Crossings as pairs of molecular bonds; boolean computations in chemlambda

I collect here the slightly edited versions of a stream of posts on the subject from the microblogging chemlambda collection. Hopefully this post will give a more clear image about this thread.

Here at the chorasimilarity open notebook, the subject has been touched previously, but perhaps that the handmade drawings made it harder to understand (than with the modern 🙂 technology from the chemlambda repository):

Teaser: B-type neural networks in graphic lambda calculus (II)

(especially the last picture!)

All this comes with validation means. This is a very powerful idea: in the future validation will replace peer review, because it is more scientific than hearsay from anonymous authority figures (aka old peer review) and because it is more simple to implement than a network of hearsay comments (aka open peer review).

All animations presented here are obtained by using the script and various mol files. See instructions about how you can validate (or create your own animations) here:

Here starts the story.

(Source FALSE is the hybrid of TRUE, boole.mol file used)


Church encoding gives a way to define boolean values as terms in the lambda calculus. It is easy:

TRUE= Lx.(Ly.x)

FALSE= Lx.(Ly.y)

So what? When we apply one of these terms to another two, arbitrary terms X and Y, look what happens: (arrows are beta reductions (Lx.A)B – – > A[x:=B] )

(TRUE X) Y – – > (Ly.X) Y – – > X (meaning Y goes to the garbage)

(FALSE X) Y – – > (Ly.y) Y – – > Y (meaning X goes to the garbage)

It means that TRUE and FALSE select a way for X and Y: one of them survives, the other disappears.

Good: this selection is an essential ingredient for computation

Bad: too wasteful! why send a whole term to the garbage?

Then, we can see it otherwise: there are two outcomes: S(urvival) and G(arbage), there is a pair of terms X and Y.

– TRUE makes X to connect with S and Y to connect with G

– FALSE makes X to connect with G and Y to connect with S

The terms TRUE and FALSE appear as molecules in chemlambda, each one made of two red nodes (lambdas) and a T (termination) node. But we may dispense of the T nodes, because they lead to waste, and keep only the lambda nodes. So in chemlambda the TRUE and FALSE molecules are, each, made of two red (lambda) nodes and they have one FROUT (free out).

They look almost alike, only they are wired differently. We want to see how it looks to apply one term to X and then to Y, where X and Y are arbitrary. In chemlambda, this means we have to add two green (A) application nodes, so TRUE or FALSE applied to some arbitrary X and Y appear, each, as a 4 node molecule, made of two red (lambda) two green (A), with two FRIN (free in) nodes corresponding to X and Y and two FROUT (free out) nodes, corresponding: one to the deleted termination node, thus this is the G(arbage) outcome, and the other to the “output” of the lambda terms, thus this is the S(urvival) outcome.

But the configuration made of two green A nodes and two red L nodes is the familiar zipper which you can look at in this post


In the animation you see TRUE (at left) and FALSE (at right), with the magenta FROUT nodes and the yellow FRIN nodes.

The zipper configurations are visible as the two vertical strings made of two green, two red nodes.

What’s more? Zippers, they do only one thing: they unzip.

The wiring of TRUE and FALSE is different. You can see the TRUE and FALSE in the lower part of the animation. I added four Arrow (white) nodes in order to make the wiring more visible. Arrow nodes are eliminated in the COMB cycle, they have only a fleeting existence.

This shows what is really happening: look at each (TRUE-left, FALSE-right) configuration. In the upper side you have 4 nodes, two magenta, two yellow, which are wired together at the end of the computation. In the case of TRUE they end up wired in a X pattern, in the case of FALSE they end up wired in a = pattern.

At the same time, in the lower side, before the start of the computation, you see the 4 white nodes which: in the case of TRUE are wired in a X pattern, in the case of FALSE are wired in a = pattern. So what is happening is that the pattern ( X or = ) is teleported from the 4 white nodes to the 4 magenta-yellow nodes!

The only difference between the two molecules is in this wire pattern, X vs =. But one is the hybrid of the other, hybridisation is the operation (akin to the product of knots) which has been used and explained in the post about senescence and used again in more recent posts. You just take a pair of bonds and switch the ends. Therefore TRUE and FALSE are hybrids, one of the other.

(Source Boolean wire, boolewire.mol file used )

If you repeat the pattern which is common to TRUE and FALSE molecules then you get a boolean wire, which is more impressive “crossings teleporter”. This time the crosses boxed have been flattened, but the image is clear:


Therefore, TRUE and FALSE represent choices of pairs of chemical bonds! Boolean computation (as seen in chemlambda) can be seen as management of promises of crossings.

(Source Promises of crossings, ifthenelsecross.mol file used )

You see 4 configurations of 4 nodes each, two magenta and two yellow.

In the upper left side corner is the “output” configuration. Below it and slightly to the right is the “control” configuration. In the right side of the animation there are the two other configurations, stacked one over the other, call them “B” (lower one) and “C” (upper one).

Connecting all this there are nodes A (application, green) and L (lambda, red).


You see a string of 4 green nodes, approximately vertical, in the middle of the picture, and a “bag” of nodes, red and green, in the lower side of the picture. This is the molecule for the lambda term

IFTHENELSE = L pqr. pqr

applied to the “control” then to the “B” then to the “C”, then to two unspecified “X” and “Y” which appear only as the two yellow dots in the “output” configuration.

After reductions we see what we get.

Imagine that you put in each 4 nodes configuration “control”, “B” and “C”, either a pair of bonds (from the yellow to the magenta nodes) in the form of an “X” (in the picture), or in the form of a “=”.

“X” is for TRUE and “=” is for false.

Depending on the configuration from “control”, one of the “B” or “C” configuration will form, together with its remaining pair of red nodes, a zipper with the remaining pair of green nodes.

This will have as an effect the “teleportation” of the configuration from “B” or from “C” into the “output”, depending on the crossing from “control”.

You can see this as: based on what “control” senses, the molecule makes a choice between “B” and “C” promises of crossings and teleports the choice to “output”.

(Source: Is there something in the box?, boxempty.mol used)

I start from the lambda calculus term ISZERO and then I transform it into a box-sensing device.

In lambda calculus the term ISZERO has the expression

ISZERO = L a. ((a (Lx.FALSE)) TRUE)

and it has the property that ISZERO N reduces either to TRUE (if N is the number 0) or FALSE (if N is a number other than 0, expressed in the Church encoding).

The number 0 is
0 = FALSE = Lx.Ly.y

For the purpose of this post I take also the number 2, which is in lambda calculus

2=Lx.Ly. x(xy)

(which means that x is applied twice to y)

Then, look: (all reductions are BETA: (Lx.A)B – – > A[x:=B] )

(L a. ((a (Lx.FALSE)) TRUE) ) (Lx.Ly.y) – – >
((Lx.Ly.y) (Lx.FALSE)) TRUE – – >
(Ly.y)TRUE – – > (remark that Lx.FALSE is sent to the garbage)
TRUE (and the number itself is destroyed in the process)


(L a. ((a (Lx.FALSE)) TRUE) ) (Lx.Ly. x(xy)) – – >
((Lx.Ly. x(xy)) (Lx.FALSE)) TRUE – – > (fanout of Lx.FALSE performed secretly)
(Lx.FALSE) ((Lx.FALSE) TRUE) – – >
FALSE ( and (Lx.FALSE) TRUE sent to the garbage)

Remark that in the two cases there was the same number of beta reductions.

Also, the use of TRUE and FALSE in the ISZERO term is… none! The same reductions would have been performed with an unspecified “X” as TRUE and an unspecified “Y” as FALSE.

(If I replace TRUE by X and FALSE by Y then I get a term which reduces to X if applied to 0 and reduces to Y if applied to a non zero Church number.)

Of course that we can turn all this into chemlambda reductions, but in chemlambda there is no garbage and moreover I want to make the crossings visible. Or, where are the crossings, if they don’t come from TRUE and FALSE (because it should work with X instead of TRUE and Y instead of FALSE).

Alternatively, let’s see (a slight modification of) the ISZERO molecule as a device which senses if there is a number equal or different than 0, then transforms, according to the two cases, into a X crossing or a = crossing.

Several slight touches are needed for that.

1. Numbers in chemlambda appear as stairs of pairs of nodes FO (fanout, green) and A (application, green), as many stairs as the number which is represented. The stairs are wrapped into two L (lambda, red) nodes and their bonds.
We can slightly modify this representation so that it appears like a box of stairs with two inputs and two outputs, and aby adding a dangling green (A, application) node with it’s output connected to one of its inputs (makes no sense in lamnda calculus, but behaves well in the beta reductions as performed in chemlambda).

In the animation you can see, in the lower part of the figure:
-at left the number 0 with an empty box (there are two Arrow (white) nodes added for clarity)
-at right the number 2 with a box with 2 stairs
… and in each case there is this dangling A node (code in the mol file of the form A z u u)

2. The ISZERO is modified by making it to have two FRIN (free in, yellow) and two FROUT (free out, magenta) nodes which will be involved in the final crossing(s). This is done by a clever (hope) change of the translation of the ISZERO molecule into chemlambda: first the two yellow FRIN nodes represent the “X” and the “Y” (which they replace the FALSE and the TRUE, recall), and there are added a FOE (other fanout node, yellow) and a FI (fanin node, red) in strategic places.



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