The beta reduction from lambda calculus is easy to be understood as the command

let x=B in A

It corresponds to the term (Lx.A)B which reduces to A[x=B], i.e to the term A where all instances of x have been replaced by B.

(by an algorithm which is left to be added, there are many choices among them!)

In Christopher P. Wadsworth, Semantics and Pragmatics of the Lambda Calculus , DPhil thesis, Oxford, 1971, and later John Lamping, An algorithm for optimal lambda calculus reduction, POPL ’90 Proceedings of the 17th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, p. 16-30, there are proposals to replace the beta rule by a graph rewrite on the syntactic tree of the term.

These proposals opened many research paths, related to call by name strategies of evaluations, and of course going to the Interaction Nets.

The beta rule, as seen on the syntactic tree of a term, is a graph rewrite which is simple, but the context of application of it is complex, if one tries to stay in the real of syntactic trees (or that of some graphs which are associated to lambda terms).

This is one example of blindness caused by semantics. Of course that it is very difficult to conceive strategies of applications of this local rule (it involves only two nodes and 5 edges) so that the graph after the rewrite has a global property (means a lambda term).

But a whole world is missed this way!

In chemlambda the rewrite is called BETA or A-L.

see the list of rewrites

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

In mol notation this rewrite is

L 1 2 c , A c 3 4 – – > Arrow 1 4 , Arrow 3 2

(and then is the turn of COMB rewrite to eliminate the arrow elements, if possible)

As 1, 2, 3, 4, c are only port variables in chemlambda, let’s use other:

L A x in , A in B let – – > Arrow A let , Arrow B x

So if we interpret Arrow (via the COMB rewrite) as meaning “=”, we get to the conclusion that

let x=B in A

is in chemlambda

L A x in

A in B let

Nice. it almost verbatim the same thing. Remark that the command”let” appears as a port variable too.

Visually the rewrite/command is this:

As you see this is a Wadsworth-Lamping kind of graph rewrite, with the distinctions that:

(a) x, A, B are only ports variables, not terms

(b) there is no constraint for x to be linked to A

The price is that even if we start with a graph which is related to a lambda term, after performing such careless rewrites we get out of the realm of lambda terms.

But there is a more subtle difference: the nodes of the graphs are not gates and the edges are not wires which carry signals.

The reduction works well for many fundamental examples,

http://chorasimilarity.github.io/chemlambda-gui/dynamic/demos.html

by a simple combination of the beta rewrite and those rewrites from the DIST family I wrote about in the post about duplicating trees.

https://chorasimilarity.wordpress.com/2015/05/04/the-illustrated-shuffle-trick-used-for-tree-duplication/

So, we get out of lambda calculus, what’s wrong with that? Nothing, actually, it turns out that the world outside lambda, but in chemlambda, has very interesting features. Nobody explored them, that is why is so hard to discuss about that without falling into one’s preconceptions (has to be functional programming, has to be lambda calculus, has to be a language, has to have a global semantics).

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