed a priori, which is an upper bound for the number of nodes and edges which enter (in any way) into the graph rewrite (as a condition or as the left pattern, or as the right pattern). The algorithm of application of the rewrites is a very important piece which is needed to make a model of computation. The algorithm is very simple, it can be deterministic or random, and consists, in the deterministic case, into the application of as many rewrites as possible, with a priority for the distributivity moves in case of conflict, and in the random case, it’s just random application of rewrites.

Here is an example, where I play with the reduction of false omega id in chemlambda

  1. Now let’s pass to the duals, the triangulated surfaces. The nodes of the triangulated surface correspond to the faces of the ribbon graph. Or the faces of the ribbon graph are global notions, because they are the orbits of a permutation. After one of the rewrites, the faces (of the ribbon graph) change in a way which has to be non local, because one has to compute again the orbits of the permutation for the new graph, and there is no upper bound on the number of half-edges which have to be visited for doing that.
  2. So triangulated, oriented surfaces do compute, but the rewrites and the algorithm of application are hidden behind this duality. They are non-local for triangulated surfaces, but local for ribbon graphs.
  3. Finally, a word of attention: these surfaces do compute not by being arrows in a category. They don’t compute in this usual, say Turaev kind of way. They compute by (the duals of) the rewrites, there is nothing else than triangulated surfaces, colored by 3 colors (red, green, yellow), there is no decoration which actually does the computation by substitution and evaluation. I don’t know why, but this seems very hard to understand by many. Really, these surfaces compute by rewrites on the triangulations, not by anything else.

ADDED: If you look at the tadpoles as pinches, then make the easy effort to see what  the SKI formalism looks like, you’ll see funny things. The I combinator is the sphere with one pinch (the plane), the K combinator is the sphere with two pinches (cylinder) and the S combinator is the torus with one pinch. But what is SKK=I? What is KAB=A? What you see in the dual (i.e in the triangulation) It depends globally on the whole term, so these reductions do not appear to be the same topological manipulations in different contexts.

Res vs objectus

Objects are evidence. If reality is the territory, then objects are on the map.  Objective reality is to be compared with bureaucracy.
If reality is not objective, then how is it? Real, of course. Evidence is a map  of the real. Passive, done already, laid further in the court, ready to be used in the argumentation.
Who makes the map has the power over reality, in the same way as bureaucrats have the power over the people.
The confusion  between res and objectus has very concrete effects in our society.
We communicate on the net via evidence. The technical solutions are these, issued from historical reasons, like wars and analytic philosophy.
We are discontent about the lack of privacy of evidence.
Objects as evidence of happiness are not the same as happiness. We are discontent because objects are not enough, when we are told that they should be.
In this setting, who controls the map making mechanism, who controls the data processing, has the power.
Ultimate bureaucracy presented as the unique way. As the only real way. A lie.