# Small graph rewrite systems (3)

Previous posts on the same subject are (1) (2). Related is this post.

In this post I update the small rewrite system SH-2.1 to SH-2.2.  If you look at SH-2.1, it has 3 rewrites: SH, GL and RM.

None of these rewrites allow two “sticks” to merge or one stick to transform into a ring.

Compare with the interaction combinators inspired IC-2.1, with the rewrites DIST, RW1 and RW2. Is true, that system is too reactive, but it has one rewrite, namely RW2, which allows two sticks to merge.

A rewrite which has this property (sticks merge) is essential for computational purposes. The most famous of such rewrites is the BETA rewrite in lambda calculus, or the $\gamma \gamma$ and the $\delta \delta$ rewrites from interaction combinators:

(figure from Lafont article).

In the oriented sticks and rings version of chemlambda we have the rewrites BETA (or A-L) and FI-FOE, with the same property.

We shall modify therefore one of the rewrites from SH-2.1.

The SH-2.2 system

We keep the rewrites SH and GL from the SH-2.1 system:

and we replace the rewrite RM with the new rewrite R2:

The new rewrite R2 needs a ring!

Let’s show that SH-2.2 is better than SH-2.1. All we need is to be able to do the rewrite RM from SH-2.1 in SH-2.2. Here is it.

Mind that the ring from the upper right graph is not the same as the ring from the bottom graph. Indeed, in the rewrite R2 the ring from the bottom is consumed and  a new ring appears from the merging of the ends of the stick with two blue nodes which sits on the top of the other stick with two yellow ends from the bottom graph.

Compared with the original RM rewrite

we have an extra ring at the left and at the right of the rewrite RM, as it appears in SH-2.2. Otherwise said the ring plays the role of an enzyme.