 Thank you for your invitation so today I will talk about an algebraic approach to the equivalence problem of MSO definable transactions on graphs of bandwidth tree widths. So this is ongoing work with Nikolaj Bojanczyk and the main result that I will prove today is that the following weaker version of equivalence is decidable. So when we have two MSO transductions are the equivalent in the sense are the outputs equivalent with respect to some equivalence relation which is not isomorphism but something weaker that we will define later. If it would be isomorphism then it would be equivalence problem. So let's recall that MSO transductions are not deterministic but in this tank functional graph transformations described by MSO formulas. An example transformation would be like this consider such formula which says that there is a walk of length 2 from x to y going through z. Variably so it transforms such a graph as you can see to a graph like this. It connects nodes whenever they are connected by a walk of length 2. So this transformation does not use the full expressivity of MSO transductions it's a simple one. And also let's say a little bit about graphs of truits k. I'm not going to define exactly what is the definition but let's just say that graphs of truits 1 are exactly trees and in general graphs of truits k are exactly those that admit a certain representation by a tree called a tree decomposition. In this case it looks like this. It's a way of packing vertices into packages and forming a tree out of those packages somehow consistent with the graph. What is relevant for the talk is that if equivalence is decidable for C to D MSO transductions when C and D are classes of graphs then they must have a banded tree with they must be subclasses of banded tree with class. So that makes our problem or the most general that can potentially have decidable equivalence. And to equivalence relation that was set to be defined later. For graphs G and H let's say they are related if there's a square matrix with real values that are possibly negative such that such an equation holds those are agency matrices and rows and columns sum up to 1. So that would be isomorphism relation if we meant non-negative integer value the axis and if the constraint would be non-negative real values that would be fractional isomorphism. So what we have here is an even more relaxed version which allows also negative coefficients. But what we will be using in the rest of the talk is the characterization by counting walks of length n for each n on graphs. If the number of walks are the same then the graphs are equivalent in this sense. So this is how it looks. And the proof is built on two theorems. Well one is that equivalence is decidable for 3 to Q registered transducers although Q can be replaced with some other rings. This result was already mentioned today. And also that MSL transductions that we are dealing with admit a characterization by registered transducers but over source graph algebra. We will define it later. So the general idea will be to somehow assimilate registered transducers over source graphs with polynomial automata. So with those 3 to Q registered transducers. So what is a registered transducer? A field of rational numbers or maybe a monoid of words or an algebra of source graphs that we will be using. So a string to A registered transducer is essentially a deterministic finite automaton. Additionally I put some registers that store so in an example let's say that we have two registers and they are initiated to 0 and after each letter they just add 1 if and only if it was an A or add 1 to register. If B was read the output function is multiplication of the register. So it's easy to see that such a transducer computes well the number of A's in the word multiplied by number of B's. Okay and let's remark that even though it's presented for string to A the equivalence result holds also for the case where trees are in the input. And as already mentioned equivalence is decidable in case A is a computable field. So for example rational numbers or field of rational functions but actually we will use another ring. Let's also say a little bit what is a sourced graph. So it's just a graph but some vertices are given names and there can be at most one vertex with each name so in this case there's a one, there's a three and there is no two. We define two operations. One is called the join. The name kind of speaks for itself. There's a join operation which essentially glues two graphs along the vertices that have the same names and the forget operation which takes away the name from a vertex and it becomes unnamed. So this is the relation and as we said the idea is to somehow simulate transducers that deal with graphs with transducers that deal with some ring. So this will be a ring of formal power series in this case or actually graphs in module of this relation. So the idea is to map a graph to such a formal power series where the coefficients are exactly the numbers of walks. And with such a definition it's straightforward that therefore graphs are related if and only if they have the same series associated. We have to extend it to source graphs somehow. So the idea is as follows, extend it. We'll use somehow the names. So let G, I, J where I and J are sources be a formal power series like very similar to this one but it counts walks from I to J that have positive length. We don't count some trivial walks. And what is important is that integers do not touch sources. So a walk that starts at one then visits someone named vertices and ends in three is okay but if you ever visit a three you have to end the walk. So in this particular example G1, 1, 2, 3, would be, well what can be the walks in here? For each length there's a unique walk. And it has to be of even length. So there's one walk of length 2, one walk of length 4, etc. So G1, 3 where there's no such thing. Oh something I didn't say about by G1 bullet we mean a walk that starts in one end and ends in any other vertex that is not a source. Yes, yes, yes. Yes, yes, that's true. So there's a mistake on the slide. Thank you for that. I'm sorry for that. It should be x plus x squared plus x to the third power. Well all the length should appear. Okay and let's not count the fourth one. So what do we precisely mean by simulating this algebra? Modulo this relation with a ring of formal power series. It exactly means that we want to reconstruct the series for joint of the graphs out of the series that we had for the graphs separately. So what can be the formula in general for this? Well the graphs, the walks from i to j in the joint of the graphs can either go entirely through G or entirely through H because they cannot touch sources in the middle. So it's just the sum. The formula is pretty simple. It's not actually true in all the cases. But it's morally speaking true. And for forget, well what can be the walk from i to j? It was either a walk from i to j that didn't touch k anyway or it touches the k because it's allowed to do it now. So it goes from i to k, then loops makes some number of loops, possibly no loops. That's why there's one. And then goes to j. So the formulas are true. Well they are clearly true for the sets of walks. Not for the series but just for the sets of walks. But then a series is just the set of walks where you replace each edge with x and you replace concatenation with multiplication essentially you just rewrite it. And this respects addition and multiplication that's why the formulas still hold without any problems there. So what are we left with right now? Now we have to show that equivalence is decidable for register transducers that will initially have polynomials in registers and can multiply and add them but also can use this cleanie plus and produce formal power series out of this. So if we analyze a little bit, we see that f plus is a rational function which has two big consequences for us. Well the first one is that it makes computation possible because from the beginning when we are dealing with formal power series, well it's not a countable object. So in principle it's not clear how to make computations there but in this case all we will get will be some rational expressions. So this is why it's a computable ring that we're dealing with. So it's not a polynomial function. This is something I think I forgot to mention that in those formulas for reconstruction it is a crucial property of those formulas that they use only addition, multiplication that they are terms over the algebra that we're using. Like it's not any function. Let's go back. So since we are trying to reduce to polynomial automata, that can update the registers using polynomial functions in a usual sense polynomial, just plus and times. This would be ideal if a function that maps f to f plus would be polynomial. So it's a rational function, a fraction of polynomials, not that bad. And this is just another problem that has to be dealt with but I would not like to expand on this topic. But we were able to prove that even with division it doesn't matter that much what ring is there, equivalence is decidable as well. So this finishes the proof of the main results. It's a way of simulating registered transducers over source graph algebra, a relation that we mentioned with formal power series. Well, it uses clean star but if you express it as a rational function and prove an additional lemma, equivalence is decidable for it as well. The future work that we plan is to use some, maybe use some already known graph polynomials because there's lots of them to obtain some equivalence relations and graphs that are closer to isomorphism. Or maybe are equal to isomorphism on some restricted classes of graphs. The first two examples that we checked were characteristic polynomial which is a characteristic polynomial of agency matrix of a graph. And also a TAT polynomial which is a generating function that counts a number of occurrences of some patterns. And it's also used for counting, like lots of things, counting colorings of a graph, counting, matching, etc. Well, the result is that they fail badly to characterize even trees. So the simplest class of graphs that we could consider. This is the result from the 70s that almost all trees, in the sense of the proportion of the trees that have the property, almost all trees have the same characteristic polynomial. Given the number of vertices, almost all of them have the same characteristic polynomial and they all have the same TAT polynomial as well. So such a mapping would not help us at all. Another example would be to construct another way of finding polynomials would be to construct them as the generating functions, maybe counting some patterns. Or maybe inductively on graph structure. Maybe it is worth mentioning that, for example, TAT polynomial admits both kinds of definitions. It can be defined as a generating function that counts some patterns but also can be defined structurally. It can be reconstructed from the polynomial of graph, it can be constructed recursively from a graph that has got deleted edge, deleted vertex, well, from subgraphs. Let's not say precisely. Okay, this is all what I wanted to say. Thank you.