I record here, to be sure that it’s read, a message of explanations I wrote today.

Before that, or after that you may want to go check the new chemlambda demos, or to look at the older ones down the page, in case you have not noticed them. There are also additions in the moves page. Most related to this post is the vision page.

“First, I have lots of examples which are hand drawn. Some of them appeared from theoretical reasons, but the bad thing used to be that it was too complex to follow (or to be sure of that) on paper what’s happening. From the moment I started to use the mol files notation (aka g-patterns) it has been like I suddenly became able to follow in much more depth. For example that’s how I saw the “walker”, “ouroboros” and finally the first quine by reducing the graph associated to the predecessor (as written in lambda calculus). Finally, I am able now to see what happens dynamically.

However, all these examples are far from where I would like to be now, they correspond to only the introduction into the matter. But with every “technical” improvement there are new things which appear. For example, I was able to write a program which generates, one by one, each of about 1000 initial graphs made by 2 nodes and follow their reduction behaviour for a while, then study the interesting exemplars. That is how I found all kind of strange guys, like left or write propagators, switchers, back propagators, all kind of periodic (with period > 1) graphs.

So I use mainly as input mol files of graphs which I draw by hand or which I identify in bigger mol files as interesting. This is a limitation.

Another input would be a program which turns a lambda term into a mol file. The algorithm is here.

Open problem: pick a simplified programming language based on lambda calculus and then write a “compiler”, better said a parser, which turns it into a mol file. Then run the reduction with the favourite algorithm, for the moment the “stupid” determinist and the “stupid” random ones.

While this seems feasible, this is a hard project for my programming capabilities (I’d say more because of my bad character, if I don’t succeed fast then I procrastinate in that direction).

It is tricky to see how a program written in that hypothetical simple language reduces as a graph, for two reasons:

- it has to be pure lambda beta, but not eta (so the functions are not behaving properly, because of the lack of eta)
- the reductions in chemlambda do not parallel any reduction strategy I know about for lambda calculus.

The (2) makes things interesting to explore as a side project, let me explain. So there is an algorithm for turning a lambda term into a mol file. But from that part, the reductions of the graph represented by the mol file give other graphs which typically do not correspond to lambda terms. However, magically, if the initial lambda term can be reduced to a lambda term where no further reductions are possible, then the finalÂ graph from the chemlambda reduction does correspond to that term. (I don’t have a proof for that, because even if I can prove that if restricted to lambda terms written onlly as combinators, chemlambda can parallel the beta reduction, the reduction of the basis combinators and is able to fan-out a term, it does not mean that what I wrote previously is true for any term, exactly because of the fact that during chemlambda reductions one gets out from the real of lambda terms.)

In other cases, like for the Y combinator, there is a different phenomenon happening. Recall that in chemlambda there is no variable or term which is transmitted from here to there, there is nothing corresponding to evaluation properly. In the frame of chemlambda the behaviour of the Y combinator is crystal clear though. The graph obtained from Y as lambda term, connected to an A node (application node) starts to reduce even if there is nothing else connected to the other leg of the A node. This graph reduces in chemlambda to just a pair of nodes, A and FO, which then start to shoot pairs of nodes A and FOE (there are two fanout nodes, FO and FOE). The Y is just a gun.

Then, if something else is connected to the other leg of the application node I mentioned, the following phenomenon happens. The other graph starts to self-multiply (due to the FOE node of the pair shot by Y) and, simultaneously, the part of it which is self-multiplied already starts to enter in reaction with the application node A of the pair.

This makes the use of Y spectacular but also problematic, due to too much exuberance of it. That is why, for example, even if the lambda term for the Ackermann function reduces correctly in chemlambda, (or see this 90 min film) I have not yet found a lambda term for the factorial which does the same (because the behaviour of Y has to be tamed, it produces too many opportunities to self-multiply and it does not stop, albeit there are solutions for that, by using quines).

So it is not straightforward that mol files obtained from lambda terms reduce as expected, because of the mystery of the reduction strategy, because of the fact that intermediary steps go outside lambda calculus, and because the graphs which correspond to terms which are simply trashed in lambda calculus continue to reduce in chemlambda.

On the other side, one can always modify the mol file or pre-reduce it and use it as such, or even change the strategy of the algorithm represented by the lambda term. For example, recall that because of lack of eta, the strategy based on defining functions and then calling them, or in general, just currying stuff (for any other reasons than for future easy self-multiplication) is a limitation (of lambda calculus).

These lambda terms are written by smart humans who stream everything according to the principle to turn everything into a function, then use the function when needed. By looking at what happens in chemlambda (and presumably in a processor), most of this functional book-keeping is beautiful for the programmer, but a a limitation from the point of view of the possible alternative graphs which do the same as the functionally designed one, but easier.

This is of course connected to chemistry. We may stare to biochemical reactions, well known in detail, without being able to make sense of them because things do happen by shortcuts discovered by evolution.

Finally, the main advantage of a mol file is that any collection of lines from the mol file is also a mol file. The free edges (those which are capped by the script with FRIN (i.e. free in) and FROUT (i.e. free out) nodes) are free as a property of the mol file where they reside, so if you split a mol file into several pieces and send them to several users then they are still able to communicate by knowing that this FRIN of user A is connected (by a channel? by an ID?) to that FROUT of user B. But this is for a future step which would take things closer to real distributed computing.

And to really close it, if we would put on every computer linked to internet a molecule then the whole net would be as complex (counting the number of molecules) as a mouse. But that’s not fair, because the net is not as structured as a mouse. Say as a handful of slime mold. That’s how smart is the whole net. Now, if we could make it smarter than that, by something like a very small API and a mol file as big as a cookie… “

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