Tag Archives: knot diagrams

Nothing vague in the “no semantics” point of view

I’m a supporter of “no semantics” and I’ll try to convince you that it is nothing vague in it.

Take any formalism. To any term built from this formalism there is an associated syntactic tree. Now, look at the syntactic tree and forget about the formalism. Because it is a tree, it means that no matter how you choose to decorate its leaves, you can progress from the leaves to the root by decorating each edge. At each node of the tree you follow a decoration rule which says: take the decorations of the input edges and use them to decorate the output edge. If you suppose that the formalism is one which uses operations of bounded arity then you can say the following thing: strictly by following rules of decoration which are local (you need to know only at most N edge decorations in order to decorate another edge) you can arrive to decorate all the tree. Al the graph! And the meaning of the graph has something to do with this decoration. Actually the formalism turns out to be not about graphs (trees), but about static decorations which appear at the root of the syntactic tree.
But, you see, these static decorations are global effects of local rules of decoration. Here enters the semantic police. Thou shall accept only trees whose roots accept decorations from a given language. Hard problems ensue, which are heavily loaded with semantics.
Now, let’s pass from trees to other graphs.
The same phenomenon (there is a static global decoration emerged from local rules of decoration) for any DAG (directed acyclic graph). It is telling that people LOVE DAGs, so much so they go to the extreme of excluding from their thinking other graphs. These are the ones who put everything in a functional frame.
Nothing wrong with this!
Decorated graphs have a long tradition in mathematics, think for example at knot theory.
In knot theory the knot diagram is a graph (with 4-valent nodes) which surely is not acyclic! However, one of the fundamental objects associated to a knot is the algebraic object called “quandle”, which is generated from the edges of the graph, with certain relations coming from the edges. It is of course a very hard, fully loaded semantically problem to try to identify the knot from the associated quandle.
The difference from the syntactic trees is that the graph does not admit a static global decoration, generically. That is why the associated algebraic object, the quandle, is generated by (and not equal to) the set of edges.

There are beautiful problems related to the global objects generated by local rules. They are also difficult, because of the global aspect. It is perhaps as difficult to find an algorithm which builds an isomorphism between  two graphs which have the same associated family of decorations, as it is to  find a decentralized algorithm for graph reduction of a distributed syntactic tree.

But these kind of problems do not cover all the interesting problems.

What if this global semantic point of view makes things harder than they really are?

Just suppose you are a genius who found such an algorithm, by amazing, mind bending mathematical insights.

Your brilliant algorithm, because it is an algorithm, can be executed by a Turing Machine.

Or Turing machines are purely local. The head of the machine has only local access to the tape, at any given moment (Forget about indirection, I’ll come back to this in a moment.). The number of states of the machines is finite and the number of rules is finite.

This means that the brilliant work served to edit out the global from the problem!

If you are not content with TM, because of indirection, then look no further than to chemlambda (if you wish combined with TM, like in
http://chorasimilarity.github.io/chemlambda-gui/dynamic/turingchem.html , if you love TM ) which is definitely local and Turing universal. It works by the brilliant algorithm: do all the rewrites which you can do, nevermind the global meaning of those.

Oh, wait, what about a living cell, does it have a way to manage the semantics of the correct global chemical reactions networks which ARE the cell?

What about a brain, made of many neural cells, glia cells and whatnot? By the homunculus fallacy, it can’t have static, external, globally selected functions and terms (aka semantic).

On the other side, of course that the researcher who studies the cell, or the brain, or the mathematician who finds the brilliant algorithm, they are all using heavy semantic machinery.

TO TELL THE STORY!

Not that the cell or the brain need the story in order for them to live.

In the animated gif there is a chemlambda molecule called the 28 quine, which satisfies the definition of life in the sense that it randomly replenish its atoms, by approximately keeping its global shape (thus it has a metabolism). It does this under the algorithm: do all rewrites you can do, but you can do a rewrite only if a random coin flip accepts it.

semtree
Most of the atoms of the molecule are related to operations (application and abstraction) from lambda calculus.

I modified a bit a script (sorry, not in the repo this one) so that whenever possible the edges of this graph which MAY be part of a syntactic tree of a lambda term turn to GOLD while the others are dark grey.

They mean nothing, there’s no semantics, because for once the golden graphs are not DAGs, and because the computation consists into rewrites of graphs which don’t preserve well the “correct” decorations before the rewrite.

There’s no semantics, but there are still some interesting questions to explore, the main being: how life works?

http://chorasimilarity.github.io/chemlambda-gui/dynamic/28_syn.html

UPDATES:

Louis Kauffman reply to this:

Dear Marius,
There is no such thing as no-semantics. Every system that YOU deal with is described by you and observed by you with some language that you use. At the very least the system is interpreted in terms of its own actions and this is semantics. But your point is well-taken about not using more semantic overlay than is needed for any given situation. And certainly there are systems like our biology that do not use the higher level descriptions that we have managed to observe. In doing mathematics it is often the case that one must find the least semantics and just the right syntax to explore a given problem. Then work freely and see what comes.
Then describe what happened and as a result see more. The description reenters the syntactic space and becomes ‘uninterpreted’ by which I mean  open to other interactions and interpretations. It is very important! One cannot work at just one level. You will notice that I am arguing both for and against your position!
Best,
Lou Kauffman
My reply:
Dear Louis,
Thanks! Looks that we agree in some respects: “And certainly there are systems like our biology that do not use the higher level descriptions that we have managed to observe.” Not in others; this is the base of any interesting dialogue.
Then I made another post
Related to the “no semantics” earlier g+ post [*], here is a passage from Rodney Brooks “Intelligence without representation”

“It is only the observer of the Creature who imputes a central representation or central control. The Creature itself has none; it is a collection of competing behaviors.  Out of the local chaos of their interactions there emerges, in the eye of an observer, a coherent pattern of behavior. There is no central purposeful locus of control. Minsky [10] gives a similar account of how human behavior is generated.  […]
… we are not claiming that chaos is a necessary ingredient of intelligent behavior.  Indeed, we advocate careful engineering of all the interactions within the system.  […]
We do claim however, that there need be no  explicit representation of either the world or the intentions of the system to generate intelligent behaviors for a Creature. Without such explicit representations, and when viewed locally, the interactions may indeed seem chaotic and without purpose.
I claim there is more than this, however. Even at a local  level we do not have traditional AI representations. We never use tokens which have any semantics that can be attached to them. The best that can be said in our implementation is that one number is passed from a process to another. But it is only by looking at the state of both the first and second processes that that number can be given any interpretation at all. An extremist might say that we really do have representations, but that they are just implicit. With an appropriate mapping of the complete system and its state to another domain, we could define a representation that these numbers and topological  connections between processes somehow encode.
However we are not happy with calling such things a representation. They differ from standard  representations in too many ways.  There are no variables (e.g. see [1] for a more  thorough treatment of this) that need instantiation in reasoning processes. There are no rules which need to be selected through pattern matching. There are no choices to be made. To a large extent the state of the world determines the action of the Creature. Simon  [14] noted that the complexity of behavior of a  system was not necessarily inherent in the complexity of the creature, but Perhaps in the complexity of the environment. He made this  analysis in his description of an Ant wandering the beach, but ignored its implications in the next paragraph when he talked about humans. We hypothesize (following Agre and Chapman) that much of even human level activity is similarly a reflection of the world through very simple mechanisms without detailed representations.”

This brings to mind also this quote from the end of Vehicle 3 section from V. Braintenberg book Vehicles: Experiments in Synthetic Psychology:

“But, you will say, this is ridiculous: knowledge implies a flow of information from the environment into a living being ar at least into something like a living being. There was no such transmission of information here. We were just playing with sensors, motors and connections: the properties that happened to emerge may look like knowledge but really are not. We should be careful with such words.”

Louis Kauffman reply to this post:
Dear Marius,
It is interesting that some people (yourself it would seem) get comfort from the thought that there is no central pattern.
I think that we might ask Cookie and Parabel about this.
Cookie and Parabel and sentient text strings, always coming in and out of nothing at all.
Well guys what do you think about the statement of MInsky?

Cookie. Well this is an interesting text string. It asserts that there is no central locus of control. I can assert the same thing! In fact I have just done so in these strings of mine.
the strings themselves are just adjacencies of little possible distinctions, and only “add up” under the work of an observer.
Parabel. But Cookie, who or what is this observer?
Cookie. Oh you taught me all about that Parabel. The observer is imaginary, just a reference for our text strings so that things work out grammatically. The observer is a fill-in.
We make all these otherwise empty references.
Parabel. I am not satisfied with that. Are you saying that all this texture of strings of text is occurring without any observation? No interpreter, no observer?
Cookie. Just us Parabel and we are not observers, we are text strings. We are just concatenations of little distinctions falling into possible patterns that could be interpreted by an observer if there were such an entity as an observer?
Parabel. Are you saying that we observe ourselves without there being an observer? Are you saying that there is observation without observation?
Cookie. Sure. We are just these strings. Any notion that we can actually read or observe is just a literary fantasy.
Parabel. You mean that while there may be an illusion of a ‘reader of this page’ it can be seen that the ‘reader’ is just more text string, more construction from nothing?
Cookie. Exactly. The reader is an illusion and we are illusory as well.
Parabel. I am not!
Cookie. Precisely, you are not!
Parabel. This goes too far. I think that Minsky is saying that observers can observe, yes. But they do not have control.
Cookie. Observers seem to have a little control. They can look here or here or here …
Parabel. Yes, but no ultimate control. An observer is just a kind of reference that points to its own processes. This sentence observes itself.
Cookie. So you say that observation is just self-reference occurring in the text strings?
Parabel. That is all it amounts to. Of course the illusion is generated by a peculiar distinction that occurs where part of the text string is divided away and named the “observer” and “it” seems to be ‘reading’ the other part of the text. The part that reads often has a complex description that makes it ‘look’ like it is not just another text string.
Cookie. Even text strings is just a way of putting it. We are expressions in imaginary distinctions emanated from nothing at all and returning to nothing at all. We are what distinctions would be if there could be distinctions.
Parabel. Well that says very little.
Cookie. Actually there is very little to say.
Parabel. I don’t get this ‘local chaos’ stuff. Minsky is just talking about the inchoate realm before distinctions are drawn.
Cookie. lakfdjl
Parabel. Are you becoming inchoate?
Cookie. &Y*
Parabel. Y
Cookie.
Parabel.

Best,
Lou

My reply:
Dear Louis, I see that the Minsky reference in the beginning of the quote triggered a reaction. But recall that Minsky appears in a quote by Brooks, which itself appears in a post by Marius, which is a follow up of an older post. That’s where my interest is. This post only gathers evidence that what I call “no semantics” is an idea which is not new, essentially.
So let me go back to the main idea, which is that there are positive advances which can be made under the constraint to never use global notions, semantics being one of them.
As for the story about Cookie and Parabel, why is it framed into text strings universe and discusses about  a “central locus of control”? I can easily imagine Cookie and Parabel having a discussion before writing was invented, say for example in a cave which much later will be discovered by modern humans in Lascaux.
I don’t believe that there is a central locus of control. I do believe that semantics is a mean to tell the story, any story, as if there is a central locus of control. There is no “central” and there is very little “control”.
This is not a negative stance, it is a call for understanding life phenomena from points of view which are not ideologically loaded by “control” and “central”. I am amazed by the life variety, beauty and vastness, and I feel limited by the semantics point of view. I see in a string of text thousands of years of cultural conventions taken for granted, I can’t forget that a string of text becomes so to me only after a massive processing which “semantics” people take as granted as well, that during this discussion most of me is doing far less trivial stuff, like collaborating and fighting with billions of other beings in my gut, breathing, seeing, hearing, moving my fingers. I don’t forget that the string of text is recreated by my brain 5 times per second.
And what is an “illusion”?
A third post
In the last post https://plus.google.com/+MariusBuliga/posts/K28auYf69iy I gave two quotes, one from Brooks “Intelligence without representation” (where he quotes Minsky en passage, but contains much more than this brief Minsky quote) and the other from Braitenberg “Vehicles: Experiments in Synthetic Psychology”.
Here is another quote, from a reputed cognitive science specialist, who convinced me about the need for a no semantics point of view with his article “Brain a geometry engine”.
The following quote is by Jan Koenderink “Visual awareness”
http://www.gestaltrevision.be/pdfs/koenderink/Awareness.pdf

“What does it mean to be “visually aware”? One thing, due to Franz Brentano (1838-1917), is that all awareness is awareness of something. […]
The mainstream account of what happens in such a generic case is this: the scene in front of you really exists (as a physical object) even in the absence of awareness. Moreover, it causes your awareness. In this (currently dominant) view the awareness is a visual representation of the scene in front of you. To the degree that this representation happens to be isomorphic with the scene in front of you the awareness is veridical. The goal of visual awareness is to present you with veridical representations. Biological evolution optimizes veridicality, because veridicality implies fitness.  Human visual awareness is generally close to veridical. Animals (perhaps with exception of the higher primates) do not approach this level, as shown by ethological studies.
JUST FOR THE RECORD these silly and incoherent notions are not something I ascribe to!
But it neatly sums up the mainstream view of the matter as I read it.
The mainstream account is incoherent, and may actually be regarded as unscientific. Notice that it implies an externalist and objectivist God’s Eye view (the scene really exists and physics tells how), that it evidently misinterprets evolution (for fitness does not imply veridicality at all), and that it is embarrassing in its anthropocentricity. All this should appear to you as in the worst of taste if you call yourself a scientist.”  [p. 2-3]

[Remark: all these quotes appear in previous posts at chorasimilarity]

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Distributivity move as a transposition (curious crossings II)

It looks that there is a connection of the following with the research on DNA topology, but I shall comment on this in a future post, mainly because I am learning about this right now. (But another main reference is Louis Kauffman.)

Let’s see.

If I am using this encoding of chemlambda nodes with crossings

 

ccdeco_1

which is a variant of the encoding from Curious crossings   then, as in the mentioned post,   the beta move becomes a CLICK between “sticky ends” which are marked with dashed lines, followed by R2b move, and also the FAN-IN move becomes a CLICK between sticky ends, followed by a R2a move.

What about the DIST moves? Let’s take the DIST move which involves an application node and a fanout node. The move looks like this:

ccdeco_surgery_1(Click on the picture to make it bigger.)

In the right column we see the DIST move with chemlambda drawing conventions. In the left column there is the translation with crossings and sticky ends.

What do we see? The strings 1-5 and 6-3 are transposed by the DIST move and a new string appears, which crosses them.

I can draw the same move like this:

ccdeco_surgery_2In this figure, the left column is as before, but the right column has changed. I just kept the order, from left to right, of the strings 6-3 and 1-5, and I wiggled the string 2-4 for this.

This is starting to look interestingly alike some other pictures from  DNA topology.

 

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Curious crossings

I’m coming back to  the post   Halfcross way to pattern recognition (in zipperlogic)    and I am going to modify the drawings a bit.

Instead of this convention of transforming chemlambda nodes into half-crossings

halfcross_1

I draw this mapping from chemlambda nodes to crossings:

cdeco_1

Remark that at the left we have now crossings. At the right we have the chemlambda nodes, each with a dashed half-arrow attached, so now the nodes become 4-valent locally planar ones.

As usual (here at chorasimilarity) the crossing diagrams are only locally planar!

Look closer: there are two kinds of oriented crossings, right? To each kind corresponds a colour (green or red) of a chemlambda node.

This is no longer a dictionary, there is no longer a bijective correspondence, because for each oriented crossing there are two possible chemlambda nodes, depending on where is the dashed half-arrow! That is the meaning of the wiggling blue arrow from right to left, it’s uni-directional.

OK then, instead of drawing this interpretation of the beta move

halfcross_2

we get the following one

cdeco_2

where the arrows are drawn like that in order to see what transforms into what (otherwise there is no need for those convoluted arrows in the formalism).

Likewise, instead of this

halfcross_3

I draw this:

cdeco_3

Funny, what could that mean?

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Halfcross way to pattern recognition (in zipperlogic)

Inspired by the Zipper logic and knot diagrams post, here is an alternate encoding of chemlambda.

This is part of the series on zipper logic (branded in this post as  “zipperlogic”).  The last post is Zipper logic (VI) latest version, detailed.

As can be seen in that post, zipperlogic is equivalent with chemlambda, but it has two interesting qualities: is more intuitive and it has the CLICK move.

The CLICK move  transforms a pair of opposed half-zippers into a  zipper, which is then unzipped with the ZIP move. While the ZIP move is equivalent with the graphic beta  move, there is no correspondent to the CLICK move, apparently.

The CLICK move becomes useful when we use other realizations of the zipperlogic than chemlambda.    In the one where half-zippers are realized as towers of crossings, the CLICK move turns out to be a pattern recognition move, and the ZIP move becomes the familiar R2 (Reidemeister 2) move, applied to that pattern.

That is why CLICK is interesting: because in order to apply moves in chemlambda, we have first to identify the patterns where these moves may be used.

Now, I want to justify this in the following.  I shall not aim for another realization of zipperlogic, but for one of chemlambda, inspired by the one of zipperlogic seen as acting on towers of crossings.

I shall use half-crossings.  Recall that in the post Slide equivalence of knots and lambda calculus (I) I wrote:

Louis Kauffman proposes in his book Knots and Physics  (part II, section “Slide equivalence”), the notion of slide equivalence. In his paper “Knotlogic” he uses slide equivalence (in section 4) in relation to the self-replication phenomenon in lambda calculus. In the same paper he is proposing to use knot diagrams as a notation for the elements and operation of a combinatory algebra (equivalent with untyped lambda calculus).

[…]

Obviously, we have four gates, like in the lambda calculus sector of the graphic lambda calculus. Is this a coincidence?

 

So this post can be seen as a try to answer this question.

But the halfcrossings which I use here are different than the ones defined by Louis Kauffman. There might be a way to transform ones into the others, but I have not found it yet.

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Here is the encoding of chemlambda by halfcrossings:

halfcross_1

Remark that each of the halfcrossings has a dangling, vanishing thread, like in the previous post Bacterial conjugation is beta reduction.

[I shall come back in later posts to the relevance of this formalism for horizontal gene transfer.]

Look at this as a new notation for chemlambda nodes and just replace  the green and red nodes by these halfcrossings in order to get the right moves for the halfcrossings.

With an exception: the CLICK move. This move consists into joining neighbouring dangling threads, in two situations, one related to the beta move, the other related to the FAN-IN move.

Look how the beta move appears with halfcrossings and the CLICK move used for pattern recognition (in the figure this s calld “pattern id”):

halfcross_2

Nice, right?

 

Now, the other CLICK move, involved into the identification of the pattern appearing  in the FAN-IN move.

halfcross_3

In a future post I shall look at the DIST moves, in this encoding.

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Bacterial conjugation is beta reduction

I come back to the idea from the post   Click and zip with bacterial conjugation , with a bit more details. It is strange, maybe, but perhaps is less strange than many other ideas circulating on the Net around brains and consciousness.

 

The thing is that bacteria can’t act based on semantics, they are more stupid than us. They have physical or chemical mechanisms which obviate the need to use semantics filters.

Bacteria are more simpler than brains, of course, but the discussion is relevant to brains as collections of cells.

The idea: bacterial conjugation is a form of  beta reduction!

On one side we have a biological phenomenon, bacterial conjugation. On the other side we have a logic world concept, beta reduction, which is the engine that moves lambda calculus, one of the two pillars of computation.

What is the relation between semantics, bacterial conjugation and beta reduction?

Lambda calculus is a rewrite system, with the main rewrite being beta reduction. A rewrite system, basically, says that whenever you see a certain pattern in front of you then you can replace this pattern by another.

Graphic lambda calculus is a graph rewrite system which is more general than lambda calculus. A graph rewrite system is like a rewrite system which used graphs instead of lines of text, or words. If you see certain  graphical patterns then you can replace them by others.

Suppose  that Nature uses (graphical) rewrite systems in the biological realm, for example suppose that bacteria interactions can be modeled by a graph rewrite system. Then,  there has to be a mechanism which replaces the recognition of pattern which involves two bacteria in interaction.

When two bacteria interact there are at least two ingredients:  spatial proximity (SP) and chemical interaction (CI).

SP is something we can describe and think about easily, but from the point of view of a microbe our easy description is void. Indeed, two bacteria in SP can’t be described as pairs of coordinate numbers which are numerically close, unless if each of the microbes has an internal representation of a coordinate system, which is stupid to suppose. Moreover, I think is too much to suppose that each microbe has an internal representation of itself and of it’s neighbouring microbes. This is a kind of a bacterial cartesian theater.

You see, even trying to describe what could be SP for a pair of bacteria does not make much sense.

CI happens when SP is satisfied (i.e. for bacteria in spatial proximity). There is of course a lot of randomness into this, which has to be taken into account, but it does not replace the fact that SP is something hard to make sense from the pov of bacteria.

In Distributed GLC we think about bacteria as actors (and not agents) and about SP as connections between actors. Those connections between actors change in a local, asynchronous way, during the CI (which is the proper graph rewrite, after the pattern between two actors in SP is identified).

In this view, SP between actors, this mysterious almost philosophical relation which is forced upon us after we renounce at the God eye point of view, is described as an edge in the actors diagram.

Such an edge, in Distributed GLC, it is always related to   an oriented edge (arrow) in the GLC (or chemlambda) graph which is doing the actual computation. Therefore, we see that arrows in GLC or chemlambda graphs (may) have more interpretation than being chemical bonds in (artificial) chemistry molecules.

Actually, this is very nice, but hard to grasp: there is no difference between CI and SP!

Now, according to the hypothesis from this post and from the previous one, the mechanism which is used by bacteria for graph rewrite is to grow pili.

The following image (done with the tools I have access to right now) explains more clearly how bacterial conjugation may be (graphic) beta reduction.

Image002

In the upper part of the figure we see the  lambda abstraction node (red)  and the application node (green)  as encoded by crossings. They are strange crossings, see the post  Zipper logic and knot diagrams . Here the crossings are representing with a half of the upper passing thread half-erased.

Now, suppose that the lambda node is (or is managed by) a bacterial cell and that the application node is (managed by) anther bacterium cell. The fact that they are in SP is represented in the first line under the blue separation line in the picture. At the left of the first row (under the blue horizontal line) , SP is represented by the arrow which goes from the lambda node (of the bacterium at left) and the application node (of the bacterium at right). At the right of the first row, this SP arrow is represented as the upper arc which connects the two crossings.

Now the process of pattern recognition begin. In Nature, that is asymmetric: one of the bacterial cells grow a pilus. In this diagrammatic representation, things are symmetric (maybe a weakness of the description). The pilus growth is represented as the CLICK move.

This brings us to the last row of the image. Once the pattern is recognized (or in place) the graph reduction may happen by the ZIP move. In the crossing diagram this is represented by a R2 move, which itself is one of the ways to represent (graphic) beta moves.

Remark that in this process we have two arcs:  the upper arc from the RHS crossing diagram (i.e the arc which represents the SP) and the lower arc appeared after the CLICK move (i.e. the pilus connecting the two bacteria).

After the ZIP move we get two (physical) pili, this corresponds to the last row in the diagram of bacterial conjugation, let me reproduce it again here from the wiki source:

 

661px-Conjugation.svg

After the ZIP move the arc which represents SP is representing a pilus as well!

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Take a better look at the knotted S combinator (zipper logic VI)

Continuing from  Knot diagrams of the SKI (zipper logic V) , here are some  more drawings of the S combinator which was described in the last post by means of crossings:

 

zipper_loop_5

 

Seen like this, it looks very close to the drawings from section 5 of  arXiv:1103.6007.

I am not teasing you,  but many things are now clear exactly because of all these detours. There is a lot to write and explain now, pretty much straightforward and a day after day effort to produce something  which describes well the end of the journey. When in fact the most mysterious creative part is the journey.

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Knot diagrams of the SKI (zipper logic V)

Continuing from  Zipper logic and knot diagrams, here are the  S,K,I combinators when expressed in this convention of the zipper logic:

zipper_loop_4

Besides crossings (which satisfy at least the Reidemeister 2 move), there are also fanout nodes. There are associated DIST moves which self-reproduce the half-zippers as expressed with crossings.

Where do the DIST moves come from? Well, recall that there are at least two different ways to express crossings as macros in GLC or chemlambda: one with application and abstraction nodes, the other with fanout and dilation nodes.

This is in fact the point: I am interested to see if the emergent algebra sector of GLC, or the corresponding one in chemlambda, is universal, and when I am using crossings I am thinking secretly about dilations.

The DIST moves (which will be displayed in a future post) come from the DIST moves from the emergent algebra sector of chemlambda (read this post and links therein).

There is though a construct which is strange, namely the left-to-right arrow which has attached a stack of right-to-left arrows,  and the associated CLICK move which connects these stacks of arrows.

Actually, these stacks look like half-zippers themselves and the CLICK move looks like (un)zipping a zipper.

So, are we back to square one?

No, because even if we replace those stacks by some other half-zippers and the CLICK move by unzipping, we still have the property that those constructs and moves, which are external to knot diagrams, are very localized.

Anyway, I can cheat by saying that I can do the CLICK move, if the crossings are expressed in the emergent algebra sector of chemlambda (therefore dilation nodes, fanout and fanin nodes), with the help of ELIM LOOPS and SWITCH.

But I am interested into simple, mindless ways to do this.

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