 The button. Okay So, hi, hello everyone Today's talk is by Barney Barney Martin We he will talk about the quantified CSP and the future of the channel conjecture Okay, thank you, Jakob. So I'm not really going to present anything new today but I'm more going to talk about how I Could imagine the future future work on quantified CSP so I'm going to give a sort of personal and historic background to the current situation Before talking about some of the monsters that zhuk has made Here's the the opposite of st. George. St. George is killing the dragon Dima is building the dragon But also there are other people involved Backwards in this work like of course Hubee and Flawan Catarina And so instead of talking just about the demise of the chen conjecture I want to talk a little bit about the future of working QCSP So normally when I'm giving this talk I start off with a definition of the CSP that probably I don't need to do that for CSP seminar But people who are working in the CSP Maybe have no Close connection to the QCSP. So I will explain to them that the QCSP is very like the CSP in its logical formulation But with the universal quantifier Somehow returned so instead of the logic being just exists and an equals The logic is for all exists and then equals and this version of the Constraint satisfaction problem this variant of the constraint satisfaction problem makes sense Makes sense logically doesn't make sense so much as a combinatorial problem like the homomorphism problem It's a generalization of the CSP very much in its logical formulation So I was writing some Survey about QCSP CSPs are ubiquitous in computer science while quantified CSPs Cannot nearly claim to be so important in applications This is probably true. I actually remember When I first heard Manuel Say that the CSP is the conics problem because of its connections to Conatorics to logic to algebra, but also many many natural problems are phrased as CSPs Whereas with QCSPs, there are natural QCSPs, but they Tend to be Boolean the Boolean one the the quantified Boolean formulas is a very natural QCSP the other the non-Boolean QCSPs are somehow not so natural because there's no way that you can restrict the constraint language and Produce relativization of the universal quantifier Of course, you can reduce relativization of the the existential quantifier by adding unary relations But because the formula is still conjunctive you can't relativize the universal variable and actually the version of QCSP where you do relativize both The universal variable and the existential variable that I hasten to add is not a QCSP in the sense that I just described has already been classified so in fact the classification for this relativized QCSP Has been done by by Hubean Manuel and the classification for the Boolean QCSP was done. Well, it was alluded to in Schaefer but the the algorithms and the removal of constants was elaborated over a number of papers subsequently And note that in the case of the CSP the relativized CSP is exactly the conservative CSP But the relativized QCSP is not the conservative QCSP because in the relativized QCSP You really can relativize the universal quantifier. You can't do that with unary relations. What I call the conservative QCSP so I said in this some I Said in this survey that what is left of the the true non Boolean QCSP is a problem I believe to be mostly of interest to theorists and I would probably stick to that What I didn't realize at the time that I was writing this is how interesting it would be to theorists. I Think it is a very interesting problem from a theoretical perspective very interesting and very mysterious unfortunately So Nonetheless finite domain QCSPs have actually been heavily studied in the literature and for a long time Have been known to reach complexities P NP complete and P space complete So there are also possibilities of complexities within P but for the purpose of this talk I will not consider anything within P so actually one can consider like a really big class of problems basically these are the problems of model checking so the CSP is a model checking problem where you fix some Finite structure and you fix a fragment of first-order logic and the fragment of first-order logic is this one here either Exists an and or exists and an equals they're roughly equivalent So why not take all of the quantifiers and connectives of first-order logic and consider all the subsets that you can have and come up with a whole collection of of of ensembles of problems a whole collection of problems where you fix the logic and then you look over all finite structures and Remarkably most of these are not so interesting. There are several that are more or less trivial The CSP is very interesting quite rich. It's rich within polynomial time But outside of polynomial time, we know it's not so rich because there's just P and NP complete so The classification is Somehow straightforward not the proof of the classification, which was not straightforward But the classification itself is fairly straightforward. You've got P and NP complete and you can Separate between them by weak mean and in the season core something like that Instead of having exists and and you could have for all and all this is the Morgan dual So you basically have a mirror of the CSP dichotomy, but now between P and co-NP completes Down here is maybe the only place apart from the CSP and the QCSP where you have a little bit of structure Because you have both P NP complete co-NP complete and p-space completes There's no great surprise that there's co-NP complete here because you've got a fragment of logic that is self-dual you've got um For all which is the dual of exists you've got all which is the dual of and No one was looking at some This QCSP Classification here that is not symmetric. It's symmetric in the quantifiers, but it's not symmetric in having and and or No one was looking at this and thinking that complexity classes other than P NP complete and p-space complete would Would it would appear it's something of a shock that they have done And it's actually that's one of the key reasons why I still think the problem Is of enduring interest to theorists because of the Strange things that we now know that are going on here. So since the resolution of the fate of arty conjecture Everything is classified here except here in QCSP and it's dual and It seems the classification here is is looking Complicated still so What I find most interesting about these problems is the interplay between first-order relational structures and first-order functional structures that are usually called Algebra's What is the interplay between the relational and the functional? Um, something that keesler wrote or it's written in chang keesler. I can't attribute it to one of the authors in particular Is this Claim that model theory equals the logic plus universal algebra. I'm not sure that's many modern model theorists would say the same thing that um, it maybe did apply to the The historic background in model theory. I rather like this Particularly also when we are studying infinite structures and then the model theory seems to play a stronger role So the interplay is something that um, every one of the seminar will be very Um, comfortable with very familiar with The first-order functional structures have their operations Working column-wise on the tuples of the relations in the relational structures um And then when we map Elements that were in the tuples that were in the relation again to something that's in the relation we get um, something that we call a polymorphism and we get these objects poll and in and I tend to write relational structures in calligraphic no notation and Functional structures in blackboard notation Although I think at some point. Yeah, this this is fine for most of the talk this way So um playing a key role in Many of these classifications is a a galois correspondence and for both the csp and the qcsp there is a galois correspondence Um, and the one that is working for the qcsp involves surjective polymorphisms instead of polymorphisms And surjective polymorphisms are not closed under composition. So the object that you get Is no longer a clone. You would have to have a clone restricted So it's surjective operation. So it it it's something slightly strange um, and so as a result of that mostly people who Have been studying qcsp tend to study The case where all constants are available Defineable in this logic for all exists and and equals and this is equivalent to restricting to the operations being idempotent. Of course idempotent operations are surjective So you basically get to a situation where everything that you are used to For the csp You can use again for the qcsp and there are no terrible surprises For the csp, it's well known that you can restrict to the idempotent case. There's no such equivalent theorem for the qcsp Um, and maybe it's not true even So there are various ways to define the qcsp. I defined it before in its traditional formulation Um, another formulation is an algebraic formulation And this is a rather non-standard algebraic formulation that is non-standard for good reason That will become apparent later So if we parameterize it by a Um by a by a clone on a finite domain Then have the input as a sentence phi in the qcsp logic. Maybe with a quality. It doesn't matter And some concrete relations That are invariant under the clone And then you ask is the sentence true on the on the on the structure b that was in of this clone And obviously in B is an infinite structure For sure, it's an infinite structure. It may not even be finitely related Um, and so there are questions already here of of encoding um So if you deal with a with an in B that is infinite, there are different ways to deal with it. You could deal with um arbitrary finite approximations of it And then say it's NP hard if some finite subset is NP hard and it's polynomial if everything is polynomial There are different ways to approach this Um, traditionally, you don't approach it by allowing actually something or some object that really is infinite Tends to approach it by studying arbitrary finite subsets, you know But I want to attack it in this slightly unusual way of allowing it to be infinite So for a long time my interest in this was brought on by Something that huby observed And let's just talk about the the growth race of Generating sets for direct powers of an algebra So for an algebra a we have this function Giving the cardinality of the minimal size generating sets of this sequence. So this is ofordinate length one. This is of two. This is of three And you ask What is the smallest subset such that under the Operate under the natural action Of the operations of the algebra you generate all of the tuple. So this is this is this method here I'm imagining that this is these are the things in the generating set And once I've got a generating set then I can actually use operations of the algebra and I can generate any tuple That's the idea here and we're interested sort of in these two possibilities one when there's a constant c such that the growth rate is bound by i to the c so it's polynomial And another where it's exponential this so the polynomial is an upper bound the exponential is a lower bound It's not immediately obvious that there isn't possibilities. There are possibilities in between. Why not sub exponential? Why not b to the square root of i? So this is not clear that there should be a gap there What is known It was originally this problem this problem was studied for groups and their groups are always polynomial. So there's some kind of gap between logarithmic and linear the first time you sort of start to get exponential behavior is with semi groups And the observation is that if b is a finite semi group then there are two possibilities either b is a monoid And you have the pgp Linear even or b has the e gp. It's fully exponential. So the polynomial case is super easy Because this is the generating set. So you can imagine When I apply The operation of the monoid along here I just repeat whatever was there and here I repeat whatever was there and there repeat whatever And that means I can actually get any tuple. You can obviously get any tuple. So the generating set is of size n times n Which is linear On the other hand Um, if there is no identity then you have this property that If that that there is a so what's going on here that for each element Um, the action of that on some on the domain Must crush the domain From size n to something smaller Because if if it weren't like that you could iterate that element and you would have to get an identity. That's the idea And not only does this hold for b it actually holds for powers of b And so then you you argue that a subset Of b to the m size r can generate no more than r That's what it started off with and that is the size that it's generating from this inequality r times n minus 1 to the m elements in b to the n. So a generating set must be of size 2n over 2n minus 1 to the m now n is fixed The size of the domain it's fixed n is the thing that is growing so here we've got fully exponential behavior So at some point Hubi was studying qc sp and he studied a particular property of constraint languages and algebras that he called collapsibility And collapsibility Is something that's a bit like what's going on here actually this is something that's called one collapsibility And after a while he studied a form of pgp That he realized wasn't explained by collapsibility And he called it switchability and his original definition is Reasonably It's not difficult, but it's not totally straightforward. Whereas this definition Is super straightforward. So we just say that an algebra has a certain we call it pgp switchable to source of distinguish it from hubi switchability And then This version of the simple version of switchability has the property that the generating set are tuples that when you read them from left to right Only change elements at most some bounded number of times some at most k times And this is like a simple form of switchability the original form of switchability was introduced as I say by hubi and it's based on something called reactive composition It was introduced in regard to the qc sp where hubi proved that switchability of the algebra actually dropped the qc sp into np this Other form of switchability. He knew drops the qc sp the bounded alternation version of qc sp into I guess np. So he knew it for the bounded alternation version, but not for the unbounded alternation and in a paper A few years back we proved that actually the two coincide the more complicated form of switchability and the Simpler form of switchability coincide and the proof here is heavily logical. It's it's it's what working through the lot through the logical characterization so this Discussion about the the the separation between pgp and egp originated in groups was generalized to Semi groups and such like and then hubi amongst others took it up for clones Um, and there was also work by keith cernes and adnazendore Um, and they made some sort of small progress and then quite remarkably zhuk Solved it in one fairly simple arguments And he proved that either there is the egp or there is the pgp and moreover the only type of pgp that you are getting for finite algebras Is this simple switchability? Um, and in fact this proof even holds in the in the non item open case, which which really smacks of some kind of witchcraft Um, so before this had happened Um in a in a set of essays or works dedicated to dexter kozum who was hubi supervisor hubi hubi wrote a An article about some meditations on on on the qcsp And he this is this is a form of what he conjectured. So a finite relational structure with it all constant So if we have the pgp, then we're in np. Otherwise The qcsp is p space complete And this is this is the original form of the kachen conjecture and in fact, it's the union of two conjectures in that original paper So zhuk is making this strong progress by proving that in facts um That there is a dichotomy between pgp and e gp. So in chen's original formulation of the conjecture, he didn't say That you have pgp and you're in np or you have e gp and a p space He didn't say that but it would surely have been on his mind And he knew it held in the domain three case the gap between pgp and e gp In the idempotent three element case He didn't state it like that. It would have been surely on his mind And so once zhuk has proved this thing it seems like This should be something that we can make progress on this should be a conjecture that's that's that's approachable And in fact, um, it wasn't known at that time But it became known later the fader vadi conjecture was proved independently by bullet often zhuk And they're saying that b is a finite relational structure expanded with all constants So if the polymorphisms have a weak knee anonymity then you're in p Otherwise your csp is np complete and that would actually turn the chen conjecture into a trichotomy conjecture between p np completes and p space completes So yeah, this is the new form of the the the chen conjecture with the knowledge of the fader vadi conjecture proved That you would have a trichotomy and it seemed, you know, like the sort of partway there so Let's talk about the life of the chen conjecture before the death of the chen conjecture um I already mentioned that I want to do something slightly abhorrent. I want to consider infinite constraint languages And infinite constraint languages can be encoded in different ways So for example, this the not all equal relation can be encoded by listing the tuples Or it can be encoded in sort of boolean logic with constants and equality And this version of the encoding is in general going to be a lot larger than that version of the encoded um And it so happens that if you choose the dnf encoding You get a form of the chen conjecture Um, you get a form with the chen conjecture that when you have the pgp this algebraic formulation of the qcsp is an mp Well that holds in in all formulations But the other way around only holds in this weird sense where we have this weird encoding And we don't get p space Hardness we instead get co-mp hardness and we can't Improve that you actually get some co-mp membership there um So then it begs the question maybe of a form of the chen conjecture where we look for finite subsets of this So if a satisfies pgp then for every finite subset of in the day we get The qcsp is in mp. Obviously that is true Um, it's this next bit that's a bit dodgy otherwise there exists a finite redux so that qcsp b is co-mp hard And this conjecture is the same as this revised chen conjecture where we replace this strange encoding with the tuple encoding Um, and it's false So if we use the tuple encoding the chen conjecture is not working But if we use this funny encoding the chen conjecture does work and this sort of compact version of the chen conjecture is Is also failing like the tuple encoding version is failing. Um, so how are we proving this? It's very simple We are using the characterisation of zhuk Um, and zhuk proved that in the idem potent case it doesn't hold in the non idem potent case But if you're in the idem potent case Then egp always coincides with some kind of relation that's definable And if you look at this relation if you look at rho it's kind of equality um so The really interesting cases happen here where alpha and beta overlap if alpha and beta don't overlap You can prove the chen conjecture you're laughing The problems arise when alpha and beta overlap And you might as well always just think of alpha being nought and two and beta being one and two so two is the overlap And then you could look at this row. It's kind of an equality relation but It accepts twos as being equal to one No, except for two. It's an equality relation So instead of looking at an equality relation. Why not look at a ternary version of an equality relation? um, that's a bit ternary so it's a bit like three-way equality But it has this problem with the element you overlap on And it turns out that zhuk's proof is easily showing that this relation also must exist in the clone And then everything is straightforward because when you turn When you look at the negation of this problem you turn three-way equality into something that's basically not all equals um so you reduce from The complement of a of an np complete problem becomes something like this and this relation Is exactly the thing that we just proved we can We can we can is in the clone. We just proved it's in the clone so it's in the clone Sorry, the relation is in the co clone. It's in the co clone But what I can't guarantee is that there's a succinct pp definition qc sp definition but I'm working in this crazy situation where we allow The relations to be encoded in whatever dnf in this In this way I don't have to worry about the The succinct definition And so I can say that it I in my new version of the problem where relations are encoded in this way I I've got I get co-np hardness here immediately. It's it's trivial It's a trivial thing um Yeah, the interesting cases happen when when the overlap is uh is non-empty And then you get this thing that's co-np hard, but also in co-np um But every finite reduct is in polynomial time. It's the thing that refutes the The old one of the alternative versions that I gave earlier So when you make a deal with the devil and start using these crazy encodings with dnf and Whatever you end up getting this co-np complete case But it's not a finite. It's not a finite constraint language. It never I never thought From this that we would find a finite constraint language whose qc sp was co-np complete No one was thinking about at the time. Um, and in fact We we were still thinking that the the the the traditional definition of the qc sp and the traditional chen conjecture Right, this is again the updated version of the chen conjecture With the proved fader body conjecture People are still thinking it should be true because If we have the bgp, we just need um We just need these polynomial size definitions and actually we can even use universal Convocation here of tor n from Be this should be something shouldn't this be possible and this um, actually these polynomial size definitions or short definitions are things that are studied by um vallstrom and lagerkivist And I know I I saw magnus is here um And it seemed like we just needed this additional trick Surely there should be this trick of how to compute Um these things it should be possible and then we could prove the the the the form The original form of the champions actually should still hold but then The things the the the the nightmare begins so We've got these simple relations on a three element domain um And you can see this second one is a bit like an encoding of and So Nought nought is nought nought one is nought one nought is nought and one one is one it's an encoding of and but it's got this sort of Except a lot of tuples that have got a two in them And this other relation we can express like this And this is a finite a very simple finite constraint language Who's on a three element domain who's qcsp is um co-np complete. So this is uh You know, this is something in auspicious this is this is quite a quite a counter example to chen conjecture but then things actually get worse because there are these um Constraint languages that have where that where the where the polymorphism clone has the e gp That's the qcsp turn turns out to be in p. So with the um With the strange dnf encoding, of course, we're co-np hard here, but without that We're in p and what does that mean? It means that yes There is no polynomial size definition here. So not only is it there isn't a polynomial computable definition jerk is proving that there isn't even a polynomial definition it cannot exist so um There are in fact these two different language basically these two up to Up to sort of autumn or up to sort of automorphisms as it were There are two different types of domain three language that have this property where where there is e gp but the um But the qcsp turns out to be in p and um, it gets worse because once you start having co-np complete qcsp's you can start building monsters more monsters dp completes um dp is the np minus np or np and co-np Theta p2 is a strange class that I often forget the definition of but it's something like p with an oracle in np of logarithmic length something like that But still um zhuk was able to Tain the madness at the three elements level There's actually only p np complete co-np complete and p-space complete there You have to go to higher domains to get these other monsters So where am I going here? So Where do the monsters originates they originate? with the co-np complete cases The co-np complete cases allow the building of the monsters and actually for the rest of the talk I want to talk about co-np membership so um I call it the olshac zhuk method because um mirek actually I think has exhibited that there must be these finite domain ones in Um, there must be these finite domain ones that are co-np complete first But the lemma I'm going to give here is djuk So on a domain of size three an operation is called Zero stable if fx zero maps to x And fx two maps to and s is a well known idempotent semilatis that maps Everything to two whenever a is different from b and otherwise it's idempotent So this s has been studied quite a bit in the literature Hubey introduced that the qcsp of inv s is p space complete there are There's a finite relational language invariant under s Where we have p space completeness and this p space completeness is actually due to pull that off But what zhuk is proving here is that when you have this zero stable operation as a polymorphism, I should mention that this one generates s Then a then an arbitrary instance of the qcsp actually drops To an equivalent pi two instance where roughly speaking you replace the universal variable variables with existential variables And set these new ones to zero so it's reminiscent in a strange way of Hubey's original ideas of collapsibility and switchability, but they drop an instance to np and this method is going to drop a method or drop an instance to Pi two and then To co-np And why is this method so important because this is in fact building the monsters not just Where we have co-np membership. It's actually also playing a key role In the polynomial time cases these ones with egp. They're in polynomial time because it's The polymorphism of this kind is first dropping the the complexity to pi two And then there's going to be a polynomial album So how does this lemma work? Okay one way The downward direction is somehow The easy direction if we have a solution of the original instance Then it is also a solution for new instance with the additional assignments where we We get the skull and functions for when we planted the universal variables for zero. So this is the simple direction Now the harder direction is the one where there is the real contribution. So Let's try to follow the proof here. So consider solutions of the new instance Such that we have yi's skull and function Involving now for the moment all of these variables because they're all before all the universal variables before And let's pose the question is is that a minimal n? Such that f of n the skull and function depends on xj for some xj bigger than its index And in particular we would like that no such n exists So we're going to assume for contradiction that such an n does exist and choose it to be minimal among all solutions Why is this really potent because if we can prove that no such n exists that we automatically get That this thing implies that thing Because the existential variables simply don't depend On the tail end of the universal variables. That's the very thing that's being proved This is the heart of the proof and it's a That it's the novel observation that gives this proof So we assume for contradiction that such an n does exist And we achieve we choose it to be minimal among all solutions So we get some solution here Where we set all of the zeros after n And here we've got some functions there Now we just redraw that tuple in a certain way. I think it would make sense. Maybe yaku for me to post my Well, there's no need. This is this is basically as it is written in the in the stock paper I agree. It's not so easy to read now But what juk is doing is he's looking for all of these variables after n And he considers all possibilities that can go there. So we're over the domain two zero one and two And then we get a huge number of solutions here And then we apply to all of these The semi lattice operation s that we had so I gave s as a binary semi lattice, but it plainly has a version that that is of arbitrary our arbitrary finite arity where it basically maps idempotently But then everything else we get gets mapped to And then we get this property here that the the action here of the Semi lattice is mapping this to see Only if it goes to see for every one of these And if it goes to a single constant for every one of these then for sure it wasn't dependent on of any of these That's the observation if it works for all of these You we can then it wasn't dependent on then and then the Otherwise There were more than one possibility that could go there and that's where we're going to use the semi lattice to say that well In fact, we could have answered two We could always have answered two there and so by this other method. It was also independent of all of the things before And then we just use a single application of the zero stable operation To generate a solution of the instance such that fm doesn't depend on on these numbers So either it didn't depend on them because it was always the same Or we could have answered two because we use the semi lattice That's the observation and then we contradict the normality of m So it's this remarkable proof that is proving That we can compress these instances into a pi two form and then we can solve the pi two form with a co-np oracle so um Where is the chen conjection going? so using um the idea of polynomially building the um The tour relations Um, give giving them a polynomial pp definition Actually allows a fairly straightforward proof that the chen conjection does hold In what I call the conservative case when the language has all unary relations It's fairly easy to say that we don't have to have the stupid form of the chen conjecture with um The nf definitions of the relations we really can give the pp definition Because the conservative case is so well behaved um And it seems like the conservative case is like some kind of Natural largest class in which the chen conjecture holds What to do with the chen conjecture in the future? Well, I don't know could we Modify pgp and egp in some way To now make the chen conjecture True may I'm obviously not the chen conjecture that has the trichotomy between p and b complete and co-np complete It would have to be more sophisticated But for me the most interesting Um direction for the chen conjecture is to try to prove generic Results about co-np membership. So how do we take The olshac zhuk method And extend it to a wider range of languages because the olshac zhuk method is also accounting For the qcsp monsters that are in p So we already have this general method to prove membership in np and it's called switchability Maybe if we then had some general method to prove co-np membership We could take out also Co-np cases and then I suppose we could try to prove dp hardness for the rest. I I'm not sure exactly what to say so, um Do we is is is is is he be with us at all today? Jacob he wasn't there at the beginning I don't know if he no, maybe he will maybe he will uh, he will see see but see the The slides at some point so in 1900 a famous mathematician you called Gave a list of problems that would sort of set the agenda for mathematics at the turn of the 20th century and In 2019 another mathematician called Came also with some problems And hubert's problems are these three problems So he originally gave this in a triple a Workshop, I think I think it was originally in 2019 And he asked For the complexity. Well, he asked for this of these three problems. So the first one hubert won Is a qcsp. It's an infinite domain qcsp The second one was the complexity of a surjective csp known as a three no rainbow Problem and the third one was some nonsense that I can't remember so Hubert two has been solved by jug Um and I'm going to stretch myself here to I can't say that it's certain but um I somehow think hubert one is in co-np And this is the this is why I'm so interested in the general methods for co-np Um, I kind of feel like it could be in co-np. So I Um, I tried to assemble a crack team um a russian Um a pole This is jug here No, this is jug here actually and this is rona here And maybe for I mean for a mathematical quest. We should probably also have a hungarian, but I didn't yet get a Get a hungarian, but maybe one day we can quest To solve for hubert one. So maybe this is um Yeah, maybe this is a bit bold because it's Not totally straightforward, but um I would I I I dream of it being in co-np and I dream of the I dream of the jug method the or shag jug method to prove it So maybe I can end here Yeah, thanks for the talk um Other some questions Comments come through examples Um barni, I think your your composition of the team is very suitable for attacking hubie three Yeah, do you um as it here's a here's a pole does does does anyone in the audience remember hubie three hubie three So I I unfortunately can't remember I can't remember what it I know it's in it's in an area. I I wasn't working in I will So it's somehow it's somehow related to one of the papers that he worked on with matt. Um, matt filiriote. I think it's it's it's a question there, but it's uh I I couldn't remember the formulation. So actually the reason I write um some nonsense is because I can't remember the formulation Not necessarily because it is nonsense Okay, I will just look look for it Maybe I can find oh, okay Um barni, maybe I can ask this question. So is is there a hope to identify some Uh class of structures Where you you have something smallish like the dracotomy or whatever No, no monsters Okay, but this this is the idea. Maybe dima has something to say. So if you We have already a generic reason for np membership Except for some monsters. So if if you could generalize this co-np membership So general co-np membership. Maybe Both of those would remove all of the monsters. You kill the monsters um And then you could have okay, then you could look for Yeah, in np or in co-np versus dp hard This would be a possibility. The other obvious possibility now is to look for um in np by um Turing reductions say Or dp hard something like that or no, so or in p Versus np hard on the Turing reduction. So that would include Co-np hard and np hard But the problem with that is you would have to know all of the monsters that are within p And the monsters within p look problematic If you actually stretched tractability to include co-np and np you might be able to kill those monsters Dima, what do you think about that? To be honest, I don't believe you can do this because um Like We still don't know whether we have uh these six classes or we have much more and My guess would be that we have more Yeah, but what about the the idea of extending this lemma to a general lemma for co-np membership Because it talks about something very similar to collapsibility. It's a collapsibility though of the existential quantifiers Not the universal quantifiers So could there be an a general generalization of this like switchability That explains all of this co-np membership Yeah, I agree. I really like this idea because uh like we need to generalize pgp And we need to do something just to find out where we can you don't have to if you could generalize this You could keep traditional pgp and egp. I think maybe Because um this method already covers the monsters that are egp but in p Okay, let me ask you another question. Um, what if we consider qcsp, which is just pi 2 What do we need to get a classification for pi 2 formulas? Yeah So to get a classification, we need something Better than pgp we have now And this idea just explains how to How to go from any formula to pi 2 formula and when we can do this Yeah, but I agree. This is a very good question. Like when can we apply this idea? How can we do this? And also, I really like your hubert one. I started to think about it So the thing about hubert one is it has some polymorphisms, so Yeah, it's But unfortunately Sorry, can I stop you? What does the relation actually mean? What is this implication? Sorry that equals it should be equals here. Sorry. I make a typo. Yeah This is equals Uh, it's a typo. Yeah, I wrote it today. Sorry Okay Now it makes sense. Um, you can continue. Sorry No, I have nothing Hopefully now zhuk is looking at this Go on The comment on this relation is that it's the last missing case to complete A dichotomy or not dichotomy a try Yeah, so well So manual you always told me that it would solve the full Classification if it's p-space complete, but you weren't sure if it were in co-np Um, but huby told me that it would either way if you could prove this is in co-np it would also be enough Right. Do you have any comment on that? Yeah, so so what what's the class of problems we are looking at we look at all structures that are preserved by all permutations Yes, all the quality languages certainly is and then Yeah, there are p-space Complete cases for example if you modify this relation so that it is for airy and it has x equals y implies u equals v Then it's p-space complete the csp would Even with this equality would still be in p, but the q csp would be p-space complete And now here we our original proof failed and we we just proved co-np Uh hardness Yeah But if you have co-np membership of this because there's a a a a a load of problems somehow between perhaps Between x equals y implies y equals z and x equals y implies u equals v um and it's Is that class of problems in between solved all right all right But huby you once told me you weren't sure if there were co-np membership here But huby when he gave this talk said it should um if you proved co-np membership here it should settle everything So He said it. He said it. Um, I asked him specifically when he gave his talk in Durham Oh But it's good because this is a nice region to revisit anyway. Um We we have with michael pinzka and huby we have only a partial classification right is still not fully explored So the question is what polymorphisms Exist on this structure that don't exist on x equals y implies u equals v Yeah, but you wipe out lots of constantly you wipe out lots of polymorphisms immediately in that paper right so Um Because we're not considering just item potent Yeah Yeah, okay, um some last chance for questions Yeah, um, maybe one question So, uh, now in for q csps. Are there any results the where the Equational structure of the algebra helps you or Does are there some identities that maybe imply pgp or something or yes in general? I yes, yes, there are classes of polymorphisms that imply pgp Um, but are they are they equations of amounts of conditions? They they're to do with sort of for example surge activity of each coordinate I'm not sure the amounts of conditions actually, but the um many item potent amounts of conditions do imply tractability like Okay, um So mouths of does um near unanimity does semi lattice in general doesn't Um, some semi lattices you can have p space complete q csps Like this while the one I mentioned actually this semi lattice semi lattice with units is tractable, but semi lattice without unit is not necessarily Thank you May I comment this on this? It's so so what is known it that the The complexity does not just depend on the variety generated by the polymorphism clone. This is known This fails Okay, thanks um, yeah, I guess That's all, um, let's thank Rani again