Looks like there is a need to make a series of posts dedicated to the people who try to use this blog as a source in order to understand graphic lambda calculus (aka GLC) and chemlambda. This is the first one.

**Sources for GLC: **

- the best source is the article M. Buliga, Graphic lambda calculus. Complex Systems 22, 4 (2013), 311-360 (link to article in journal) (link to article in arXiv)
- you can see the GLC page (link) here, which has been updated many times after chemlambda appeared, but it is unmodified starting from the section “What is graphic lambda calculus?” and there are links to many others posts here which explain GLC as it is, that is before chemlambda.

**GLC is a graph rewriting system for fatgraphs** made of trivalent or 1-valent nodes. The trivalent nodes used are A (application), L (lambda), FO (fanout), epsilon (for dilations). There is one 1-valent node, T (termination). Loops with no nodes and arrows, i.e. oriented edges with no nodes are accepted as well. There is no algorithm proposed for the reduction of these graphs.

The graph rewrites are:

– local ones (i.e. involving only a finite number, a priori given, of nodes and edges)

- graphic beta move, which is like Lamping graph rewrite, only purely local, i.e. there is no limitation on the global shape of the graph, thus it is somehow more general than the beta rewrite from untyped lambda beta calculus; a possible interpretation is C= let x=B in A rewrites to x=B and C=A

- CO-COMM and CO-ASSOC rewrites for the FO (fanout) which are, due to the orientation of the edges, really like graphical, AST forms of co-commutativity and co-associativity
- local pruning group of rewrites, which describe the interaction of the trivalent nodes with the T node; incidentally T and FO interact like if T is a co-unit for FO
- a group of rewrites for the dilation nodes, which are those of emergent algebras, which involve the fanout FO and the dilation nodes

– global rewrites:

- global fan-out
- global pruning

In section 3 of the article on GLC is given an algorithm of conversion of untyped lambda terms into graphs which is a little more than a modification of the AST of the lambda term, in such a way that the orientation of the edges is respected. That is because the lambda node L has one incoming edge and two outgoing edges!

I prove then that GLC can be used for untyped lambda beta calculus, but also for emergent algebras and also for a variant of knot theoretic graphs where there is no need for them to be planar. Finally, I show that there might be other “sectors” of graphs which may be interesting, regardless of their meaning (or lack of it) with respect to lambda calculus.

The problem of GLC is that it has these global rewrites. Can these be replaced by local rewrites?

That’s how chemlambda appeared.

I was aware that some applications of global fanout can be done with local rewrites, but not all. Also, the interest in GLC was not extended to the interest into emergent algebras, to my dismay.

On the other side I became obsessed with the idea that if the global fanout can be replaced by local rewrites entirely then it should be possible in principle to see the rewrites as chemical reactions between individual molecules. A bigger problem would be then: would these reactions reduce the graph-molecules under the dumbest algorithm among all, the random one? Nature functions like this, so any algorithm which would be global in some sense is completely excluded.

This led me to write the article: M. Buliga, Chemical concrete machine (link to DOI) (link to arXiv version). This is **chemlambda v1**, which is actually a transition from GLC to something else, or better said to another research subject in the making.

**Other sources** for this mix glc-chemlambda v1:

- M. Buliga, L.H. Kauffman, GLC actors, artificial chemical connectomes, topological issues and knots, (link to arXiv version)
- M. Buliga, L.H. Kauffman, Chemlambda, universality and self-multiplication, (link to arXiv version) (link to MIT Press article)

**Chemlambda v1, or the “chemical concrete machine”, is a graph rewriting algorithm** which uses the trivalent nodes A, L, FI (fan-in), FO, and the 1-valent node T and only local rewrites:

- the graphic beta rewrite (between the nodes L and A) and the fan-in rewrite (between the nodes FI and FO)

- the CO-COMM and CO-ASSOC rewrites

- two DIST (from distributivity) rewrites, for the pairs A-FO and L-FO

- local pruning rewrites (involving the 1-valent node T)

- and elimination of loops

As you see, all rewrites except CO-COMM and CO-ASSOC are alike the ones from Lafont article Interaction combinators, except that the graphs are directed and the nodes don’t have a principal port for interaction. Here are the interaction combinators rewrites

In the chemical concrete machine article are mentioned Berry and Boudol chemical abstract machine and Fontana and Buss alchemy, but not Lafont. My fault. From what I knew then the beta rewrite came from Lamping or better Wadsworth, the fan-in came from Turaev (knotted trivalent graphs), and the DIST rewrites came from the graphical version of linear emergent algebras, or even better from the Reidemeister 3 rewrite in knot theory.

In this article there is no algorithm for application of the rewrites. (You shall see that in chemlambda v2, which is an artificial chemistry, there are actually several, among them the deterministic greedy one, with a list of priority of the rewrites, because there are collisions otherwise, and the random one, which interested me the most.)

However I suggest that the rewrites can be seen as done in a truly chemical sense, mediated by (invisible) enzymes, each rewrite type with it’s enzyme.

I proved that we can translate from GLC to chemlambda v1 and that global fan-out and global pruning can be replaced by sequences of rewrites from chemlambda v1. There was no proof that this replacement of the global rewrites with cascades of local rewrites can be done by the algorithms considered. I proved the Turing universality by using the BCKW system of combinators.

The chemical concrete machine article ends however with:

“With a little bit of imagination, if we look closer to what TRUE, FALSE and IFTHENELSE are doing, we see that it is possible to adapt the IFTHENELSE to a molecule which releases, under the detection of one molecule (like TRUE), the ”medicine” A, and under the detection of another molecule (like FALSE) the ”medicine” B.”

The chemlambda page (link) here is reliable for chemlambda v1, but it also contain links to newer versions.

## One thought on “Graphic lambda calculus and chemlambda (I)”