# Example: if-then-else in the chemical concrete machine

… along with a short discussion about what the chemical concrete machine can compute. I start with this and then go to the example.

Until now I proved that the graph rewriting system called “chemical concrete machine” can do combinatory logic. Therefore it can compute anything (a Turing machine can).  I shall be more specific, because to the eyes of somebody who is not used with functional programming, this might seem outrageous.

It can compute anything which can be computed by using any programming  language based on lambda calculus, like Haskell or Scala. And then, some more (as regards only the things those languages can do without using any syntactic sugar, of course). The procedure is straightforward, namely translate the (essentially) combinatory logic terms into graphs and then let the enzymes (moves) to do their magic. I shall give a bit later the example of the if-then-else structure.

There are, of course, interesting questions to ask, among them:

• do exist real molecules and real enzymes which react one with another like in the chemical concrete machine formalism? Here I need your help, dear chemists.
• is this an example of a sequential or parallel computation?
• what evaluation procedure is used in the chemical concrete machine?

The second question is the most easy to ask: is parallel computation.  The molecules (graphs) are mixed in a reactor with a choice of enzymes and then all possible reactions can occur in parallel, at all times. (Realistic models will have attached a probabilistic part, but there is nothing special here, because the chemical concrete machine, initialized with molecules and enzymes, is just a chemical reaction network, so any procedure to attach the probabilistic part to a chemical reaction network can also be used in conjunction with the chemical concrete machine.)

But we can also prepare the initial molecules such that the computation is sequential. It suffices to use zippers. More later. But for the moment, it is worthy to mention that the chemical concrete machine (as well as the graphic lambda calculus) are already proven to be more powerful than lambda calculus. Why? for at least two reasons:

• lambda calculus, or combinatory logic, are just sectors of the formalisms, i.e. they correspond to only a part of what the formalisms can do. There are other sectors, as well, for example the tangle diagram sector.
• they are naturally parallel, as long as we use only local moves, as is the case for the chemical concrete machine, for example. Indeed, I wrote that a graph is a “molecule”, but this is only a way of speaking, because molecules could be better identified with connected graphs. But in these formalisms the graphs are not supposed to be connected: any (finite) collection of graphs (i.e. molecules) is also a graph. The moves being local, there is no interference appearing in the simultaneous application of several instances of the (local) moves in different places, for different molecules (connected subgraphs) or even for the same molecule, as long as the places where the moves are applied are different one from another. On the other side, lambda calculus and combinatory logic are naturally sequential.

The third question, concerning the evaluation procedure will be also explored in further posts.  Care has to be taken here because there are no variables in these formalisms  (which translates in less demand of different species of real molecules only for the need to name variables). So it is about the order of moves, right?  The short answer is that it depends, sometimes the computation done by the chemical machine can be seen as greedy evaluation, sometimes as lazy evaluation.

Let me make again the point that somehow the chemical concrete machine formalism should be seen as part of the beautiful idea of algorithmic chemistry. So, it’s not so unearthly.

Finally, it is well known that lambda calculus and Turing machines are the two pillars of computation. For historical reasons chemists seem to concentrate only on the emulation of Turing machines (pleas correct me if I’m wrong).  The main idea of algorithmic chemistry, as far as I understand, is that a sufficiently complex chemical network has the capacity to do lambda calculus. But, if you are determined to use only Turing machines for chemical computing, then, supposing that algorithmic chemistry idea is true, you have to translate the natural language of lambda calculus into the Turing machine frame. This is a tarpit. Very fast, it becomes very hard to follow.  Instead, why not use lambda calculus as it is, for the real powerful applications of chemical computing, and in parallel use one of the excellent languages for simulating in silico chemical computing.

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The if-then-else construct has already been explored in graphic lambda calculus,  see Teaser: B-type neural networks in graphic lambda calculus (II)   in . Here I shall do a much simpler example, just by making the right translations, as explained in the beginning of this post.

In lambda calculus, there are terms called TRUE, FALSE and IFTHENELSE, which are the Church encodings of the booleans true, false and if-then-else. Theassociated graphs in the chemical concrete machine are:

Take other two molecules A, B, with one exit each, in particular you may think that they correspond to terms in lambda calculus (but it is not mandatory). Then IFTHENELSE TRUE A B should become A. In the chemical concrete machine, with only beta + enzymes, look what is happening:

Along this chain of reactions, there is no other choice than the one from the figure. Why? Because essentially at every step there is only one reaction site available to the enzyme beta+ (of course, in the region of the reactor represented in the figure). The result is, unsurprisingly, compatible with the lambda calculus version, with the exception that A and B are not supposed to be (graphs corresponding to) lambda terms. They can be anything, as for example, from the family of “other molecules”.

In lambda calculus IFTHENELSE FALSE A B should become (by reductions) B. In the chemical concrete machine look what happens:

The previous remarks apply here as well.

With a little bit of imagination, if we look closer to what TRUE and FALSE are doing, then we can adapt the IFTHENELSE to what I’ve called a B-type NN synapse and obtain a molecule which releases, under the detection of a certain molecule, the medicine A, and under the detection of another  the medicine B.

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Here is what I believe that  UD is doing. This can be surely multiply optimized, but the basics is: put the z buffer in the 3d space.

I. Preparing the database.

1. Imagine a 3D cubic lattice which is as big as your 3D scenery bounding box.  Take coordinates aligned with the lattice. Put the camera at coordinates (0,0,0) and surround it with a cube C. The faces of the cube C will serve as the cubemap we want to construct. Each face is covered by pixels of a given resolution. We have already the following parameters to play with, given in the units of the coordinate system chosen: the step of the lattice, the dimension of the cube C, the resolution of a face of C.

2. render, by any means you like, the lattice, as seen from the (0,0,0) pov, on the 6 “screens” -faces of C.  We have 6 view frustra, the discussion will be the same for each of them. In order to render them you need to put small balls, or squares facing the relevant face of the cubemap, or whatever, at the lattice nodes, so we have another parameter here, the diameter of the ball or square.  As a result of the rendering you now know the following things:

• for any lattice atom you know which pixel from which face it corresponds. You may do better for lattice atoms which are inside the cube C, namely “ignore” them, say you attribute to them a IGNORE label, otherwise you attribute to each lattice atom the 2D coordinates of the pixel from the cubemap and a information which says which face of the cubemap the pixel is,
• you have information about the scale, which you attach to the lattice atoms, like this: if two neighbouring lattice atoms project on the same pixel then attach IGNORE to both. If the ball/square/whatever of a lattice atom projects on more than one pixel then attach to it  a number SCALE approximately proportional with the square ROOT (thanks ine) of the number of pixels it projects (or the dimension of the bounding box of the pixels)

Of course, you don’t want to take a huge lattice, with very small balls. That’s all in the parameters choice.

3.  Take now your database of 3D points, i.e. the real one, which you want to render eventually, UD style. I shall ignore, for the sake of the argument, how is the database implemented: as an octree or otherwise, or even if the database is made by 3D points or by some more complex objects, like polygons.  Put this database in the coordinates chosen first, such that, for example, if you are working with octrees, the cells of the lattice correspond with some level of the octree.  Attach to each node of the lattice the supplementary information: the points from the database which are within the ball surrounding the atom, or else the label VOID. Alternatively, think that any point from the database is inside some cell lattice and “project” it on each corner of the the cell lattice (i.e. attach to each lattice atom a list of points from the database which are in the neighbourhood of the lattice atom, the nil list corresponds to VOID)

4.  Let’s see what we have, supposing for example that we use octrees. We add the 3D lattice to the 3D database of points (by using lists of points as explained at 3.) and for any lattice atom we attach also the information as explained at point 2.

How to compress this efficiently? There is a new parameter here, namely the level of the octree and also how is the color, for example, information stored in the octree. Of course, this is a matter of recursion, namely at points 1-3 we may take finer and finer resolutions and lattice steps, and so on, starting from a very gross lattice and resolution, etc, then trying to figure a correct recursion procedure. That’s work to be done, is not trivial but it is somehow straightforward once you figure it.

II. The pipeline and rendering.

The problem is that we want to be able to get very very fast, from the database constructed at (I), only the points which are needed for realtime rendering, when the camera is at coordinates (x,y,z).

This problem splits into two different ones:

• at the start of the realtime UD rendering, we want to be able to cull something close to the minimum number of 3D points, when camera is at (0,0,0). According to the information given by Euclideon, a good algorithm should   able to do this in about 1s.
• then, we need a procedure to take what we need from the database when we change the pov from (x,y,z) to (x+1,y, z) (or alike). This should be much more faster, allowing for realtime rendering.

As a preparation, let’s remark that:

1. if the camera is a (0,0,0), then we already know where each lattice point projects, is written in the database. So we just need to start from a pixel, at a given resolution (reccursion again), and to choose from the database only the lattice atoms which are CLOSE, in the decreasing order of SCALE, and from those the real 3D points which are neighbours (of course we have to use the octree structure). We get for each pixel a number of points of the order of log(dimension of the world).
2. if the camera is at (x,y,z) we also know where each point from the 3D database projects, because we read it from the data attached to the lattice atom which, translated by (x,y,z), is the neighbour of the point. We get also the SCALE parameter from this.

1. We use remark 1 to solve the first problem, namely what comes through the pipeline from the huge 3D database to the pre-rendering buffer B, when we start with the camera at (0,0,0). The buffer contains about (number of pixels) X log(dimension of the world)  3D points, along with pixels coordinates where they project and with SCALE.  This is fed directly to the rendering procedure, which you can choose freely, but it is almost trivial.

2. What happens when we move the camera? We update the pre-rendering buffer B, only by updating the pixel and scale information for the 3D points in the buffer D, getting only by a translation (addition) the relevant data from the huge database, in case here are two things which might happen: there is a point which exits the buffer, or there is a hole in the image.

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Is this making any sense? Please let me know, because I was asked repeatedly what do I really believe.

# Example: decorations of S,K,I combinators in simply typed graphic lambda calculus

Continuing from   Simply typed graphic lambda calculus  , let’s look at decorations for the S,K,I combinators.

We work in the lambda calculus sector of the graphic lambda calculus, which contains graphs in $GRAPH$ which are associated to terms in lambda calculus by the algorithm explained here.

We want to decorate the arrows of such graphs, according to the rules given in the following figure:

Decorations are denoted by letters $\sigma, \tau$ … or by letters $a, b, c$ …. (what’s in a name?).  These decorations are called “types”. There is only one operation, called “$\rightarrow$“, which takes a pair of types $(\sigma, \tau)$ and returns the type $\sigma \rightarrow \tau$.

In algebraic terms, here’s what is happening. We start with a graph in $GRAPH$ and we decorate all it’s arrows with different letters from an alphabet (of types). Then we associate to it the magma with:

• generators being the (decorations of the) arrows
• relations being the ones from the rules of decorations given in the previous figure.

If the magma is free,  then we say that the graph is well typed. That means we can find a finite collection of types, with no relation between them, such that all arrows can be decorated by some element of the free magma generated by that collection of types.

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Remark 1:  Please, don’t let me reinvent the wheel.  Surely somewhere in the big work on type theory, there is something analogous done already. Just send me citations and I shall add them with due acknowledgements.

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Remark 2: This is exactly the same idea which is used in knot theory, where we associate to a knot diagram it’s knot  quandle. But with a twist: while in knot theory we look for quandles which are not free (we cannot eliminate all relations between the generators), in this simply typed lambda calculus we concentrate only on those graphs which have free magmas associated.  Please look at the right, lower corner from the previous figure, where is given the rule of decoration for the $\varepsilon$ gate. (We are not going to use it in relation to the lambda calculus sector.)  As you see, this is compatible with the trivial quandle with the operation $xy = y$.

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Remark 3:  I am not going to enter or take the extremely interesting path of cartesian closed categories , basically because my strong bias against anything cartesian. The initial source of this bias comes from the experience with sub-riemannian geometry, where any application of cartesian ideas, like slicing a problem until it becomes 1 dimensional, then solving it by 1d calculus and analysis techniques, then reassembling the slices, leads always to “rigidity results”, which have the form: if you try then you arrive at a contradiction. Don’t get me wrong, there are amazing and extremely deep such results, but they are always in the form of a proof by contradiction.

Remark 4: The point of view from this simply typed graphic lambda calculus is surely related to the Hindley-Milner type system.

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Let’s see how this works for the I, K, S combinators, then for the $\Omega$ combinator.

The procedure is the following:

• we number the nodes of the graph (arbitrarily)
• we put arbitrary, different labels on each arrow of the graph
• we write the relations which appear from each node, according to the rules of decoration
• finally, we examine the generated magma, to see if, eliminating some generators by using the relations, we can arrive to prove that’s a free magma.

In the next figure this is done for the $I$ combinator (the identity) and for the $K$ combinator.

We obtain the right types for identity and $K$ combinators (there are no contexts, that’s for later). It was easy, they are “well typed” according to the definition from here.

Now, let’s look at the $S$ combinator:

We have to work a bit, but not too much:

(For the convenience of the readers, I added in the figures the usual notations for combinators and types. )

We obtain again the right types for $S$ as well as for the variables involved.

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Let’s try to decorate  the combinator $\Omega = (\lambda x . xx) (\lambda x. xx)$.

We obtain the following magma presentation:

which cannot be free, because of relations (4) and (6). So $\Omega$ is not well typed, exactly like in the simply typed lambda calculus.

# Simply typed graphic lambda calculus

Let me prepare the path towards discussing about types and spaces. In this post is a first proposal for a simply typed graphic lambda calculus, which is compatible with simply typed lambda calculus.

types are decorations, in the same way as elements of an emergent algebra are decorations.

Motivation: Remember that emergent algebras appeared as an effort to understand the formalism of decorated binary trees which appeared first in Dilatation structures I. Fundamentals, section 4. Various efforts have been spent for this, as witnessed in Computing with space, sections 3-6,  until finally arriving to the formalism of graphic lambda calculus, as exposed in  arXiv:1305.5786 [cs.LO] . The main “trick” of graphic lambda calculus is that it eliminates variables, therefore it eliminates decorations of graphs!

So, the program is that now we go back from the other side, logic, where types are decorations, as you will see, in order to connect with emergent algebras, which deal with the problem of describing spaces in a purely relational way, without any appeal to points (well, in the graphic lambda calculus formalism).

Interesting thing will appear, surely.

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In this version of simply typed lambda calculus we introduce types (just a bag of names  $\sigma, \tau$, … ), with one operation, denoted by $\rightarrow$.  We use types as decorations of arrows of graphs in $GRAPH$, according to the following decoration rules:

Remark how the decoration of the $\varepsilon$ gate corresponds to the trivial quandle.

I don’t claim that I can decorate any graph in $GRAPH$ by using these rules.  But when I can do it, then under any move from graphic lambda calculus the decoration remains compatible with the decoration rules. Indeed, this is proved in the next figure:

The pruning and the FAN-OUT moves are trivially preserving decorations.

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In another post we shall see why is this compatible with the simply typed combinatory logic.

# Chemical concrete machine, pit stop

I think it’s pretty clear already what the chemical concrete machine is and that’s a fair idea about the things it can do, in principle. It is time to prepare a more conservative presentation, aka an article or two,  based  on the  list:

There are lots of other features which were not yet discussed here, as well as surprising paths which demand future investigations.

I need as much input as possible from those who are interested in this subject. Because there are many ways to tell a story, depending on the story teller and on the audience. As you know, I am a mathematician who wants to stir an interdisciplinary collaboration on this subject which seems to fall in the algorithmic chemistry  or biological computing fields.

So I need help from you, in two directions:

• to calm down my math habits, for the sake of presenting a reasonably comprehensible story to non-mathematicians
• to tell me which parts need more development, which are more interesting, so to say.

# Chemical concrete machine, detailed (VI)

Let’s continue from  the post  Chemical concrete machine, detailed (V) , by adding some more facts and remarks.

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We have seen that in order to not have to apply a controlled sequence of CO-ASSOC molecules (i.e. moves), it is better to introduce a composite move, saying that a certain simple molecule is a propagator.

With this move the formalism of the chemical concrete machine is exactly as powerful as without it. The reason is that the move called “PROP+” can be seen as a sequence of CO-ASSOC moves and DIST+ moves.

Let’s accept this move (i.e. like if there is an associated enzyme to it) and let’s revisit the proof that the W molecule is a multiplier.

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Besides the B, C, K, W molecules (which correspond to the  B,C,K,W system  ), other molecules are also interesting: in the following figure are presented the molecules F, I, S’ and S.

The molecule F may represent FALSE in the Church encoding of the booleans, along with K which encodes TRUE.  The molecule I is the identity and the molecule S corresponds to the combinator S from the SKI combinator calculus. which is

a computational system that may be perceived as a reduced version of untyped lambda calculus. It can be thought of as a computer programming language, though it is not useful for writing software. Instead, it is important in the mathematical theory of algorithms because it is an extremely simple Turing complete language.

These correspondences are established by using the algorithm for transforming lambda calculus terms into graphs in the graphic lambda calculus, then by using the dictionary between the gates from graphic lambda calculus to the essential molecules from the chemical concrete machine.

What about the S’ molecule?  It is a version of the molecule S, i.e. it transforms into S by this reaction:

Also, there is a proof, analogous with the one for W, of the fact that S’ is a multiplier, by using the PROP+ move. This proof is better than the proof for S which was given at the end of the post   Local FAN-IN eliminates global FAN-OUT (I) .

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# Chemical concrete machine, detailed (V)

is a variant of combinatory logic that takes as primitive the combinators B, C, K, and W. This system was discovered by Haskell Curry in his doctoral thesis Grundlagen der kombinatorischen Logik, whose results are set out in Curry (1930).

In this post, I shall explain which are the correspondents of the B, C, K, W, combinators in the formalism of the chemical concrete machine, then I shall prove that they are multipliers. In a future post I shall prove that  their correspondents in the chemical machine make the machine Turing complete.

Remark: In a sense, this was already established, by using the S, K , I combinators, in the post Local FAN-IN eliminates global FAN-OUT (I) . You only have to pass from the  black and white drawing conventions of graphic lambda calculus to the drawings coloured with red and green of the chemical concrete machine, and then use the result which says that combinatory logic can be implemented in graphic lambda calculus with global FAN-OUT. All in all this gives that in the chemical concrete machine the combinatory logic can be implemented without any global move.

However, I think is more instructive to see first why other combinators than S, K, I are multipliers (which is a reformulation of the result mentioned in the remark).

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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 . )

Now, let’s prove that they are multipliers!  The proofs for B and C are very much alike, therefore I put here only the proof that B is a multiplier:

By the conventions of the chemical concrete machine, I mention here the enzymes which are involved in the reactions, instead of writing the moves, like in the graphic lambda calculus.

The proof that K is a multiplier is the following:

Notice how, in both cases, the reactions seem feasible, in the sense that there are no cases, they can be accomplished by a linear process: pour some DIST enzymes, then some FAN-IN, for B, or DIST, LOC PRUNING and then FAN-IN, for K. More than that, remark that the order of reactions do not matter, i.e. they can be accomplished by pouring all the needed enzymes at the same moment.

For the W combinator (molecule), things get a bit more complex, as was the case for the combinator S:

There is a reaction (or move) which needs explanations. I called it DISENTANGLE (CO-ASSOC) reaction. It is this:

It can clearly be done by a controlled succession of CO-ASSOC moves (reactions). From the point of view of the feasibility in the real world (provided a real implementation of the chemical concrete machine will appear), it seems hard to control the exact order of applications of CO-ASSOC moves which gives the DISENTANGLE move as an effect. So, probably,  we shall need a “disentangle enzyme” dedicated to this.

As an alternative, remark that for proving that W is a multiplier we need an application of the DISENTANGLE composite move, described in the next figure:

For practical (or theoretical as well) purposes, it is enough to take this as a move. If you are lost and don’t understand what is the correspondent of this move in lambda calculus, is like this:  for any term $T$, if you apply a FAN-OUT gate to $TT$ then you obtain two $TT$.  (Recall that’s practically invisible in lambda calculus, because there is no FAN-OUT, instead all variables have names and the pattern matching works by some magic outside that formalism.)

In other words, what would get rid of needing a controlled sequence of CO-ASSOC reactions for multiplying the molecule W is this:  assume that the molecule which is connected to the “y” essential molecule (i.e. to the input of a FAN-OUT gate) is a “propagator”. Propagators are defined in the next figure:

Propagators are different from multipliers, because they are molecules with one selected input and one selected output which “propagate along a FAN-OUT gate”, while multipliers are multiplied by a FAN-OUT gate. Propagators can serve in fact as labels, or names, which propagate along a tree of FAN-OUT gates.

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