I-don’t-always advantage of the chemical concrete machine

… over other computing formalisms, is contained in the following wise words:

WHEN_FAN_OUT

Indeed, usually a FAN-OUT gate is something which has a variable as an input and two copies of it as an output. That is why FAN-OUT gates are not available in any model of computation, like for example in quantum computing.

But if you don’t use variable (names) and there’s nothing circulating through the wires of your computer model, then you can use the FAN-OUT gate, without  impunity, with the condition to have something which replaces the FAN-OUT behaviour, without it’s bad sides.  Consider graph rewriting systems for your new computer.

This is done in the chemical concrete machine, with the help of DIST enzymes and associated moves (chemical reactions). (“DIST” comes from distributivity.)

In graphic lambda calculus, the parent of the chemical concrete machine, I proved that combinatory logic can be done by using local moves available and one global move, called GLOBAL FAN-OUT.  This global move is what is resembling the most with the behaviour of a usual FAN-OUT gate:  A graph A connected to the input of a FAN-OUT gate is replaced by two copies of the graph.

That’s bad, I think, so in the chemical concrete machine I arrived to prove that GLOBAL FAN-OUT can be replaced, as concerns graphs (or molecules, in the chemical concrete machine formalism) which represent combinators, with successions of local DIST moves (and some other local moves) .

It is possible exactly because there are no variable names. Moreover, there’s something almost biological in the succession of moves: we see how combinators reproduce.

As an illustration, the following is taken from the post  Chemical concrete machine, detailed (V) :

Here are the four “molecules” which represent the combinators B, C, K, W.  (Via the red-green vs black-white change of notation, they can be deduced from their expressions in lambda calculus by the algorithm described here . )

bckw_2

Let’s see how the “molecule” K behaves, when connected to a FAN-OUT gate (green node with one input and two outputs):

bckw_6

The “reproduction” of the molecule B is more impressive:

bckw_3

In the formalism of the chemical concrete machine, \delta^{+} is a distributivity move (or “enzyme” which facilitates the move in one direction, preferentially), and \phi^{+} is a FAN-IN move (facilitated in one direction).

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See more about this in the Chemical concrete machine tutorial.

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This makes me believe that, as long as we don’t reason in terms of states (or any other variables), it is possible to have FAN-OUT gates in quantum computation.

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