 this is a joint work with Emanuel Kironski in Jan Ottopp And it is about modallogic Let me start with the definition of modallogic and the syntactic definition is very easy it's just a propositional logic with two modalities square, which is read as it is necessarily that and diamond, which is read as it is possible that the fact is that the decision is very easy, and while this syntactic definition is very easy, giving the precise semantic is a little bit harder, and the first approach to give this semantic was by giving the proof system for this logic, and this proof system was based on a proof system for propositional logic, and it was extended by one rule to axioms, and then we say zelo si očetavali, da vi svetič, že formula čistos v modu logicu, bo ime čas jo početno. In zelo, da je tamo zelo, da je tako početno potrebojga mikrovala. Poza to, konč trusti početnih logicov. Vse ob avvrjajoällen, Johnchar ne nevisimo. Proste, da smo predstavili v počubsih v modu logicu. Izgleda vi eventskih semactic v tem od mojce na noženju tudi tako kot krippke strakče. Krippke strakče je vse izgleda direktivne grafje, kaj je vse izgleda frame in valočnje propozivnje varajev. Vse je tudi frame in tudi je krippke strakče. Formula square phi je tudi tudi izgleda krippke strakče, in se vse glas flor je zadovala, ...,... jefle se nekaj glas flor zadovala. To se vsega, da there is a delovan konnekčen v oppunničnoče, in zpočencije nismo prvno... Vseznik nekaj avstrikin, potrebiti v bojri, ...,... je dve vsezniki vštrike vseznik. Zelo, ta axiom, diamond-diamond-p, ima diamond-diamond-p, korrespondi do klasov tranzitivne strane. V moj povedu, taj poved, kaj je tudi, tudi, nekaj je tudi, tranzitivne strane, je povedu, da diamond-diamond-p ima diamond-p, tudi je nekaj rizč. Zelo, da bo srednji srednji, da je tudi vsega, zelo je vsega, tudi je vsega, zelo je vsega, vsega, tudi, tudi je klasov tranzitivne strane, tudi je vsega, vsega, vsega, tudi je zelo, vsega, tudi je Neoparte Church si te i ta vsega. 3. 3. 4. 5. Zelo, to vsega rever zelo je. Prometo penguini. v tezno problem! Na skul Иvacчost iz frames K vse mladi dopravijo vso mnoka mnoka.sa mnodalje modela dve. Vse blali k frame vzbale VAMDA v sem vse vzbale v zelo. Na sve simultaneously vzluvati s globalizem s vzbadalej. Kaj smelimo, da bom izgleda model, in vzne, da vz quadranti tudi je uvršena o potrodi. Prejdeš tudi, da so tudi v vršenah v zelo stavili tega. Tudi tega obršena se bi z taken tri prikorajovano. Svoje zupaj, če je bilo, da si razpracujem, kako vzrenem, in je vzныхam. So you may think all those variables are quantified universally. Natural question, thing I ask, is are global cases always decidable for classes defined by 1.0 formules with free variables? In v 1996, zelo je vsezat, da vsezat, da se vsezat, da je vsezat, da je vsezat, da je vsezat, da je vsezat. Mormura je tukaj, je tukaj nekaj barajbolj, tukaj nekaj barajbolj, tukaj nekaj barajbolj. Tukaj je prišljeno 15 ljudi ljudi. V tem paperu z MFS-a. Zelo smo nekaj barajbolj, ali tukaj je tukaj biti ljudi. začeli z njih samo poživati števnji del, zelo tudi je ševega vsega prišljena. Naša najboljacka je, da vsega vsega vsega vsega vsega je vsega vsega vsega vsega vsega vsega vsega vsega vsega vsega. ki je vse zelo privalične, da je zelo, da je semplj, a je to vse semplj, ker je tudi vsezljena z 6 liter. Vsezljena je, da je vsezljena za vsezljena z vsezljena. In zelo, da je vsezljena za vsezljena. in lahko lepo vseh teželji skupaj v početku, pa načinaj vseh teželji skupaj, kaj je začal ta nekaj model, ali se pa bi, da se je zelo, in vseh teželji, nekaj katero je, izvrata, da se je tudi izvrata vseh teželji skupaj. In izvrata je, da svoje teželji skupaj poskupaj, z njal nal, z vsev, zhledajščjev. Ko je vsev. Vsev, človek, zelo. Vsev, človek, zelo, svojev, svojev. Ko je svev, človek, svojev, svojev, svojev. To prijev se da je kot nekaj počin v komplečnosti. Vse, nekaj, nekaj, nekaj. in očilji Repeat. Zelo, ta v verması, še zelo se povedila,ovanja še se zelo v trakoje, bar so BJZ zelo se povedila. Tama je zelo, zelo se povedila, tako, doolla bolo neko vodin, ko je vzela vodin, tako, da nekaj goten je, tudi je konnetil do zvoje in zvoje, potem je to nekaj razliko, posetil je pojavlje v tjih triangla. Zelo, kaj smo nekaj zelo, počekaj smo neskratil taj drugi problem, si je konek. Tak je konek, tudi je dobrovitek, Prejznaj, da se tukaj traktur izgleda vse formula. Vse, kaj smo zelo počke, nekaj nekaj zelo, zelo je preseksesor, tukaj zelo. Vsleda je izgleda vse zelo. Tukaj zelo izgleda za tukaj. There is also a model formula that makes every model of this formula look locally like this grid. And this formula says that each word has at least three successors. Oh, we used nine axlars variables P 0, 0, P 0, 1, P 0, 2, and so on to P 2, 2, And this formula says that each word has a horizontal successor, vertical successor, and diagonal successor. And this part says that it cannot be connected to it horizontal predecessor, vertical predecessor, and diagonal predecessor. And how does it work? Well, let's consider any word a. This part, again, says that it has three successors. And this first-order formula says that because here we have a word and two successors, then there has to be at least one more edge here. But this edge cannot be from this word to this because this is the vertical predecessor of this word. Similarly, it cannot be from this word to this because this is the horizontal predecessor, and this word is the horizontal predecessor of this word. And therefore, the only possibility for another edge here is this dotted edge here. And in the same way, we can prove that this edge here can be from a h to a d. And this is how we enforce that our models locally look like grids. And once we have a grid, it's really easy to prove that such a logic is undecidable. Simply, we encode some kind of domino problem in it. And for the local satisfiability case, we use trick presented by Edith Hessembler and Schnur. We simply observe that our model formulas work, even if all those words here in this grid are reflexive. And the trick is that we add one irreflexive word. We guarantee in this model formula here that such a word exists. We simply say that this initial word satisfies r, and all next words satisfies not r. So the word that satisfies this formula cannot be reflexive. And in using the first-order formula, we can say that this irreflexive word has to be connected to all reflexive words. And then the question about satisfiability of phi in all those points can be translated to the question about satisfiability of square phi in this particular word. Because now square says that we go to all other words. And that's all about our undecidable result. And now a few words about the satisfiability. Recall that universal whole formulas are formulas of the form phi1 and phi2 and phij, where each phii is a whole clause. And that means that it is a disjunction of literals of which at most one is positive. And here's one example. We usually write set of whole formulas as implications. And this part defines. It says that each model is transitive. This is a whole formula with no positive literals. And, of course, there are more examples here. And what is interesting, that there is only a finite number of universal whole formulas with three variables. Well, of course, we can write something stupid, like p and p and p and so on. But if we exclude this kind of formula, then there is only finite numbers of such formulas. And this number is still very large. So it is impossible to just check each of them. And, okay. So the idea beyond our proof of decidability is that we observe that model logic is decidable due to three-moder property. And in our case, there is no set property. We want to get as close to this three-moder property as we can. And we define the closure of three with respect to universal whole formula phi as the smallest structure that contains t and satisfies phi. The smallest with respect to the number of edges. And one of our technical lemma says that if model formula phi is satisfied in some class defined by horn formula, then there exists a tree such that phi has a model based on the closure of this tree. So what we are going to do is that we are going to study the possible shapes of the closures of three, depending on the universal formula. So we define four simple classes of frames and we associate each universal horn formula with one of these classes. And we show that closure of any tree with respect to the universal horn formula belong to the class associated with this formula. And then we show an algorithm, we show algorithms for all these classes. And now I'm going to give you at first some examples for each classes. Then formal definition and then a few words about algorithm. So the first case is very simple. This formula here, it defines frames in which all words except for maybe the first one are reflexive. And another possible formula is this one that says something, but if we apply this formula to this tree, we don't have to add any edges, so simply the closure of this tree is this tree. And in this very simple case we call this class of models semi-trees. We say that a frame is semi-tree, and it is contained in the reflexive symmetric closure of this tree. So we can add only edges that make some words reflexive and some edges symmetric, and that's it. And in this case we can use standard algorithms to solve the satisfability problem. And in case of local satisfability it leads to p-space completeness. And in case of global satisfability it leads to x-time completeness. Second class is a little bit different. This formula here, it defines transitivity. So all frames, all closures of all trees with respect to this formula are transities. And our second class is called class of transitive trees, and we say that a frame is transitive tree, it contains tree. It is contained in the reflexive transitive closure of this tree, and it contains long edges just to distinguish it for semi-trees. And another by very easy modification of the standard algorithms we can show that in this case both local and global satisfability problems are p-space complete. The third class is more interesting. Consider this formula. It says that if x is connected to y and y is connected to z, then z is connected to x. So for this word it means that from this word there is an edge to the first word, and so on. And it turns out that while this model, which is a closure of this tree, seems to be not regular, then we can rearrange the words here and obtain three words with this property that every word from this first part is connected to every word from the second part. Every word from the second part is connected to every word from the third part, and every word from the third part is connected to every word from this first part. And in this case, we call it free partition, and we simply say that a frame is a true partition if its words can be partitioned into three independent sets, but there are edges between corresponding sets. And what we do here is that we show that each of those three sets can be reduced to the set with polynomial size. And if you have only three sets with polynomial size, then it leads to polynomial model property, and it leads to NP completeness. And our last class of frames is a little bit more complicated. Consider this formula here. It says that each word, except for words without predecessors, or without a predecessor that contains predecessors, are reflexive. So all those words here are reflexive. And this part says that the successors of reflexive words are connected. And it makes those models look in the following way. We have some three at the beginning of the model, but then we have some number of clicks. All those words here are connected in every possible way. Another example here is as follows. This formula says that all words, except for the words without successors, are reflexive. And this formula says that if a word is... Oh, sorry. For each word, each edge that leads to the word that is not... Okay, sorry, once again. Each edge that leads to the word that is not the last word, it contains some successors, then such edge is symmetric. So at first we add those symmetric edges here, but not here and not here. And this part says that if a word is in symmetric connection with some other word, then it is connected to all other words. And it means that what we obtain here is a big click here, and only those words without successors are not in this click. So to sum up, our definition of the fourth class is probably unreadable, or read it anyway, is as follows. A frame is a click union. If it can be partitioned into head, some number of clicks and tails, where head is a tree above those clicks, then we have some clicks and some tails that can be outside of this click, but both this head and tails are of head at most, too. And in this case, we prove that it is enough to consider only polynomial number of clicks with polynomial size, and it leads to the polynomial model property, and it leads to NP-competence. So to sum up, we proved those three results, and the complexity for the further result is presented in this table. So thank you for your attention.