Because the behavior of his body can be written really as okay? Or by B, 20 that remember. So stay based systems tend to take to use an algebra will events. They basically give up operations, the namely events and they use operations to determine how to combine different behaviors. So if you combine this with another machine here, which is easily a C or a P then now this format changes my phone. I've not exactly closing off. So it's AB CRV know the you get an A and full of Ib then he's followed by. These are and that can be dependable. So they're kind of the nice thing about it is about debates is, I mean, no, this example system doesn't. They have an algebra event. And then so they compute things, you take two things, you and put an operation, we get the behavior of the composite. So this very attract and many, many languages that I mean, my inform of models that are based on how to provide events, you know, better do that. And I mean you may need help from Obamacare. I'm sorry. CBN. CSP Google's, I'm going to sequential programming, there's CCS a spike outdoors that, you know, of a lot of models and they use very much stuff, right? And that's what appears a lot of people. It's the start nature quite naturality. This led to building actual systems and build Lego building out of specifications. So for example, control systems, they are interested in the events. They don't need they don't need to clearly about, you know, hey maybe case I'm bigger, don't worry about that. We want to invest data communication systems, don't know they've got languages, like, Estelle. They've been built all on top of event models, and those are the kind of becoming international standards for telecommunications other. So there's an entire conscience there. They basically say no state is fundamental state, is it? And that appears to me is a state is easier for understand. If I tell you that, I have a very good X in the program. You can just, yeah, that's your state eggs, whatever value is. And that's your state. It's much easier to think about about it. Now the problem was with kind of a little quote about events or estate events, represent the state change, No States. Transition one possible state foundation. Is that you move from this state here to speak alpha back into when we went to a If you are releasing about this state state has never changed. So is that that's kind of difficult for us to in the state models? That's a cork, you know. So we can have changes in the state. I'm not really invisible to. And so that can of course discomfort but generally the state is much easier to deal with and you can see the state in terms of places. And level in the system is a control cell to the data set in a good focus on the. I mean in that case basically they're looking at the data stream which separates. Okay? Is a state has to change. You're not going to compute, we need to stay generated, change the state. So how do we change things? So basically, they decided that the best. The first one could be them does is that the feministic Assignment. So that the determination of assignment as it was selected, as a way to change that, basically, to compute Now that a couple of issues here. First of all the y, why something that is atomic. But if something is not the topic, it becomes very difficult to the many systems, you know, automation is critical, you know, transaction systems. You know, we basically you know, we assumed that transactions are atomic. I mean that was feedback about other complexion in the nutrition. They're not invited by other transactions in this system, operating systems without obesity of operations. When allowing them to register your employment, registering one single and clock or do we have this one beaten at a time if I've been wanting to try that or visit into our build the entire registering outdated environments in one step, then it's at the register level. Does that very important thing. I mean, the effective way you can write the circuitry, you know, being able to not to do that one. Step, there's things that assignment is something that we understand very well Simon by express express expression. I mean, there's an expression of value and then aside the value, we were very much had an assignment, why you connect plus one. Basically, you compute the expression, x plus one, and you take the value between your the cardiology of the entirety and change. So you know if the system is able to view full pages, memory pages in one single operator from a corporation then you know, talk about kilobytes of data that moving, you know, atomically in radius sign. If you can only move one more, So you can choose a number right here. Okay? So atomic basically means that this it's all of nothing scenarios. These are high succeed and make all the changes or no, I don't make anything. So in some sense when I'm executing this operation, whatever you wanna be transactions up the memory and whatever. Nothing else can happen, largely In reality, lots of things are happening, But from a logical point of view this atomic. If nothing could impact the, if I started this thing and exactly this code, I know that that's fake and doesn't matter. What else happens in the world? That's part of which is all and that's exactly it database transaction. So a silent easy way for us to use it in every product. So that's why they stop objectives as a message was because turned out when we used concurrency and permission is of primary problems. So, if you take, if that I can see and you say x equal x y, + 1. When you write that line of code, you want to think in that all of that is going to happen. The way it sleeps them down. You need the value of why you are the concerned by one and put the next, okay? What you really doing? That's obesity assumption. You have assuming that it's upon me, If somebody else is shared, why would you you read? Why is three You had one before You take the four and you put into x but the time you put the four into x y is actually 133. So the relationship between the y when you finish operation, you should have 124 units. But you have four because somebody's moved it very well underneath doing, unless you are using the computation. And the problem is that many anything I actually imagine to literally value the problems with see where things were not the way you normally expect and because of, but things are not computed the way you expect them. I mean, to order the variables with different in the computation become comparative of optimizing things and I was getting the wrong results. So so anyway, so we are very most this concept very variables and have a similar segment that can multiply this value. So, they saw that. That's also due to importance. Now we've done this thing is longitude missing. Slow control. What is flow of control? When you when you write this equation, the program, if you start in the first line because the first line, then you move to the next line. The next line. The next one you might have a look, but it's always you. Kind of know what the next line is going. If the default database, if anyone that you know, the state of the variable are you know what you're going to not atomistic is when you're trying to basically the order which things happen, it's it's not controls. So for instance, you got any two processes, Venus processes and they this is opening system, the schedules when this process is going on as they can get the CPU, you don't know exactly how they going to be. I'm going to be getting something to be next. We need execution, and that's what makes it more common. So it makes them debugging because really you don't, you don't have this. Projectable you can run the program once and you get an arrow and you're not even getting it's not and don't know why. Well, it had to do as a opening system. Scheduled things in a different order because somebody else was logging here at that time and slowed down this process. And therefore, this whole thing was done before that saying, and that's why you do there. But if you are done in different orders. So we use one of them as a way to model cutters in comparison systems. That's only when we begin to I mean we can't do models that use access exclusive productivism and actually our model can do that to those say that, but but it's not the way we can think about component system because we don't have control over the time. We don't have that control over the old that induces. So we always assume that things are happening in some order that we do not control and because we only we don't control, we think is not the can be fully deterministic at the operating system, which just we don't we don't don't think a lot of problems. I just wanted the mystic and images, they just happened. I mean if you have input, if you have I know and we are all inputting data to the same, for the same file. But what I am looking into controls that because we are, you know, doing it in front of another episode, he has really fashioned. So that's why again, monetary is when his present many products. So the arguments basically 100%. So control is a one, another problem, with mist, or download. The now we can work with our builders one, but we don't know the order interesting are happening. And the other thing is that forever, we always thought we all are model. Concurrency is a one to ten basic instead of leaving of state. So, therefore became interested in fundamentals. But what is really? So that's, that's why they want to notice anything. The other thing is it not an interesting thing to flow control. Are not under. Whether the flow of control points are the from broad structure that sequence, do I have? And that's our flow of control in some that's to use that consumer code to express algorithms in English. But basically this big claim that this is just a left over from sequential thinking, it's a leftover from, you know, when we first sized on machines, when we first developed models of computational machine, every single sequence. So the so the flow of control was a socialist sequential language. It is not necessary. It's not fundamental. If you're dealing with complex so replace any law control with. So that's going to be a fundamental very different way. You're going to have to think and wrap your hands are on the in a very different way that you ever done. Because of this I think is the ratio of simply So synchronous, if you can think of it almost like a longer the line of things which are happening in Rockstar, That's it. So a lot of super computers What they do. There are many different machines. So there are synchronous in that sense, they all run into own schedules and but the wrong computation the way it works is that if machine and since everybody done every time and everybody is done, they extend information and they start again. So, that's a synchronous motive execution at the ground and the high general level. However, that machines that are actually built to be to be synchronous, like this is stalling machines, exotic arrays. So there's a machine, the basically, have memory the processors and the processors can be nice. So some processes are turned off in different tasks but the processors at all working on the same kind of control. So everybody stored really to read, whatever they need Computer, the whole computer whatever they need After better. They all are developing me And it's all done, really synchronously. The circuit is the last person. The last thing that we go to about the chip design, what's on the chicken. One of the most important things on the cheap is apply. So all the gates, they fire when the plant says go, So there are all symptoms. So, basically, the their response is that even if we are not able to express those synchrony may simply we're linking our mind. So, I have to be able to do this, and that's fundamentally. So, what happens? They use this concepts to build the formula. The social is a formal operation of phonologics. The formal logic of God allows us to think about the programs and that all the things that they do they like forensics. I mean, once you have the phonological and have problemization, this is to code and this is through screen. That means that we put, for instance, this formal logic to write a span is a formal notation that I could program, that's a connection between the two. If you have a program and you have a spare, your my one to prove that the program needs to play all of these things can be done, is formalization the other thing that you can do and you know very much about some but very zero. It's got that you can disappoint the specifications start to leave J program in a stifle as you find the buses. So, basically, Saturday, specification of the system which is fairly high level, Very magicalistic. And then there is a specification that you prove that is equivalent to the previous one, it just more restrictive. It's more deterministic because less number and I keep on doing that and you restrict the number of solutions to the point where the look at the spec and the solution becomes your obvious. Anybody can just take this, let's write the code because it's so low level. So detail just kind of the simple solution and the simple thing is that I need to get out of the class. Okay, so that's my goalie. I'm here. I need to get out. So that door there, many thousand different ways. Now, I can do that. The spec only says that you move from the desk to the gopher. Now, if you put a constraint that you can download through the rows, then all the solution that goes through, the rows are now eliminated. The only way to do is go this way, it still satisfied. The original condition requirement. The environment was moved from a test to the door like I can still do that by no idea limited. The way I'm, I can put another limitation and say, well, you know, another robot and only take, you know, 1990 degree angle. So therefore I start here and I go straight and when I get here, I take the right turn. Then I go there, see now in the same space but we reduced and both it becomes more constrained and then you do or spending your concerning you are trading here to get so this is a really quality of formal methods. They would like basically write a formal specification and the code from it in the code is automatically recording It does. All of that. In some situations, you can actually write one of specifications. This is a compiler in the quantity of the code correctly from the, you can do them a lot of things like that. But you can do is access most of the efforts practical airports today have to be more like this, you've got a program and you got some specification of partial specifications and you ask the question that is pronounced, that is quite described. But usually you ask the question and this problem is like this is profitable. And then machines compute over the over, your code is the examine the entire state space and then it tries to give you an answer yes or no, that's in the problem. Sometimes they make and say about, I don't. Okay, so what you need? What you gonna need to learn in the place? Do it in the form. There's a lot of things on budget and, you know, in the form limitation is something that we're going to introduce questions. No, I want to point out that And what I'm doing is actually very job. We have using a particular patient, using a particular providing to think about progress or so. But I'm settings. We can just gradations of it as a meditations other ways of thinking about them. So, like, for instance, I was, you know, I, I can use this kind of specification or build a real problems, I mean, if I considering the problems, you know, in the proper way and very careful having it. So real, you know, seems into my you know when they actually told him I'm gonna take a lot of. So, So let's talk a little bit about another picture. To the way, I'm gonna float in one of gently production and I'm expensive. Even kind of just the basic structure and then it takes some type of problem and we sold the problems just without having a lot of knowledge and just took a sense of how things are approached. And then we go about and say, okay, now this is the most systematically through the hospital. So, there's a basically have a program. And the program has a number of sections. It has a declarative section as they need section. As always. Considered a sign section, 10 minutes. This is a keyword that you use, then you're fitting the back. And this kind of reflects this thinking. So, declarations are basically just standard huggable sky degrees. We don't have, I mean, we can monitor, this is whatever will just try to stay consistent with the class but it's not really important that you have to say Initialization information is there or anything has to be initialized. But want to, we have to write down whatever initial. This is really assumptions about variables that we make about the initial state. If we have one variable and the variable is x and it's an integer. And the initial values zero is very same, but we don't have to have a specific number. Maybe the all program works for any adversity that is dead. So we can, we have flexibility to specify initial conditions over yourself. The other thing is that the initial conditions and I really going to become you know a credit card because you need so we take them square particular, creatives critical coordinate which is the initial conditions of the problem. Always is so princess this to ourselves the planet because we have this very variable between the same problem and that is there. So we're going to write in consistent. Things always is more like markers. If you find different, you find definitions that are made of code simple. So, if you want to forensive, you have another AA. And you want to check if it sorted, you know, you can write an expression which is chapter, but the expression is about this norm. So, what you do in the only section you define sorted as a predicate and then you just you can everybody use it in your program and say sorry to be so to see he gives you answer. Yes. Or no disorder. So it's it's simplifies a code, it's not less than not necessary, or it's just for elegance of Now, the sign section. It's set of assignments. Things. That's interesting thing. No it's not a sequence outside. Like we are used to sets. So basically says, it's not in the environment, it makes more sense than ever said, doesn't order secrets. So the set is a more natural concept and then we have an execution model. Basically takes assignments. Except why is it Remember the summer thought? So you pick up on any fire. There's nothing else in the entire world. So nothing happens in and so the the way the competition works is that you actually get a assignment plan, inevitably more control. So that's a basic model. No, another position, the logic we have because it's going to be next week, you're going to go more detail on on the notation. But let's just start with a simple problem and give you some examples of sol, you know, before we actually explain the form. So the the problem is for the earliest meaning So we have three people and they want to meet but they want to meet as soon as possible. The time I'm old and they open. And you know the truth is now you can say that English and maybe we agree that we have meaning or maybe somebody has a different expectation. So for instance, you are going to be side result. Asking us that, you know, consider weekends because people don't need to be making their work. You decides if you're gonna include weekends, dedicated citizen. So you see you're going to write a different form so that's why we need precisely became no at this point where we don't have the mechanisms to do that, but we will use more regular locations to try to communication with the moment specification and family have to models things to my relationship with us. That's what we need. So For almost specification, this is healing one is given. Well, first of all, we need to represent time. So, one option is moving inside is basically So if somebody can mean that I, as we be difficult, okay? So he's not saying it's real client and 505 and particularly he just I'm going zero and that images. So it's our own, we only do my best if you want to have a more defined to find the approach, then we can define it more at more deeper. This is kind of a simple way to model them and this is generally used in assimilations. And The thing is that we need to model schedules. So, the schedules basically, you can say, we have three different people. So f is the first schedule and G. The second statement in the third schedule. All. So, that's going to tell us one seeds of these animal enough. But we really want to find out when somebody can meet next. So there one little one way to basically specify that is if you feel monitors function that and they say, well I'm proposing that we needed T the schedule for filing me. I can meet in the falling. The first building is time about the so the what it returns is functional returns something is greater than or equal to the same. So if I can meet a clear eternity, so let's say teaspoon, I can mean a ton of frequency but the economy. Feel true about I returned 40. So that's the property of all this of all these functions which is basically and non-decreasing, there are monopoly, the only can go up and find the scary. And after you meet at 12 we can't say no I can be the 10. That's kind of it's more about it. And now that integer functions that that's a very comfortable constraints on schedule, they'll see that it's very important is a, if I can meet at a time, you know, has and he told me to commit at that time. Like, you should always say that you change. So when you say have T respond of myself, Because if you don't know that, you can make a certain time, then when I give you that time again, you should give me that number pocket. So now we find a way to models schedule. And this is a, I mean, this is, you know, this is a trillion problem, but this is a way, you know, people have to save it on any specifications for specification, They have to reduce things to mathematics. So, this always And that's a function will come of the crazy using of FRP because of So everybody can meet the value. So that was actually the column forever working with attend and reversing the second economy. Okay? Girl, you're probably not solvable if there is no common income and that's a very easy thing to miss in the specification and then run into problems with the pillow. So basically, we have to say that there is a z How to explain this former asset, I don't have because you have very specific notation in terms of formulas, but basically, it says there is a common meaning. There is a time when you, okay? So these are the things that are given to us, you know, given in English, we formalize it in using mathematic conversion and more. We need to do is design a program. And what he wanted, he wanted to computing the final number of steps. So we don't want this program to run forever wanted to get a result in your father. Number says, and we want the results to be called R to be the minimum. Z. So you don't need to fully understand the data and semantic details. We want to talk to them a little bit later but basically says this is a minimum test that's that's what we want to achieve. No. So in order for us specification and you know, we can quite use not as equipment but use magnetization. And to know that the issue is that we need to come up with solutions this problem. And so how and we do have this program structure. So I'm using this kind of introducing rotation, but how would you do it how to solve this problem? Feeling a problem, but ways to think about something else. And some of the solutions of yes, a while and then say if the T equals G, Filter, everything. So basically, you're saying ghosts statics which people zero, Okay. And then with the second appointment, so it's equal 0 and basically he joined going on and keep on checking testing people 1 to 345 and you know, one another power is through, that's one of those. Okay? But that's kind of a sequential natural sign door and we probably she's not going to transfers like that. I'm not really part of our vocabulary in order to avoid doing that Google, We may do the same thing. But when, using going other ways to think about, We could do is repeatedly set baby bowl at the PGFD or agency and then the program will get stock why they're all equal. Yeah. Okay, so when I was a example of a solution would be like a read. So, the four robots like this, you know, I propose Democrat and I send it to you. You say 12? She says 14. She sent to him. He says, 16 I get 16. I said yeah sure. So I passed back 60 you say yes. You say. Yes. Yes. And then we'll stop at 16. So now many many different books another way is bidding. Okay says what? I I get it said and then you come back and says, no, I can make it 12 and that means, no, I can meet at 16. So, you know, asynchronously different people are feeling and you just beat the next number. The blog, what has been big so far. So, you know, you can extend and you're ready to twelve. But somebody else comes in first and it's 14, then you're gonna respond to that and go to 18 or whatever. So that many different ways to to solve the problem. So let's see some solution that use it. The notation as it was done quite understandable because okay, some issues that you know those surface. So, the same program. Remember if you fancy what you're doing in the bag just like okay probably down. And So I'm going to ignore the declare in all these are not going to use always and I'm going to assume that we know how to. So one solution by the sensor values sign, so that's the same statement. And I only want a Samsung five quarter income In this class colon equivalents assign economics test. So don't make sure your consistent for this name. You search that 0 you and B. And, and you. So basically when this formula does, it looks at all the values between 0 and z these. And it's we know that there is a power reading time. So that's a z different that you use here and you know, so and we check that it's a the particular view, it's a meeting time. And then we take a hold of that is that you can compute this way. The only find a number of just that when you see values that accepted, as you guys become a little bit, Inevitable. So this is only one statement. Couple of interesting things about this mean is a great website meaning of. Thanks It says very so it's not a low frame, not a finding program to course and feelings about and may appear to them here too. Initially, that is really nothing good. Why? Because assume that we never see, you know, at least one you know one alpha bomb on the some some problems and the caramel economy. So that's looks really bad, but this isn't what you need to understand in this models is that if you ride the United curve. But any information, what can you see by the interpreters code? There's going to be about version. So this program, this program is exactly the program that needs to be suggested. You start at zero and you can invite my one and keep on testing them. And as we find the first one, you said, So that's a way this would be actually coded or so, even though it looks like we need to know the meaning value for the scanning and I truly don't. It's we can think of a patient can be successful and just keep on testing these. So, once really telling that, oh, and this is not good If we do no see So we have z processors some processors, then it can compare your voice right Now. Wait and the amount yet get in a long time of Z. We get that. Yeah. So I know this looks a bit confusing, but the message that is important to understand, here is, you are used to performance of the code based on kind of you know very various methods team in the unity. The program tells you what you're going to compute. How when, and where? So, for long for a while, we're going to develop programs and depend on the architecture. At some point, we're going to say weird. We actually go around this program. So computers, how many computers are we CPUs are using? How are you using the how they connected? How is that going to start impacting us and two things We can write the code? So it maps better from this architecture more naturally, or you can basically write the code and then we do another analysis and find out what the best architecture oriented codes. And then the concept of performance and, you know, space, whatever performers is really working. Only after we understand the relation between the code and the architecture was a problem because what there's an allocation to an architecture. So we're going to have to learn how to describe architectures, how to map programs to architecture and then how to draw conclusions about program. And other thing that you probably going to be positive here but it's shouldn't happen. Just what is this? What is this program finish? One other stuff. Community programs. Never stop. They ran to infinity assumption. Is that a smart machine can tell the program? Can't copy it anymore and stops. So we always have to infinity because most programs come from as I'm not supposed to stop any so that. So, there's a different that we it is. And get What is different example of the most used by the Not his power equals 0. It is not assigned here. Something quality is an equation expression. This is towards so we can use it later on as a liquid as a large reservation assign. Allergic, maximum pump half of r, gene or par patient. And So now I have a prayer which is really pretty much. I mean explicitly, the question this we basically several we know how to how we put it compiling machine but this is really explicitly. Simple, I'm gonna have equal 0 again, we give 0 to this 3. We take the maximum dose and we get a new number. Then when this number is this, I'll get another number. And if we reach a common problem, if this is a common weak time, only figure this number back, nothing changes, that's in checkers, hasn't changes nothing changes. Now, the program runs whatever the infinity but naturality is finished computing. The result in so many steps. And so, the concept that we introduce is possible fix point, It's called and every program is an expression. FB was, you know, the and he turns out that a unity model is so clever. FP is computed by basically taking this assignment 700 replacing with an equal. Basically says r equal maximum of FR. Think so that is this was an assignment. The completing six person when he finished completely expression, they assigned by their son. This is an electrical expression. It says, r is equal to the maximum this rate. And so, fixed point is the concept where the state normally changes, because once read this page, this is time to have my effect anymore. So, fixed point is really how many think about every nation is now actually having the system. Terminated Now, sometimes we do need to terminate the program to determine the face of suddenly face. Then we have to write a code for definition connection. We need Dr. Test, instead of that he has phase one is done. Now, you can work in place since because all obligation, but there is no building and then you can very easily show that the that are it's really the hardest computed here. It is really the accommodating time because if our is the maximum space, they must have some orders. And they say that HR is written in GR. I mean, can be anyone without loss of generality, we just choose this much greater than or equal, and this has to be people who are, and then this has to be equal better. Okay, so inside this formula because you can prove format not divided yet. R is the colony of what you are missing in. This is the fact that how do we know that he's as low as? You know, our equation tells us that is the earliest reading time because we we started from zero and about but we countries are intuition and it's not true. So to do that proof, we talked about technologies but we have new guided and want to use that in Guardian. We're going to show that anybody lower than our cannot a combinating time. Therefore, I guess the decision loss. Okay, so you see it's a different way of approaches. Any questions, Yes? What? Why not minimum for not mean? That makes why, by the early east meeting time is not the mean of the schedule. We have the program. Get the code. Yes, the code tells us the fixed point is this? Oh, that's only thing we can draw from the code at this point. Is fixed. And people know that this is true that R is equal to this. Then by using this formula, we know that everybody has to be equal to our oh, that implies. It is a common meaning that, but there's nothing in this formula to tell us. That is a earliest man. And so that means that our professing complete, we come to that just by by the, by using the fixed point. And I'll give you an example. And then gladly has that question. Okay. So that's we have this timeline here and instant is 110 is 221, right? Possible within touch. So your information tells you then when I pick up them. Yes, minimum yes, mind version. Tell me that it depends on what my initial values. If we actually pick up an initial value of our equal to 86. We're going to get this number and this is not, the meaning is a closest to work that each other but that's not a definition of many the minimum wage relative to zero. So you see, we have to figure out a way to bring the initial values into the into our thinking and the way to bring the initial of the thing is to run through a construct, called the evaluator, you can properly. There's always persistent. It's almost there forever. Very good question, but you see that she wants me really want to try to kill formally. There are always supplements. It's very easy to wag your hand and but if you don't know, if you put it wrong in here, you this is not giving And now, let's go. Some more systematic analysis of motivation. So what we learned today is so far is that all we have a notation is called. We have gonna have to have a lot of it. If we have not seen it, you can have to learn it and use it. Learn this. The logic open. So I have another question. What's the meaning of the notation? Double color. Get into that just next. Okay. It's I'm so slightly confused. Why? It's programmed of ours. We are in Utah and Well, here I think the video you don't think the maximum because you can eat a 10. He can be the 12 and I needed 14. The next minute come is 14 is not 10 o'clock. The next, the next time that somebody who's willing to here to the minimum global wise. So that's like this built in and here is why we do have to initialize us because it the program basically didn't care what that you could start a house with a million 23. And it still used that I know or here the heavy initial properly. Otherwise, it would. So we also learned a little bit about the fact that we're going to have at some point the backtrack and start thinking about their processors that were used, so that's going to affect what. So that means that they need to be a model processors. And I need to, to think about the code, it was expecting the processors that we are using, okay? So, That's stuff with the question that important. It's a very elegant, very gentleman here that's already in and it's using is hard to the unity, but I use it in papers and in classical because it's just there's a video living. So when you understand it, so the rotation is called this the quantified. Is a general notation for quantified experience. So, it does a bracket. Like this is not a smaller than it's just a widely bracket. You can find it that type of wood I'm operator. And very means. Orange another phone. That's the general form had generally, the operator is associative and communities. So the order journey doesn't matter Variables is a quantified variable which means that doesn't variables are not variables in the program that you know, variables using it under a quantifier. The same thing that we use affordable x such that y is equal to x x. It's a universal quantified, you know, variable it's a dummy variable, is nothing to do, is a program variable inside them and problem, okay? So, let's take some examples. So evaluators plus the very The range is one, okay? And Next. Now, plays a dummy, very good. So, this is not something that can exist in the program, right? It's quite important. Now X is I'll be fine. So what is it? Well in this case it is a program variation It's another way that is in the program. So what it is basically says, it's a form for everybody of David Martin contributing suspect. So, you're going to be the next one. Three. And technically, for reals, are supposed to square brackets. I'm gonna be very loose about that, and sometimes we're using in some type of butter, forget use, you should know. So, that's a person you do. So basically for every value of K, the satisfied is probably and other people who are doing, that's what this property is, You compute this expression being? That's it After that return to this operator and you have operator in between. So what do you get? Some of SK from, not some. Okay, I couldn't get some of J. If this expression here was retailer but it's okay. So some of what little extra on that. Oh, so all the values you names. So I'm just getting exonics through. So this is a, I'm summing in a rate into using this expression. Now, you are used to seeing. Okay, do you want to know the message so much? Right? So that And this is exactly what can normally use this under the So we just blast what? We use signal and there's nothing 393 in the passenger. Now, the interesting thing is that you can use a lot of. So for instance, you could use science. But I do didn't start, but actually made this multiplication. Okay. Multiply them right now because I told elements of the reading and if you actually say something like start okay, Colin one, I'm going. Okay, what do you get? 1 times 2 times 8, times 4. That's what, is that? So good, so you can use almost any operator. So let's say some other issues. Let's say that if you want to check people under razor for years no variants. And okay, he's a dummy. Very good one and okay. Xk. And Now remember, the expression here has to max operator because we can't put the plus between two of the other to streams you can just for the plus between it. This expression here, it's a logical expression. It's a problem. In terms of volume bullions comparable to the end. So you can say and you can say x 0. So it's the same thing is basically, thanks x1 0 0 and basically says area all the elements over a view. So sometimes we're gonna, you know, write them like simpler than that, it doesn't say legs equal zero to either the whole array of, you know. Okay, so we're doing just short notice. But you need to understand this mot because it's very terrible and very far. Now we use again but this is really the same thing as a, there's nothing you can use at all for all K. Everybody is equal to zero or and everybody, and you're going to the same set, If you want to check that, you know, there is a list one. Okay, then you basically would use a door 4k. So there's this x 0 or x, 1 of xn is equal to 0. And so that solubility in any basically like this, when you can understand anything which means exactly something special just an order fun because we want to play with things. So let's say that you want to build a set of all the values in your ranking, how do you build sets, whatever operations are set? The unit intersection doesn't plus minus the unit center. It's actually so we can place an issue union. Okay. But I have to say in the future sometimes when we know exactly what's the range here, we know the case in a simple brain tumor, just we in our hand but I think we should, we should be precise. Okay? So Thanks K. Just call it. We found necessarily know what x of k is. Well, union works on what on sex. So unless x cases said this doesn't work, I can't take the value again, and add, put them all together into our set. Then we have to make it an element, integral do this. So this is a set containing xk and then we take the union over of those. No Sometimes it's kind of pain related like that. So something is really tricky to say so, okay? Thanks. So basically, where students set is an operator that takes two things and makes because of intersets and makes them and to the union between them. You want the sequence? You can be careful because the sequence is ordered so sequence. Gay. You have to make an assumption here that there is an order of evaluation of these values and in this case it's natural and water would be wanted once again. But he had two variables becomes, it can become confusing. So, you have to profess some explanation, So this would allow you to make an sequence. Accelerating to the extent. As long as things, I mean, place it or a footnote which has a used is All right, that's pretty good, right? So, let's say that I have So we're basically saying, we say the array sort of the location trade needs. XK is more than XKO. So if case 3, so the index is 345 and this is a 7 and this is 18. We can say it's sorted at 3. This is for is not sort of this forward. So what it is in just a definition. So we use this, this is specifically like, you know, the way to make a list of definitions, the order doesn't matter, guessing it is a separate definition. So I basically would be fine. So okay was respect to x is only makes sense if x exists this and equals sine number or another distraction, this is equal. It's a definition. This is a predicate of that which is defined as this formul. They're just looked like they're out. Yeah. Something. That's, it's a definition which is one me. I'm roll it. He's really implement. So you could have written this so you can write, okay? Okay. So we can do the same structure to define now, the macros or definitions of predicates that we want to use. We could even do use the same answer with the. Okay box is big we are going to use unity. We're going to use two through symbols across box is going to be used. Every time you have at least an order list. Like I said, fire bar, he's made to be used to create synchronous things. You can type two things and you build, put them together and the executive one single step. So this is this is really interesting. After the book allows you to use parameter by another person, okay? So this is very different statements. This is general, at least of whatever you want can be used in box, so it's not equations. I think they need we can describe the uses as a list of equations. You can just follow us. There's a list of definitions inside. The sound is a list but it's really safe. It's not important soon, okay? It's not working. It's not one of so we can use, for instance, the the box here, to create statements over a bunch of steps. So here is k again. And now we have to do a statement here. See here. We had a definition, so there's no confusion understanding here. If you like a statement, there's no computers definitions. So, an example of a statement by these something like of x, okay? Comma came from us, one for people. K. So we So this is a statement, this is a range. So it creates 10 statements, when I see they can run in the in our program. Sometimes you can create n squared. And cube statements, okay? It doesn't mean that. So basically says for every instance of K take this text substitute, the correct number and you got the sustainment. So this way instead of like and I can, you know, and lines of code which is like one expression like this and they get equivalent of 10 lines of code. Each one of these to one level. And so, basically what this says, is computer escape as one computer escape. Well first of all, check this condition is so it's this condition is true so they are actually not sorted. You know, the sort of then computer escape password complexity and assign any. So is really you want to check what it's called. The God then you computer all the expressions here and the number of variable assignments has to match the number of expectations. So you have two variables here. You have two expressions there. So x k. Plus 1 goes into x. K x k, does mix it, and everything that's economy which means that your first in the current state, you evaluate all these parts, we have only the expression here. If the expression is false nothing, this is true that we know it and it's about all the expressions here and write them to cash. Then when you're done with all that expression, because maybe, you know, and cute expressions, that were computing. When you're done, it's all of that. Then in one single step, you make all the assignments of the so this. So, let's get all the bacon to the memories all day. You can also build any complications statements. Okay, so when that disappears again, our advice came Who, how many statements are there? Even under the parallel. Bar is like hard cement and producings, and you put the final body between them. They're not two things anymore. They're one. So by the body is hard, I've had been symptoms so everybody because once in a second, doesn't matter how big how big it is in my shrink and expand but it's one step. So what this statement does basically copies, keep us on as people want to escape. So that's a shape like and in such a zero in the last position. Now you can see what's going to happen is if there is a good shift, is every shift we put more, more more zeroes into the array. At some point, the holiday is real, but the end doesn't. And another thing he's a constant, how so you have. Okay. So here. Okay, so they all have values. One, seven, two, three. In the same synchronous step. These guys have to seal this guy gets asleep. These guy gets a tube. He's not getting seven. Oh, so everything shifted. And then you see Next time you get to zero again. That doesn't even matter because it's already, zero changing thing, but this zero is going to propagate individually. The whole shift register becomes it. So important thing is there's all one statement, One secret understandably and all this happens. Same time to have a nice evenness. She registered would have to synchronize it, and put flags and so on. But for a synchronous, you know, electronic circuit level and one of those for the idea though is that the array remains the same size. Yeah. It's a few days and testing is not as if you decide K okay. For every K. What was the 457? You know, you get relations have like that. More advocate. We go outside with a friend and plus, or something. So he gets so whatever he say, we have a standard like this. And for every day we have a statement like this also. So that our instincts like this, it doesn't matter to you. If they do the same thing, if they would give them batteries the same variable in the output and the update, that would matter alone, that will be illegally. But a remarkable statement signing 0 is always coming up, doesn't make any difference. It's still 0 that you get. So widely we just say okay hold that coin equals 0, he is not there, he couldn't understand that by the state and instances We could have done something else but I wanted to do it as part of this concert, We could move these outside so we could do final five. Thanks and 20.0. There's only once then we let this, but without M of them or just one since it's really the same thing as okay questions. Yes. Even if the parallel is outside of the blue works, you can use contacts on this to create one construct. And you can do two of them or three of them and you can, you're gonna see like Iran. You can do crazy things like glue, glue 11th things that you can do is one statement in the end, but you cannot do. You can also do. But because you're saying every single is good and here I say no, I like to you. That's not possible. That's not something. So once you have good 30 seconds it will categories. The big steps And it counts as one statement not multiple times, okay? All right. Let's take a look. So right now, conducted this time, I'm going to print somewhere. In video. See whatever that all comes from, in a pajama. This two one, half more. Really. Still happy. Okay. Plus one equals. Came close one. Shipping that way. Two more one to the ticket value and the last issue. So this is indexed. Yeah I got it because that's a a assignment so we need to do that. So all in all time okay for one to end so you do that for all them and other ask you want to do that you want to do that. Oh, I want it. Let you guys know. This is 0 1 to. I think there one base. If I had. Yeah, you next one. Oh yeah. Open them after one and connect. So, For the heat. And because that sounds here we need to do that. Okay, from one to seven. One, two, three, four to four more times so you can use the reversal question, it's no subjects. So more and for one thing, and this is it. There's one, oh, one. Oh, that's one in this thing here. Yeah, Also here. Maybe all of that and already ready? Sure, you have accessories. Here possible songs. I'm cleaning me. All right, then all kind of. Okay, for this when I was looking at the only one that I felt kind of weird in, my brain was the four all. It's all kind of strange to use it that way. Just because if we're in inserting something in between, I don't know if that necessarily would get the same value as an hand. Yeah, Tomorrow is the national forest. And and for all operation, that is the order with operation patterns a lot, the other with the follow operations that give you new one, two, three, all the that, or to the founder actually. And oh, it doesn't matter. Okay, there's a computer all the values and it's put an end in between them. That's the definition of it just feels weird for me, I guess, I'm not sure why. I think my branch just doesn't like, just again, hearing that you Forward first, unity have some similar variety to how well, this function language now. I mean, it can describe harder but it's not because I think however, everything is terrible. Yeah, so we're gonna see later on that again, if you need to describe hard, like, very long, yeah. Oh yeah, okay, so we need to do a couple of things and then Just kind of introducing the taking someone from here. Now, we need to get together easy to work and we need to do this two things, whatever we know is one over program, so we can discuss it. And then we need to talk about another design so you can ask questions about that. Okay, so, so the program that we're going to look at. Is matrix inversion transposition Recently discount. So, we have metrics like this position. Me that you clicked from the. So this they just change them back is going to change that. This is changing that and the diagonal doesn't change because what transportation, okay? It's just a trigger example. I'm going to make it, I mean, I'm going to make it make a solution, I believe. So do not irritate the solution and we'll talk about what's nice and nothing solution. Because you saw it like this imp. So the first of all, we have a declaration No prepare. So basically in the greatest section, you can use whatever type, you integer, boolean, array set of sequences sequence of sets As long as you define, but then they mean it's fairly going to use integers and who is probably never going to use closed. So we have and define as an in. Then there's a box. Basically says I have a set of things so I have one more thing here. So, and is defined an array. Going to want to him costing. None of us key under the rain. You so n is a size of the rate that are avoid working with. It's a constant going to see that doesn't change program was used here. It has to be a constant using this here and then, and is a related to transpose and B is just housekeeping to to make sure we know who starts. And so we don't transmose the voice. Then they can have always No, let me say something else. Since if we think of anything of constant outside of the program, we don't do that which is business. So don't really need to prepare that just kind of again pushing things unnecessary, okay? M&B our grades that they're going to modify the program. So we need that. Now you have things which are quantified drops can see by that. Those are not very when the program those such modifications so you don't have to declare those Always, we can do something like this box. Sorry. So what I'm saying that I is going to be from want to end is really going to refer to those are. Okay. It's going to say. There are no high has been trusted and so we have any final communities. Just first, where we use the blue line to actually for housekeeping to check there. Something was transportable. So we're going to look at the value of the building array. So we're going to basically say them maybe pretty. So this, my since it's the for all, this must be something that is true or false. So assume that it's true, if it was transparent. So basically says, everybody in the whole eye, there's been transport. That's what this says. Okay, so we're gonna now, use it in the code. Don't do this. I mean, I'm doing it just for illustration purposes and ugly thing to do when you know, I can write the code so much more now. But again, this lady can be a left side of the equal. If after that in the equal time so this is in the right here. Yeah. Okay. All right. Okay, bro, it's okay. Let's be. So then initial So this shows you Is that my referral war in that? I don't know. No, it's just came up as a definition. Well, we find these are. Okay. Because first, you know anything, yeah, it's a definition that we just made so to use it in the code later. So the only section is always a, first of all, is always just a set of markers can be defining any order, but you can't create sanitary, cursive and you can't create things which are circle definitions. So, one thing that you're going to have really problem with on the right code, is you're going to feel the temptation to write recursion? There is no mechanism for precaution in this language. So you're going to have to deal with it. So solve the problem in a different way. Okay so so these are we really want to have here. We have end definitions for one or two lowercase one. Okay to grow. Okay, So then they can be used whatever. We need the but they like macro definition, not necessary, if I can just replace a code in the definition in the code, and it's still the same thing because we're going to check if our okay eye. Well, I can say if this is true, the same thing. So just pure convenience, makes it called cleaner, whatever. Not in this case in this case actually you know, just take a second illustration. So now the next thing is they initial reception so and is equal to 10 box. Oh, that's a great thing. You cannot, I mean the vocaliser to use power bars all over the place. You only can use fire bars to create magazines. You cannot use them to create least lang just going to get confused. So we know always sectionally means section. In declared section, I always have to see boxes. I can never see part of those. Okay. That's kind of a possible. So the box I'm from a J colon foreign and and one Drake form. So basically what this says is that you can run in and JPMine or and so I get pairs of IJ so you can get one, one, two, three, three, and then. Okay, to be right, should never come out there on the board and always simple forms. So notice what we have done here is very for every DIJ, every location, in that ID, medicals initially. 10 dollars. Pride J. Same by and, and for J's and that would call him. And I J people. It's ugly and it's actually a couple of things notice here any is equal to 10. I'm visualizing them. So basically, I saying that you can write this program overlay that is temperature. It is silly. So this is really overkill. So what we want to do here, basically saying and is going to that more. So now it was for any size of it and you see how the flexibility of this equation, it can do with this restricted specific value. We can write the operator. Now this here for every ING it makes BIJ false. Now a couple of things. First of all, this could have been written as not. The type J people So VIJ if it's a boolean like that. Another actually more or less than that. We could have written drugs, B or false for B equals 0. Because if we make that human is 0 and 1 our fall center. So you see this is a bit much much cleaner. You're gonna distribute that much money. All right. Clear. Okay, so KFC flexibility. You can see more element notation What's happening here and he's a very more programming. The majors that you're going to transport, what in the world is MIJ, it's not declared anywhere. And like I said this, you know. Anyway since is a constant think of it like is a parameter of the program and transpose of M. So you could sing that basically F is just another thing outside of the problem. So what happens is that this language does not have any product so to visualize for something to search the girls you have to to assume a function f that is going to be you know they will initial values and that's from outside of the program you only use it initialization and then using improves. So what happens here is two cents. If I need to refer to F in the pros, whatever is a national configuration of the matrix. I write it like this where F comes from outside of the pro, it's a constant function in the transition values that come from outside of the program. Like I said you could have practically say we don't do that but you could have said transpose and cover. As soon as a globally given to you from outside, if you don't care about your initial value for of them, you can play, you can basically take it out. What happens if you text this right now? So you figure what it says is that I'm not getting any particular additional values, whatever, initial values, are I programs to convert for those initial lines? I'm leaving the initial guitar If I want. For instance, to limit the initial values to be only positive strictly fast examples, then I would replace it. So these they get around option B. I could say, actually, I should be, I could say 10 greater than 0, who says that all the batteries in them are integers are greater than zero. And if I want to say that any nation initializes, that I can say, m equal to whatever is defined in external as a, as a concept outside of the problem. So now you didn't know you see that, you know, this is simpler. This is simpler and this is so what? I don't hear could have been simplified with three eyeliner questions on that. It just asking about the order of nuclear always doesn't matter. You have to make sure that there is no circular definition. So you can define x in terms of y and actually put a definition of y after x doesn't matter. It's not that there is one order in which we could evaluate. So what do you say x? And then go and pick up the definition of y and use it inside the definition of. So no similarity, no equation and doesn't know what it doesn't. Everything is a set. That's a nice thing about this thing because I found this in alternatives. I want this answer. Sign is worth computing business. The sign has to be a set of assignments. So, I'm going to give you one example and then we're going to discuss. So I'm gonna say, if I want to be one one before anyone one country, if one. So, if B11 integral, So really this values. So this question here, okay? V 1 1. It's had nothing in it, in initialize INJ to one. All right. Now, we're gonna hit the problem here, whose eye and Jane. After programs, we need to be better out of its. So, when we think here, Can you see what is actually does? Is again sadly and I'm intentionally fully understand the complications of. I put an initializing here. I'm going to say hi to him one and then living simple. And in there box. I comma J bypass on the body. Different particle came high and the price more than, Okay. So what's happening here, with notice that three values here variables here, That's three values there. So we checked. It VR1 is just false and then if it's false and initialism to one in one and I'm additional as we want to model, the we also make big one result transporting, anything because since when the diagonal automatically orientation transport. So what's happening this time? What is time increasing? Time is high because one injury gets worse. So what I'm doing, I'm moving one row up. If this current row has been completely transformed and I reset. So I move one off and then it set up the J to the beginning of the so we can see kind of someone like I see, you know basically it's it's almost like a sequential program coating. Here you go. You're going on this row and this is done because this row. So this this increase, I okay and go to the news now visible. That's what I'm looking for down there but it's really sequential and the box vehicle and You know, how to change. On this thing. One does it moves Jane along this line? If you had enough, you know, transpose position you want to know, Okay, so this game is doing this motion and that sermon is moving this. And then, finally, and James 1, and I J. Beast the time for me from and eyes by J. Come on. Two coming through here. So this demo basically received doesn't transposition, these two and inverted and these two inverted order and then in March the full defense causes, so if you transfer this to my post bullion and then transport again, So this is our from a couple of other places. The way the system works is that you feel because there's an opening system In orbiting system. Basically, you have now so many. So many statements, we have here to say four statements, the opening system, basically picks up a statement, the Mexico, then peace of Arabic statement, energy and does this forever to infinity. That's what operating system. The problem is that it will be possible for operating system to decide that it just doesn't like this thing and never exited. There's no way or from wherever people who succeed, So we have to make some assumptions on the offering system. What to do we make that weakest possible assumption, We also operate system to be what is called, weekly effect. We furnaces that property system can choose statement specifically at any time. Many already thoughts any number of times but you cannot deny execution to anyone particular. So it could execute this time at a million times. It's from once. This one works. This is doing a tabs but as long as it doesn't stop. So point of the importance of already described something concept of fairness, we need to be everything processes fairness doesn't exist and find a sign because if you stop at the 20 steps, how do I know this company system would have with a child selected at the statement in a step 21? So only in the internet sequences of a service and this is weakness assumption. So we're very conservative the best of the same. No car. I don't care what of any system. It provides me with policing providing ways. My program would. Now if I take a long time because opening system is blessing. God bless us, but we'll be gonna be aware or process very conservative And the most possible circumstances we would get Couple of things. Maybe it probably did, because I maybe And anything, that's the case before. This is eventually initialized Once. This is visualized, then they can execute. And this is going to have effect so many times, executed until there always finished. And this when I have a fact, when I always done and you can move to the maximum, and this is going to have a fact. Whenever I am James pointed. Okay. Then what elegance solutions. So, for instance, info What we've done here. We created 10 squares. One quarter deposit, combination of INJ and for each one of them would check in the values of transport month, either and not in transport within the transportation. So, that pair is transport once and never. Again For this it's using the conversation narration of statements creating once and for every transposition and then making sure that and this works even for the I equal to J in the diagonal, it's been worse and it has the best position. So we don't have to, I enjoy here. I'm no longer variables that don't be variables. People around at this quantifiers and the federalists n squared. Statement and operations are fixed upon fixed upon fix of 1601 to someone. Now, replicate something different. What, once then? In one single statement that would make me sust. So here they can be too portion of this statement. They do the same thing but they are assigning the same values so it doesn't really matter. So this is one single statement and that's all the transposition together in one single the first in the first selection. After the first selection, all the matrix beads all through. So, nothing is going to happen. After the singularization real estate, he only did beat up for triangular, do maybe the 49? Well, there are the solutions for modeling, okay? But this is to explain the show to you the how we can get more and more during the style rather than discussion. This is something really mechanics. But this is kind of the first one for how many people stand And notice that after the first statement you know this video becomes true. Now, we need it. So what you could have done is just create a variable date which is a whole and basically, if you So we don't need another way one full edition of because you're thinking everything is Now, there was a number of version don't really have problems so I'm not going to use it. We were very okay, we can defend one because in this relationship and what we're going to do, we're going to implemented by one whenever the assignment plus 1 even, okay? Is smaller than emphasize. And then in parallels will, the lifestyle is favored right down my very Portland. All the restrictions on ING. I'm gonna find and plus G. Okay. Corn corn. Okay. Hi. We don't need a polymerase. So what's really going to happen with other? You should say this is a hormone That's going to happen. Is that? Remember case too? I asked Jay has to be equal to 2. So this is 1 and 1, I plus J's 2. So this is the first response, Then in the same step we implement K by 8 plus, So now instead, First one is one, plus two. So from the next step, you know, for free In the next step forward, really from here, to the top single single assignments across energy. So you can really come up with a clever and in the notes, which I put on the earth. And then I learned in that example of Any questions at this point? So, the definition of our okay. Is that a double colon before the VIK would be being a variable? Yes, our okay. Is a definition. Yes. And is that a double color? Oh yeah. Oh, okay. Some and think that the book actually has a rule of thumb and you skip, the range is implied because that will call it. But, you know, just he's only Like that. After people be there here. Yeah, that's not. Okay, you have a question and the dash in the last night. Is that abbreviation that I? That after IJ. No, the that's not. Yes. Has been right. Oh, actually together. So the homeowners you have to address as couple of parts and I think to make sure you solve one part at a time. They're very different and it's been problem, but, you know, somebody emissions and you need to be very, very careful to get the first version, right? And move to the next and Okay, so game of life. They? So, basically, what happens is that? Remember to, So, we assume that this of him, Okay, and this is supposed to simulate a, you know, an environment in which life exists. So if you have a one here and there should have somebody who's 0, 0 means it nobody there. So the basic name is played in the four language. It's you create generations of the generations of the. So, you have some people who And you move this them synchronously even from generation zero position one. How based on a couple rules, First of all, is that an organism drives you guys more than four members and fewer than three. So death. Now the neighborhood is viewed as this you don't get diagnosis. So you can see this one organism because every one neighborhood, which is two identity, the definition. So the idea is, if your isolated economy, it can reproduce yourself. If your crowd is going to die because you're overcome Now, there is a verse Address is of course in the cell that doesn't have anybody yet, okay. And she requires to have three things. So if we take this cell here, it has one two, three members. So in the next generation, there will be diverse there and the process is deep again and again and again. So the very simple rule based on the number of neighbors Jesus is alive or die. Or get born questions. What is the station was? Which can have any configurations, so you don't don't give it a nation. Say what you guys. No, there are three problems entering a song so just to make it to make it incrementally, more difficult. So, the first problem, so I don't know how the game is played, okay? So that's a basic approach. You move from generation 0, to 1 and so on. And sometimes this game can last, whatever there are something called, seraphores where you move from all configuration to the next and just flip flips that sometimes they don't very interested. So, you know, it depends on the initial conversation but to make the problem simpler. Initially, you know, one then more one and So that's not a way to normally answer normal game. The whole board changes in spontaneous. I want to first develop a rather program in which you know price as you can do this because you're done. It's a new generation into another one. They're done here. Career. So he's synchronous. So whatever you're going to do is probably going to use a box and healthy. It was saying was about it anymore. Second is synchronous generation. Which is a normal game. So, everybody confuses their neighbors and in our single step, everybody the dies, survives of dollar, or something's worth. Okay. So, most likely you want to have to use it on a bar because you have to have synchrony. So the whole thing happens to simultaneously. Okay, so that's a part. The first reason, most difficult part. One colonial event. So call me is defined over set of cells, would they have neighboring, their neighborly churches. So see, look at this. Three, they touch each other. So they are really synchrony. So the way you have to solve it is is you choose a colony and then you all need synchronously? Only that colony nor the entire world. Now once y'all did economy, the configurational current is changes because you know, something to me, die. And the next configuration, it may be two colonies. What it was one or because of a birth, you know, you can get, you can get two colonies to come together. So the the gas are pretty simple. Okay? It's the same same thing. You just have to choose your quality. You then you have to know what colony are the water going and then you'll do the updates and that colony synchronously on all the members of the garden. And remember who you use that be to keep housekeeping, the flag, double in, that's going to be helpful. In this case to keep some, some housekeeping of, which colony are you talking about? It's no, there's no complicated part comes to the fact that. Okay, so this is the colony but the birth constantly from the on cell to generally your life. The voice is being placed in social chart empty. So in reality well, new to the comedy. You have to be a little bit more discriminating and basically, to All the cells on the boundary. Which I think because that's the only place when a birth would take place if you don't do that. No personality. So when you final colony is really a connected group plus their empty neighbors. That's a definition of economy now. See this neighbor here. May have, you know, then maybe another colony over there so there may be an element here. Okay. Here. So Monday just because of these two who belong to other colony and because of this who belongs to his own colony, the county is three. So the person's going to take place and then the place. And then what is going to take place and then going to be onto the other colony because you're going to be negative part. This is the most difficult part of the home. Yes. Don't combine them, right? Or none that sell it because the other guys may be dying here in this dice. Think that this one gets born but separated. So the thing is that you need to select the colony which is a connected connected set of cells and themselves. And you need to expand the colony to all the neighbors that are empty and then when you compute birth, you are not only counting things up in your colony, but your counting seems that around the empty cell, so which may be your colony and solid. Maybe somebody else is called, so you are dying. You update your colony of life cells class. All the empty cells surrounding. Questions. So when we assume at the end of our of our matrix, is that just now neighbors. No neighbors. So what's so very easy to to simplify your life? Is if your, if you drive the border which is ones and then it's you and you don't come, you don't update the border. But this way, when you like your formulas for computing, neighbors the same same array. So you just count the numbers. So there are these things like the first problem. Picking up a cell in the update counting. How many neighbors I choose for three redundancy In the second. You just take the first problem and you apply it to everybody in Belgrade. In the third, you need to select a colony. So why don't we just make a call? You specific point in the colony? Because once we say like a point then come to the college. So it's like a colony. You basically development understanding of who is part of the company and then you apply the rules in parallel only to people who are in the country and, you know, was in the colony because some some, some some housekeeping that could struggle, whose was in a particle. Not trying to go left to right from colonies. I think that remember many things are not intending system, so you're basically it's okay to create n squared assignments and the whole brain system is going to choose one and then maybe you're lucky day that might be the winning assignment and that you use that because it as a seed, for the car. So, here is not a terrorism to make sure that random choices. Do not worry about this, real fairness. We not try to choose things sequentially. User always section and an elegant way that you live for counting because I did And to travel when he was by the body. And and what we're going to do next time, remember all the hormones that you I have to have them in the morning because I'm going to probably death in So we're going to actually look at how to do this on point. When naturally just doesn't program the code one point and later on, we're going to look at us. The most mental health is Very, very Any questions. But in question. Okay. Remember to get help? Never to and then four, if you actually want me to look at partial sol, then maybe like this questions and answers to and it has two missing and I come up at the PDF because it did something. It was crazy. This. Okay, you do the cross modes, give you examples of Here. Labor lot of money. I wanted but also look at them or they don't have no clean. It was gonna try to like oh not all about Toyota. Yeah. Yeah, there are just kind of I know, it's okay. It's good news, which is and move it. Well what? Yeah, so much usually pretty good. Is not different. I said here, if you wanted to see any of my liked news about here, every class in the morning. I'm just really proud of myself. No, it's just directly If you wanted to swim like I was friends doing other classmates fun. We can directly talk about it around the beginning but found here. Yeah, no. Thank you. Thank you, man. I'm like, oh, you're doing this. So we need a product process so far so I don't know a lot of people yet. Yeah. I know. I told you a work full time but I take one class this semester. Yeah. Where do you work? Yeah, Good evening. I don't think so, stop. The plants here said, it's actually. You see them in this way, we select a cell and there's That I think. You know. Oh, you don't want to.