The rewrites are:

With these rewrites we can do some lambda calculus reductions, by using the same parsing of lambda calculus terms to (chemlambda) graphs as in GLC. For example

A rewrite which is introduced later in the article is DISENTANGLE (as a pattern it appears later as SHUFFLE)

With this supplementary rewrite I can prove that the BCKW combinators system

can be implemented, in the sense that the combinators can duplicate (example here for B)

and that all needed reductions among B,C,K,W combinators can be done (example here for K)

This graph rewrite system has conflicting rewrites, but it has no algorithm of application of rewrites. Especially the status of DISENTANGLE is unclear. This will be solved in chemlambda v2.

The first published (in the usual ways) article which contains the name "chemlambda" is joint with L.H. Kauffman, in the ALIFE-14 conference, [arXiv:1403.8046]. Here Kauffman pushes again for a more topological version [Kauffman' slides] which mixes tangle diagrams with chemlambda nodes, in order to be able to do computations (in the logical sense).The emergent algebras part of GLC, which is still present in chemlambda, used such tangle diagrams manipulations for differential calculus computations, instead. In these articles the emergent algebras aspects and interest are hidden.

Together with Kauffman, we experimented a lot with the power of chemlambda v1. Our essays on paper were not going too far, due to the lack of a program which could make much easier such exploration. In the background this was because of a lack of a clear rewrite application algorithm.

During this period I became aware of the existence of chemlambda quines. See as an example the tedious manipulation of reductions for the predecessor of a Church number taken from [this chorasimilarity post]

Meanwhile Kauffmann started to use Mathematica for doing chemlambda v1 reductions. He introduced the "Arrow" 2-valent node, which is useful for easy application of several rewrites in the same time.

Chemlambda v2

is an artificial chemistry, in the sense that it is a purely local graph rewrite system together with an algorithm of applications which can be done by local machines. See [arXiv:2005.06060] for more explanations.

Motivated by the first programming essays by Kauffman for chemlambda v1, I wrote the needed programs for chemlambda v2, in awk and html+d3.js used for visualization. The first chemlambda project site contains demonstrations made with these programs. The github repository is chemlambda-gui.

The latest chemlambda project site, which contains several repositories from various contributors, is this. With the help of contributors, now we can reduce chemlambda molecules directly in js, as you can see in the more recent demonstrations.

Chemlambda can now be compared with Interaction Combinators and there is a lambda calculus to chemlambda parser and reducer.

The chemlambda v2 rewrites solve the problem of the curious DISENTANGLE rewrite by introducing two fanout nodes (FO and FOE) instead of one and by a modified set of graph rewrites. These are the following. We use the mol notation for clarity and we describe the rewrites by giving the LHS and RHS patterns:

and the COMB rewrites which eliminate Arrow nodes.

The DISENTANGLE rewrite is no longer needed, because it's effect can be achieved by the SHUFFLE move, which is a sequence of two chemlambda v2 rewrites: FO-FOE and FI-FOE.

Interaction Combinators, as they appear in chemlambda v2 programs

For completeness, here are the graph rewrites of Lafont' Interaction combinators, as they appear in the chemlambda v2 js programs, chemistry IC in chemistry.js: FRIN and FROUT 1-valent nodes are introduced (also in chemlambda) for the free half-edges. We use also the Arrow 2-valent node and the COMB rewrites for eliminating them.

All LHS patterns, of chemlambda and IC, are made of two nodes connected through their active ports. The main difference between chemlambda v2 rewrites and IC rewrites is that several chemlambda nodes have two active ports, while IC nodes have only one active port.

It is possible to modify the chemlambda v2 chemistry such that there are no conflicts. We obtain a chemistry called "dirIC", or directed interaction combinators, explained in
[Alife properties of directed interaction combinators vs. chemlambda. Marius Buliga (2020), https://mbuliga.github.io/quinegraphs/ic-vs-chem.html#icvschem]
over the same set of nodes as chemlambda v2. With the chemistry dirIC, there is a translation between IC nodes of Lafont to chemlambda nodes: As argued here, the existence of conflicting rewrites in chemlambda v2, compared with dirIC, creates much more interesting alife behaviours.