 So yes, I'm Chad, I'm a graduate student in Tallinn, Estonia, Tallinn University of Technology, and this talk is about a proposed sort of foundation for these ledger systems that we see in blockchain technology. And I'll just start off by explaining sort of what I mean by that this is probably a bit different than most of the talks. And the thing to start off with the thing I'm going to assert is that these distributed ledger systems have two parts and the two parts that they are sort of orthogonal to each other they interact a little bit, but not a lot. And the two parts of course the distributed consensus protocol and the ledger that you use the distributed consensus protocol to maintain. So, think about the second part in this time care about the consensus protocol we're going to talk about the algebra of these ledgers. And in particular, we're going to see that the algebra of these ledgers is to be found in something called a monoidal category. So we'll have a sort of correspondence between a notion of ledger and what we call a symmetric monoidal category. And before we start again we're going to talk a bit about what algebra is because this is a bit subtle, and I'm going to lean on this later we're setting up an analogy. And so the first point is that some so if we think of addition, for example, there's a sense in which seven plus three, say and three plus seven are the same thing. So they're both 10 was sort of denote the number 10 and they're equal in that way but in another sense seven plus three and three plus seven are completely different things and we just have to look at them to figure out why we sort of got these terms and the terms are different but they might be the same. For some notion of the same. This is sort of what algebra is about or at least that's the perspective that we're going to take here. So in particular we're going to have sort of our, our ledgers as things like three plus seven and seven plus three, and we'll be able to talk about when they're sort of the same like when this sequence of events on the ledger would be the same thing. So we have to talk a bit of math first but you don't. Mostly we have to look at these pictures because these pictures are good enough, they're sort of a fully formal or rigorous syntax for the things we'll be talking about. And I think they're a lot more intuitive so we're using them pretty heavily. Right so a category is a thing like an algebraic structure that consists of a bunch of objects, and a bunch of morphisms and morphisms are things that are from some object to some other object and we'll sort of draw them like this on the left. And then we have to draw them from F from a to be in our category. And if I have morphisms where the thing that one is to matches the thing that another one is from that I can compose them to get another morphism is why I could sort of stick F and G together to box wire things like this in the middle. And then they satisfy some rules so this composition has to be associative and it has to have a unit, whoops, which is just sort of the wire with no box and so on so a category is a bunch of things that you can compose. We're going to talk about categories with a bit more structure in particular monoidal categories, which are ones with a monoid structure on the whole thing. So this means that if I have a morphism F and morphism G I can put them sort of beside each other to get one that does both of those things in parallel if you like. And then a monoidal category is symmetric if I can cross these wires in a way that makes sense. So that's very quickly what a monoidal category is. And the way that we're going to read these diagrams and in our monoidal categories is as the whole category really is as a theory of resource convertibility, where our objects the things that our morphisms are from and to are like collections of resources. And our morphisms are going to be ways to transform the collection of resources that they are from into the collection of resources that they are to. And so, for example, we might take our sort of basic objects to be just like bread dough water flower and oven to set up a sort of naive theory of resource convertibility for bread. And then we might have our morphisms generated by one that allows you to mix water and flour and transform it into dough here on the left. And one that allows you to bake dough in an oven to get back some bread and the oven, you're in the middle, and one that allows you to need dough to get more dough on the right. And so that's sort of the thing and once we have this we can. So I'm hiding a lot of algebra here we're going to generate the free monoidal category on this data. And in that we'll be able to construct things like this this will also be a morphism there. So basically we can read it kind of intuitively as describing the transformation of well two bits of dough and another into two bits of bread and an oven by baking one piece of dough in the oven and then baking the other piece of dough in the same one after another. And so that's how we'll be thinking of these morphisms in our monoidal category is as transformations of resources. The syntax part the three plus seven is not the same as seven plus three part. And the algebra part will be that to the axioms of a monoidal category, more generally, our resource theories will make two of these string diagrams equal. In case the sequence of events they describe is sort of the same sequence of events in the sense that it would have the same effect on the resources involved. So for example in these two large diagrams on the left, we see we're mixing and needing water and flour sort of at the same time in parallel twice and then we're baking the resulting dough in sequence one after the other. Whereas on the right, we mix and need and then bake and then after we bake the first look we mix and make the second look so these are strictly speaking different procedures, but they would have they would do it the same thing. That's according to our theory, they might not do the same thing if you actually baked bread but that's okay it's a model. And so in this way we can read these string diagrams I'm going to the word I'd like to use is material histories these string diagrams tell us about the material history of some stuff it's a little slice of that. And from material history is exactly what a ledger is, in particular, a ledger like say the Bitcoin ledger is the material history of all the coins on it. That's the point of a ledger. And a transaction as well a transaction is just a string diagram and we can sort of append a transaction to our ledger expressed in this way by sticking it on the bottom. So I have three big ones and I wanted to do something with the bits of bread, maybe the oven on the bottom while I could stick more string diagram on the bottom by composing another sort of string diagram of mine. That's how we would do that by composition. And then like I was saying with that picture before equal transactions are sort of these material histories that would have the same effect, according to our, our occasional theory, and similarly a ledger is just like a big transaction so two ledgers would sort of lead to the different states of the system if they were equal in the theory. So that's, that's main idea one probably the most important idea is that the algebra of these ledger systems that we see in distributed ledgers distributed ledger technology is to be found in what we call symmetric and thing to sort of. Okay, so one thing that we care about in distributed technology, especially is ownership. You know, most of the systems that I'm all of the systems that I'm aware of care mostly about modeling who owns what or who is responsible for what. So the second part of what I'm going to talk about is how you can model ownership in a nice way in these symmetric monoidal categories, which I'm just going to call resource theories. And in particular we'll get a nice graphical calculus that will be a lot like the one we saw before with diagrams like this, but with additional information allowing us to denote ownership and change of ownership in a sort of rigorous way, which we'll talk more about at the end for now we're just going to use the diagrams. And so to set this up we're just going to assume a set of things that are capable of owning things. So these could be public private key pairs that's very cryptography thing but I'm just going to call them Alice Bob Carol and so on. And we're going to associate each of these with a color because our, we're going to use the color in our diagrams. So Alice is blue blue and Bob is yellow and Carol is red for me. So I want to go to the next slide there. And so the idea is that we're going to start with one of these resource theories one of these symmetric monoidal categories and we're going to construct a new one by coloring everything in, or sort of by overlaying color on everything and our diagrams, where a color corresponds to ownership by one person and something is shaded blue that's going to mean that it's owned by Alice. And formally this corresponds to us having objects like x Alice and why Bob that sort of x owned by Alice and why by Bob, and also we'll have combinations of these so we could have one object that is excellent by Alice and why by Bob and so on. And morphisms are just, you know, again, there are there are colored in morphisms from our original resource theory. The idea being that anything you can do to resources you can do to resources that you want. That's kind of the, that's whatever she means. And so a blue morphism will be Alice transforming some of her resources, as opposed to just a resource transformation happen. What we've already introduced in doing this is a difference between sort of Alice having an X and Alice having a Y and Alice having an X and a Y, it's sort of a silly formal difference but we have to mediate it. And this, this is something that crops up in for example the Bitcoin system where there's a difference between having two $1 coins and one $2 coin because you can't pay Bob $1 with two, sorry with one $2. And so we'll introduce morphisms resource transformations that allow each person each sort of different color to split apart and combine the collections of things that they own. So just just the intuitions given by these diagrams here below. And they have to satisfy a number of axioms and these axioms are just things that you would hope are true but will corresponds to a sort of canonical mathematical structure. The two on the top here are asking that, well, we'll just read what they say is that if Alice does F and G to two collections of her things separately and then combines them that's the same thing as if she had combined them first and done F and G to the different parts after. I'm realizing that the one right is the same diagram that should be the diagram on the left upside down and that would say something about splitting and doing things. Combining and splitting must be associative and so on. This is just what you would expect to be true about collections. And then we need some axioms to deal with empty collections to be sort of fully formal it turns out to work out much better if you have empty collections as well. So this is just at any time from from nothing Alice can manufacture a collection of nothing and vice versa can get rid of. Are we doing for time pretty good. We've got these colored string diagrams denoting sort of material histories of things owned by people, of course ownership changes over time so we're going to introduce some new resource theory more morphisms that allowed the owner of a thing to change so for any two colors of people and a sort of thing here X, we introduce a morphism that just, you know, in the way we read this is that x belongs to Alice and here the color changes so now x belongs to Bob, it's like a change of ownership thing. And this has to satisfy axioms as well just to make sure that it's coherent with respect to all of these collection management axioms we introduced earlier. So if I have some X, if I have some things X and some things why I combined them into one collection, I have some x and y and then I give that to Bob, this ought to have the same effect on all those things like that. It's the same thing as if I had given them up to Bob first and Bob combined them, and so on and so forth for empty collections and splitting collections apart. Whoops. And then the last axioms will ask for is that that sort of here stating that the ownership of a thing is somehow immaterial and that if I perform a transformation of some things and then give them away that's the same as I had given them away and somebody else had done exactly the same things after. And finally we ask that. I know this is a lot of axioms. We'll talk about what they mean mathematically. So we have axioms where if Alice gives something to Bob and Bob gives it to Carol that says the same effect as Alice just giving it to Carol that's here on the left, and then if Alice gives something to herself that's the same effect is nobody doing anything. That's your bottom right. So when we have all these axioms and all these equations we can represent or we can sort of represent and derive things are in our equation theory, like this, and this is sort of the point of all of this, where we might read this diagram on the left, as the transforms well you know an oven belonging to Alice, two bits of dough belonging to Carol, and an oven belonging to Bob into bread belonging to Carol and ovens belonging to Alice and Bob. The one on the left describes the sequence of events in which Carol gives one piece of dough to each of Alice and Bob, then Alice and Bob bake that dough in their ovens before giving it back to Carol. Whereas the sequence on the right describes another way of doing this where Alice and Bob each give their ovens to Carol Carol bakes the dough into bread and then gives the ovens back. So there are different things, but they should really have the same effect. And indeed in the equation theory that you can find the details in the paper. But in the equation theory we've sort of been building with these colored diagrams we can derive that the corresponding morphisms are equal so like really formally these have the same effect, which I think is kind of neat. So that's, that's sort of the, the main thing I suppose I guess I have five minutes left and we'll spend the rest of the talk talking a bit about more of the math so we did this thing with all of these colors and all of these color components to be a bit normal we could also do this by asking for a resource theory, we'll call it x, just a symmetric model category, a set C of colors and we'll define the colors of X C of X to be well the free product category on X cross C. And with some additional morphisms given here in sequence calculus form subject to 18 equations just corresponding to all of these things I've been talking over for a while now, you know these can all be written down fully formally in the notation you see here as opposed to the diagrammatic one but I'm not going to put that on the slides. So, right. And once we do this, our equations and our additional morphisms and our whole setup is largely characterized as health that so for any symmetric model category X, this is our original resource theory, and any set C of colors, you know, we've sort of our axioms insist exactly that there's a thing that we call a strong symmetric monoidal functor from X, our original resource theory to see effects. And the idea is that anything that is blue, anything belonging to Alice is sort of in the image of this factor, if that means anything to you. And then that's sort of our collection and splitting apart axioms. And then almost all of the rest of our axioms tell us that for any two of these functors these sort of owned by Alice functions and owned by Bob factors. That was what we would call a monoidal and co monoidal natural transformation between these two functors. And that's that sort of this thing where what we call the components of the natural transformation are given by this change of color and our axioms ensure exactly that it has these properties. So if this means something to you that's great, if not the point I'd like to make here is that these are very canonical things it's kind of remarkable that we can use this canonical structure to model this our ship thing in a nice way. And then something, one final math fact is that the, so there are two axioms that we don't need to get that previous proposition, the bottom two on this slide. And these axioms turn out to give us exactly that there's an equivalence of categories between our original resource theory X and our new colored one with ownership, see effects. In fact we get a whole bunch of equivalences but they're all equivalent to each other. And the sort of the point of this is that we know we haven't made a mistake in adding ownership, we know that we haven't accidentally destroyed or created any of the categorical structure. It's sort of the same thing structurally but with colors, which is nice to sort of good. That's all I wanted to say. If you found this interesting I'd encourage you to have a look at the paper. It takes a while to get used to sometimes, but beyond that I guess we can move to questions. Thanks for your time.