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

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

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 . )

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

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

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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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.

# Local FAN-IN eliminates GLOBAL FAN-OUT (II)

As I wrote in   Local FAN-IN eliminates global FAN-OUT (I) , the introduction of the three moves (FAN-IN and the two DIST moves) eliminates global FAN-OUT from the lambda calculus sector of the graphic lambda calculus.  In this post we shall see that we can safely eliminate other two moves, namely R1a, R1b, as well as improving the behaviour of the crossings from the $\lambda$-TANGLE sector.

The equilibrium is thus established: three new moves instead of the three old moves. And there are some unexpected advantages.

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Proposition.

Proof.  (a) Done in the following picture.

The proof of (b) is here:

Finally, here is the proof of (c):

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The $\lambda$-TANGLE sector of the graphic lambda calculus is obtained by using the lambda-crossing macros

In Theorem 6.3   arXiv:1305.5786 [cs.LO]  I proved that all the oriented Reidemeister moves (with the crossings replaced by the respective macros), with the exception of the moves R2c, R2d, R3a and R3h, can be proved by using the graphic beta move and elimination of loops.  We can improve the theorem in the following way.

Theorem.  By using the graphic beta move, elimination of loops, FAN-IN and CO-COMM, we can prove all the 16 oriented Reidemeister moves.

Proof. The missing moves R2c, R2d, R3a and R3h are all equivalent (by using the graphic beta move and elimination of loops, see this question/answer at mathoverflow) with the following switching move, which we can prove with FAN-IN and CO-COMM:

The proof is done.

# Local FAN-IN eliminates global FAN-OUT (I)

For being able to build  a chemical concrete machine (see the posts  A chemical concrete machine for lambda calculus  and  Why build a chemical concrete machine, and how?) we have to prove that  universal computation can be attained with only local moves in graphic lambda calculus. Or, the lambda calculus sector of the graphic lambda calculus, which gives universality to graphic lambda calculus, uses the global FAN-OUT move (see theorem 3.1 (d)  arXiv:1305.5786 [cs.LO]. Similarly, we see in proposition 3.2 (d), which describes the way combinatory logic appears in graphic lambda calculus, that again global FAN-OUT is used.

I want to describe a way to eliminate the global FAN-OUT move from combinatory logic (as appears in graphic lambda calculus via the algorithm described here ).

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There are reasons to dislike global moves in relation to B-type neural networks (see the last post    Pair of synapses, one controlling the other (B-type NN part III) ). Similar concerns can be found in the series of posts which has as the most recent one Dictionary from emergent algebra to graphic lambda calculus (III) .

In this first post I shall introduce a local FAN-IN move and two distributivity moves and I shall prove that they eliminate the need for using global FAN-OUT in combinatory logic. In the next post I shall prove that we can eliminate two other moves (so that the total number of moves of graphic lambda calculus stays the same as before) and moreover we can recover from distributivity and local FAN-OUT moves the missing oriented Reidemeister moves from the $\lambda$-TANGLE sector.

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Definition. The local FAN-IN move is described in the next figure and it can be applied for any $\varepsilon \not = 1$.

• as you see, in the move appears a dilation gate, what can this has to do with combinatory logic? As I explained previously, the properties of the gates are coming through the moves they are involved in, and not from their name. I could have introduced a new gate, with two inputs and one output, call this new gate “fan-in gate” and use it in the FAN-IN move. However, wait until the next post to see that there are other advantages, besides the economy of gates available, in using a dilation gate as a fan-in.
• the FAN-IN move resembles to the packing arrows trick which is used extensively in the neural networks posts.  This suggests to use as a  fan-in gate

the green triangle gate and as fan-out gate the red triangle gate. This would eliminate the $\Upsilon$ gate from the formalism, but is not clear to me how this replacement would interfere with the rest of the moves.

• the FAN-IN move resembles with the dual of the graphic beta move, but is not the same (recall that until now I have not accepted the dual of the graphic beta move in the list of the moves of graphic lambda calculus, although there are strong reasons to do so):

which is needed in the emergent algebra sector in order to make the dictionary to work (and related as well to the goal of non using global FAN-OUT in that sector).  This latter move is in fact a distributivity move (see further), but we are free to choose different moves in different sectors of the graphic lambda calculus,

• I know it is surprising that until now there was nothing about evaluation strategies in graphic lambda calculus, the reason being that because there are no variables then there is noting to evaluate. However, the situation is not so simple, at some point, for example in the chemical concrete machine or for neural networks, some criterion for choosing the order of moves will be needed. But it is an important point to notice that replacing global FAN-OUT (which could be seen as a remnant of having variables and evaluating them) by local FAN-IN has nothing to to with evaluation strategies.

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Definition: The distributivity moves (related to the application and lambda abstraction gates) are the following:

• the first distributivity move is straighforward, an application gate is just doubled and two fan-out moves establish the right connections. We see here why the “mystery move” can be seen as a kind of distributivity move,
• the second distributivity move is where we need a fan-in gate (and where we use a dilation gate instead): because of th orientation of the arrows, after we double the lambda abstraction gates, we need to collect two arrows into one!

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Combinatory logic terms appear in graphic lambda calculus as  trees made by application gates, with leaves one of the combinators S, K, I (seen as graphs in $GRAPH$.  I want to show the following. [UPDATE: made some corrections.]

Theorem.   We can replace the global FAN-OUT move with a sequence of local FAN-IN ,  DIST, CO-COMM and local pruning moves, every time the global FAN-OUT move is applied to a term made by SKI combinators.

Proof.  First, remark that a sequence of  DIST moves for the application gate allows to reduce the problem of replacing global FAN-OUT moves for any combinator to the problem of replacing it for S, K, and I. This is because the DIST move for the application gate allows to do the FAN-OUT of trees of application gates:

Now we have to prove that we can perform global FAN-OUT for I , K, S combinators.  For the combinator I the proof is the following:

For the combinator K we shall also use a local pruning move:

Finally, the proof for the combinator S is the following:

Now we are going to use 3 DIST moves, followed by the switch of arrows explained in   Local FAN-IN eliminates GLOBAL FAN-OUT (II) , which is applied in the dashed green ellipse from the next figure:

And we are done.

UPDATE: At close inspection, it turns out that we don’t need to do switch (macro) moves. Instead, if we go back at the point we were before the last figure,  we may use  first CO-ASSOC and then perform the three FAN-IN moves .

# Fan-out moves: CO-COMM, CO-ASSOC, GLOBAL FAN-OUT, LOCAL FAN-OUT

This is part of the Tutorial: Graphic lambda calculus.

Here are described the moves directly related to the $\Upsilon$ gate.

• CO-COMM move.  This is the local move depicted in the following figure

It means we may permute the outputs of a $\Upsilon$ gate. The name means “co-commutativity”, because the diagram resembles to the one of the commutativity property, with the exception of the arrows orientations, which are in opposite directions (hence “co-“).

• CO-ASSOC move. This is a local move, described by the next figure

It has the following effect: by using CO-ASSOC moves, we may move between any two binary trees formed only with $\Upsilon$ gates, with the same number of output leaves. The name means “co-associativity” and the explanation is similar to the previous one, with “commutativity” replaced by “associativity”.

•   GLOBAL FAN-OUT. This is a global move, because it involves a modification of an arbitrary number of nodes (gates) and arrows.

Precisely, the move acts like in the following picture: if $A$ is a graph in $GRAPH$  then we may replace the graph $A$ connected to an $\Upsilon$  gate by two copies of $A$.

GLOBAL FAN-OUT implies CO-COMM  (namely two GLOBAL FAN-OUT moves have the effect of one CO-COMM move).  The move CO-COMM is not useless though: we may choose to work in a sector of the graphic lambda calculus which uses CO-COMM but not GLOBAL FAN-OUT.

There is a variant of this move which is local.

• LOCAL FAN-OUT. Fix a number $N$ and consider only graphs $A$ which have at most $N$ (nodes + arrows). The $N$ LOCAL FAN-OUT move is the same as the GLOBAL FAN-OUT move, only it applies only to such graphs $A$.

LOCAL FAN-OUT does not imply CO-COMM.

Important remark: The gate $\Upsilon$ is really a fan-out gate only in the sense described by the FAN-OUT moves. In the absence of one of this moves, the gate cannot  be described as a “fan-out”. The “fan-out” is in the moves, not in the gate.

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