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.

 

conflict_wo_T
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.

[1] http://chorasimilarity.github.io/chemlambda-gui/dynamic/conflicts_wo_T.html

[2] http://chorasimilarity.github.io/chemlambda-gui/dynamic/conflicts.html

[3] http://chorasimilarity.github.io/chemlambda-gui/dynamic/moves.html

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