 zvonim o sodaturno obheženju za vjelji vsov nradični fragments in nekome zvoljote informacije. Prigno, da vam je kroznjem nekaj nekajne plo. Se je plo v predstavnem z vsem, da se vziželim zdaj, evaž sem zdednjen in drugih pasjev, na dobro, mobilize propoje, to je nekaj nekaj sanji vsov, Zelo, to je teorem do Gredel, Vardi in Fokon collatijs. Tukaj to je izglednje površenje, kaj sem izglednjala, da se zbava vsega vsega vsega vsega vsega vsega vsega vsega vsega. Vsega je odpočena, da se vsega vsega vsega vsega vsega vsega vsega vsega. In ja vse moj sem vzelo, da so na dvoje vizila vzelo, in za to, ko se naredil, nekaj nekaj, je tega vzela, je ona vstalo tega vizala. In to se prišel, da je delalno. In druga noga vzela je vzela. Zvom nekaj je, da se vzela vzela, nekaj se vzela, nekaj je vzela, nekaj si je vzela. Vzela se, da je vzela, nekaj si sem vzela. do vsočenju systemu zarobnjenju in načinov. Zden je je bošan, da bošan je zelo jen vzal, da njo pričo se krečajo, da vedno se izpravim. K rečenju, pa bošan vzal, da se zelo vzal, da imamo, da imamo, da imamo, da se izpravimo vzal, da ne lahko se nasičim vzal. then the satisfiability problem becomes undecidable and for this we require three equivalence relations and possibly I will sketch the proof of the decidability of the two variable logic with two equivalence relations, so we will be very close to the border between decidable and undecidable cases. In the first thing is two variable fragments of first order logic. If by two variable fragment of first order logic, I mean the fragment in which we use only two variables, x and y, but they may be reused, so the quantifier depth of the formula may be larger than two. We do not allow for function symbols, because they quickly lead to undecidability. We can use, of course, equality symbols. As you know, from the morning session, the satisfiability problem for first order 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 fragment? 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 ghetto's class is decidable with equality. However, 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's class with equality is undecidable. So Scott's 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 at such a model and then verifies it using model checking, which was also introduced today in the morning. The bounce on the size of the model was later improved by ghetto qualities 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. And this is the first thing I want to show you, but before let me also give some other motivations for variable logic. So one of the motivations is that this is a maximal fragment of first of the logic with respect to the number of variables, which is decidable. Another thing is that many formalisms used in computer science, such as model, temporal, description logics and other things, are essentially two variable logics. Of course there are some exceptions, for example in temporal logic we use of the operator until, which whose natural translation requires three variables. There are also some additional constructions very often, but the core of the language 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 pairs 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 model there is at most one element satisfying p. We just write that for all xy, if px and py, then x is equal to y. In fact, this is a formula which calls that this gubbels class is on the side of a quality. There is also an example for using variables. We may say that each element satisfying p is connected to an element satisfying q by r path of length 3. In a natural way we reuse the 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, but we cannot do it with two variables. The argument is as follows. In a moment we will see that every satisfiable f of 2 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 there exists an r successor of this element, 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 strict less than relation. 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 transitive. 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, let us think that we have unary predicates p0, p1 up to pn minus 1, and they, using those predicates, we may encode numbers. Each element encodes a number from the range from 0 up to 2 to bn, in a natural way, as a 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. As it just says that there is a point in which one of the elements encodes one and the other encodes zero, 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. Of course, having such a formula, we now can easily say that a model contains all the numbers from 0 up to 2 to bn minus 1, by saying that there exists an element encoding zero and for each element there is an element encoding a number greater by one. I will use the notion of atomic types. 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. 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. 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 r, xx. It is 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. For example, the two type has two free variables x and y and we may say that p is true at x, q is not true at x, not r, x, x and in y p is not true, q is not true, r, y, y is true, r, x, y is true and not r, y, x is true. In our constructions we will consider only signatures consisting of unary and binary relations symbols. 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. If there are no relations of a greater r, it is enough, we know the structure. The theorem I want to prove is that every satisfiable formula is the most exponential with respect to phi. However, because I am 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 lemma 4 here. So let phi be an effold to formula and let a be its arbitrary model. Then there is a 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 is the slight 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 without touching one types to obtain a model. The first step in the construction is establishing a normal form. We show that we can concentrate on formulas of a simple shape. In this case, the shape of this formula 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 exist 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 even 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. Vokabulari. Vokabulari signature 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 a m of phi on this slide which says that we can get rid of symbols of parity greater than two. Construction also. So the proof of this normal form goes in the following way. We start from some formulas, sub formulas of phi which are of the form quantifier variable and quantifier free formula. And we substitute such sub formulas by unary predicates. For example, here we have just an example. We have a formula of the form there exists y, that's something, both x and y. We introduce fresh variable p and we try to enforce that for all x, p of x is equivalent to exist y side 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. And we proceed analogously with sub formulas of the other possible shapes and we repeat the process from down to up until we get formula in normal form. So 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, each of them 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 in the original structure. Similarly the connections between d and d and e. We only slightly modify the connections between c and d. So here is how the proof works. Recall the number four. We have a formula which says that for all xy 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 is called c here. What is c? So in c we put all the elements from the original model whose one types are rarer in this model. I mean whose one types are realized at most m times. Then we take all realizations of such one types to this fragment c. And for all of the remaining one types we choose m arbitrary realizations of them. So here on 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. Respectively two yellow element and three green elements in the original model a. All the remaining types, I mean pink, this should be gray I think 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 if we consider the substructure induced by this set c then 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 we consider the blue element in the lower right corner. In the original structure we had the so-called witnesses for this element and these formulas of type for all exist. 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. In the model of formula phi we had m witnesses for the blue element and these formulas of type for all exist. 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 choose witnesses for all the elements from 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 for all subformulas for all exist. 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 im realizations of each type which were realized at least in times. Of course e the elements from e have the elements from e can have witnesses outside this small fragment only if these elements have types which were realized and we can modify the connections between e and c. For example if you consider the gray 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 in c to provide all the witnesses for elements from e because for each element we may require at most im witnesses. So in fact this finishes the construction. We have 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 im realizations where m is linear in the size of the formula. The number of one types is exponential. The number of one types are to construct the 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 two choices. 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 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 phi. And it is a model of phi because in fact all binary connections are taken from a so the formula for all x, y, phi, 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. This verification as I told is made using model checking. 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. Next time. A lower bound is easy to obtain using this trick we can't take up to to the n. I'm going to skip. In the further part of my talk I will need one more simple simple observation. Let me define a notion of a royal type and a king. These are the notions which appear in the original paper by Gredel Khojtis in Pakti. So what is a king? Consider a structure. A king 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 xy if px if y 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. This 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 do it safely because we know that there were at least two realizations of the yellow 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 for all 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 this pattern. OK, so 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 define let me first define the guarded fragment first of the logic. So the guarded fragment was introduced by Andrei Kavanbentem 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. This logic is closed under Boolean 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. 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 exists to x such that 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 x is connected to y or by e we could also try to say here 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 that there is a royal type in a model we see that it is not easy we will see in a moment that it is not possible. It can be also observed that this description logic concept which I showed at the beginning of the talk is translated in fact into the guarded to variable fragment because this formula cxy and cy x are atomic formulas and are guards. A short review of results on the guarded fragment so Eric Gredesh proved that guarded fragment also has the finite model property every satisfiable formula has the finite model the complexity of satisfiability problem is deterministic doubly exponential in the general case and if we are strict the number of variables to some constant in particular to then the problem becomes x time 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 decided about one of these extension you will you will hear the talk by Dietmar. 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 and if you want to construct models from them we are enforced to interpret some binary symbols as equivalence relations so we simply have a list of 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 decided will be undecidable to obtain a decidability 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 final satisfiability problem both of them appear to be complete for non deterministic exponential so today I would like to to show you only this first theorem the general satisfiability problem is decidable in next time before just a short 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 is we have found that model problem analogously in none of the guarded fragment and two variable logic we cannot say that the 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 and here this is an example of a formula which has only infinite models yes we have two equivalence relations e1 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 for simplicity similarly each element satisfying q is connected by e2 to an element satisfying p and if you consider this first three three sub formulas we may be the natural model model for them yes 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 an 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 sorry 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 about 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 e2 class is trivial it consists only of it is just a single element 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 contracts for example for the element q we would like to reuse one of the earlier elements satisfying p as a witness for the third 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 fragments of equivalence guards we have to take into account these infinite models observe that in fact this guarded fragment with two variables and equivalence guards may be seen as an extension of full two variable logic because every first of the formula for example this formula in normal form may be translated to two variable guarded fragment equivalence guards by using one of the equivalence relations as an artificial guard artificial relativization of modifiers so instead of saying for all xy phi xy we say for all xy which are connected by one phi xy and similarly for these formulas of type for all exist this way 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 we get that the satisfiability problem for guarded fragment with two variable equivalence guards 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 it I will explain these three unravelings in a 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 when I say that there exist some elements satisfying some there exist 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 relations 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 for all exist formulas for all x if alpha of x where alpha is in guards there exist y and again either it is connected to x by an equivalence relation e i there is some number of equivalence relations or it is connected to x by a non-equivalence relation by a relation which is not required to be interpreted as 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 because we say that for every formula in guarded fragment with equivalence guards we can compute a formula which is a disjunction of formulas 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 next in next time so we can simply compute yes yes they are very similar the first one is guards of the form r i x y and the second r i y y x so ok they could be written as a single formula ok so going back to this lemma we show that for every formula we can compute a disjunction of normal formulas and in our proof we can simply compute such a disjunction of normal formulas because we are going to use not deterministic exponential time so there is no problem with this and we can simply check each of them separate yes where 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 sub superscript ek which denotes the fragment of this formula consisting of of those conjuncts 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 in this 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 this formula 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 picture there is six elements and what is the meaning of colors here so red connections represent this is an equivalence relation e1 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 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 from normal form yes normal form of the form exist x such that something yes 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 is required for which can be seen from this yellow element so we take its e1 class yes consisting of three elements so we add this orange element and this 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 yes 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 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 yes for all x for all y if are x y then something so here this is so splitting such pink element 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 to them by nonequivalent connections for example from this orange element we add a copy of red element yes and connect this red element with orange element in exactly the same way as they were connected in the original model yes we do not add copy of even class this red class for the orange element because such a class is constructed on the earlier elements for the pink element we have we have its red class built earlier so we just add its blue class and and so on we do it for all elements and we continue this process usually infinitely because for example in this example we see that there will be infinite path yes yellow orange red yellow and again orange element and so on so starting even from a finite 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 oh no no consider for example this pink element yes I look at the original in the original model and I I do not add an additional copy of red equivalence relation because this pink element in this tree has its red equivalence class yes but I add a copy of blue class and green class for for for the case of those black arrows I also provide them so for example for this for this green element which comes with its red relation yes to this tree red equivalence class the only thing which it may need it's this black arrow to blue element ok so such a structure it's still a model of sign yes because all all non trivial connection binary connections in the model I taken from from the original model yes non trivial I mean those containing some positive binary atoms and for all elements of course we take all the required witnesses yes because every element in this and every element in this model of say red type can see can see exactly the same the same types of elements which which were seen by red by red element in the original model yes can be seen and in fact is connected by the same types so the problem now is that those classes in in the original model and in 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 I'll say infinite number of elements yes as on this picture however what do we want from such an equivalence class we want it we want from it only to provide witnesses to the formulas from phi i blue is so in the case of blue connections formulas for from phi i2 sorry sorry consider just formulas which have i2 in guide so from this from this class we only want to provide witnesses to the yellow element for formulas of the form for all x exists y connected to this yellow element by i2 yes but in fact we can hear just to reuse the construction for two variable logic yes 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 go look at this normal form the purpose of the i2 class of the formula is just to provide witnesses for the elements of the fourth type yes with i2 as a guard the class in this i2 class we should also satisfy all the universal formulas which have i2 as a guard and those universal formulas which have non-equivalence relations as guards we do not have to we do not have to bother about formulas which have some other equivalence relations in guards because this formula psi cannot speak about about another equivalence relations so in fact what we do we look at such a such a i2 class for example as a model for fo2 formula constructed from this just from this conjunct containing i2 and all the remaining universally quantified conjuncts without equivalence relations yes and we assume that this fo2 formula says that for all xy i2 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 we may have from one point we may have many black arrows but what is the purpose of these black arrows to provide witnesses for formulas of the form 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 element in this tree we have some number of small classes the number of these classes is of course linear with respect to size of the formula and some number also linear of of the elements which are called free witnesses free meaning non-equivalent connected so 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 an 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 connected 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 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 how to construct e2 type for this class and how to construct free witnesses for this class for this one type we in fact need only exponential number of elements exponential 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 trilike model of this kind obviously if a formula has a model then it has a trilike model we saw a moment ago so for every we can choose a realization of a type take its class 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 this existential formula exist x at something and provide classes and witnesses to those elements using the seed we have an element we take its class from this from the pattern its class its free class and so on free witnesses then for the elements on the first level of the tree we proceed analogously if we need any e1 class we take it from this and so on this is quite easy observation and it follows the satisfiability problem for this logic is next time complete because what is the procedure now just place a seed for a model and check this is a proper seed checking the properties of a seed is quite 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 formulas yes, I think this is all I wanted all I want 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 yes, it is crucial if we allow to equivalent symbols to appear outside guards then the problem becomes on the side so in fact I will show you I think here we had from the yellow element we had connections to a pink element two equivalent 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 yellow element there is no red connection and again it is not dangerous because because this normal form formulas cannot use in the third conjunct they cannot use equivalent symbols in psi you think about those regarded fragment or something like this so I am not I am sure that with transitive relations we cannot do so because this results generalized to transitive relations we can prove a similar result with transitive relations in guards so I am sure that those regarded fragment with transitive relations is undecidable in guards transits undecidable I think that with equivalence relations it will also be decided the restriction to the guarded fragment this original guarded fragment is important 3 fo2 if 3 with 3 equivalence relations is undecidable no fo3 sorry fo3 of two equivalences no fo2 equivalent relations is undecidable that is an old result of the 70s fo is undecidable yes fo2 equivalent relations with one equivalent relations is decided in having a elimination of quantities you think about the case in which equivalence relations is the only symbol in the signature aha, yes thank you