 V odvršenju, zato sem pričel, da pravimo včasnje včasnje vrste logične in neko vanjev vrste. Počut kaj sem pa nami težiš. Zato je zelo pravimo na bolje deli. Zato začel včasnje. Zato sem počut ... ... na modučenje vrste logične vrste logične. V sej, to je teorem do Gredel, Vardi in Fokon kolatih, in to sem izgovoril po nekaj nekaj nekaj rezultat. Zelo sem izgovoril, da je to vzpečniko tega, to je vzpečniko vzpečniko, zelo sem izgovoril, da je to tudi izgovoril, da nekaj nekaj revoljnosti so revoljnosti. Prijezaj, da izgledaj... Ja počutal, nažah, da se hrbi se od separateda pravdu, o sp היiko. Prijezaj, ki je zmah, da je v seženjuhabenju modučne vse, če je zpite. Tadejo, ki je način, začin da je vsak počut, je, ki so vstajal, bo tako splete, tak je bo, način, da je, ki je, sprana, je, da je, začin, začin, da je vseč, vzouto, da je izgledajstv poslednji sistem vzouto in zelo. Svajte občas, da je čest, da so ima del tudi nekaj in vzouto, da je začič nekočno izvizivno. Zato pa bomo da razložinili, da imamo počke, da imamo, da so imamo, da imamo da, ki imamo počke, da imamo počke, čest nič nekaj, da imamo počke, Зв proporциjačnji v vrhece rovok. I potem zato sajtebilila predstavljeva nasiljama. A za to je dialo trajno three equivalent relationalne. In pevne את res sem pozdravil te začelično razljavljenje of tovaraible materijali v te doskud vratelji. Skupelji so, da na stranju z vrhece in naprati kad ma držite. Ok, zato. The first thing is two variable fragment of first of the logic. S trakmačnih varibih vstodologworkersih, drožem sem prav, v krištih nekaj neži gumov, x in y, vzbeni bomo vverega so delali, vse se potrebnava pomez, leč lažneざ, ne lahko sem hranila poce om, boh počel si v saci sobo stabilnih doko trval po celu, boh se da pomaga tudi značen ​ecstorskost. As you know, from the morning session, the satisfiability problem for first of the logic is undecidable. This is a classical result by church during a ghetto from 1930s. In fact, to show undecidability, we need only three variables. It follows, for example, from undecidability of Kar-Murwang class. It has the set of sentences, which have the prefix of quantifiers for all x, exist y for all z, and then there is a quantifier free formula. What about two variable fragments? The first result in this area was by Scott, who reduced the satisfiability problem for F02 to the so-called ghetto class. It has the class of formulas starting from a prefix of existential quantifiers then exactly to universal quantifiers and again a prefix of existential quantifiers. At those times, in the 1960s, it was believed that the ghetto class is decidable with equality. However, the ghetto wrote only the proof for the case without equality and just claimed that this proof can be extended to the case without equality. It appeared later that it is not true and the satisfiability problem for ghetto class with equality is undecidable. Skos argument in fact worked only for F02 without equality. The first proof, which worked for the full case of F02 was given by Mortimer and this proof was by showing that every satisfiable formula has a model of doubly exponential depth with respect to length of the formula. Then you can simply, the procedure solving the satisfiability problem just guesses such a model and then verifies it using model checking, which was also introduced today in the morning. The bump on the size of the model was later improved by ghetto class in Vardy and they proved that every satisfiable formula has a model of exponential size, which in fact leads to next time completeness of the satisfiability problem. This is the first thing I want to show you, but before, let me also give some other motivations for variable logic. One of the motivations is that this is a maximal fragment of first order logic that affects the number of variables, which is decidable. Another thing is that many formalisms used in computer science, such as modal, temporal, description logics and other things, are essentially two variable logics. Of course, there are some exceptions. For example, in temporal logic we use the operator until, whose natural translation requires three variables. There are also some additional constructions very often, which is very often two variable. For example, here is an example of description logic concept, which naturally translates to two variable logic with unary predicates and binary predicates. Some examples of formulas of two variable logic are for example a formula, which states that the relation r connects all parts of elements. We may say that each element satisfying q is not related by r to an element satisfying p. We may say that a relation is anti-reflexive using just one variable. We may say that a relation is symmetric. These are very natural formulas. An interesting formula is that one which says that in a modal there is at most one element satisfying p. We may try that for all xy, if px and py, then x is equal to y. In fact, this is a formula, which calls that the disgubbles class is on the side of a quality. We may also... There is also an example for using variables. We may say that each element satisfying p is connected to an element satisfying q by a path, r path of length 3. In a natural way, we reuse variables to go from an element in p to an element in q. An example of a property, which is not expressible in two variable logic, is that a binary relation r is transitive. We hear today that in first of the logic, we cannot say that a relation is a transitive closure of a relation. Of course, we can say in first of the logic that r is transitive. This is using three variables, and the argument is as follows. In a moment, we will see that every satisfiable F2 formula has a finite model. If we assume to the contrary that we have a formula, which expresses transitivity of r, it tests in every model of this formula, r is interpreted as a transitive relation. Then the following simple formula, which says that r is transitive for each element that exists, an r successor of this element is, and none of the elements is related to r by itself, is satisfiable because it can be satisfied in the set of natural numbers, and if we interpret r as less than relation, strict less than relation. So there is a model of this, but it is easy to see that there is no finite model. At none point we cannot reuse an earlier element as this r successor, because then an element would become r related to itself because of transitivity. OK, so these are some examples. One more example, in two-variable logic, I told you that we will see that every satisfiable formula has a model of at most exponential size with respect to the formula. Here we see that this result will be essentially optimal, because we can enforce exponential models in a very simple way. Having two variables we may say, we may... OK, let us think that we have unary predicates, p0, p1 up to pn minus 1, and they, using those predicates, we may encode numbers. Yes, each element encodes a number from the range from 0 up to 2 to the n. In a natural way, about the binary encoding, and having two variables it is easy to write that an element encodes a value greater by one than another element. Here is an explicit formula, this is of quadratic length with respect to the size, with respect to n. Yes, it just says that there is a point in which one of the elements encodes 1 and the other encodes 0, all positions to the right are ones in one of the elements, zeros in the other elements, and all the positions to the left are identical. So, of course, having such a formula, we now can easily say that a model contains all the numbers from 0 up to 2 to the n minus 1. By saying that there exists an element encoding 0, and there is an element encoding a number greater by 1. I will use the notion of atomic types. So, what is an atomic one type? Atomic one type of an element in a structure A is a complete description of the structure on this element. So, we simply say this one type simply says which unary predicates are true in this element, and which binary relations are true on this element and this element again. So, for example, if we have two unary relations, p and q, and a binary relation r, an example of one type may be px, not q, x, and not rx. Yes, a complete information about the element. Similarly, we have a notion of an atomic two type which gives a complete information about a pair of elements and connections between these pair of elements. So, for example, when the two type has two free variables, x and y, yes, we may say that p is true at x, q is not true at x, not rxx, and in r, p, p is not true, q is not true, r, y, y is true, rx, y is true, and not r, y, x is true. Yes. So, in our constructions we will consider only signatures consisting of unary and binary relation symbols. So, note that under this assumption to completely specify a structure, it is enough to specify its domain, the list of one types of elements and the list of two types of elements. Yes. If there are no relations of a greater r, it is enough, we know the structure. So, the theorem I want to prove is, the theorem was in Vardy, which says that every satisfiable formula has a model of size at most exponential with respect to phi. However, because I'm going to use this result in the following parts of my talk, I will prove a slight variation of these results. A slightly stronger result in fact, namely that namely, I'm a four here. So, let phi be an effort to formula and let a be its arbitrary model. Then, there is model a prime of phi of size exponential with respect to the length of the formula, such that its domain is a subset of the domain of the original model a and the one types of elements are retained. So, this slide generalization of this lemma 3, because in this lemma 3 we just simply show that there exists a model of exponential size and here we also say that from every model we can choose a substructure and slightly modify it one types to obtain the model. The first step in the construction is establishing a normal form. We show that we can concentrate on some simple formulas of a simple shape. In this case, the shape of these formulas in normal form will be like this. We have a universal quantified formula for all x, y and something about x, y and this something, this phi 0 is quantifier 3. And then we have m formulas of the form for all x exists y, phi i of x, y, and again this phi i are quantifier 3. And we can prove that for every fo2 formula we can compute a formula in normal form such that phi prime is satisfiable if and only if phi is satisfiable. Moreover, we can show that every model of phi can be expanded to a model of phi prime. Expanded because phi prime will use slightly richer vocabulary. Vacabularity will be extended by some unary symbols. And in other direction every model of phi prime if restricted to this original signature will be a model of phi. In fact there is also lem of phi on this slide which says that we can get rid of symbols of rt greater than 2. I will skip the proof of this fact. This is quite easy construction also. So, the proof of this normal form goes in the following way. We start from subformulas of phi which are of the form quantifier variable and quantifier free formula. And we substitute such subformulas by unary predicates. For example here we have just an example. We have a formula of the form there exists y that's something we introduce fresh variable p and we try to enforce that for all x p of x is equivalent to exist y psi of xy. And it can be easily checked that this equivalence can be enforced using a conjunction of two formulas of the shape which appear in our normal form. So, we proceed analogously with subformulas of the other possible shapes and we repeat the process from down to up until we get formula in normal form. Now we restrict our attention to formulas of normal form. What is the plan of the proof of the theorem? We start from an arbitrary model possibly infinite. We distinguish in this model three subsets of elements exponentially bounded size and the elements from the subsets will become the domain of our small model. In fact this new structure a prime will be the connections between elements from C and D will be taken from the original structure. Similarly the connections between D and E. We only slightly modify the connections between C and D. How the proof works? Recall in the number four we have a formula which says that for all x, y something and some formulas that for all x exist y. Let us take an arbitrary model A, this model will appear here a fragment of this model will appear on this picture and the first fragment of this model is called C. In C we put all the elements from the original model whose one types are rare in this model. I mean whose one types are realized at most m times then we take all the realizations of such one types to this fragment C and for all of the remaining one types we choose m arbitrary realizations of them. In this picture we have one types are represented by colors so we have one realization of a red type which means that in the whole model there were only one realization of this we have for example two realizations of yellow type, three realizations of green type so there were respectively two yellow elements and three green elements in the original model A. All the remaining types pink this should be gray and blue were realized more than m times so we have chosen m realization of them and put them to the set C. So this is the first fragment of the universe of our new structure. Of course if we consider the substructure induced by this set C then it obviously satisfies this universal contract of our formula phi because this is just the substructure of the original model. But the second the formulas of the second kind may be not satisfied because so consider the blue element in the lower right corner. In the original structure we had so called witnesses for this element and this formulas of type for all exists. A witness for a conjunct of this type and an element is an element which together with this first element satisfies phi i of this y. So of course in a model of formula phi we had m witnesses for the blue element this formulas of type for all exists. At most m because sometimes an element may serve as a witness for more than one subform. Of course not necessarily all of them were members of C. So what we do, we extend we define a new set D which contains witnesses for all the elements for C. So we choose witnesses for each of the elements at most im of them and we collect them to the set. And we repeat this step once more. So we repeat this step once more. We choose witnesses for elements from D and collect them into a set E. And now consider an element from the set E. Again in the original structure it has witnesses for all subformulas of the form for all exists. Some of them live in this fragment C plus D plus E but some of them are not members of this fragment. But now observe that we can modify the structure induced by C plus D and E to fulfill all these requirements of the elements from E in the set C. Because in C we have at most m realizations of each type at least m time. Of course the elements from E can have witnesses outside this small fragment only if these elements have types which are realized at most m times. So we can modify the connections between E and C. For example, if you consider element and its gray witness instead of using this gray witness outside C plus D and E as a witness we can use one of the gray elements in C. Simply by copying the connection between these two gray elements to this new. We have enough elements in C to provide all the witnesses for elements from E. Because for each element this finishes the construction. We have a structure consisting of at most exponentially many elements. Because how many elements we have in C. For each one type we put in C at most m realizations where m is linear in the size of the formula. And the number of one types is exponential. The number of one types are one type you simply say whether each symbol from a vocabulary for each symbol of the vocabulary we say that p of x or not p of x. So the number of one types is exponential with respect to the size of the vocabulary. Of course we may restrict our vocabulary to the symbols which appear in the formula 5. So this is exponentially many times linearly many. In the set D for each element from C we add at most m elements. Similarly in E for each element from D we add at most m elements. So the total number of elements in C plus D plus E is exponential with respect to the size of formula 5. And it is a model of 5 because in fact all binary connections are taken from A and 5 by 0 x, y is satisfied. And for each element we have witnesses. We took care of this for this. So if we know that every satisfiable formula has a model of exponential size we may conclude that the satisfiability problem for two variable logic is in the next time. Why? We guess a model and we verify that this is a model. Tomorrow in the morning we saw that model checking problem is p space for first order logic. But it is not hard to see that if we restrict our attention to two variable fragment that it becomes then the problem is in p. So we simply guess a structure in nondeterministic exponential time because we have exponentially large structures and we verify that phi is true in this structure. A lower bound is easy to obtain using this trick with counting up to 2 to the n. I am going to skip. In the further part of my talk I will need one more simple observation. Let me define a notion of a royal type and a kink. These are the notions which appear in the original paper by Gredel, Coetis and Pagdi. So what is a kink? Consider a structure. A kink is an element whose one type is realized only once in this structure. And such a type is called royal. Recall that we can enforce royal types. We can say for example that there is at most one p in a model. For all x, y, p, x, p, y then x is equal to y. And the observation in M9 says that if we take an arbitrary model of a formula phi then and an arbitrary non-royal type realized in this model this type is called t here then we can to this model a we can add an additional realization of this type t. And the proof of this fact is also very simple. Consider a model a and let us try to add an additional realization of the yellow element. These colored edges between elements represent two types. Connections between elements. So we can add an additional element of yellow type and connect this element with all the elements in the original structure a with the exception of this realization of yellow type. Exactly as this element was connected to those elements. They are the same colors. In this way observe that we will definitely we will ensure that the new element has all the required witnesses. And of course we won't violate this for all properties because we take our two types from the original model. The only thing which is left to do is to connect the new yellow element with this original one. But we can do it safely because we know that there were these two realizations of the yellow color in the original model. So we simply take this connection and connect the new element with the original one using the same type of connection. So definitely we won't violate this formula this way. Observe that it's important here that this yellow type is not royal. Because for royal type we wouldn't have this pattern. This way we prove that every satisfiable formula in two variable logic has a model of exponential size. Now I would like to consider another problem. I want to show that something called guarded fragment restricted to two variables is decidable with equivalence relations. Let me first define the guarded fragment first of the logic. So the guarded fragment was introduced by Andrei Kavan Bentem and Nemeti. And in guarded fragment in general we do not restrict the number of variables. So this is the definition of the whole guarded fragment. In this fragment quantifiers have to be relativized by atomic formulas. So if we want to say for example for all y where y may be here a couple of variables for all y phi of xy then we have to put an atomic formula called guard here. So for all y such that and here we have an atomic formula something happens. And analogously for existential formulas. So this is the only restriction and this logic is closed under bullet operators and all atomic formulas belong to guarded fragment with my use of quality. Here are some example formulas for example the formula defining symmetry of a binary relation is guarded. For all xy if x is connected to y by r then something. So these blue atomic formulas are guards here. As you see in the third example as a guard we may also use equality. So in fact if we just quantify a single variable the guard is not necessary because we always have equality. Some formulas which are not guarded are for example the first formula which says that there exist two x satisfying p and for all yz if rxy rxy is not a proper guard here because it does not contain set. The next formula defines transitivity of a binary relation it is not a guarded formula because it is relativized by a conjunction of two formulas and not an atomic formula. The third formula which says that for all xy if px and py then for example there is x is connected to y by e we could also try to say here that x is equal to y. This is not guarded again because this formula relativizing a quantifier is not atomic. So as you see in the guarded fragment we cannot enforce a royal type in a model. We see that it is not easy at least. We will see in a moment that it is not possible to talk. It can be also observed that description logic concept which I showed at the beginning of the talk is translated in fact into the guarded to variable fragment because these formulas cxy and cyx are atomic formulas and a short review of results on the guarded fragment so Eric Gredesh proved that the guarded fragment also has the finite model property every satisfiable formula as a finite model the complexity of satisfiability problem is deterministic doubly exponential in the general case and if we restrict the number of variables to some constant in particular to then the problem becomes exponent complete for singly exponential deterministic time. There are many interesting extensions of two variable guarded fragment and two variable guarded fragment and the whole guarded fragment in fact for example by fixed point operators constant transitive relations and so on which are decidable. About one of these extensions you will hear the talk about so what I am going to show you I am going to show that the two variable guarded fragment it is the restriction of the guarded fragment to variables is decidable when extended by equivalence relations what do I mean we consider just two variable guarded formulas in which if we want to construct models from them we are enforced to interpret some binary symbols as equivalence relations so we simply have a list of equivalence relations and a list of symbols for binary relations and they have to be interpreted as equivalences unfortunately if we consider such a language the problem will be decidable to obtain decidability so we have to restrict the usage of those special equivalence symbols only to guards this is why I called it two variable guarded fragment of equivalence guards so as I told you I want to consider first the general satisfiability problem and then the finite satisfiability problem both of them appear to be complete for non deterministic exponential so today I would like to show you only this first theorem that the general satisfiability problem is decidable in next time before just a short motivation for this logic so one of the main drawbacks of guarded fragment and two variable logic is that we cannot express transitivity of a binary relation I explained explained to you why the argument for guarded fragment of course is analogous to the argument for two variable logic we have found that model proper analogously in none of the guarded fragment and two variable logic we cannot say that a binary relation is an equivalence relation and I will show you in the moment a formula which is on the infinite models of course transitivity and equivalence relations are very natural and it would be nice to have them in some practical applications so this is why we consider this guarded fragment with equivalence guards here this is an example of a formula which has only infinite models yes we have two equivalence relations represented on this picture by red color and E2 represented by blue color and what we say we say that there exist an element satisfying P and S we say that every element satisfying P is connected by E1 to an element satisfying Q not equal to X similarly each element satisfying Q is connected by E2 to an element satisfying P and if you consider this first three three subformulas we may be the natural model for them we start from an element in P and S then we choose an element, we add an element satisfying Q and connect it to this element this first element by E1 then we choose a new element satisfying P and connect it to the previous element satisfying Q by E2 and so on in alternative fashion we construct infinite chain to enforce that this chain has to be infinite we add three additional formulas the first one says that each pair of elements connected by E1 it says that in every E1 class there is at most one element satisfying P we enforce it by saying that for all pairs of elements connected by E1 PX and PY implies that X is equal the trick we used about which I talked earlier similarly in E2 classes we allow only for at most one element satisfying Q and because of this first element in P and S we say that for each element satisfying S its E2 class is trivial it consists only of it is just a single and now if we would like to try to reuse at some point one of the earlier elements we will violate one of these three lower conjunctions for example if for the element Q we would like to reuse one of the earlier elements satisfying P for the witness for the cert formula then two elements satisfying Q would become E2 connected would become members of the same E2 class which is forbidden by the fifth formula the last formula is just because the reason for the last formula is that sometimes we could try to reuse the first element as a witness so we forbid this explicitly so this formula is satisfiable but only in infinite models so if we want to show decidability of two variable guarded fragment we have to take into account these infinite models observe that in fact this guarded fragment may be seen as an extension of full two variable logic because for example this formula in normal form may be translated to by using one of the equivalence relations as an artificial guard artificial relativization of modifiers so instead of saying for all x, y we say for all x, y which are connected by 1 similarly for this formula of type for all exists we say if this F02 formula is satisfiable then we may construct a model of the second formula consisting of just one equivalence class of V1 relation and in the opposite direction if there is a model of the lower formula then we can take its arbitrary equivalence class and it will be a model of this two variable formula so as a corollary that the satisfiability problem for guarded fragment is next time hard because it follows from next time hardness of F02 how can we obtain the upper bound so we will see that every satisfiable formula has a model of a tree-like shape and then I will argue that such tree-like models can be made regular and their existence can be very easily checked so I will explain these three unravelings in the moment of course we will need some kind of normal form as in the case of two variable logic to make things simpler and this normal form is very similar so we have some we may say that there exist some elements satisfying some there are some elements whose one types are defined by alpha x and psi x and then we have universal formulas for all x, y something but they have to be guarded and I decided to distinguish two kinds of such formulas one with equivalence relation in guards these are the formulas for all x, y e, i, x, y then something and the second kind of formulas are the formulas with symbols which are not required to be equivalence to be interpreted as equivalence similarly for all exist formulas for all x, if alpha of x where alpha is in guards there exist y and again it is connected to x by an equivalence relation e, i there is some number of equivalence relations or it is connected to x and non-equivalence relation by a relation which is not required to be interpreted as equivalence as an equivalence recall this restriction that equivalence relation appear only in guards so we cannot have symbols e, i in those formulas psi this is crucial for the proofing part this is just a statement that we can restrict our attention to normal for formulas it is not as nice as in the case of two variable logic because we say that for every formula in guarded fragment with equivalence guards we can compute a formula which is a disjunction of exponential number of formulas in normal form this is a disjunction but it is not harmful because we are going to show that the program is in next time so we can simply compute yes? the first one uses guards of the form r, i, x, y and the second r, i, y, x ok they could be written as a single formula ok going back to this lemma we show that for every formula we can compute a disjunction of normal formulas in our proof we can simply compute such a disjunction of normal formulas because we are going to use in next time so there is no problem with this and we can simply check each of them separate we can check if one of these formulas is satisfied so in fact now we restrict our attention to formulas in normal form sometimes I will use a notation phi with superscript ek which denotes the fragment of this formula consisting of which either speak about the relation ek or so about those formulas in which ek is a guard and about all universally quantified quantified conjuncts with this non equivalence so phi i is a fragment of the formula which doesn't speak about about equivalence relations different than ek and doesn't speak about these formulas for all x exist y and then a guard which is not ek so consider now consider now a model of a formula in normal form this is a picture there is six elements and what is the meaning of colors here so red connections represent this is an equivalence relation blue connections the equivalence relation e2 green connections the equivalence relation e3 there are also two black arrows which represent the connection by a binary relation which is not an equivalence which does not contain an equivalence relation yes what we are going to do we are going to unravel this model into a tree which is still a model of of the formula phi how we obtain such an unraveling we start from elements which satisfy those existential formulas in the normal formulas of the form x exist x such that so we can start from such elements assume that there is just one one such element here say yellow element and let us build on the first level of the tree all the information which can be seen from this yellow element so we take its e1 class consisting of three elements so we add this orange element and this pink element we take its e2 class consisting of pink element and the blue element and we take its green class so we add one red element observe that we have now two copies of the pink element from the original model but there is no danger here because we put no connections between these pink elements so guarded formulas cannot forbid such two elements here in other words we connected the two pink elements by a binary free type there are no binary connections here so having two such elements here cannot violate formulas of the form for all x for all y because they can speak only about connections element for all x for all y if are x, y then something so here so splitting such pink element in two copies is not dangerous here and we repeat the process we now consider the elements from this first level for example and we add and we add their classes and the elements connected by non-equivalent connections for example from this orange element we add a copy of red element and connect this red element with orange element in exactly the same way as they were connected in the original model we do not add copy of e1 class this red class for the orange element because such a class is constructed on the earlier elements for the pink element we have its red class built earlier so we just add its blue class and so on we do it for all elements and we continue this process usually infinitely because for example there will be an infinite path yellow, orange, red yellow and again there will be an orange element and so on so starting even from a finite model we will usually obtain an infinite model ok so this is the first step of our construction unravel the model into a tree sorry consider for example this pink element I look at the original in the original model and I do not add an additional copy of red equivalence relation because this pink element in this tree has its red equivalence class but I add a copy of blue class and green class for the case of those black arrows I also provide them so for example this green element which comes with its red relation red equivalence class the only thing which it may need it's this black arrow to a blue element ok so such a structure is still a model of sign because all non trivial connection binary connections in the model are taken from the original model non trivial those containing some positive binary atoms and for all elements of course we take all the required witnesses because every element in this model of say red type can see exactly the same types of elements which were seen by red element in the original model can be seen and in fact is connected by the same types so the problem now is that those classes in the original model and in its tree unraveling could be very large or even infinite for example we may have the following situation in the original model the blue class may contain of say infinite number of elements as on this picture however what do we want from such an equivalence class we want it only to provide witnesses to the formulas from phi i blue is e2 in the case of blue connections formulas from phi i2 sorry formulas which have e2 in guide so from this class we only want to provide witnesses to the yellow element for formulas of the form for all x exist y connected to this yellow element by e2 but in fact we can hear just to reuse the construction for two variable logic because such a single class may be seen as a model for two variable formula without guards ok, this is not clear sorry ok, let us look at this normal form the purpose of the e2 class of the formula is just to provide witnesses for the elements of the fourth type with e2 as a guard the class in this e2 class we should also satisfy all the universal formulas which have e2 as a guard and those universal formulas which have non equivalence relations as guards we do not have to bother about formulas which have some other equivalence relations in guards because these formulas psi cannot speak about other equivalence relations so in fact what we do we look at such a e2 class for example as a model for f2 formula constructed from this conjuncts containing e2 and all the remaining universally quantified conjuncts without equivalence relations and we assume that f2 formula says that for all xy e2 xy and we just apply the construction I presented for two variable logic this way we obtain a model of this formula small model of this formula or like here for example just by removing some elements and maybe changing some connections but changing in a safe way in such a way that all elements from this class will have the required witnesses inside this class so we apply this construction for all classes in our model this way obtaining a model with small classes similarly of course from one point we may have many black arrows but what is the purpose of these black arrows to provide witnesses for formulas for all x exist y connected to x by non-equivalent connection there is a bounded number of such formulas so we can simply choose only some small number of witnesses linear with respect to the size of the formula as we did for two variable logic so then for all elements in this tree we have some number of small classes the number of these classes is of course linear with respect to the size of the formula and some number also linear of the elements which are called free witnesses, free meaning non-equivalent connection this is the statement of the theorem about the shape of such models, we have small classes and in fact we obtain the property that the intersection of an e1 class and then e2 class for example contains at most one element because we we constructed the model in such a way that we split the elements if necessary if there were two connections from an element to another element e1 and e2 then we simply constructed two copies of this element and connected one of them by e1 and one by e2 ok, so it's also not hard to see that such models may be constructed in a regular way what I mean by a regular way we may define a seed for a model what is a seed it is just a set of one types which are going to be realized in this model and for each one type a pattern which says how to construct e1 class for this type e2 type for this class and how to construct free witnesses for this class so for each one type we in fact need only exponential number of elements exponentially bounded information classes and free witnesses and I claim that a normal formula is satisfiable if and only if it has a seed for a true-like model of this kind obviously if a formula has a model then it has a true-like model we saw a moment ago so for every we can choose a realization of a type take its classes and take its free witnesses to the seed in the opposite direction if we have a seed then we simply start from elements satisfying these existential formulas exist x at something and provide classes and witnesses to those elements using the seed we have an element we take its class from the pattern e2 class, e3 class then for the elements on the first level of the tree we proceed analogously we need any e1 class we take it from this and so on this is quite easy quite easy observation and it follows that the satisfiability problem for this logic is next time complete because what is the procedure now just quest a seed for a model and check that this is a proper seed I think the properties of a seed is quite easy I do not show all the properties which are required but they are quite natural for every element the seed should provide all the required witnesses without violating for all formulas yes, I think this is all I wanted to say tomorrow I want to concentrate on the finite satisfiability problem because as I mentioned this way we construct usually we construct infinite models so the finite satisfiability problem will be slightly harder but it is crucial if we allow to equivalence symbols to appear outside guards then the problem becomes on the side so in fact I will show you I think here is such a moment from the yellow element we had connections to pink element two equivalence connections E1 and E2 and we can split this element into two copies and connect one of these copies only by red relation and the other one only by blue relation so here between this pink element and the other element there is no red connection and again it is not dangerous because this normal form formula cannot use in the third conjunct they cannot use equivalence symbols in psi you think about loosely guarded fragment or something like this so I am sure that with transitive relations we cannot do so because this result generalized to transitive relations we can prove a similar result with transitive relations in guards so that those regarded fragment with transitive relations is undecidable I think that with equivalence relations it will also be decided the restriction to the guarded fragment this original guarded fragment is important F02 with three equivalence relations is undecidable F03 with two equivalences F03 sorry F03 with two equivalences no F03 with two equivalences F03 with two equivalences F03 with two equivalences with one equivalence relation is decided in elimination of quantities you think about the case in which equivalence relation is the only symbol in the signature aha yes thank you