Tracelet Algebras
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Behr, Nicolas
Stochastic rewriting systems evolving over graph-like structures are a versatile modeling paradigm that covers in particular biochemical reaction systems. In fact, to date rewriting-based frameworks such as the Kappa platform are amongst the very few known approaches to faithfully encode the enormous complexity in both molecular structures and reactions exhibited by biochemical reaction systems in living organisms. Since in practice experimental constraints permit to track only very limited information about a given reaction system (typically the concentrations of only a handful of molecules), a fundamental mathematical challenge arises: which types of information are meaningful to derive and computable from a stochastic rewriting system in view of the limited empirical data? Traditionally, the main focus of the mathematical theory of stochastic rewriting theory has been upon the derivation of ODE systems describing the evolution of averages and higher moments of pattern counts (i.e. the concentrations of molecular species). In this talk, we present an alternative approach based upon so-called tracelets. The latter are the precise mathematical encoding of the heuristic notion of pathways in biochemistry. We demonstrate a novel mathematical concept of tracelet algebras and highlight a computational strategy that permits to derive structural, high level insights into the dynamics of pattern counts. In view of the focus of CAP on combinatorial aspects, we will illustrate this mathematical approach with an analysis of planar rooted binary trees in a rewriting-based formulation utilizing the Rémy generator. Pierre Boutillier et al., ”The Kappa platform for rule-based modeling.”, Bioinformatics 34.13 (2018): pp. 583-592. Nicolas Behr, ”Tracelets and Tracelet Analysis Of Compositional Rewriting Systems”, Electronic Proceedings in Theoretical Computer Science 323 (2020), pp. 44-71
 Thank you very much, first important question you can see the slides, hopefully. Yes, we can see you. Yes. Can I start? Should I wait for the minute? No, no, it is a... You can begin now if you want, if you wish. Thank you very much, Jaan, also the organizers for this invitation. It's always very nice to be at CUP now virtually, but I hope at least this way some more people can participate. And the talk I would like to give today is about a topic which I think is quite the spirit of CUP, because it is about this type of research which tries to make sense of the deeper links, the mathematical links between approaches to manipulation of graph-like structures and computer science, mathematical physics and mathematics. So, I mean, of course, every of these disciplines has many, many examples of situations where it is meaningful to look at these types of manipulations. For example, the very simplest such is drawn here at the bottom. It's just when you manipulate, for example, in physics, in distinguishable particles, this would be a model of chemistry, for example. Then, of course, in social network science, maybe you look at processes which can be formalized as, you know, manipulating locally a network graph, like, for example, the rewiring and edging here, but then there are also many, many more sort of Baroque examples. I mean, trees, for example, can be seen as a type of graph which has additional structure, like particular incidences, for example. And my personal motivation is really from organic chemistry and from biochemistry, where you encounter these beautiful types of theories, where, for example, you formalize proteins and other molecules through these type of writing steps. And sort of my main motivation for this work is that I'm originally a mathematical physicist, but I'm now working in computer science, and I discovered that there's relatively little cross-path between these three disciplines that would allow you, for example, to take a continuous time Markov chain theory and directly apply it to one of those organic chemistry simulation techniques in biochemistry, say, for example. And, of course, one of the main motivations is to ask, so in these type of very complex systems, where can you attack with combinatorics? So what are the combinatorial structures, of course, beyond the data type of these graphical structures that one could use to analyze the systems? And I'm just putting this here because, at least for bio and organic chemistry, it is really the case that, I mean, sometimes you have abstractions which are made because you can then, at least as a toy model, talk about these systems. But at least in the case of bio and organic chemistry, these are really the state-of-the-art techniques to simulate these systems. So in Kappa is a framework where you have these sort of circuit side graphs, there's a special type of graph, you know, everywhere it takes some sites at which you can link to other vertices. And in organochemistry, it's even more fantastic, it's really that the molecules you draw by hand also for chemical reactions are a data type that you can then formulate transformations on. And so these are the two main frameworks. So if you, I mean, I would say a little bit about this later on, and it so happens that there's really a very formal semantics for these type of theories. And that is good because that is really, I mean, it's very mathematical formulation. But that's a plus. What's the problem is that if you then look really at a realistic scale example, and here's a little piece of the human metabolic network, in this huge graph, every part of this network is, I mean, an enormous complexity, and you have a very many different types of molecules that participate in these reactions. And here's only drawn like the macro molecules, or the larger molecules. And then you have enormously many reactions. And so the real question is all of these transitions, these manipulations are fired at random. So what is it really you can say about these systems? And especially if you look at biology, which is still a very fast evolving field, of course, I mean, it is nice that you have this interesting data type, which is quite good also at, you know, encapsulating new knowledge you have about just individual transitions. But you can see immediately, since every of these green blobs, these agents, as typically on the order of seven to 10 sites, this is an extremely highly combinatorial data structure. And then if you start manipulating, it is very unclear of what you can actually extract as information. So main problem is to understand the function of such systems, because nature has given them, this is the abstraction that is very accurate. So what can you say about the function of the systems? And so, so to just motivate how one could approach this, I'm going to take an example of the transformation system, which is of course much simpler than these biochemical reactions, but it's complicated enough to explain the main sort of root of attack for these systems. So here's a system where you have an input state, which is just some graph, now performing transformation. So the language for these transformations always looks similar, you have little pieces of evolution, if you will, each of those is asking for some input drawn at the bottom in this little cartoon, and some, then it produces some output. And the output here is, so the input is here, some two vertices, the dashed lines indicate that they are kept throughout the transformations are identically kept, and then you link them up with an edge. That's this local manipulation. And the second part of this framework of the semantics is that you to apply such a rule, you have to exhibit a match, and a match is nothing but an embedding of this input motive into the graph. You see immediately that, I mean, here I've drawn a very, very small graph, and I mean, this rule is sufficiently simple, but already here you have quite a high number of possible matches. So part of this sort of encoding is that it's usually quite simple to write the rules, but I mean, there's usually quite many ways of applying these rules. But anyways, so then here, applying the rule amounts to locally, then, you know, running this transformation, which here amounts to inserting this edge in these two vertices. So let's do a few more of those steps, another of such exactly the same rule applied at a different place. Now, of course, I mean, I'm only showing some very simple rules, you could also have one which unlinks, for example, like this one. And let's take another one to link and another one to link. Okay, so now I have given you a small sequence of the transformation system. And very nicely, not only for this particular case, but I mean, for an enormous variety of possible things of graph like structures and their manipulations, they are all covered by a mathematical theory called categorical writing. And nicely enough, it is very close to implementation sources. I mean, this is something you can, you can put very directly into algorithms. So it's sort of the source code for these sort of manipulations. But sort of the typical problem is now, I mean, going into this picture of what happens in biochemistry, how could you understand it? So imagine you were given these type of transformation, you had a little vocabulary, say just the linking of edges and the unlinking. But now you are asking, so of all of these possibilities, what is the likelihood of if you fire these say at random with same properties for sake of argument, of seeing a triangle appearing? Yes, so I mean, you're trying to track how many times to see a triangle that was really produced, I mean, newly produced, or maybe deleted through this application of sequences. And so what classical writing theory doesn't have much of an answer to is how to approach this and utilizing this feature of writing. So what one can do, and this is now I will motivate this more later on, but sort of the the starting point of this combinatorial analysis I would like to propose has to do with, of course, you can track how actually these rules were applied. And here in this picture, you see that these are simply just the, I think, five steps one after another drawn works vertically. And whenever you you attack with a second, third, fourth rule at a position where you have previously already applied a rule, then this is marked with these light blue lines. And the red lines mean if you touch something that is in your original graph, but not interacting. And so immediately, it is obvious that there's some combinatorics in how you can sort of plug together these, these transformations and acting with each other versus how how they are acting on on this graph state. So again, like if you if you now look at this problem of finding triangles, you discover something very important, namely that sort of producing a triangle is an inherent feature of of this above sequence of rules. I mean, because they are plugged together in a way that they are guaranteed to produce a triangle, there might be other ones that accidentally are produced by just simply linking up a V shape say with one edge. But that is definitely already one guarantee you can give this sequence will produce a triangle. And the other point of interest is, so these two parts of the transformation sequence highlighted in orange, of course, they are happening, this is one possibility to apply, but the two actually do not contribute anything to the production of the triangle. In fact, they they they only sort of they produce an edge and deleted later on, so that that actually doesn't contribute anything to this count. So what one needs is some sort of mechanism, how one can analyze this combinatorially. And the way to do it is to first put to as first class citizens, so to speak, these interactions of rules. So in some shape, to produce these type of diagrams with a recursive or other generative mechanism, and to reason about their combinatorics. And these objects is, I would I call tracelets. Okay. So the plan of the talk is that I would like to start from something which I think is quite close to interest of many people in the audience, which is certain type of HOPF algebras, which have been introduced by Gerard Duchamp, Carville-Penson and others, which give exactly sort of the blueprint of this type of combinatorial construction. And then pretty much the second part of the talk is giving you a little background on categorical writing theory, just just enough to demonstrate how this how this works. And then to show you how you get to tracelets and in particular some notion of algebras that replace the C's diagram of algebras in the general case. Okay. So for me personally, I started working on this about five years ago. And I was studying this paper by Gerard Duchamp, Carville-Penson, and their collaborators from 2011, which was called combinatorial algebras for second quantum theory. Okay. And so from this idea, I developed my very first version of some notion of graphical algebras, graph transformation algebras together with one sort of an host and colleagues. So here really, so I'm presenting to you sort of a slight reformulation of Gerard and Carville in a language which then later directly generalizes to the case of reviting, but it's completely equivalent. So the idea is if you look at the manipulation of just graphs that have you don't have any vertices, then what can I mention, there's some very elementary things one can do. So I mean, we are always only looking at isomorphism classes of graphs. So in this sense, you know, vertices are indistinguishable. And then there are two transformations one can do elementary ones you can create, you can add some vertex or you can remove a vertex. Now, the starting point of getting to some notion of diagrams is that from these elementary building blocks, you can assemble larger diagrams. And they will typically look as follows, you have some occurrences of these particular sort of little elements of the creation and deletion. I should say that each diagram, you have vertices characterized as output and some as input. So in the pictures, because I mean, the creation only has one vertex and I draw these little dash lines to indicate whether it's an output or an input. Yes. So they are read from bottom to top the diagrams. And then sort of the second information in the diagram is how some of these outputs, some of these creations are then wired into inputs or some of the deletions. And so more mathematically speaking, I'm giving two sets, sets of input and output vertices and a relation between them, which should be one to one. So that and also I'm only considering again, sort of essentially isomorphism classes of these type of diagrams. So more concretely, I'm looking at equivalence classes under, you know, joint permutations of sub vertex sets, which preserves incidence structure, situation. Okay, so these are the diagrams. And now this beautiful idea from the aforementioned paper of Sharon Cavill was that you can now enrich the mathematics by, sorry, you can build the mathematics over these diagrams or equivalence classes by constructing a vector space that whose basis is indexed by these equivalence classes. So for each diagram, d, little d, you have a basis vector, which I always write delta of d. And beautifully now, and this is of course, you will immediately see how this glinks to rewriting. Now one can construct on this vector space, a binary operation, sort of called the diagrammatic composition. And the idea is that if you are, if you give two diagrams, you, I mean, first the one, the one and then the two, you can sum over all ways you can consistently wire together these diagrams, meaning you have some, some, like this drawing on the right here is for example, the one and the two, this blue m12 are the two lines that link some inputs to some, some outputs, some inputs. And then performing the composition amounts to forgetting that these were two diagrams wired together, sort of seeing the whole thing as one diagram, and of course, taking equivalence class. And so this gives a very nice composition operation. And the first important result is that this operation to get, I mean, the vector space together with this binary operation gives the associative unical algebra called diagram algebra, whose unit element is the equivalence class of the empty diagram, or I mean the basis vector associated to the empty diagram. So that, that for me was sort of the first, I mean, the literature I knew was the first description of such diagrammatic operation, which, you know, reproduces some, some combinatorics of these composition structures. Okay. And then sort of now looking more closely, we see that actually our little catalog of elementary diagrams wasn't complete, because sort of one of the connected, so essentially elementary diagrams are the connected components of such larger diagrams, which are of course the creation and deletion of vertex diagram, but also the only other connected component you can get is creating a vertex and then deleting it again. So these are the three elementary sort of, sort of little mini diagrams. And if you simply introduce, so now we need a notation, which means sort of pasting this jointly such, such elements, but I mean up to isomorphism. So I mean, so, so happens to be the composition along trivial overlap in this definition. And it, of course, immediately by looking at these diagrams, how they already drawn, you see that the equivalence class of any given diagram is given is completely characterized by the numbers of occurrences of these three elementary letters. Yes. That's simply by the construction. And now that, of course, is already an interesting sign where one could get some combinatorics. And in particular, sort of the, so when one looks very closely, one can define something called, I mean, as usual, we now have a binary operation. So we can define a bracket, a commutator. I'm just using the ordinary commutator symbol for, of course, a commutator in this diagram algebra. And so these three elements have the interesting property that if you take the Lie algebra form by these three elements and the bracket, the only non-trivial commutation is also, I mean, you first create and then delete, you have one more option, namely the e pattern, then the other way around. So I mean, that's the only commutation relation you have. So these three, the basis vectors of these three diagrams form the Heisenberg Lie algebra. This is exactly the commutator structure of Heisenberg Lie algebra. And now one can take sort of very nice results from the mathematical literature, which is a Poincare-Bekhoff-Hitt theorem. So if you form the universal enveloping algebra of the Heisenberg Lie algebra, which is to say, you basically, you can tensor arbitrary basis vectors together of these, you know, chosen from these three types, modulo relation, which is created by the ideal of, you know, exactly created by this commutation relation. There is a very nice basis for this space, which is, I mean, giving some arbitrary order on the elements, but in particular the one where you just sort first the string of all the v plus, the v dagger occurrences and just all the v's and then all the e's. Then you can write a basis for this universal enveloping algebra of the form shown here. This is a Poincare-Bekhoff-Hitt theorem. Now, so the interesting thing is, of course, this is again just characterized by the triple of integers, the numbers of how many times each of the, of these elementary basis vectors, of course. And so, recalling that there was this notation of this transunion of diagrams and that sort of the characterization of an equivalence class of a diagram is exactly by the numbers of connected components of each type. So it's motivated that you can actually find an isomorphism between elements of this diagram algebra and elements of this universal enveloping algebra, the Heismic-Lea algebra. And indeed, it's exactly just like for every distance transunion, you produce the ordered tensor product and conversely, going the other way around, you forget the order, essentially. But more than just sort of a coincidence, it turns out that this is actually also an isomorphism of algebras. So this exactly preserves the algebra product from D into U and the other way around. So that's sort of the first nice observation. But of course, since we have some people very well vested in Hopf algebras here, it's of course well known that this Heismic-Lea algebra, universal enveloping algebra is a Hopf algebra. So the question is where do you get a co-product from? Because I mean, one could ask, is then also the diagram algebra or Hopf algebra? And indeed, there's evidently this idea that you have some diagrammatic structure. It's already characterized through connected components. So if you really dissect a given diagram into connected components, now need some notation. So effectively, every element of the diagrams, every possible diagram is just the disjoint union of these connected diagrams. So in the notation, the only strange thing, but it's for convenience, is that you say disjoint union over empty set is just the empty diagram. It's just to make this formula more consistent, notationally nice. And now the co-product is just the sum of all ways to partition parts of the diagram into the left tensor factor and parts of the diagram into the right tensor factor. Now, this definition gives you indeed a co-product. And indeed, this isomorphism I showed before extends to Hopf algebra isomorphism. So you can indeed define a Hopf algebra structure on these diagrams. And I spare you, of course, all of these many axioms. But this was, I think, one of the core results in the work of Dushan Benson also. Okay. And I apologize for the typos this morning, so I had to write this by hand. So if one now looks closely, I mean, so far we have just talked about high-level properties of these diagrams, but one can simply just write down now the product of two elements in the diagram algebra. Again, they are characterized just by the numbers of occurrence of these elementary patterns. And the main point here is that there occurs a very interesting coefficient, which is the number of matchings of, so the first diagram has K1 outputs, so vertices created. The second one has two many inputs, so vertices that will be deleted. And now the sum is of all ways of pairing those, with forgetting the order of the pairing. So this is exactly the combinatorial coefficient, the number of ways of doing this. But if you look now at the famous Heisenberg-Weil algebra, or more precisely its representation, this number vector basis. So, I mean, of course, everyone knows this from quantum mechanics, for example. So there's an algebra defined over vector space indexed by just natural numbers, which has two generators and their representations act as follows. So a dagger on the vector n gives you the vector n plus one, and a, the annihilator on vector n gives you either zero of the field, if n was zero, or n times n minus one, if not. And indeed, the normal order elements in this algebra, normal order means first of all annihilators and all creatives, have precisely this above number coefficient as in the algebra structure. And for Jean-Pierre-Pencet, that was the motivation to study this diagram algebra, because as you saw before, the coefficient is purely produced by combinatorics of matching in these diagrams. And now one can immediately think about, so we will now simply, I use different letters now. So before these were called v dagger and v for radius creation and deletion. I now use capital A dagger and A in the other algebra, just to say these are, I mean, exactly the same diagrams, but seen in a different algebra. So for each, there's a little dictionary where you say with phi bar, you forget of a diagram in this hopf algebra structure, all the parts which are these great and then the deep pieces. This way you get a diagram and on the right hand side. And if you embed such a diagram in the larger algebra, you just have hopf algebra diagram without any of these create and then delete patterns. And it turns out that this completes the picture very nicely, because now this gives you an algebra H, again defined over diagrams. And it's exactly sort of a diagrammatic encoding of the Heisenberg Weiler algebra. So it's this algebra with the only commutator A with a dagger commuted gives the identity. And finally, sort of a little piece of the information is also that you have a representation, which is exactly just that, you know, to such a basis vector and H, you can assign an action on number vectors through representation. And so everything put together gives you a very nice picture of the structure of Heisenberg Weiler algebra, mostly explained combinatorially. And also even the representation, I mean, there's another way of seeing how the representation can even be seen as a combinatorial action. And so in summary of this part, I found this back then extremely interesting, because sort of all of the combinatorics of how these transformation steps interact is coded in these diagrams. And you then postpone to a later step through this representation row, acting on states. And this is exactly the type of deciphering that is needed to make progress in these transformation systems. Okay, and now I just wanted to show you one sort of, this is still a bit experimental, but I mean, sort of the key point about Heisenberg Weiler algebra is that you can formulate chemical reactions in a language where you do not look at the internal structure of molecules. So you just count abstractly number of occurrences of different molecules. And so one can quickly write down a continuous time Markov chain for it. So I mean, this is now in this Parkman-Fock basis where annihilation is d by dx, creation is multiplication is formal variable, and you are tracking the probability distribution of states with n particles coded as monomial xn. Anyway, so and so this is a birth-death process. And if you draw these pictures again, the birth-death process chooses when to jump, when to perform transitions, either the creation or dilation. And again, of course, we could imagine we now look at the picture in this in this Hopfer algebra. And there you would track when a creation was followed immediately, or maybe at a later point, but connected with the deletion. So it's impossible to put a measure on any possible time point for this. But what you can do is you can put a box of time capital T, and then just characterize the content of these transitions by exactly the classification into connected components. So creation, deletion events, and events where you create and then delete. And so there's a little trick. I mean, so far there is no Markov chain theory for the software algebra in that form to be developed. But here, at least when you create, you can simply make this a different type of molecule. And then when you delete, you can make sort of a little artifact produce some third type of molecule. And this way, if you now track the numbers one, and you run all of this machinery, in the end, you can indeed get a nice expression, which tells you a little bit more about this dynamics, saying essentially, of course, you stabilize on a cross-off distribution, ultimately. But you also see that in the time limit goes to infinity, you grow linearly with the number of these create delete events. So that's the dynamics of the system. And I mean, this is by no means a full theory yet. It's just like one very first motivation that presumably also Markov chain theory, one can accept some interesting information, only that, of course, this key particles have very little structure. And so I return to the question. So how could you now approach this for the much more complex situation, where you're not describing just vertex graph, but you actually describing graph and hitting all of this combinatorial complexity. And so just to recall, the idea is now pretty similar. So we will mostly focus on first of all, classifying combinatorially how to interact with writing steps. And sort of the ultimate goal would be to then understand if you want to count, for example, triangle patterns that are produced or created or deleted by such sequences, how can you how can you give a measure of likelihood on that? That's the ultimate goal. And again, we would like to somehow find a way and this will be through commutators, of course, to sort of drop out some of the possible contributions, which do not actually influence the column. And the only sort of the real obstacle for this was that if you, I mean, this is a perfectly valid sequence of events, but at first sites, what complicates this is. So one could imagine these diagrams, the semantics will just be pairing of subgraph, you know, before we just paired vertices to vertices in these other diagrams. But here, it turns out it's a little more intricate because actually, it seems you're pairing half edges. And that is sort of the real problem for this. I mean, how to come up with a concrete generalization. And so after a lot of experiments, so I came up to formulations in categorical writing theory. And so, I mean, this is maybe a little bit of an exotic theory, but the nice thing about it is this completely formal. And it can immediately be implemented in algorithms. So I'll tell you a little bit about this. And then afterwards show you how you can produce these sort of the analog of these diagrams. Okay. So the full generality of this theory is quite intricate because in general, you know, only look at graphical structures, but also some which are sort of constrained additional structural constraints, for example, in trees. So and I mean, so this is a paper which will jointly version Caribbean, which should be published soon in the applied category theory general compositionality. And so I just want to show you a little part of the theory enough to get the idea, hopefully. So before we had transformations were essentially specified as partial maps between inputs and outputs of vertices, this generalizes now in. So we take a category typically, it should have some some nice properties called adhesivity. But sort of the important thing to notice that any, so for example, graphs or such a category, we can have undirected graphs, multi graphs, and all sorts of variants of specifications. But main point is, you take such a category, and you can formulate partial maps and quotation marks as spans of monomorphism, so spans of embeddings. And so in the most general framework on top, can put some conditions, which essentially constrict, constrain how you can apply these rules. And again, to make a meaningful theory, we, I mean, this would in general be a class, even for graphs, even for good categories. So we typically also have to quotient by some notion of isomorphism as we had to do in the diagram case. Okay, but this is sort of the analog of these little building blocks of the transitions that one can use. And sort of unfortunately, then there's a little bit of, I mean, this is very general, it covers pretty much all of the known case of such transformations, but for that, it's also a little bit technical. It is just to say sort of the analog of finding an input pattern in a state X is now finding an embedding of I into X. And then how to from that, you know, apply the rule how to unroll this transformation depends on your semantics. And it's typically performed either in the double push out semantics, or in the CESC push out semantics. And the first one is essentially trying to compute some notion of set complement to get this object K bar. And the second one is using some construction called final pullback complement. So I mean, in these graphic graphic theories, push out is typically gluing together along common overlap, pullback is finding intersection of two objects, push out complement is roughly like the set complement is just that, for example, if you have a graph, and you just try to delete vertex, then it depends on whether this incident edges. So there's some some difference. But other than that, it's roughly the idea of applying these steps as in the graphical description. But so the key point is that there exists a notion from the pure rewriting theory of how to compose two steps, how to how to interact with two rules. And so this is very much like in the graphical language for these diagrams, you, you have two rules, and you try to find an overlap of one of the output of the first rule, with the interface, the input, I two of the second. And so this is again coded as a partial overlap. Now you glue together along this partial overlap, and then you just run the rules. That's essentially what this says. And of course, there's some, some technical details for how to do this with conditions. But I just wanted to show you, I mean, I think intuitions are pretty good here, also for the graph case. So I'm now drawing the diagrams from right to left instead of bottom to top to be more in parallel with this mathematical notation. So here I'm drawing a picture where I take two vertices, I link them up with an edge and make an additional vertex and an edge. And then I use one of the vertices to link to it, to ask for it to an edge to the incident. And then, you know, delete that edge makes new edge pattern. And so exactly how to interpret this diagram is to say, Okay, so I'm encoding here a sequence of events where sort of the intermediate state, this is shape where you glue together the patterns along the common overlap. And then you run essentially the one rule in the forward direction and the other in the backward direction. And now this is exactly the analog of these diagrams you saw before in this graphical case. So it's a sequence of events, which was produced through interaction of these two rules. And sort of the overall input motive here on the very right top bottom is seen necessary pattern you need to find in any state to apply the sequence. So this is sort of the very first example of such diagrammatic calculus. So there is one complication, maybe which prevented this from being useful from the start, because just graphs are rarely very interesting. Normally, most of these applications, you want to have some more structured species. And for example, one very nice species, I think, which is also good communication grounds with the communaturist is, for example, planar rooted binary trees. And so if one wants to, you know, formalizes in this context of writing, the first step to note is that you can write, I mean, not only trees, the forest, of course, planar rooted binary forest, you can see them as some type of graph, which are typed. So you can take the slice category over some type graph. So every edge is given one of three exclusive types, either being left, right, or the bottom. And so that in itself is sort of a first step. But then of course, you see, it's not only a graph which is colored by these three types of edges, but also it has some a lot of structural properties. And of these, I'm just listing some here. So for example, you never have two leaves directly incident, and you never have two of these root edges and so forth. So I mean, this is not very pretty. You can formalize this as conditions. But of course, this complicates a bit the story. It can be done algorithmically. So at least that's a plus. And so in the end of today, so transformations in this language you can do on trees can be made on can be formalized as writing rules. But I mean, with the expense of those calculus, that is to be said, but in principle, it is now completely formalized how to do these transformations. Okay. And now finally, so the idea is how do you now get to combinatorics? So how do you get to calculus on these graphical writing steps? And the idea is that so as in these pictures in the introduction, you want to reason about all possible ways of applying and different transformation steps, where each step is chosen from some finite vocabulary of possibilities, let's say, for example, vertex creation, vertex deletion. So you want to classify the ensemble of all possible trajectories, let's say for a fixed input state x zero. And the strategy which is very efficient is to first classify all possible ways how these n steps can interact with themselves. So sort of the minimal context you need to fire a given transition to apply a given sequence. And then as a separate step ask how many ways are there then to apply the overall of such input to the state x zero. And so this is precisely this idea of of tracelets completely formalized in this categorical writing theory now. Okay. And so the idea is that if you again look, so this is now the analog of one of these diagrams from before in the graphical setting, where you now have three steps there in the sort of shaded boxes. They are drawn from right to left. And this wiring diagram indicates sort of a possible sequence how these three steps could interact. And you can produce from it now this type of tracelet. So the idea is that each of the wires codes for an overlap. And now for example, you can take you can zoom in on the first two and compute how they interact. Producing this little subsequence of events. And now you see that indeed the overlap of this third diagram with what you just produced is now a proper graph. So there's no half edge or stuff like that. So you can glue together. And you essentially just complete sort of the full sequence of events that is minimally coded in these three rules as the bottom sequence. And it looks like this is and finally to obtain what I call tracelets. I mean, it turns out the only information you need to retain is just the outer hull of this diagram. So it looks precisely like just a sequence of transformation steps. But the specialty is that it only contains enough information to permit the sequence to occur. Not all I mean, this could be happening in a much larger context. And so this is exactly where you gain something in the complexity. And at first sight, it seems this is sort of asymmetric. But one can show that that it's equally possible to build up this trace that from the same diagrammatic overlap structure, just in a sort of first computing what what how three and two overlap and then also overlap this one. So that's that's sort of part of this calculus. Okay, so I mean, it's just to say that these diagrams can be completely coded as just sequences of applications. And on those now you can do a combinatorial calculus. So sort of the what's communatorily interesting is that now you are given your vocabulary. So it's just the abstraction of saying sort of the top parts are the colored bars are the individual rules. And so you can build up all possible trace, let's say here of length four by recursively composing or iteratively iteratively composing your letters. And so in each of these steps, of course, you can perform analysis. So this is exactly this philosophy of the diagrams that you can you can reason transitively so to speak on on compositions. Okay, and so I just want to briefly show how this looks like in practice. So a trace that of length one is just, you know, the special case where you have just one rule, it really only needs its input. So that's sort of the sort of the trivial case. And sort of what I showed just in pictures looks in reality, like you are inductively building tracelets of length n plus one from trace of length n and traces of length one. And so it is just to show you that that even in the case of trees, this doesn't look pretty, I admit, but it can be encoded in the algorithm. So I mean, this is a fully formalized structure. And sort of one of the interesting features of these tracelets is that you have at the bottom of this diagram, the sequence of steps. And so you can read out the composite effect of that sequence. And in the language of the Sean Benson, these hope diagrams, this was exactly this operation of evaluating the net effect, how this acts in the Heisenberg-Weil algebra. So this is now generalized here as reading out the net effect of the sequence of transformations. So that is nice. That's called an evaluation. And this evaluation is also compatible now with composition. So you can, indeed, so the analog of diagrammatic composition is now composing these tracelets. And again, it is only important to take an overlap of the output of the tracelet with an input of the next. And so this is very exactly analogous to this diagrammatic composition. And finally, sorry, maybe just to say, so there is now this operation of composing two tracelets, drawn with this Vetch notation at the bottom here. And this composition, now, of course, if we go with analogy, should be associative in some sense. And indeed, sorry, and indeed, it is associative in the sense that just as in the Sean Benson case, sort of the number of ways to wire two diagrams together and then with a third are exactly isomorphic to the number of ways first wiring the third and the second and then the result with a first. So that is, that is a property this structure has. Now finally, to put everything together. Yes, now precisely, this aforementioned characterization that if you have a sequence of transformations, you can equivalently count all possible ways of performing these transformations by first counting the number of ways you can compose up these tracelets, and then together with number of ways to applying them. So that's sort of, it's just to say this now exists for all of these writing theories, including, of course, hope of case. Okay, and sort of the final piece of the puzzle is then how to actually get from here to algebras, because we want to analyze everything using commutator relations. One thing one has to do from the start, I mean, we already had to go to isomorphism classes for rules to even obtain a set of equivalence classes. And again, there's something like this also for tracelet. So sort of everything is constructed with push-offs and push-off complements and so forth. So you have to quotient by isomorphisms. Something less trivial is that so you do traces are slightly too large. I mean, normally do not want to keep every bit of this information. In particular, you do not want to keep like in this diagram here, information of order when the steps are completely exchangeable up to the effect. And so that's called shift equivalence. And finally, there's one oddity which is that simply formally the trivial rule sort of intuitively should leave a transformation sequence invariant, but formally, I mean, it just produces an n plus one length sequence with some repeated parts. So you can define an equivalence that simply quotients out by such occurrences. And if you put all of these together, indeed, then oops, sorry, sorry, yeah, here, you get to exactly now finally the construction of tracelet algebra. Basis elements are labeled by equivalence classes of these tracelets under the aforementioned equivalences. And now the product is precisely as in the digital pencil construction, the wiring together in all possible ways, the overlaps in all possible ways of the tracelets. So this is now the tracelet algebra product. And you can give an action of tracelets on states precisely by this aforementioned trace characterization that if you have a sequence coded in a tracelet, you can apply the entire sequence by finding embeddings of the overall input into a state. And this gives you a representation of this algebra. So the theorem here is that not only does this tracelet give rise to an associative unit of algebra, but moreover, this row is indeed a representation, which means that you can, if you if you now want to do combinatorics on numbers, if you want to count number of ways of applying writing sequences, you are free to first you can partition your problem by this bottom right outcome of the equation into first trying to characterize the number of overlaps of tracelets, which is very advantageous, because now you can use relations such as commutators and so forth. So this is sort of the final outcome. And I then realized while preparing the talk, this was already much too much information. So let me just conclude on reproducing the special case of discrete diagrams. So here you see on the left, the elementary sequence of creating a verdicts and deleting a verdicts. This is how it looks like in a tracelet. So this is the analog of the diagram of this generator E and the combinatorial hopf algebra. And on the right, you see exactly why you need this shift equivalence, because I mean, creating a verdicts and deleting, I mean, forgetting and taking equivalence classes the same as, you know, in quotation marks first deleting and then creating. And this is in the tracelet language exactly obtained through such an equivalence. So it's not, you know, a trace value, but we label the elements of algebra by equivalence classes. And finally, then, indeed, the type of relation you can then derive are precisely commutation relations. And those are key to the analysis in these combinatorial arguments. Okay. And I mean, I wanted to speak about planar-rooted binary trees, but I think I'm out of time. So it's just to say that in these calculations, and I also showed this last year at CAP, one can now start to see combinatorial simply counting arguments on why certain commutators have the form they do. So that's sort of the main motivation for this work, but I do realize I'm out of time. So it's just to say sort of one of the, so this will be a forthcoming paper for beginning of next year, and one of sort of one of the cases will be an explanation of why sort of, you know, you see certain commutator structure in these planar-rooted binary tree computations. Okay. But so let me conclude to stay in the time limit. So I've given you a quick tour of a new concept which I call TraceLed, which is intended as a generalization of this Dishon-Penzen et al. construction of diagram algebras. It seems to be very useful. I mean, I have a sort of, I'm developing an implementation of this with an Z3 SMT solver, which is available online, but I mean, this is work in progress. And sort of the mathematically interesting question is maybe whether you also have some Hof-Algebras structure on these TraceLeds, which for some cases I know one can demonstrate, but in general is sort of a research question. And sort of the long-term goal of this work is to bring combinatorics also into these bio and organochemical reaction systems, which through the TraceLeds now will boil down to enumerative combinatorics on TraceLeds. Okay. And with this, I would like to thank you for your interest and thanks a lot for your time. Thank you very much, Nicolas, for this is, are there questions? Yes, I have one question. What about directed graphs? Do your construction applies to this? Yes. Sorry, I just showed underrated for the sake of the diagrams. No, no, so it applies for any type of graph you can formalize categorically. So even for simple graph or for multi-graph or for hypergraphs or for attributed graphs and so forth. No, it was just for the pictures. So, directed graphs in particular are appreciative. You know, you can formalize them as appreciative. And so any adhesive category, so any appreciative gives you an adhesive category. So that's, yeah. Okay. And in particular, for the directed graph case in this paper, in this archive footprint, we even have the Hof-Algebras structure. So may I ask a question? Yes. Yes. Other questions? Yeah, I have a question. Yes. Nicolas, could we describe the construction of a penrose tiling with your graphs? Is it possible? Aha. It's like simple chemistry, after all, but the dimensions. So, I mean, the only question about this is whether you can generate your tiling in some process which only asks local information. So, I mean, I'm not quite familiar with penrose tiling, but I mean, if you can describe. You have just several tiles of forms and there are just rules, quite simple rules, how to glue them together. Interesting. Yeah. Yes, so that sounds very much like an example you could study. Yeah, I think all quasi crystals, because it is much more interesting than crystals, because they are, of course, periodic, but this is up-periodic. Yes. So one of the things I could imagine is you might, for example, ask in such a quasi crystal, what is the average occurrence, say, of particular subspecies? No. But probably with your technique, it may be described in a very simple way. Yeah, so in particular, if you have a model for how these local manipulations are fired. Of course, of course. Yeah. I have a little comment on this, because all of your stuff, it's like it's really chemistry, it's bullion, it's no special structure. And of course, in penrose tiling, you get things, like a plane. And in principle, one can, for example, there are these universal equations equivalent to universal Turing machines, you do dominos in two dimensions, like square tiles, and each side is colored by some color, and then you try to build things since it's a universal Turing machine. Of course, you can start to glue things, you get some pieces of planes, so you get eventually some kind of space like structure, abstract story. Yeah. I mean, one of the things we tried was, for example, polygons, so you can have these sort of tissue models, which are simply just polygons glued, but I mean, without gap in the plane, and you can sort of divide the polygons, you can sort of insert triangles by expanding vertices and so on. And this is also in the realm of these writing techniques. And so you can use it in, there is a problem with capsid, you know, the viruses, which have icosahedral symmetry. They are surrounded by capsids. These are just the coat proteins that come together, and they have five-fold and six-fold symmetry. And of course, they build these capsids, and again, there are very simple rules that make them glue together. Yeah, I mean, so the main motivation here is that for these types of theories, if you sort of the motivation is if you have a good, if you have a problem, which you think is of this nature, it is relatively quick to check how to formalize it into the writing. So I mean, it's essentially to see whether you need more information than what you can expand in a small local neighborhood of the rule or not. If not, there's a very good chance you can. So what you get in quotation marks for free are, for example, commutation relations. So you can, for example, ask the pattern, the counting of the pattern is implemented as just an identity transformation of that pattern. And so you can, for example, ask how many of the occurrence do you have more or less before, after applying this transformation? And so these commutators really carry a lot of information. And it's much smaller information than if you were to analyze the entire structure. So that's sort of the main, you get sort of average information. Yeah. Are there questions, remarks or comments? Can I ask one little question? Have it been studied which information on the solgebres are carried by some homologies of the solgebres? Like, hoaxed homologies or whatever, something like that. I mean, I mean, these algebras have not been even written down. So, I mean, so the question is, I mean, the answer is no from my side. I would not be surprised if you look at special examples, maybe you could, I mean, it would be more that you recognize the trace lip stuff in it rather than, I mean, I don't have a good answer. Sorry. I think it's a good question, especially because last year at CUP, we had this nice talk by, we had this nice talk about graph homologies or homologies, I forgot, which also could be formalized through writing. So, I mean, if you have, actually, if you have a good case, which you think it could be of this elementary graph like structure, I would be very interested because I would like to play with it. Yeah, it should be possible in general kind of point of view, saying that they should carry some information. Interesting. No, I would be very interested if you could have a good sign, Yelin, maybe. Okay. Thank you. Any other questions, remarks or comments? Oh, thank you, dear Nicolas for this amazing and huge expansion of our paper with Favère Boisac and Carol Penzon. And I have a small question. Are your variants graded? Do you have each time some way to count vertices or edges so that? Yes, exactly. So, the reason these formalized rise, that I mean, so in my formalism, they are filtered. So, it seems as though in principle, you could count the occurrences of these connected components in some system. So, I mean, if you give grade two to the, you know, this e-pattern, which consists of two subdiagrams and grade one to each of the others, they will give you a grade. But the thing that generalizes is the filtration by essentially the cardinalities of the interfaces. And then this through this filtration, I mean, always when you compose, you exhibit a little bit less of the interfaces to the outside world. So, that decreases the filtration degree. So, I mean, it's clear that this by composition gives a decreasing sequence. I mean, at most, you have the same filtration degree if you don't connect overlap, because it's simply the sum of the overlaps of the interfaces and otherwise it decreases. And this happens to be compatible as a core product. So, yeah. Ah, okay. Thank you very much.
 Thank you very much, first important question you can see the slides, hopefully.  Yes, we can see you. Yes.  Can I start? Should I wait for the minute?  No, no, it is a... You can begin now if you want, if you wish.  Thank you very much, Jaan, also the organizers for this invitation. It's always very nice to be at  CUP now virtually, but I hope at least this way some more people can participate.  And the talk I would like to give today is about a topic which I think is quite the spirit of CUP,  because it is about this type of research which tries to make sense of the deeper links,  the mathematical links between approaches to manipulation of graph-like structures and computer  science, mathematical physics and mathematics. So, I mean, of course, every of these disciplines has  many, many examples of situations where it is meaningful to look at these types of manipulations.  For example, the very simplest such is drawn here at the bottom. It's just when you manipulate,  for example, in physics, in distinguishable particles, this would be a model of chemistry,  for example. Then, of course, in social network science, maybe you look at processes which can  be formalized as, you know, manipulating locally a network graph, like, for example, the rewiring  and edging here, but then there are also many, many more sort of Baroque examples. I mean,  trees, for example, can be seen as a type of graph which has additional structure,  like particular incidences, for example. And my personal motivation is really from organic chemistry  and from biochemistry, where you encounter these beautiful types of theories, where, for example,  you formalize proteins and other molecules through these type of writing steps. And sort of my main  motivation for this work is that I'm originally a mathematical physicist, but I'm now working in  computer science, and I discovered that there's relatively little cross-path between these three  disciplines that would allow you, for example, to take a continuous time Markov chain theory  and directly apply it to one of those organic chemistry simulation techniques in biochemistry,  say, for example. And, of course, one of the main motivations is to ask, so in these type of very  complex systems, where can you attack with combinatorics? So what are the combinatorial  structures, of course, beyond the data type of these graphical structures that one could use  to analyze the systems? And I'm just putting this here because, at least for bio and organic  chemistry, it is really the case that, I mean, sometimes you have abstractions which are made  because you can then, at least as a toy model, talk about these systems. But at least in the case of  bio and organic chemistry, these are really the state-of-the-art techniques to simulate these  systems. So in Kappa is a framework where you have these sort of circuit side graphs, there's a  special type of graph, you know, everywhere it takes some sites at which you can link to other  vertices. And in organochemistry, it's even more fantastic, it's really that the molecules you draw  by hand also for chemical reactions are a data type that you can then formulate transformations on.  And so these are the two main frameworks. So if you, I mean, I would say a little bit about this  later on, and it so happens that there's really a very formal semantics for these type of theories.  And that is good because that is really, I mean, it's very mathematical formulation. But that's a  plus. What's the problem is that if you then look really at a realistic scale example, and here's a  little piece of the human metabolic network, in this huge graph, every part of this network is,  I mean, an enormous complexity, and you have a very many different types of molecules that  participate in these reactions. And here's only drawn like the macro molecules, or the larger  molecules. And then you have enormously many reactions. And so the real question is all of  these transitions, these manipulations are fired at random. So what is it really you can say about  these systems? And especially if you look at biology, which is still a very fast evolving field,  of course, I mean, it is nice that you have this interesting data type, which is quite good also  at, you know, encapsulating new knowledge you have about just individual transitions. But you  can see immediately, since every of these green blobs, these agents, as typically on the order of  seven to 10 sites, this is an extremely highly combinatorial data structure. And then if you  start manipulating, it is very unclear of what you can actually extract as information. So  main problem is to understand the function of such systems, because nature has given them,  this is the abstraction that is very accurate. So what can you say about the function of the systems?  And so, so to just motivate how one could approach this, I'm going to take an example of  the transformation system, which is of course much simpler than these biochemical reactions,  but it's complicated enough to explain the main sort of root of attack for these systems.  So here's a system where you have an input state, which is just some graph, now performing  transformation. So the language for these transformations always looks similar, you have  little pieces of evolution, if you will, each of those is asking for some input drawn at the bottom  in this little cartoon, and some, then it produces some output. And the output here is,  so the input is here, some two vertices, the dashed lines indicate that they are kept throughout  the transformations are identically kept, and then you link them up with an edge. That's this local  manipulation. And the second part of this framework of the semantics is that you to apply such a rule,  you have to exhibit a match, and a match is nothing but an embedding of this input motive into the  graph. You see immediately that, I mean, here I've drawn a very, very small graph, and I mean,  this rule is sufficiently simple, but already here you have quite a high number of possible matches.  So part of this sort of encoding is that it's usually quite simple to write the rules, but I mean,  there's usually quite many ways of applying these rules. But anyways, so then here,  applying the rule amounts to locally, then, you know, running this transformation, which here  amounts to inserting this edge in these two vertices. So let's do a few more of those steps,  another of such exactly the same rule applied at a different place. Now, of course, I mean,  I'm only showing some very simple rules, you could also have one which unlinks, for example,  like this one. And let's take another one to link and another one to link. Okay, so now I have given  you a small sequence of the transformation system. And very nicely, not only for this particular case,  but I mean, for an enormous variety of possible things of graph like structures and their manipulations,  they are all covered by a mathematical theory called categorical writing. And nicely enough,  it is very close to implementation sources. I mean, this is something you can, you can put very  directly into algorithms. So it's sort of the source code for these sort of manipulations.  But sort of the typical problem is now, I mean, going into this picture of what happens in biochemistry,  how could you understand it? So imagine you were given these type of transformation, you had a  little vocabulary, say just the linking of edges and the unlinking. But now you are asking, so of  all of these possibilities, what is the likelihood of if you fire these say at random with same  properties for sake of argument, of seeing a triangle appearing? Yes, so I mean, you're trying  to track how many times to see a triangle that was really produced, I mean, newly produced,  or maybe deleted through this application of sequences. And so what classical writing theory  doesn't have much of an answer to is how to approach this and utilizing this feature of writing.  So what one can do, and this is now I will motivate this more later on, but sort of the  the starting point of this combinatorial analysis I would like to propose has to do with, of course,  you can track how actually these rules were applied. And here in this picture, you see that  these are simply just the, I think, five steps one after another drawn works vertically. And  whenever you you attack with a second, third, fourth rule at a position where you have previously  already applied a rule, then this is marked with these light blue lines. And the red lines mean  if you touch something that is in your original graph, but not interacting. And so immediately,  it is obvious that there's some combinatorics in how you can sort of plug together these,  these transformations and acting with each other versus how how they are acting on on this graph  state. So again, like if you if you now look at this problem of finding triangles, you discover  something very important, namely that sort of producing a triangle is an inherent feature of  of this above sequence of rules. I mean, because they are plugged together in a way that they are  guaranteed to produce a triangle, there might be other ones that accidentally are produced by just  simply linking up a V shape say with one edge. But that is definitely already one guarantee you  can give this sequence will produce a triangle. And the other point of interest is, so these  two parts of the transformation sequence highlighted in orange, of course, they are happening,  this is one possibility to apply, but the two actually do not contribute anything to the production  of the triangle. In fact, they they they only sort of they produce an edge and deleted later on,  so that that actually doesn't contribute anything to this count. So what one needs is some sort of  mechanism, how one can analyze this combinatorially. And the way to do it is to first put to as  first class citizens, so to speak, these interactions of rules. So in some shape, to produce these type  of diagrams with a recursive or other generative mechanism, and to reason about their combinatorics.  And these objects is, I would I call tracelets. Okay. So the plan of the talk is that I would  like to start from something which I think is quite close to interest of many people in the  audience, which is certain type of HOPF algebras, which have been introduced by Gerard Duchamp,  Carville-Penson and others, which give exactly sort of the blueprint of this type of combinatorial  construction. And then pretty much the second part of the talk is giving you a little background  on categorical writing theory, just just enough to demonstrate how this how this works. And then  to show you how you get to tracelets and in particular some notion of algebras that replace  the C's diagram of algebras in the general case. Okay. So for me personally, I started working  on this about five years ago. And I was studying this paper by Gerard Duchamp, Carville-Penson,  and their collaborators from 2011, which was called combinatorial algebras for second quantum  theory. Okay. And so from this idea, I developed my very first version of some notion of graphical  algebras, graph transformation algebras together with one sort of an host and colleagues. So here  really, so I'm presenting to you sort of a slight reformulation of Gerard and Carville  in a language which then later directly generalizes to the case of reviting, but it's  completely equivalent. So the idea is if you look at the manipulation of just graphs that have  you don't have any vertices, then what can I mention, there's some very elementary things one  can do. So I mean, we are always only looking at isomorphism classes of graphs. So in this sense,  you know, vertices are indistinguishable. And then there are two transformations one can do  elementary ones you can create, you can add some vertex or you can remove a vertex. Now, the starting  point of getting to some notion of diagrams is that from these elementary building blocks,  you can assemble larger diagrams. And they will typically look as follows, you have some occurrences  of these particular sort of little elements of the creation and deletion. I should say that each  diagram, you have vertices characterized as output and some as input. So in the pictures, because  I mean, the creation only has one vertex and I draw these little dash lines to indicate whether it's  an output or an input. Yes. So they are read from bottom to top the diagrams. And then sort of the  second information in the diagram is how some of these outputs, some of these creations are then  wired into inputs or some of the deletions. And so more mathematically speaking, I'm giving two sets,  sets of input and output vertices and a relation between them, which should be one to one.  So that and also I'm only considering again, sort of essentially isomorphism classes of  these type of diagrams. So more concretely, I'm looking at equivalence classes under, you know,  joint permutations of sub vertex sets, which preserves incidence structure, situation. Okay,  so these are the diagrams. And now this beautiful idea from the aforementioned paper of Sharon Cavill  was that you can now enrich the mathematics by, sorry, you can build the mathematics over these  diagrams or equivalence classes by constructing a vector space that whose basis is indexed by  these equivalence classes. So for each diagram, d, little d, you have a basis vector, which I always  write delta of d. And beautifully now, and this is of course, you will immediately see how this  glinks to rewriting. Now one can construct on this vector space, a binary operation,  sort of called the diagrammatic composition. And the idea is that if you are, if you give two  diagrams, you, I mean, first the one, the one and then the two, you can sum over all ways you can  consistently wire together these diagrams, meaning you have some, some, like this drawing on the right  here is for example, the one and the two, this blue m12 are the two lines that link some inputs  to some, some outputs, some inputs. And then performing the composition amounts to forgetting  that these were two diagrams wired together, sort of seeing the whole thing as one diagram,  and of course, taking equivalence class. And so this gives a very nice composition operation. And  the first important result is that this operation to get, I mean, the vector space together with  this binary operation gives the associative unical algebra called diagram algebra,  whose unit element is the equivalence class of the empty diagram, or I mean the basis vector  associated to the empty diagram. So that, that for me was sort of the first, I mean,  the literature I knew was the first description of such diagrammatic operation,  which, you know, reproduces some, some combinatorics of these composition structures.  Okay. And then sort of now looking more closely, we see that actually our little catalog of  elementary diagrams wasn't complete, because sort of one of the connected, so essentially  elementary diagrams are the connected components of such larger diagrams, which are of course the  creation and deletion of vertex diagram, but also the only other connected component you can get is  creating a vertex and then deleting it again. So these are the three elementary sort of,  sort of little mini diagrams. And if you simply introduce, so now we need a notation, which means  sort of pasting this jointly such, such elements, but I mean up to isomorphism. So I mean, so,  so happens to be the composition along trivial overlap in this definition.  And it, of course, immediately by looking at these diagrams, how they already drawn,  you see that the equivalence class of any given diagram is given is completely characterized by  the numbers of occurrences of these three elementary letters. Yes. That's simply by the  construction. And now that, of course, is already an interesting sign where one could get some  combinatorics. And in particular, sort of the, so when one looks very closely, one can define  something called, I mean, as usual, we now have a binary operation. So we can define a bracket,  a commutator. I'm just using the ordinary commutator symbol for, of course, a commutator  in this diagram algebra. And so these three elements have the interesting property that  if you take the Lie algebra form by these three elements and the bracket, the only non-trivial  commutation is also, I mean, you first create and then delete, you have one more option,  namely the e pattern, then the other way around. So I mean, that's the only commutation relation  you have. So these three, the basis vectors of these three diagrams form the Heisenberg Lie algebra.  This is exactly the commutator structure of Heisenberg Lie algebra. And now one can take  sort of very nice results from the mathematical literature, which is a Poincare-Bekhoff-Hitt theorem.  So if you form the universal enveloping algebra of the Heisenberg Lie algebra, which is to say,  you basically, you can tensor arbitrary basis vectors together of these, you know, chosen from  these three types, modulo relation, which is created by the ideal of, you know, exactly created by  this commutation relation. There is a very nice basis for this space, which is, I mean, giving  some arbitrary order on the elements, but in particular the one where you just sort first  the string of all the v plus, the v dagger occurrences and just all the v's and then all the  e's. Then you can write a basis for this universal enveloping algebra of the form shown here.  This is a Poincare-Bekhoff-Hitt theorem. Now, so the interesting thing is, of course, this is again  just characterized by the triple of integers, the numbers of how many times each of the,  of these elementary basis vectors, of course. And so, recalling that there was this notation  of this transunion of diagrams and that sort of the characterization of an equivalence class  of a diagram is exactly by the numbers of connected components of each type. So it's motivated that  you can actually find an isomorphism between elements of this diagram algebra and elements of  this universal enveloping algebra, the Heismic-Lea algebra. And indeed, it's exactly just like for  every distance transunion, you produce the ordered tensor product and conversely, going the other way  around, you forget the order, essentially. But more than just sort of a coincidence, it turns out that  this is actually also an isomorphism of algebras. So this exactly preserves the algebra product  from D into U and the other way around. So that's sort of the first nice observation.  But of course, since we have some people very well vested in Hopf algebras here, it's of course  well known that this Heismic-Lea algebra, universal enveloping algebra is a Hopf algebra.  So the question is where do you get a co-product from? Because I mean,  one could ask, is then also the diagram algebra or Hopf algebra? And indeed, there's evidently this  idea that you have some diagrammatic structure. It's already characterized through connected  components. So if you really dissect a given diagram into connected components, now need some  notation. So effectively, every element of the diagrams, every possible diagram is just the  disjoint union of these connected diagrams. So in the notation, the only strange thing, but it's  for convenience, is that you say disjoint union over empty set is just the empty diagram. It's  just to make this formula more consistent, notationally nice. And now the co-product is just  the sum of all ways to partition parts of the diagram into the left tensor factor and parts  of the diagram into the right tensor factor. Now, this definition gives you indeed a co-product.  And indeed, this isomorphism I showed before extends to Hopf algebra isomorphism. So you can  indeed define a Hopf algebra structure on these diagrams. And I spare you, of course,  all of these many axioms. But this was, I think, one of the core results in the work of Dushan  Benson also. Okay. And I apologize for the typos this morning, so I had to write this by hand.  So if one now looks closely, I mean, so far we have just talked about high-level properties  of these diagrams, but one can simply just write down now the product of two elements in the  diagram algebra. Again, they are characterized just by the numbers of occurrence of these  elementary patterns. And the main point here is that there occurs a very interesting coefficient,  which is the number of matchings of, so the first diagram has K1 outputs, so vertices created.  The second one has two many inputs, so vertices that will be deleted. And now the sum is of  all ways of pairing those, with forgetting the order of the pairing. So this is exactly the  combinatorial coefficient, the number of ways of doing this. But if you look now at the famous  Heisenberg-Weil algebra, or more precisely its representation, this number vector basis.  So, I mean, of course, everyone knows this from quantum mechanics, for example. So there's an  algebra defined over vector space indexed by just natural numbers, which has two generators and  their representations act as follows. So a dagger on the vector n gives you the vector n plus one,  and a, the annihilator on vector n gives you either zero of the field, if n was zero, or n times n  minus one, if not. And indeed, the normal order elements in this algebra, normal order means  first of all annihilators and all creatives, have precisely this above number coefficient  as in the algebra structure. And for Jean-Pierre-Pencet, that was the motivation to study this diagram  algebra, because as you saw before, the coefficient is purely produced by combinatorics of matching  in these diagrams. And now one can immediately think about, so we will now simply, I use different  letters now. So before these were called v dagger and v for radius creation and deletion. I now use  capital A dagger and A in the other algebra, just to say these are, I mean, exactly the same diagrams,  but seen in a different algebra. So for each, there's a little dictionary where you say with  phi bar, you forget of a diagram in this hopf algebra structure, all the parts which are these  great and then the deep pieces. This way you get a diagram and on the right hand side. And if you  embed such a diagram in the larger algebra, you just have hopf algebra diagram without any of these  create and then delete patterns. And it turns out that this completes the picture very nicely,  because now this gives you an algebra H, again defined over diagrams. And it's exactly sort of a  diagrammatic encoding of the Heisenberg Weiler algebra. So it's this algebra with the only  commutator A with a dagger commuted gives the identity. And finally, sort of a little piece  of the information is also that you have a representation, which is exactly just that,  you know, to such a basis vector and H, you can assign an action on number vectors through  representation. And so everything put together gives you a very nice picture of the structure  of Heisenberg Weiler algebra, mostly explained combinatorially. And also even the representation,  I mean, there's another way of seeing how the representation can even be seen as a combinatorial  action. And so in summary of this part, I found this back then extremely interesting, because  sort of all of the combinatorics of how these transformation steps interact is coded in these  diagrams. And you then postpone to a later step through this representation row, acting  on states. And this is exactly the type of deciphering that is needed to make progress in  these transformation systems. Okay, and now I just wanted to show you one sort of, this is  still a bit experimental, but I mean, sort of the key point about Heisenberg Weiler algebra is that  you can formulate chemical reactions in a language where you do not look at the internal structure  of molecules. So you just count abstractly number of occurrences of different molecules. And so one  can quickly write down a continuous time Markov chain for it. So I mean, this is now in this  Parkman-Fock basis where annihilation is d by dx, creation is multiplication is formal variable,  and you are tracking the probability distribution of states with n particles coded as monomial xn.  Anyway, so and so this is a birth-death process. And if you draw these pictures again, the birth-death  process chooses when to jump, when to perform transitions, either the creation or dilation.  And again, of course, we could imagine we now look at the picture in this in this Hopfer algebra.  And there you would track when a creation was followed immediately, or maybe at a later point,  but connected with the deletion. So it's impossible to put a measure on any possible  time point for this. But what you can do is you can put a box of time capital T, and then just  characterize the content of these transitions by exactly the classification into connected components.  So creation, deletion events, and events where you create and then delete. And so there's a little  trick. I mean, so far there is no Markov chain theory for the software algebra in that form  to be developed. But here, at least when you create, you can simply make this a different type of  molecule. And then when you delete, you can make sort of a little artifact produce some third type  of molecule. And this way, if you now track the numbers one, and you run all of this machinery,  in the end, you can indeed get a nice expression, which tells you a little bit more about this  dynamics, saying essentially, of course, you stabilize on a cross-off distribution, ultimately.  But you also see that in the time limit goes to infinity, you grow linearly with the number  of these create delete events. So that's the dynamics of the system. And I mean, this is by  no means a full theory yet. It's just like one very first motivation that presumably also Markov  chain theory, one can accept some interesting information, only that, of course, this key  particles have very little structure. And so I return to the question. So how could you now  approach this for the much more complex situation, where you're not describing just vertex graph,  but you actually describing graph and hitting all of this combinatorial complexity. And so just to  recall, the idea is now pretty similar. So we will mostly focus on first of all, classifying  combinatorially how to interact with writing steps. And sort of the ultimate goal would be to then  understand if you want to count, for example, triangle patterns that are produced or created  or deleted by such sequences, how can you how can you give a measure of likelihood on that?  That's the ultimate goal. And again, we would like to somehow find a way and this will be through  commutators, of course, to sort of drop out some of the possible contributions, which do not actually  influence the column. And the only sort of the real obstacle for this was that if you, I mean,  this is a perfectly valid sequence of events, but at first sites, what complicates this is.  So one could imagine these diagrams, the semantics will just be pairing of subgraph,  you know, before we just paired vertices to vertices in these other diagrams. But here,  it turns out it's a little more intricate because actually, it seems you're pairing half edges.  And that is sort of the real problem for this. I mean, how to come up with a concrete generalization.  And so after a lot of experiments, so I came up to formulations in categorical writing theory.  And so, I mean, this is maybe a little bit of an exotic theory, but the nice thing about it is  this completely formal. And it can immediately be implemented in algorithms. So I'll tell you a  little bit about this. And then afterwards show you how you can produce these sort of the analog of  these diagrams. Okay. So the full generality of this theory is quite intricate because in general,  you know, only look at graphical structures, but also some which are sort of constrained  additional structural constraints, for example, in trees. So and I mean, so this is a paper which  will jointly version Caribbean, which should be published soon in the applied category theory  general compositionality. And so I just want to show you a little part of the theory enough to  get the idea, hopefully. So before we had transformations were essentially specified  as partial maps between inputs and outputs of vertices, this generalizes now in. So we take  a category typically, it should have some some nice properties called adhesivity.  But sort of the important thing to notice that any, so for example, graphs or such a category,  we can have undirected graphs, multi graphs, and all sorts of variants of specifications.  But main point is, you take such a category, and you can formulate partial maps and quotation  marks as spans of monomorphism, so spans of embeddings. And so in the most general framework  on top, can put some conditions, which essentially constrict, constrain how you can apply these  rules. And again, to make a meaningful theory, we, I mean, this would in general be a class,  even for graphs, even for good categories. So we typically also have to quotient by some notion  of isomorphism as we had to do in the diagram case. Okay, but this is sort of the analog of  these little building blocks of the transitions that one can use. And sort of unfortunately,  then there's a little bit of, I mean, this is very general, it covers pretty much all of the  known case of such transformations, but for that, it's also a little bit technical. It is just to  say sort of the analog of finding an input pattern in a state X is now finding an embedding of I  into X. And then how to from that, you know, apply the rule how to unroll this transformation  depends on your semantics. And it's typically performed either in the double push out semantics,  or in the CESC push out semantics. And the first one is essentially trying to compute some notion  of set complement to get this object K bar. And the second one is using some construction called  final pullback complement. So I mean, in these graphic graphic theories, push out is typically  gluing together along common overlap, pullback is finding intersection of two objects, push out  complement is roughly like the set complement is just that, for example, if you have a graph,  and you just try to delete vertex, then it depends on whether this incident edges. So there's some  some difference. But other than that, it's roughly the idea of applying these steps as in the graphical  description. But so the key point is that there exists a notion from the pure rewriting theory  of how to compose two steps, how to how to interact with two rules. And so this is very much like  in the graphical language for these diagrams, you, you have two rules, and you try to find an  overlap of one of the output of the first rule, with the interface, the input, I two of the second.  And so this is again coded as a partial overlap. Now you glue together along this partial overlap,  and then you just run the rules. That's essentially what this says. And of course, there's some,  some technical details for how to do this with conditions. But I just wanted to show you,  I mean, I think intuitions are pretty good here, also for the graph case. So I'm now drawing the  diagrams from right to left instead of bottom to top to be more in parallel with this mathematical  notation. So here I'm drawing a picture where I take two vertices, I link them up with an edge  and make an additional vertex and an edge. And then I use one of the vertices to link to it,  to ask for it to an edge to the incident. And then, you know, delete that edge makes new edge  pattern. And so exactly how to interpret this diagram is to say, Okay, so I'm encoding here  a sequence of events where sort of the intermediate state, this is shape where you glue together  the patterns along the common overlap. And then you run essentially the one rule in the forward  direction and the other in the backward direction. And now this is exactly the analog of these diagrams  you saw before in this graphical case. So it's a sequence of events, which was produced through  interaction of these two rules. And sort of the overall input motive here on the very right top  bottom is seen necessary pattern you need to find in any state to apply the sequence.  So this is sort of the very first example of such diagrammatic calculus. So there is one  complication, maybe which prevented this from being useful from the start, because just graphs  are rarely very interesting. Normally, most of these applications, you want to have some more  structured species. And for example, one very nice species, I think, which is also good communication  grounds with the communaturist is, for example, planar rooted binary trees. And so if one wants to,  you know, formalizes in this context of writing, the first step to note is that you can write,  I mean, not only trees, the forest, of course, planar rooted binary forest, you can see them as  some type of graph, which are typed. So you can take the slice category over some type graph.  So every edge is given one of three exclusive types, either being left, right, or the bottom.  And so that in itself is sort of a first step. But then of course, you see, it's not only a  graph which is colored by these three types of edges, but also it has some a lot of structural  properties. And of these, I'm just listing some here. So for example, you never have two leaves  directly incident, and you never have two of these root edges and so forth. So I mean, this is not  very pretty. You can formalize this as conditions. But of course, this complicates a bit the story.  It can be done algorithmically. So at least that's a plus. And so in the end of today,  so transformations in this language you can do on trees can be made on can be formalized as writing  rules. But I mean, with the expense of those calculus, that is to be said, but in principle,  it is now completely formalized how to do these transformations. Okay. And now finally, so the  idea is how do you now get to combinatorics? So how do you get to calculus on these graphical  writing steps? And the idea is that so as in these pictures in the introduction, you want to reason  about all possible ways of applying and different transformation steps, where each step is chosen  from some finite vocabulary of possibilities, let's say, for example, vertex creation, vertex  deletion. So you want to classify the ensemble of all possible trajectories, let's say for a fixed  input state x zero. And the strategy which is very efficient is to first classify all possible ways  how these n steps can interact with themselves. So sort of the minimal context you need to fire  a given transition to apply a given sequence. And then as a separate step ask how many ways  are there then to apply the overall of such input to the state x zero. And so this is precisely this  idea of of tracelets completely formalized in this categorical writing theory now. Okay. And so  the idea is that if you again look, so this is now the analog of one of these diagrams from before  in the graphical setting, where you now have three steps there in the sort of shaded boxes.  They are drawn from right to left. And this wiring diagram indicates sort of a possible  sequence how these three steps could interact. And you can produce from it now this type of  tracelet. So the idea is that each of the wires codes for an overlap. And now for example, you can  take you can zoom in on the first two and compute how they interact. Producing this little subsequence  of events. And now you see that indeed the overlap of this third diagram with what you just produced  is now a proper graph. So there's no half edge or stuff like that. So you can glue together. And  you essentially just complete sort of the full sequence of events that is minimally coded in  these three rules as the bottom sequence. And it looks like this is and finally to obtain what I  call tracelets. I mean, it turns out the only information you need to retain is just the outer  hull of this diagram. So it looks precisely like just a sequence of transformation steps. But the  specialty is that it only contains enough information to permit the sequence to occur. Not  all I mean, this could be happening in a much larger context. And so this is exactly where you  gain something in the complexity. And at first sight, it seems this is sort of asymmetric. But  one can show that that it's equally possible to build up this trace that from the same diagrammatic  overlap structure, just in a sort of first computing what what how three and two overlap  and then also overlap this one. So that's that's sort of part of this calculus. Okay, so I mean,  it's just to say that these diagrams can be completely coded as just sequences of applications.  And on those now you can do a combinatorial calculus. So sort of the what's communatorily  interesting is that now you are given your vocabulary. So it's just the abstraction of saying  sort of the top parts are the colored bars are the individual rules. And so you can build up  all possible trace, let's say here of length four by recursively composing or iteratively  iteratively composing your letters. And so in each of these steps, of course, you can perform  analysis. So this is exactly this philosophy of the diagrams that you can you can reason  transitively so to speak on on compositions. Okay, and so I just want to briefly show how  this looks like in practice. So a trace that of length one is just, you know, the special case  where you have just one rule, it really only needs its input. So that's sort of the sort of the  trivial case. And sort of what I showed just in pictures looks in reality, like you are  inductively building tracelets of length n plus one from trace of length n and traces of length  one. And so it is just to show you that that even in the case of trees, this doesn't look pretty,  I admit, but it can be encoded in the algorithm. So I mean, this is a fully formalized structure.  And sort of one of the interesting features of these tracelets is that you have at the bottom  of this diagram, the sequence of steps. And so you can read out the composite effect of that sequence.  And in the language of the Sean Benson, these hope diagrams, this was exactly this operation  of evaluating the net effect, how this acts in the Heisenberg-Weil algebra. So this is now  generalized here as reading out the net effect of the sequence of transformations. So that is nice.  That's called an evaluation. And this evaluation is also compatible now with composition. So you can,  indeed, so the analog of diagrammatic composition is now composing these tracelets. And again,  it is only important to take an overlap of the output of the tracelet with an input of the next.  And so this is very exactly analogous to this diagrammatic composition. And finally,  sorry, maybe just to say, so there is now this operation of composing two tracelets,  drawn with this Vetch notation at the bottom here. And this composition, now, of course,  if we go with analogy, should be associative in some sense. And indeed, sorry, and indeed,  it is associative in the sense that just as in the Sean Benson case, sort of the number of ways  to wire two diagrams together and then with a third are exactly isomorphic to the number of ways  first wiring the third and the second and then the result with a first. So that is,  that is a property this structure has. Now finally, to put everything together.  Yes, now precisely, this aforementioned characterization that if you have a sequence  of transformations, you can equivalently count all possible ways of performing these transformations  by first counting the number of ways you can compose up these tracelets, and then together  with number of ways to applying them. So that's sort of, it's just to say this now exists for all  of these writing theories, including, of course, hope of case. Okay, and sort of the final piece  of the puzzle is then how to actually get from here to algebras, because we want to analyze  everything using commutator relations. One thing one has to do from the start, I mean, we already  had to go to isomorphism classes for rules to even obtain a set of equivalence classes.  And again, there's something like this also for tracelet. So sort of everything is constructed  with push-offs and push-off complements and so forth. So you have to quotient by isomorphisms.  Something less trivial is that so you do traces are slightly too large. I mean,  normally do not want to keep every bit of this information. In particular, you do not want to  keep like in this diagram here, information of order when the steps are completely exchangeable  up to the effect. And so that's called shift equivalence. And finally, there's one oddity  which is that simply formally the trivial rule sort of intuitively should leave a transformation  sequence invariant, but formally, I mean, it just produces an n plus one length sequence  with some repeated parts. So you can define an equivalence that simply quotients out by such  occurrences. And if you put all of these together, indeed, then oops, sorry, sorry, yeah, here,  you get to exactly now finally the construction of tracelet algebra. Basis elements are labeled  by equivalence classes of these tracelets under the aforementioned equivalences. And now the  product is precisely as in the digital pencil construction, the wiring together in all possible  ways, the overlaps in all possible ways of the tracelets. So this is now the tracelet algebra  product. And you can give an action of tracelets on states precisely by this aforementioned  trace characterization that if you have a sequence coded in a tracelet, you can apply the entire  sequence by finding embeddings of the overall input into a state. And this gives you a representation  of this algebra. So the theorem here is that not only does this tracelet give rise to an  associative unit of algebra, but moreover, this row is indeed a representation, which means that you  can, if you if you now want to do combinatorics on numbers, if you want to count number of ways  of applying writing sequences, you are free to first you can partition your problem by this  bottom right outcome of the equation into first trying to characterize the number of overlaps  of tracelets, which is very advantageous, because now you can use relations such as  commutators and so forth. So this is sort of the final outcome. And I then realized while  preparing the talk, this was already much too much information. So let me just conclude on  reproducing the special case of discrete diagrams. So here you see on the left, the  elementary sequence of creating a verdicts and deleting a verdicts. This is how it looks like  in a tracelet. So this is the analog of the diagram of this generator E and the combinatorial  hopf algebra. And on the right, you see exactly why you need this shift equivalence, because  I mean, creating a verdicts and deleting, I mean, forgetting and taking equivalence classes the  same as, you know, in quotation marks first deleting and then creating. And this is in the  tracelet language exactly obtained through such an equivalence. So it's not, you know, a trace  value, but we label the elements of algebra by equivalence classes. And finally, then,  indeed, the type of relation you can then derive are precisely commutation relations. And those are  key to the analysis in these combinatorial arguments. Okay. And I mean, I wanted to speak  about planar-rooted binary trees, but I think I'm out of time. So it's just to say that in  these calculations, and I also showed this last year at CAP, one can now start to see  combinatorial simply counting arguments on why certain commutators have the form they do. So  that's sort of the main motivation for this work, but I do realize I'm out of time. So it's just  to say sort of one of the, so this will be a forthcoming paper for beginning of next year, and  one of sort of one of the cases will be an explanation of why sort of, you know, you see  certain commutator structure in these planar-rooted binary tree computations. Okay. But so let me  conclude to stay in the time limit. So I've given you a quick tour of a new concept which I call  TraceLed, which is intended as a generalization of this Dishon-Penzen et al. construction of  diagram algebras. It seems to be very useful. I mean, I have a sort of, I'm developing an  implementation of this with an Z3 SMT solver, which is available online, but I mean, this is  work in progress. And sort of the mathematically interesting question is maybe whether you also  have some Hof-Algebras structure on these TraceLeds, which for some cases I know one can  demonstrate, but in general is sort of a research question. And sort of the long-term goal of this  work is to bring combinatorics also into these bio and organochemical reaction systems, which  through the TraceLeds now will boil down to enumerative combinatorics on TraceLeds. Okay.  And with this, I would like to thank you for your interest and thanks a lot for your time.  Thank you very much, Nicolas, for this is, are there questions?  Yes, I have one question. What about directed graphs? Do your construction applies to this?  Yes. Sorry, I just showed underrated for the sake of the diagrams. No, no, so it applies for any type  of graph you can formalize categorically. So even for simple graph or for multi-graph or for  hypergraphs or for attributed graphs and so forth. No, it was just for the pictures. So,  directed graphs in particular are appreciative. You know, you can formalize them as appreciative.  And so any adhesive category, so any appreciative gives you an adhesive category. So that's,  yeah. Okay. And in particular, for the directed graph case in this paper, in this archive footprint,  we even have the Hof-Algebras structure. So may I ask a question? Yes. Yes.  Other questions? Yeah, I have a question. Yes. Nicolas, could we describe the construction  of a penrose tiling with your graphs? Is it possible? Aha. It's like simple chemistry,  after all, but the dimensions. So, I mean, the only question about this is whether you can  generate your tiling in some process which only asks local information. So, I mean,  I'm not quite familiar with penrose tiling, but I mean, if you can describe.  You have just several tiles of forms and there are just rules, quite simple rules,  how to glue them together. Interesting. Yeah. Yes, so that sounds very much like an example  you could study. Yeah, I think all quasi crystals, because it is much more interesting than crystals,  because they are, of course, periodic, but this is up-periodic. Yes. So one of the things I could  imagine is you might, for example, ask in such a quasi crystal, what is the average occurrence, say,  of particular subspecies? No. But probably with your technique, it may be described in a very  simple way. Yeah, so in particular, if you have a model for how these local manipulations are  fired. Of course, of course. Yeah. I have a little comment on this, because all of your stuff,  it's like it's really chemistry, it's bullion, it's no special structure. And of course,  in penrose tiling, you get things, like a plane. And in principle, one can, for example, there are  these universal equations equivalent to universal Turing machines, you do dominos in two dimensions,  like square tiles, and each side is colored by some color, and then you try to build things  since it's a universal Turing machine. Of course, you can start to glue things, you get some pieces  of planes, so you get eventually some kind of space like structure, abstract story. Yeah.  I mean, one of the things we tried was, for example, polygons, so you can have these sort of  tissue models, which are simply just polygons glued, but I mean, without gap in the plane,  and you can sort of divide the polygons, you can sort of insert triangles by expanding vertices  and so on. And this is also in the realm of these writing techniques. And so you can use it in,  there is a problem with capsid, you know, the viruses, which have icosahedral symmetry.  They are surrounded by capsids. These are just the coat proteins that come together,  and they have five-fold and six-fold symmetry. And of course, they build these capsids, and again,  there are very simple rules that make them glue together. Yeah, I mean, so the main motivation  here is that for these types of theories, if you sort of the motivation is if you have a  good, if you have a problem, which you think is of this nature, it is relatively quick to check  how to formalize it into the writing. So I mean, it's essentially to see whether you need more  information than what you can expand in a small local neighborhood of the rule or not. If not,  there's a very good chance you can. So what you get in quotation marks for free are, for example,  commutation relations. So you can, for example, ask the pattern, the counting of the pattern is  implemented as just an identity transformation of that pattern. And so you can, for example, ask  how many of the occurrence do you have more or less before, after applying this transformation?  And so these commutators really carry a lot of information. And it's much smaller information  than if you were to analyze the entire structure. So that's sort of the main, you get sort of average  information. Yeah. Are there questions, remarks or comments? Can I ask one little question?  Have it been studied which information on the solgebres are carried by some homologies of the  solgebres? Like, hoaxed homologies or whatever, something like that. I mean, I mean, these  algebras have not been even written down. So, I mean, so the question is, I mean, the answer is no  from my side. I would not be surprised if you look at special examples, maybe you could, I mean,  it would be more that you recognize the trace lip stuff in it rather than, I mean, I don't have a  good answer. Sorry. I think it's a good question, especially because last year at CUP, we had this  nice talk by, we had this nice talk about graph homologies or homologies, I forgot,  which also could be formalized through writing. So, I mean, if you have, actually, if you have a good  case, which you think it could be of this elementary graph like structure, I would be very  interested because I would like to play with it. Yeah, it should be possible in general kind of  point of view, saying that they should carry some information.  Interesting. No, I would be very interested if you could have a good sign, Yelin, maybe.  Okay. Thank you.  Any other questions, remarks or comments?  Oh, thank you, dear Nicolas for this amazing and huge expansion of our paper with  Favère Boisac and Carol Penzon. And I have a small question. Are your variants graded? Do you  have each time some way to count vertices or edges so that?  Yes, exactly. So, the reason these formalized rise, that I mean, so in my formalism, they are  filtered. So, it seems as though in principle, you could count the occurrences of these connected  components in some system. So, I mean, if you give grade two to the, you know, this e-pattern,  which consists of two subdiagrams and grade one to each of the others, they will give you a grade.  But the thing that generalizes is the filtration by essentially the cardinalities of the interfaces.  And then this through this filtration, I mean, always when you compose, you exhibit a little  bit less of the interfaces to the outside world. So, that decreases the filtration degree. So,  I mean, it's clear that this by composition gives a decreasing sequence. I mean, at most,  you have the same filtration degree if you don't connect overlap, because it's simply the sum of  the overlaps of the interfaces and otherwise it decreases. And this happens to be compatible  as a core product. So, yeah. Ah, okay. Thank you very much.
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