 Okay, so it's time to stop. Let me introduce the today's speaker, which is the William DeMau of Charles University currently. He will talk about the homomorphism problem for Boolean structures. So I'll see all yours. Okay, thank you. Can you hear me okay? Yep. Okay, great. So today I'd like to talk about this problem we've been working on, homomorphism problem for Boolean structures. So this is a general homomorphism problem, not just for relational structures, but for structures that could have relations and operations. And I first want, before I get into this, I want to thank the organizer, Jakub. You've done a great deed organizing these lectures, because this is really amazing. It was my first Zoom lecture ever. I'm very excited. Thank you for inviting me to do this. And also I want to thank my colleagues, Libor Bardo and Antoine Moté, because they have contributed significantly to this work. And without them, I'm not sure I would be able to do it. And well, maybe I would, but it would take many times longer. So I really appreciate working with them. Okay, so let's get into this. What is this problem we're talking about? So we've managed to prove this new dichotomy result for general structures. Well, I'm not quite general, but for Boolean structures. So if you consider an arbitrary finite signature S, so S could have relation symbols, as well as operation symbols. And then if you take a Boolean structure B, the problem of deciding if you're given another S structure A, is there a homomorphism from A to B? This is either NP or NP complete. So maybe it seems like, well, that's already been done by Schaefer, right? This dichotomy has already been proved by Schaefer for Boolean structures as well, not only for Boolean relational structures, but what we've done is extended this to arbitrary structures with relations and operations. So let's talk about the background of motivation for this. So we're trying to study a generalization of the finite template CSP over an arbitrary finite structure. So where we allow both relation symbols and operation symbols in the signature. So this is generalization of Schaefer's dichotomeric theorem for relational Boolean structures from relational Boolean structures to arbitrary Boolean structures. So we've only done this so far for Boolean structures, that's why we're calling it a generalization of Schaefer's result. But still, there's no general theory for CSP over arbitrary structures, right? So what we've done so far is only for arbitrary structures, assuming that the template is Boolean. So the universe only has zero and one. Okay, so the main results that we have is a generalization of Schaefer's dichotomies to structures with relations and operations. And the theorem that we proved is that if B is a Boolean structure with finitely many relations and operations, then the CSP of B is NP or it's NP complete. And to say this more precisely what we actually proved, we'll need a definition which most we are probably familiar with. So we say that an n-ary function is essentially unary if there exists an I possibly constant function g such that this, we have this equation holds, right? So it just means that either f is constant or it depends on one variable of its arguments. Okay, so right, so then the theorem that we proved is that if B is a Boolean structure, then the CSP of B is NP if B as an operation that is not essentially unary, or there exists a polymorphism of the graph of B, which I'll explain what that is in a minute, that is constant or not, not essentially unary. And if all functions in B are essentially unary, then the CSP of B is equivalent to the CSP of the graph of B. So I'm not sure if everyone's familiar with what this graph of B is, but let me just try to explain what that is. So we have this structure B, whoops, okay, let me try that again. So B is the structure with some universe and some operations and relations maybe. And then the graph of B which I'm calling script G of B, that's the structure with where the universe is the same. But the operations, well, it only has relations. And the relations are the relations of B, as well as the relations you get from viewing the operations as relations, right? So if you take an operation F, let's say, and it goes from B to the N to B, then we will instead view that as a relation that is a relation in, so we'll view that, we'll view the relation of F as a relation in B to the N plus one, where you have N two pulls of B that you plug into F and then you get some output and that output is the N plus first element of the of the two pull that's in the relation. Does that make sense? I'm not sure I explained that very well, but in any case, what we're doing is we're, when you view an operation as a relation, that's what we're considering the graph, this graph of B. Okay, I hope that's clear. If it's not, please feel free to stop me. Okay, so let's move on. And so let me explain how this result is a bit stronger than what I stated. So what did I state? I said that we have this result that the CSP of B is either in P and B complete for these Boolean structures, but actually we've proved something a little bit stronger, which says that under the above conditions on B, which is that B is this Boolean structure, so it's the universe is zero and one, and it has finite, so finite, finitely many operations and finitely many relations, and these are finite parity. Then in that case, there's at most a polynomial number of homomorphisms from each instance of CSP B to B. Okay, so not only do we know that it's in PR, NPR and B complete, but also we can put a bound on the number of homomorphisms. And we can effectively enumerate these. In fact, that's how we prove that this is in NPR and B complete. So that gives us this polynomial algorithm for CSP of B. Okay, so what has been done already in this problem? Of course, a lot of work in CSP has been done for the constraint satisfaction problem for relational structures, but not much has been done for structures that have relations and operations. The only work we know of is this work of Federer and Madeline and Stewart, this paper from 2004, dichotomies for classes of homomorphism problems involving unary functions. And they show that purely algebraic CSP generalizes to the relational one in the following sense. Each CSP of a finite relational structure is equivalent to a CSP of a finite algebraic structure, module polynomial time during reductions. So in other words, they look at relational structures and they say, oh, this is actually equivalent to if you made this into an algebraic structure. But module polynomial time during reductions, not many to one reductions. So that's one caveat. But we will show that in general, there's no polynomial time reduction from CSP of the graph of B to the CSP of B. But I'm going to use this abbreviation for the duration of this talk. FMS refers to Federer, Madeline and Stewart, and they prove that they prove this reduction from the CSP of the graph of B to the CSP of B when all operations of B are essentially unary. Oops, sorry about that. Okay, so their theorem, which I'll call the FMS theorem, says that for a structure B whose operation symbols are at most unary, the CSP of B is equivalent to the CSP of the graph of B by a polynomial time menu one reductions. Okay, so for this one, they actually achieve this menu one reductions result. So this is a really nice result. But the serum doesn't extend to structures with operations of higher arity. So only when as I mentioned here, oops, sorry about this. Yeah, so for these structures where the operation symbols are at most unary, they have this result. But if you consider structures with operations that have higher arity, this result doesn't doesn't obtain. And we'll give an example where we have just a two element algebra with a single binary operation for which the result fails. Sorry, Graham, can I stop you for a minute? There is a question from Katarina. She's asking about the difference with the value setting. Is that an operation can output a Doppler rather than a single value? Is that correct? Can you repeat that question? The difference with the values, I think that is that an operation can output a topo rather than a single value? Okay. Should I ask a question? Can you can you tell me what is an operation exactly? Sure. Yeah, well, what's the difference with that with a function? This was my question. Okay. Yeah. So an operation, I mean, we're only considering if you have so an operation on, sorry, this is my first time using this, an operation on a set B is just a function from some power of B to B. All right. So, okay, okay. I got, okay. Not a general operate. It's not a general function. An operation has domain, some power of B, an operation on B has domain, some power of B, and codomain is B. Yeah, sure. Okay. Thank you. Okay. Thanks for the question. Okay. So shall we resume? Okay. All right. So I'm going to go through some background now. So the the border between P and NP complete in Schaefer's dichotomy theorem can be described in terms of polymorphisms. For a binary relational structure B, CSP of B is in P, if B has a polymorphism that is constant or not essentially unary. And I think did I already define essentially unary? I don't think I did. Sorry. I want to make sure that I did. Yes. Okay. Good. So we covered what essentially unary means. Okay. Okay. And otherwise it's NP complete. Okay. And the discovery of the role of polymorphisms for relational CSP as obviously had a big impact on CSP in general, as you can see in some results that I'm citing here. And these made possible to conjecture this borderline between P and NP complete CSPs over non-billion domains. And these were important for generalizations to infinite domains and promised CSPs, value CSPs. And we don't have analogs of any of these results for the non-relational setting. So as soon as you put in some operations and your structures, the theory is still wide open. So I think this opens up a lot of interesting questions for future work. Okay. So let's talk about some reductions. Here's a fact that a map from a structure A to B is a homomorphism, if not only if it's a homomorphism from the graph of A to the graph of B. This is a very easy to prove result. And therefore you get that the CSP of B is reducible to CSP of the graph of B, where the reduction is a map from an instance of the CSP of B to the instance of you just take the graph of the instance. Okay, that's sort of obvious. And so just to review some definitions that we're going to need in the lemmas that we use in our proof, quantifier-free primitive positive formula in a signature S is just a conjunction of atomic S formulas. And two S structures in B over the same domain are called term equivalent. If for every operation of the first structure, there exists a term, an S term, such that this operation is equal to that S term for every tuple. Okay. And conversely, for every operation of the latter structure, you can find a term such that this operation is equal to that term. Okay, so we're going to use this term equivalence in our lemmas that I'm about to go through. So the first one is that... Can I distribute for a second? Yes, sure. To the term equivalence of these structures, it does not say anything about the relations. You just, you forget about the relations and you just compare the algebra. Yes, it's basically just about the operations. Okay, thanks. Okay, so if you have a signature and you have some structure in that signature and you have some finite set of identities in that signature, such that B models all over those identities, then for every instance of CSP of B, we can compute in polynomial time an instance, Y, such that Y models those identities. And the instance X has a homomorphism B if and only if it has a homomorphism, if and only if Y has a homomorphism to B. Okay, that's the first result we'll use. And the second is that if you have two structures that are term equivalent, and you suppose that every relation of A has a quantifier free primitive positive definition in B, then CSP of A has a polynomial time menu on reduction to CSP of B. Okay, so does this make sense? Okay, as a corollary, we retain this algebraic invariant for complexity of CSP of finite structures. So recall, a partial operation is a partial polymorphism. So what a partial a partial operation is, it's just this operation that's defined on some parts of the domain, but possibly not all of the domain. And it's a partial polymorphism if for all relations are and for all tuples in the relation. If you plug those tuples in to this function H, wherever it's defined, then it's in R. Right, so it doesn't have to be defined everywhere, but if it is defined, then it belongs to the relation. And of course this, I'm like expecting it's assumed that this evaluation of H on all of these tuples is evaluated component-wise. Right, so H is an n area operation or an n area function from R to B and R to the end of B. And then we've got these tuples in R. And when you plug them into R to H, you just take it component-wise. And the result, if it's defined, has to belong to R. Okay, there's this result of Romo from 1981 that says, if you've got two finite relational structures such that every partial polymorphism of B is also partial polymorphism of A, then the relations of A have a quantifier-free P definitions in B. Okay, so we're going to use this result. And the corollary of this is that for a finite structure B, the complexity of the CSP of B depends only on the clone generated by the basic operations of B and the set of partial polymorphisms of the relational reduct of B. So the relational reduct is just throw away the operations and just consider B with its relations. Okay, so in the Boolean case, the complexity of CSP depends only on the clone generated by the basic operations of B and the total polymorphisms of B. So we don't have to consider the partial polymorphisms in this case. Sorry, the total polymorphism of the graph of B. Okay, so this is invariant is weaker in the sense that it contains less information than the previous one. And indeed, every polymorphism of the graph of B is a polymorphism of the relational reduct of B. So in that sense, it's weaker. So it's unclear to us whether this weaker invariant is enough to separate the tractable and the NP-complete problems for structures with bigger domains than the Boolean case that we've so far considered. Okay, so let's get to the main theorem that we want to show here. And that is the dichotomy theorem for Boolean structures. So if you have a Boolean structure, then the CSP of B is in P. If B has an operation that is not essentially unary or the polymorphism, the polymorphism clone of the graph of B has a polymorphism that is constant or not essentially unary. Otherwise, it's the CSP is NP-complete. And the proof relies on this classic result of post, which is that if you have a clone of operations on 01, if this clone contains operations that an operation that is not essentially unary, then it contains one of the operations meet, join, majority, or minority. Okay, so let's go through the steps of the proof. They're fairly straightforward. So we won't get too technical. So I hope you'll bear with me. Okay, so if the Boolean structure has an operation that's not essentially unary, then by post theorem, the clone generated by the operations of B has either a semi-latest majority or minority operation. And by lemma two, which we already went through, we can assume that B has such an operation in its language. And we prove that the presence of such an operation puts the CSP of B in P. So it will be tractable. So in fact, we prove a stronger result in this case for each finite instance X, there will be at most a polynomial number of homomorphisms from X to B. Okay, so I'll go through each case. Maybe I'll go through the first couple of cases and you'll get the idea and we won't have to cover all of them. But there's only a few cases. So let's begin. So the first theorem is that if we have this Boolean structure, notice we've got this structure here. It's on the universe is zero one. And we have S is some operation. And then we might have other operations and relations. And in this case, we're going to assume that the operation is this meat operation, or maybe it's this join operation. If the structure has either of those, then the CSP of S structure is in P. And that's where we're going to show now. Okay, so if you fix a structure as an instance, then this structure X satisfies, you can find this structure X prime that satisfies all the equations that the original instance satisfies, such that if you have a homomorphism from X to B, then you also have a homomorphism from X prime to B and vice versa. So this is I probably let me let me try to say that again. So if you have an instance X, you can find an instance X prime that satisfies all the relation or all the identities that be satisfies, and such that there's a homomorphism from X to B, if not only if there's a homomorphism from X prime to B. Okay, and by lemma one, we can assume that that S, the S above, sorry the S, so let me make this clear, we can assume that the S interpreted in this structure X is a semi lattice operation because of this fact that you can find you can find a structure that satisfies the identities that B satisfies and has a homomorphism to B, if and only if the original instance has a homomorphism to B. Okay, so if we define this lesser equal, did you have a question? Okay, X lesser equal to Y, if and only if X is equal to S of XY, then for all H from A to B, the set, this inverse of one, the inverse under H of one is a principal filter in in this structure X lesser equal, right? So a principal filter is this upward closed set that's closed under S. So we can see that by saying that by seeing that if X is in the inverse image of H of one under H and X is less than Y, then H of Y is, well you can see these equations, right? So what we show is that each homomorphism, if you can find a homomorphism, it corresponds to one of these principal filters, and of course you have at most the number of elements in X, many principal filters, so there are most that many homomorphisms from X to B. Okay, so this is the idea is that you consider, if you have a homomorphism from A to B, and then you consider its inverse image, so it takes some, it takes some elements to one, and then you consider the inverse image of one, and then this is a principal filter in this structure X with this lesser equals, and so if you want to count the number of homomorphisms, all you have to do is consider the number of principal filters, and of course this is bounded by the number of elements in X in the domain. So this is how we know that we can find a bound on the number of homomorphisms from X to B. Can I have a question? So you are using this reduction here, I assume that this reduction doesn't change the number of homomorphisms significantly, going from X to X prime, is this correct? That's correct, so yes that's a good question, so we're assuming as I think this is Alex Kozda who's talking now. Yes. Yes, so that's a good question. So what we're doing now is going from an S structure S, X, so that's the input instance, and then we want to now work with the structure X prime that satisfies these identities, and Alex's good question is well if you start working with X prime, then now you have to know that you haven't changed the number of homomorphisms, and yeah that is correct. We have verified this and I'm sorry I can be a very clear answer why this is so, but maybe someone, one of my co-authors could come to my rescue at this point and tell me. The point is that X prime is obtained by factoring X by a congruence, and it's a congruence that satisfies that every homomorphism factors through this congruence, so that you don't change the number of homomorphisms at all. Like the kernel of every homomorphism from X to B would contain this congruence that we factor with, so that's the number, so that there is a bijection between the homomorphisms from X to B to the homomorphisms from X prime to B. Thank you. Thank you very much Anton. This is my co-author Anton. It's explaining that if we consider homomorphisms from X to B, and then we have this other X prime to B, and there won't be more homomorphisms from X prime to B than the real way from X to B, so yes thank you for clarifying that for us. Okay good. So right, so we do similar things in the majority case where instead of having this semi-ladis term, we have a majority term, which is some term that satisfies these identities that most of you are probably familiar with, and permeate these inputs, and we do similar things with this, and things seem to work out fine, right, so we fix a structure with a signature, and we have M that's a ternary operation, we're not sure what it is, but again we can assume that we have the input instance satisfies the same equations that the template satisfies, so we can assume that M is indeed a majority operation, and it satisfies these equations, and using these facts we're able to show that you have a bound on the number of homomorphisms, again by the fact that you have inverse images of these homomorphisms, must be principal filters in the lattice, and so there's only so many that you can have, right, because there are only so many principal filters that you can have, in fact that's bounded by the number of elements in the domain. Okay, so that's the idea with the majority case, and the affine case is similar, what I'm calling the affine case, if the operation is x plus y plus c, you got a similar result, so okay, so the proof of the main theorem is that if you've got b, a Boolean structure with finitely many relation and operation symbols, then CSP of b is in p or in p complete, and the proof is first, if you, the first case, if b has an operation that is not essentially unary, then CSP of b is in p by these, sorry lemmas, these were lemmas one, two and three that I just showed you, and then if b has only essentially unary operations, then b is term equivalent to b prime, whose operations are all unary, and the CSP of b and CSP of b prime are polynomial time equivalent by lemma two, and then the theorem of Federer and Stuart, and I'm forgetting the other author, Antoine, do you want to help me on this? All right, so the paper by Federer at all is showing that the CSP of b prime and the CSP of the graph of b prime are polynomial time equivalent, okay, so by Schaefer's dichotomy theorem, the CSP of the graph of b prime is in p if the polymorphism clone of g of b prime contains an operation that's essentially, not essentially unary, and otherwise it's np complete, okay, so finally the clone, the the polymorphism clone of the graph of b prime and the polymorphism clone of the graph of b are the same, and that yields the final result, okay, so I'll show an example now where this is the example that we were struggling with for a while to try to show that this result of, of, oh, Madeleine, Federer and Madeleine and Stuart doesn't doesn't apply in general if you if you take structures with operations that have higher arities than than one, and so this structure, we have we found a structure where the CSP of the graph of that structure does not reduce to the CSP of the structure, and it's very simple, but it took a while for us to prove, or at least for me, I think maybe my co-authors could have done it much quicker, but so if you take this structure that's just on zero one, and you have this binary operation where it's, I guess this is called the shepherd stroke, I don't know if that's what it's called, but it's it's very easy to describe this is the operation table, it looks like this, so zero times zero is one, and otherwise, oops, sorry, this is a mistake, okay, let me try that again, okay, so this operation is one only if both arguments are zero, otherwise, the the result is is zero, okay, so if you consider the relation induced by this operation, what do you have you have these triples zero, you have zero zero one, right, this is this first tuple in the relation, and then zero one zero, right, that's the second, so you construct the relation from this this operation, and you get this, and it's clear that that none of these operations, the the join meet majority or minority is a polymorphism of this of this structure, this relational structure, right, okay, and of course the constant map into zero one is also not an endomorphism B, so the CSP of the graph of B is NP hard, right, and on the other hand this operation, this operation, if you take this operation, x, y goes to x dot y dot x dot y, that's exactly the join, x join y, and okay, and so the the results we've gone through so far imply that this will be actually tractable, so the CSP of this structure is tractable, and yet the CSP of the graph is not tractable, the graph of the structure is not tractable, okay, so let me move on and in conclusion discuss some open questions and future work, of course this is brand new stuff and we haven't really even sketched the surface of CSP for general structures, I mean the the initial question that I was interested in because I'm I'm not really a specialist in relational structures, but I'm more a universal algebraist, so I deal with algebras with just universes and operations, and so for me it's very interesting to know when when do you know you have a homomorphism from one algebraic structure to another, and that was the initial question that got me interested in this, but then in working with Levor and Anton it's clear that when you throw in relations you have an even more general and more interesting question, when are there homomorphisms from these general relational structures to to another general relational structure, so this obviously we haven't we haven't made very much progress, all we've done is we've managed to to answer this question for Boolean structures, but of course there's a lot of open questions when you consider bigger domains than Boolean, and so let's discuss a couple of these, and so this raised a number of questions for for developing in general theory of CSP of arbitrary finite structures, not just relational structures, and the reduction from CSPs of relational structures to CSP of algebras in Federer, Madeline, and Stewart is a polynomial time-turing reduction, and we'd like to know uh does this generalize to or can we can we get a polynomial polytine many one reduction instead of just uh the stirring reduction, that's the first question, and then another question is that we showed for a finite structure uh be the complexity of CSP of B depends only on the clone generated by the operations of B and partial polymorphisms of the relational reductive B, and maybe the complexity of CSP of B is captured by the polymorphisms of the relational reductive B, and that's something that is unclear at this point, and finally for the relational for the Boolean dichotomy of above that we that we laid out, the main step is establishing that as long as B has an operation that's not essentially unary, then for all instances A there are only polynomial many more because it's from A to B, right, so it's generally unknown what other properties of finite algebras imply such polynomial bound, and that that would be an interesting question to resolve, okay, so there are also some more questions relating to um width and um relational width and and so forth, but I didn't discuss the results in our paper that have to do with that, so I think I'll just leave these on the slide and you can consider these if uh if that's what you're interested in, so okay, and I'd like to invite now my uh co-authors to give any comments or input that they would like to because I think they they might have noticed some areas that I that I was uh not very clear on or that I lacked uh sufficient um insight into that they would like to clear up, okay, thanks for the talk, thank you, there's some comments, questions, remarks, corrections, I'm sorry, what happens if you have a purely a operational structure, so a homomorphism for algebras? Right, so do we know the answer or anything? Yeah, that's a good question, um in fact the the in the federal medallion and stewart art uh paper they they proved that in fact this is more general than the relational problem, so you can represent this relational structure uh can be reduced to a csp of algebraic structures, so this is somehow more general or more difficult than the relational structure csp. Don't they also give a reduction in the other direction showing that the csp of an algebra reduces to the is equivalent to a csp of a relational structure up to turing reductions? Yes, so somehow it's the same pure relational structures and pure finite algebras, yes, but again their their paper is mainly only considering these structures with unary operations, right, so in this case they show these are equivalent. No, so sorry, do we have the economy or not? That's somehow a bit confused. A dichotomy for algebras? In this case where we're just talking about Boolean structures, yes. No, in general. No. What do we know in general? I'm confused as well. I'm confused as well because I thought there is no reduction, like it's not known whether every algebraic csp is equivalent to some relational csp. Yes, that's correct. Only for unary algebras is what you said. For unary algebras, yes, so every relational csp is equivalent to an algebraic csp where you only need like two unary operations. Yes. The framework of just pure algebras can be marginal. Okay, so it's open, is that what you say? It's open, yes. Okay, interesting. And what does the dichotomy look like for Boolean algebras? That should be somehow like for Boolean algebras. Yeah, for Boolean algebras it's simply if there is an essential operation, then it's easy. That's already proved. And if there is none, then it's also easy, right? Yes, it's always easy. Yes, yes, it's always easy. Okay, well, then yeah, it's a good question whether there is a reduction going the other way. Yes. Because I suppose this paper of Federer and others, it's from the times when... It's prehistory, it's medieval age. Yeah, essentially if you want to say you can forget about this problem, you can prove that csp dichotomy reduces to it, and that's that. This means you can forget about it, it's too difficult. So maybe they didn't even think about the other direction. Okay, interesting. Maybe let me just comment, because William told me, told us on this last slide, can you put the last slide please, William? Yeah, no, no, the last this one. Yes, there are some questions about width. So I just want to clarify why these questions are there perhaps. So for relational structures in general, we know that if it has bounded width, then it has relational width to three. All the algebraic things, pure algebraic things are solvable actually by local consistency. So they have bounded width. But the width is strange, and they do not have provably width to three. So it's also one thing which is in the paper. So there is no collapse of width, at least not that brutal as for relational csp. Yes, thank you for having that. Some more questions? I guess it would be interesting to have a specific example of an algebra for which the csp is open, the complexity of the csp is unknown. You know, some three-element algebra something. It would be good. Yeah, I mean this is, yeah, renewed to me, this question is new. So we just managed to do it for just Boolean structures. And now from here we can consider other more general, hopefully more general results. Some more questions? So by the way, your deposition side of your dichotomy says that you always have a polynomially many insolutions. Is that it? Yeah, there are most polynomial many homomorphisms. Well, no, that's not correct. It's only in the case where the algebra has an essential operation. Because there is, I mean, the relational case is part of this, right, the purely relational case. And it's not true that you only have polynomially many homomorphisms. Well, not in general, but in this one case where it's tractable. Well, there are some tractable cases that come from the relation or purely relation. The thing is that you have polynomially many homomorphisms provided that your target algebra has an essential polynomial, an essential operation. Okay. So it's an example which has many solutions. I think there are some results in CSP about the structural restrictions, which classify when you have polynomially many solutions to something like this. So I have a question about this actually, because it kind of feels like when we are talking about Z2 modules or vector spaces, then we should have many homomorphisms. I'm not quite sure, but you're saying that it's polynomially this case too, when it's algebra with some Maltsev operation, let's say, and we are mapping one to the other. It's still polynomially many. Well, there is only like one Maltsev algebra here, right, which is Z2. And so the number of homomorphisms between vector spaces is very small, right? It's determined on the basis. Okay, this Z2 example is actually cute because Z2, like relational Z2 is very unbounded with, right? And the Z2 algebraic is bounded with. But it has polynomial many, but we don't know in general, for example, for Maltsev algebras. I don't think we know it. Yeah, we have, Kompatcher mentioned that from some results, it is actually true for, I don't know, nil-pult and super nil-pult and something like that, or a billion at least, but even Maltsev in general is open. So I begin to see why it's not exponential. It's because the input algebra is not given, like 0, 1, 2, 15th power. It's given by a list of elements. So I cannot go exponential. I guess that's a yes. Yes, that's correct. Yeah, yeah. But like even if you go 0, 1, 2, 15, 15th power, then there is only like 2, 2, 15 of them? Yeah, okay. Yeah, yeah. Yeah, which is linear. Yeah, sure. Okay. If there are more questions, I want to thank my co-authors for joining me in this project. I think it was really fun to do and they've made it a pleasure. It was really good working with them and they're both Antoine and Libor are great people to work with. So thank you very much. I'm sorry if I didn't do the work justice in this talk. Okay, I guess that's all with the questions. So thanks again.