That would be great. Thanks. Otherwise, you're good to go. Yeah. He'll figure. OK, everyone. Kristen will continue her lectures from earlier today. Great. Thank you so much. Thank you for coming back for the next installment. So in the first lecture, I introduced, oh, thank you. That's much better. In the first lecture, I introduced super singular isogenic graphs and explained how they could be used to construct a cryptographic hash function. And so the goal in this lecture is to explain a little bit the background and the context of super singular isogenic graphs as what we call expander graphs, and also to explain another application, which is the key exchange application, which is actually the one that's in the standardization process for the NIST PQC competition. So can everybody hear me at the back? Is it good? OK, excellent. Thanks. OK, so let's start by what do we mean by expander graphs. The title of our original 2005 paper introducing super singular isogenic graphs into cryptography was called Cryptographic Hash Functions from Expander Graphs. So it was not just SIG, super singular isogenic graphs, but in general, expander graphs that we were proposing. In particular, the other main graph that we focused on was the LPS, Lubosky-Philip Sarnak graphs. So what is an expander graph? So as I said in the first lecture, the notation that I'm using is G equals V comma E, where V is the set of vertices and E is the set of edges for the graph. And we are concentrating on the case of k regular graphs, where each vertex has k edges coming out of it. So an expander graph that has N vertices, which has expansion constant C, is a graph such that if you take any subset U of the vertices, such that the size of the subset is less than or equal to half of the vertices, then if you take the boundary of U, which means all vertices that are connected to U, but which are not in U, then the boundary of U has to be at least size C times the size of U. So that expansion constant just kind of from an intuitive point of view just gives you how much the any set is growing when you add the boundary to it. So and then you keep adding the boundary to it. And it grows and grows. So that's what we mean by an expander graph with expansion constant C. And so you can get some properties of the graph from actually doing some linear algebra on the adjacency matrix of the graph. And that is actually connected to the expansion constant. So I'm not going into a whole course on graph theory here, but just trying to tell you some of the relevant facts. So the adjacency matrix of a graph, let's call it A with entries Aij, is given by so the ijth entry of the matrix is given by the number of edges from the ith vertex to the jth vertex. So if there's n vertices in the graph, this is an n by n matrix. And so I guess I use the notation A of L just if we're talking about the L isogenic graph. That is the edges are L isogenes of degree L, where in our context L is always coprime to the characteristic. So these are all separable isogenes. So the adjacency matrix of an undirected graph is symmetric because there's no difference between edges from i to j and edges from j to i. So since the matrix is actually symmetric, all of its eigenvalues are real. And so for a connected k regular graph, we have several facts which are very useful, which is that the largest eigenvalue is actually k, where k is the degree of the regular graph. And all of the other eigenvalues are strictly smaller. So if you look at the notation I'm using here, where k is the largest eigenvalue, let's see, I don't have a pointer, but I think the cursor goes there. So this difference between k and the next largest eigenvalue is that difference is usually called the spectral gap. And so when you look at this formula, this next formula, which gives you a lower bound on this expansion constant. So c is greater than or equal to this expression here, two times this spectral gap divided by this quantity here. What you can see is that this spectral gap is related to how good of an expander this graph is. So you'll have a whole, that's a whole area of study thinking about this spectral gap. And in particular, you want this spectral gap to be kind of as large as possible. And so we have what we call Ramanijan graphs, which refer to being optimal in a particular sense. So it's a little bit nuanced. It's not quite as clear cut as that. We don't necessarily have a lower bound on this spectral gap, but what we have is asymptotically the Alon-Bopanatha theorem, which says that if Xm is an infinite family of connected k regular graphs with the number of vertices tending to infinity, then the lim inf of this mu one will be bounded below by two square root of k minus one. So what that means is not that it's always greater than two square root of k minus one, because that would give you an absolute lower bound than on mu one, which gives you an upper bound on the spectral gap. But at least in the limit, it says that this quantity mu one, the second largest eigenvalue is going to be bounded below by two square root of k minus one. So in our case, k is equal to L plus one, where we're looking at the graph with L isogenes. And so what you're looking at is a lower bound on the mu one, which is two times the square root of L. So now what I'd like to do is I definitely am not gonna give you the proof that these graphs are Riemannian, which involves a lot of very sophisticated number theory and theorems. But just to give you the idea of why these, we know that these graphs are Riemannian. So actually this Riemannian property, what you can, since k minus one is equal to L in this case, what we're really just asking is that in order to be Riemannian, that mu one should be less than or equal to two times square root L. So the way this comes about is that if you look at the vector space S2 of P, which is the vector space of weight two cusp forms of level P, it turns out that the action of the Hecker operator, TL, is actually given by the Brant matrix, I'll just call it B of L, but which is actually equal to the adjacency matrix of this graph. So what ends up happening, and behind the scenes what's happening, and we'll talk more about this tomorrow in my next lecture, is that there's an alternate description of this graph. The super singular isogenic graph that we've proposed for use in cryptography has nodes which are elliptic curves, which are representatives of their isomorphism class, and isogenes, which are the edges between them. But we could describe that graph in a different way, which is in terms of the endomorphism rings of these super singular elliptic curves. As I mentioned, the super singular elliptic curves have endomorphism ring, which are maximal orders in a rank four, maximal orders in a definite quaternion algebra. And so if we flip over and we think of our graph instead of being elliptic curves with isogenes, we actually think of it as being maximal orders in quaternion algebra with connecting ideals between them that are of a certain norm, in this case norm L. Then we have a different description of our graph, and we actually have a different way to attack the underlying cryptographic problems. And so unfortunately, or fortunately, one or the other, it's very hard to actually make this correspondence. So we know that these two graphs are the same, but we don't know how to match them up. Because over on this side, we can say a lot about its properties and we can also actually get an attack on this side, which I will explain tomorrow. But so we are actually using the description of the graph over on this side to claim that it's Ramanijan. And these graphs, when we introduced them into cryptography, we called them Pizer graphs. Because Pizer was the one that described them in, I'm sorry, I can't remember, it was the late 80s or early 90s and proved the Ramanijan property when P is congruent to one mod 12. But another name for the graphs, they are often called Mestra graphs. Because Mestra recognized that these graphs, well, he may or may not have recognized that they were Ramanijan. But anyway, he was using them in the theory of modular forms. He introduced what was called the method of graphs for computing bases for spaces of modular forms. So you'll have some people calling them Mestra graphs. Some people calling them Pizer graphs. And then in cryptography, we're calling them super singular isogenic graphs. So the Ramanijan property comes when you recognize this fact that the Brant matrix is actually the same as the adjacency matrix for these graphs. And we have Deline's proof of the vacanjecture that the eigenvalues of this matrix actually satisfy the Ramanijan condition. So you can see there's a lot of deep theorems from number theory going into that to establish that these graphs are Ramanijan. So that being said, honestly, we don't really use the Ramanijan property in cryptography. The fact that these are optimal expander graphs means that the output of this hash function is basically as close as we can get to being uniformly distributed. So that's the main outcome that I want to talk about a little bit next. Just a very side note is that when we first proposed these graphs for use in cryptography in 2005, we were not actually working on the elliptic curve graphs that we use now today. With Yalgorn and Dennis Charles, we were actually working on a higher dimensional analog of the graphs, which you can get from looking at what we called super special abelian varieties and super special orders in abelian varieties. So our construction, which is in a different paper from 2007 is actually constructing families of Ramanijan graphs from higher dimensional abelian varieties. But there were so many problems with making this explicit if you wanted to use this for a hash function construction that we actually just kind of simplified things and went back down to dimension one and looked at elliptic curves for the cryptographic applications. But there's now starting to be more and more interest in the isogenic community in developing higher genus and higher dimensional analogs of these cryptographic constructions. So in particular, genus two. And the problems that we had with our graphs in higher dimension, even dimension two, were that we would be looking at abelian surfaces where we didn't really have good invariance that we could use to label them. So we didn't have, there are igusa invariance, but igusa invariance are pretty hard to compute. Number two, we didn't actually have in 2005 a really good way to compute isogenes between a billion varieties. Since then that problem has also been kind of solved. So there's the AV isogenes package from Damia, Robert and Lercier and their collaborators. So that problem has kind of been solved. But just in terms of efficiency for cryptographic purposes that are very abstract to mention two analog of these graphs is still very unwieldy to work with. And so what we've seen is some newer proposals at least one that I know of, for example, from Florey and Smith's, which proposes to use essentially the Rosenhain model for genius two curves as the nodes in the graph and isogenes, but then you don't have as many of the nice properties automatically at least as we have for our graphs, which is being connected, expand our graphs that are optimal in the sense of being Ramanujan. Okay, so that part about the higher dimensional analog was just a little bit of an aside here. So now back to using these objects for cryptographic purposes. So to avoid some kind of bias in the output of these hash functions, it would be nice if the output was kind of uniformly distributed. And the expansion constant for expander graphs gives you a very explicit way to relate how well you approximate the uniform distribution to the expansion constant of the graph. So the better your expansion constant, the more closely you can approximate the uniform distribution in a short number of steps as you take a walk through the graph. So another thing I should have probably said earlier today is that this idea of using expander graphs for doing all kinds of things and proving theorems in complexity theory and number theory is not at all new. So computer scientists have been using walks in expander graphs as a way to approximate uniform or uniformly random outputs for a long time. And so the thing that we did that was different when we introduced these graphs into cryptography was, well, first of all, introducing specific graphs, the isogenic graphs. But second of all, we were really trying to get this cryptographic property that it's hard to find collisions or it's hard to find pre-images. And those are both tied to being able to find a walk, you know, a path in this graph. So, but if you think about it from like the complexity theory point of view, the computer scientist point of view, they have a very different point of view than what a lot of people working in isogenic based crypto do right now. Which is just think about it from a common sense point of view. Let's say you have a three regular graph, like I've told you is our main object of study here, a three regular graph. And once you start your walk, you're not allowed to backtrack. So you cannot go backwards. So every time you get to a node, you only have two choices for your next step. So that means basically if you take n steps, you will visit and you don't have any collisions, like you never hit a repeat. If you take n steps and you don't backtrack and you hit no collisions, then you will have visited two to the n vertices. So what that means is that when you think about the diameter of these graphs, like diameter means it's kind of like the, whatever the maximum over the minimum of, so given any two pairs of points in the graph, what is the minimum distance between them? And then now take the maximum over all pairs. So that's the diameter. Like if you give any two nodes in the graph, then there will be a path, there exists a path which is no bigger than the diameter. So given what I've just said, that if you take a walk of length n, you'll visit two to the n vertices, you can see that roughly speaking, you expect the diameter to be roughly the log of the number of vertices. And so having an optimal expander graph, having this Riemannian property gets you kind of as close as possible to this theoretical understanding. So in general, you'll have the diameter will be somewhere between log of the size of g and two log of the size of g. And I believe that it's still a conjecture that for these graphs that I think this is due to Lubatsky-Philips Arnak, that that expansion constant is between one and four thirds. And actually within our Win Four group, we did a lot of computation. Yana in particular did a lot of computation on this. And it's still hard to see whether the expansion constant is tending towards four thirds or closer to one. So anyway, this is all kind of background about the properties of this graph. And the reason that I focus on this for a few minutes here and a couple of slides is that thinking of SIG, the super singular isogenic graphs abstractly in terms of expander graphs and kind of the more the combinatorial point of view gives you different insight than if you just think of it in terms of elliptic curves. And that's kind of the point of view that I've always had because we started with the point of view of proposing expander graphs in general for cryptography. So now I'd like to move on to the other applications of super singular isogenic graphs. In particular, in 2011, Zhao and Defeo proposed a key exchange from super singular isogenic graphs and encryption in a paper in 2014 with Zhao, Defeo, and Plou. And in 2016, there have been a number of signature schemes that have also been proposed. I just picked out a couple of them here. Galbraith, Patine, Silva proposed a signature scheme in 2016 and then SQ sign is a signature scheme that was proposed in 2020. So I will talk about the signature application tomorrow. But my goal right now is to talk about the key exchange application. So this is the actual scheme that is proposed in the NIST PQC competition, which is now in the fourth round. It's called Psyche, super singular isogenic key exchange based on Zhao, Defeo, and Plou's work and paper. And so the only thing is that in their original paper, which was quite long, you had a lot of different hardness assumptions being introduced, actually five different hardness assumptions, none of which were known previously and they had not been stated previously and they were not related to other known hard problems. So that made it a little bit weird and a little bit hard to understand the security of this key exchange. And so what I'll explain in this lecture today is that in a fairly intuitive way, you can see that the security of this key exchange also relies on the same hard problem that the hash function relies on, which is the hardness of finding paths in this graph. Okay, so first let me just give you a picture. It's like a diamond here and these arrows are isogenes but they are not a single step in the graph. Each of these arrows is a whole bunch of steps like n steps in one case and m steps in the other and then n and m. So within the graph, each of these top and bottom of this diamond corresponds to n plus m steps in the Li-Sogyny graph. So here's how they propose to do key exchange. For those of you, it's like Diffie-Hellman. So for those of you that know Diffie-Hellman, so each party, they usually say Alice and Bob would pick a secret integer and there will be a known, a publicly known point on the elliptic curve and in the group of points on the elliptic curve, Alice would compute like A times P and Bob would compute B times P not sharing the integers A or B and then they would make it public, basically share with each other. Alice shares A times P and Bob shares B times P and then they each know their own secret. So when they receive this other point from the other person, they can use their secret and multiply times the point they received and in the end, both parties can compute A times B times the point P and any eavesdropper just sees A, P and B, P and all they can do is compute A plus B times P. So Jao de Veilplut's key exchange is very analogous to this but instead, Alice, what Alice is gonna do is going to pick a secret isogeny. So let me see if I can get my cursor here. A secret isogeny, which we're calling phi sub A and Bob is gonna pick a secret isogeny, phi sub B and so what happens is that they each are going to compute this curve, E sub A or E sub B and make that public or share it with each other. We assume when I say make it public, meaning if they share it with each other you can assume there might be an eavesdropper. So it might as well just assume that information is public and then there's gonna be some, actually some auxiliary information. So this is something that can potentially make this problem easier than just the hardness of finding paths but using that auxiliary information both Alice and Bob can compute these other isogenes phi prime sub A and phi prime sub B and they can both compute this other elliptic curve E sub AB and use the J invariant of this final curve E sub AB in order to establish their common secret. And so I like to describe it that way so you can see how it's analogous to Diffie-Hellman key exchange except this time it's with isogenes. So let's be a little more specific about how this works. So the key exchange setup is gonna be, so E is gonna be a starting point. It's a super singular elliptic curve over Fp squared but P has to be of a particular form in order for this to work. So we're gonna assume that P is actually, you have two small primes, LA and LB and these are basically the degree of isogenes in the two isogeny graphs that Alice and Bob are gonna work in. But just to make things easier, you can just assume LA is two and LB is three. That is the most efficient instantiation of this system. So really what you're saying is that P is equal to some large power of two times some large power of three plus one. So and it needs to be prime of course which won't always be the case. But for reasons that I'll explain in a minute these powers M and N also need to be somewhat similar to each other for security reasons. So in order to exchange the key given this setup what they need is they need to work on the torsion points on the elliptic curve. So something that I didn't explain earlier about ECC is that in general for the implementation of classical elliptic curve cryptography you wanna have a curve over a finite field Fp where the curve has either a prime number of points on it or a close to a prime number of points. If the order of your group is smooth then you have a tax which will run in time proportional to the square root of the largest prime factor. So for security reasons you really want the order of the elliptic curve to be prime. And here we have something similar which is that in the so over the base field in the classical elliptic curve case you'll take a generator of that prime sub either prime order group or prime order subgroup the largest prime that divides the order of the group. And that'll be your point P. So you could call that like if the order of that group is M you can call that an M torsion point. Well if you wanna do your operations over the base field and as I explained earlier today I mean the bigger the field the more costly it is to compute in it. If you wanna do it over the base field then you actually need the point to be defined over the base field. And so that's an issue here too and that is that the reason we choose P to be of this form is so that we actually have that LA and LB to these powers divides P minus one. So that's why P is picked to be of this form. So we're going to take points that are P sub A and Q sub A for Alice and P sub B and Q sub B for Bob which actually generate the relevant torsion subgroups for these elliptic curves. So E join like two to the M and E join three to the M three to the N. So now Alice and Bob in order to do their key exchange are going to pick random integers. So A pick Alice picks M sub A and sub A and Bob picks M sub B and sub B and uses Veluz formulas or there are more efficient ways to compute this isogeny which I'm not gonna talk about but I think that there's some exercises kind of going into this. That A uses Veluz formulas to compute the isogeny which is taking E and quotienting by the subgroup M A times P A plus N A times Q A and then Bob will do the same thing. So what you end up with is that Alice and Bob have each computed their new elliptic curve. They can use Veluz formulas to compute the new J invariant and the model for these curves E A and E B. And then using the knowledge of these points the images of these points like the images of Bob's points under Alice's isogeny and the image of Alice's points under Bob's isogeny that's the auxiliary information which is provided. Now they can each use that to then apply their secret integers to create the next subgroup that they need to quotient by. So E A will be, or Alice will be quotienting E A by subgroup basically M A times phi B of P A plus N A times phi B of Q A, et cetera in order to compute the curve E A B and then Bob will do kind of the analogous thing on his side and they will both end up computing the same curve. So one way that you can see that it's the same curve is by actually just seeing that by quotienting twice again these are a billion groups you're quotienting by two subgroups here that you have actually done the same job on the top and the bottom. You've actually quotiented by the same subgroup and that's why you end up at the same curve. So now let's think about the security of this key exchange. So clearly if you can find the isogeny for example phi A then you can, so I'm sorry if you can find the path between E and E A then you can break the key exchange. So one thing to note is that the walks on each stage of the key exchange here are only roughly half the diameter of the graph. Okay, so let's go back to this expression for P. So P is, think of it as two to the M times three to the N plus one. And so if you, let's say that this were just two the two and three are pretty close to the same size. So just thinking about the size of things. If you just take two to the N plus M plus one, that's roughly P. So if you take the log like the log base two of the size of the graph which I told you was P over 12 the log here is basically M plus N. And I told you that the diameter of these graphs is roughly the log of the total number of vertices. So what we're saying is that in general the length of a path to get from one vertex to another random vertex is going to be M plus N. And if M and N are roughly the same size then in each stage of this diamond you're only going about half the diameter of the graph. And so that's why I said I like to think about these things from a kind of combinatorial point of view and that is that just go back to my description where I said in a three regular graph with no backtracking if you take a walk of length N you only get at most to two to the N new vertices. So if you only take a walk of length half log of the vertices you are not visiting most of the points. You're visiting basically like one over square root of P of the points. And so if P is like two to the 256 or in this application P needs to be bigger if it's 500 bits or 750 bits if you take two random vertices in the graph it's very unlikely that there is a path of length half the diameter between them. So what that means is that if you are given these two points E and EA and if you can find a path between them then it's overwhelmingly likely that it was the path that was used to set up the key exchange. So what that shows you is that path finding is most likely with very, very high probability enough to break this cryptosystem at worst you could just run your algorithm again and try to get another path if that was not the path that was used in the key exchange. So this was formalized as part of a paper in 2017 from my Win 4 group with Anna Kostas, Brooke Feigen, Micah Messer and Anna Puskas. And it's a little bit more formalized here basically saying that if you can do path finding in these graphs, the LA isogenic graph and the LBI isogenic graph, then in either one of those graphs then you will be able to break the key exchange with very high probability. The probability being basically the probability that you failed being basically one over square root of P. So like two to the minus 256, for example. So the nice thing then is that gives us kind of a clean story. It's not maybe quite as clean as that. Certainly path finding has to be hard in order for this key exchange to be secure. If you can find an efficient algorithm for path finding in the two isogenic graph or the three isogenic graphs then you will be able to break the key exchange. However, just like Nadia's comment about the non-equivalence of factoring and breaking RSA, especially in this case, there's some auxiliary information that's given in this protocol which is the image under these isogenes of the public points of your person you're trying to do the key exchange with. So it is possible that there are easier ways to break this key exchange that do not involve finding paths in these graphs. Okay, so just to kind of restate what are the hard problems in the supercyclic singular isogenic graphs that we're relying on. So to avoid collisions where, like the picture I showed this morning very briefly, a collision would be two paths from the same starting point to the same ending point. So that's a cycle. So that's hardness of finding cycles in these graphs. Finding, so you could state that as either the difficulty of producing a pair of super singular elliptic curves, E1 and E2 and two distinct isogenes of degree L to the N between them, which are, by the way, not equal to just multiplication by L to the N over two. Or you could say it in terms of finding an endomorphism which actually corresponds to going along one path and going back along the other path. So that would be an endomorphism of the initial curve E which has degree L to the two N which is not multiplication by the L to the N map. So those are two different ways to say what is the hard problem in terms of elliptic curves that corresponds to collisions. And then another way to say what the hard problem is that corresponds to finding paths is given two super singular elliptic curves just to find an isogeny of degree L to the N between them. So just as a little side comment, there is a question in the context of the hash function whether you fix the length of the path. Like as I mentioned, the hash function can be generically thought of as a function which maps bit strings of length M to bit strings of length N. And so once you've done that, that pretty much fixes the length of the path. And so that's why I've stated this here as finding an isogeny of degree exactly L to the N. But you could think a little bit more generally about what if you're allowed to find paths of different lengths, not just length N. So that could potentially be an easier problem. Okay, so then the last thing that I want to talk about today it's a little bit late and we've been talking and listening for a long time today. So I think I will try to end a little bit early. But so the last topic I wanted to address is what do generic attacks on this cryptosystem look like? So when we say generic attacks in cryptography or we often call them square root attacks, we're often thinking about the like, Pollard row or baby step, giant step or kind of birthday, basically birthday paradox attacks. And so what we mean by that is that if you kind of just assume that everything is random and you apply these algorithms where you do some kind of either random steps or some kind of deterministic rule even that like approximates random behavior, that you'll end up with these square root attacks which heuristically run in usually the square root of the group size. So because we know of generic attacks for almost everything including this, which I'm just about to tell you, that's what also makes us to set the parameters the way we do. So for the hash function, at the time when we proposed it if we wanted a kind of US government minimum security which is 128 bits, we wanted to set the size of the graph to be basically 256 bits so that the best generic attacks would run in time roughly two to the 128, making it roughly to, that secure two to the 128. So since the number of nodes is like I said the Eichler class number, which is P over 12 roughly that means that that's why we set P to be 256 bits for the hash function. Now what you could see from what I explained about the key exchange is that P needs to be at least like twice as big for the key exchange because the walks, those walks are only half as long, half as long as the diameter. Whereas in the hash function, the walks are length fully the full diameter of the graph. So you can think of basically a birthday attack strategy on this problem which is that you randomly, you have two nodes in the graph, you're trying to find a path between them. So you take a random walk from both sides from both ends until it hits each other. So, and then given the kind of reasoning of the birthday attack, this will run in time roughly square root of the graph size. So in other words, kind of starting from both ends and kind of a meet in the middle kind of attack. So the generic attacks have been known since the beginning and since we first proposed these graphs, we considered a lot of different kind of strategies for attacking them. And the main kind of, I would say most promising strategy that we have today is that instead of thinking of these graphs as being the nodes are the elliptic curves and the edges are the isogenes, if you can somehow move over to the other description of this graph, which is in terms of maximal orders in a quaternion algebra and connecting ideals between them, we actually have an algorithm for a relatively efficient algorithm for attacking the graph in that setting. And so what that leaves us with is the fact that the security of these crypto systems is based on actually being able to do this association of these two graphs, which is called computing the endomorphism ring. So for every node, if you can actually just compute its endomorphism ring as a maximum order in the quaternion algebra, then you'll be able to use the attacks over here on this side and kind of get back an attack on the original problem. So luckily, like I said, luckily or unluckily, the problem of computing the endomorphism ring seems to be just as hard as computing the paths in the original graph so far, anyway. But there's a lot of interesting work in the community going on now on that particular problem. So that will be the topic of my lecture for tomorrow. And so for now, I'd just like to stop and thank you for listening and see if there's any questions. There was somebody that had a question this morning that didn't get to ask it, but they're still here. Okay, I don't see any, so let's thank Kristin again.


Dear learners, today I will take a video class for you. The unit known 6. The unit is Modern Classification of Propositions, Simple and Compound, Singular and General. Modern Classification of Propositions. Simple and Compound, Simple and Compound, Singular and Singular and General. In this unit we basically discuss Modern Analysis of Propositions, Propositions and Sentences, Modern Classification of Propositions, Simple Propositions, Compound Propositions and General Propositions. Now you will see there learners. This unit introduces you to the Modern Analysis and Classification of Propositions. Modern Logicians do not agree with the traditional definition of Propositions. They define Propositions as a statement which is either true or false but it cannot be both true and false. A proposition is not identical with sentence. Similarly, a proposition may express a fact but it is not a fact. Only indicative sentences may be judged to be either true or false. If proposition may have any number of constituents and these constituents are combined in various ways. The mode of combination expresses the logical form of proposition. Modern Logicians have classified proposition into Simple, Compound and General. Modern Logicians have classified proposition into Simple, Compound and General. Now there are learners. In Modern Logic, proposition means a statement which is said to be true or false. For example, Asako was the king of Pataniputra. 3 plus 6 is equal to 9. So Critics was a philosopher. No man is immortal. Truth or falsity can be ascribed to every one of them. Every one of them can be either true or false. But may God help us is not a proposition because the properties, truth or falsity cannot be ascribed to it. In Modern Logic, the component of a proposition that is subject and predicate are known as constituents of a proposition. Again I repeat there, learners. The components of a proposition that is subject and predicate are known as constituents of a proposition. Subject and predicate in Modern Logic known as constituents of a proposition. You see, there are learners. Truth or falsity of a proposition is called truth value. The truth value of a proposition is determined by the fact. If a proposition represents a fact as it is, then the proposition is true. Otherwise it is false. But a proposition is different from a fact. Now you see there are learners the difference between a proposition and a sentence. There is a clear difference between a proposition and a sentence. Do a proposition is expressed in the form of a sentence it is not identical with a sentence. We can point out following points as their differences. You see there learners the difference between proposition and sentence. You see a proposition may be understood as that to which the truth or falsity can be ascribed. So in case of proposition we ascribe truth and falsity. But truth and falsity cannot be ascribed to all kinds of sentences. All grammatical sentences are not propositions. Only indicative sentences can be just to be either truth or false. Indicative sentences can be said to be true or false. Indicative sentences, imperative sentences, operative sentences cannot be just to be either true or false. For example, have you ever gone to Kolkata? Then may God help us? Truth or falsity cannot be ascribed to these propositions. So they are not propositions. The same proposition may have different verbal expressions. Therefore the same proposition may be expressed by different sentences. But various sentences have a single underlying meaning which is called a proposition. Now the learners you see modern classification of proposition. Modern physicians have broadly classified the propositions into elementary propositions and non-elementary propositions. Elementary propositions are of two kinds, simple and compound propositions. And non-elementary propositions include all general propositions. Let us discuss all types of propositions. Now the learners you see what is simple propositions? A simple proposition expresses a simple fact and it cannot be analyzed into further propositions. Moreover a simple proposition makes an assertion about an individual, a person, a place, a theme, a country and so on and so forth. For example Sima is doing a course in logic. Simple propositions have four forms. You see first one is you can say that is subjectless proposition. Second one you can say subject predicate proposition. Third one you can say relational proposition. These are the examples of subjectless proposition. These propositions are also known as the explanatory propositions. The propositions such as it rains, it thunders are known as impersonal propositions. Such propositions have no logical subject. In these propositions the thinker asserts something but the statement is not fully expressed. These propositions give information and therefore they are regarded as proposition. Now the learners the second one is subject predicate proposition. A proposition which asserts that a quality or an attribute belong to something is called subject predicate proposition. For example Sima is intelligent, this paper is white. Subjects of this type of proposition is a singular term. The subject predicate type of proposition is represented as SP. It indicates subject predicate proposition. In predicate logic small letters ABC are used to symbolize individuals and capital letters ABC are used to symbolize attributes. For example Sam is intelligent the symbolic expression of this proposition is I. Sam IS means Sam is intelligent. Here the symbol I stands for intelligent and S stands for Sam. Sam is intelligent can be symbolized as I capital S small. I S means Sam is intelligent. I is capital S is small. S indicates attribute and not S indicates person and I indicates attributes that is capital letter. So it indicates Samal is intelligent. The learners now you see what is relational proposition. Relational proposition asserts a relation between two or more constituents. There are various ways to express relations. We may use the verbs drink, love, enemy, heart or words greater than smaller than etc. to express various relations. For example the lecturer teaches a course in English. He owes me 5 rupees for a bag. So in a relational proposition the relation proceeds from something to something else. This is called a direction or sense of relation. The term from which the relation proceeds is called a referent and the term to which the relation proceeds is called a rhythm. So in case of relational proposition we find the direction of relation where we find the relationship between referent and rhythm. In the above example which we already mentioned the lecturer teaches a course in English.  Therefore their learners in relational proposition we find relatum and relatum and what? Relatum and referent. Now you see their learners what is class membership proposition. So class membership proposition asserts that an individual is a member of a class. For example Shibaji is a hero, Rabindranath Thakur is a poet, Gurgenstein is a philosopher. This class membership proposition is symbolically expressed as R-E-P. It is R-E-P-E indicates the relations you see belongs to. So this class membership proposition is symbolically expressed as R belongs to P. So you see here R stands for Rabindranath Thakur and P stands for a class of poet. In symbolic logic the capital letters A, B, C are used to symbolize the class and small letters A, B, C are used to symbolize to individual. So this is all about class membership proposition. Now their learners you see what is compound proposition. So you see their learners compound proposition contains two or more simple propositions as these components. For example Socrates was philosopher and Russell was a mathematician. This proposition is a single proposition. Simple propositions are combined by various ways to form a compound proposition. It means that compound proposition provides two or more simple propositions. Now you see compound proposition again is of four kinds their learners. Compound proposition again of four kinds one is conjunctive proposition. Second one is disjunctive proposition. Third one is alternative proposition and fourth one is implicative proposition. So these are the four kinds of compound proposition. Now you see what is conjunctive proposition. Now you see what is conjunctive proposition. So the compound proposition in which two or more simple propositions are combined by the word N is known as conjunctive proposition. Suppose you see Manmohan Singh is the Prime Minister of India and Barack Obama is the President of USA. This conjunctive proposition is symbolically expressed as P.Q. When two simple propositions are combined by the word N then it is known as conjunctive proposition. Here we use the symbol N it is written as dot P dot Q. Now you see what is disjunctive proposition. Now dear learners the compound proposition in which two propositions are combined into one by the word or either or in the inclusive sense is called disjunctive proposition. In the inclusive sense for example either logic is interesting subject or it is a scoring subject. Logic is interesting subject or it is a scoring subject. The components of disjunctive propositions are presented in the form of disjuncts. This disjunctive proposition may be symbolically represented symbolically stated as P.H.Q. P implies or P indicates that logic is interesting either logic is interesting or it is a scoring subject. So both these propositions are joined by the symbol P.H. This P.H.Q. it indicates the disjunctive proposition like either logic is interesting either logic is interesting or it is scoring. Now dear learners you see the third one that is alternative proposition. So this is very very important one. You see the compound proposition in which two propositions are combined into one by the word or either or in exclusive sense not in inclusive sense is called alternative proposition as for example the man is either literate or illiterate. It indicates that the two propositions cannot be true in the same time. That's why it is used in the exclusive sense. P alternate Q either man is literate or illiterate either man is rich or poor. In this exclusive sense we use the symbol that alternate we use. So you see their learners here P and Q stand for two alternative and the symbol the latino art stands for or in this exclusive sense not in inclusive sense. The meaning of or in its exclusive sense can be expressed as P alternate art Q dot negation P dot Q. Now you see their learners the fourth one that is implicative proposition implicative proposition. The compound proposition in which the two simple propositions are combined by the word if then is called an implicative proposition. For example if rain comes then the price of food grains will rise. If rain comes then the price of food grain will rise. The component proposition which follows if is called the antecedent or the implicance. And the component proposition which follows then is called the consequent or the implicate. So in case of implicative proposition we find two parts one is called antecedent that is implicance and the another is called consequent that is also called as implicate. It in an implicative proposition one component proposition implies the other the proposition which implies the other is implicance and the proposition which is implied that is consequent part it is called as implicate. Now their learners so this is all about compound propositions. Now we will discuss what is general proposition. Now you see their learners general propositions are non-compound propositions. They are all about classes such propositions either affirm or deny the existence of proposition the existence of something or a property of the whole universe and the relation between two classes. So there are three kinds of general propositions their learners first one is existential general propositions. Existential general proposition and second one is one predicate general proposition. Second one is one predicate general proposition and third one is general proposition general proposition asserting relations between asserting relations between two classes. First one is existential general proposition, second one is one predicate general proposition and third one is general proposition asserting relations between two classes. Now their learners now you see what is existential general propositions. An existential proposition directly affirms or denies the existence of something. For example tiger exist in predicate logic the tiger exist can be symbolized by using existential propositions such we can Ex in predicate logic we symbolize the tiger exist as ex as dx is indicates that tigers exist. Then one predicate general propositions you see one predicate general propositions either affirm or deny a property or an attribute about the whole universe for example everything is mortal nothing is permanent. The first proposition is symbolized as you see one predicate general proposition everything is mortal we can symbolize as x dx is indicates that everything is mortal and the second proposition is also implies nothing is permanent. Nothing is permanent means you can say x implies px nothing is permanent it indicates that in predicate logic we symbolize like that we use the universal quantifier x for all and we use the symbol that ex for existential quantifier means tigers exist. Now you see their learners general propositions asserting relationship between two classes. Here you see this proposition states that one class is wholly or partially included in or exclusive from another class. So here we find all the four propositions of A, E, I, O all lions are animals no man is perfect and some students are clever some philosophers are not scientists. Here the traditional categorical propositions you see traditional categorical propositions A, E, I, O is there. Means general propositions asserting relationship between two classes here all lions are animals all lions are animals then no man is perfect then some students are clever some students are clever and some philosophers are not scientists some philosophers are not scientists so these are four categorical propositions which are included in general propositions asserting relationship between two relations between the two classes. The traditional categorical propositions belong to this category so you see we can symbolize here above the four examples like all lions are animals we can symbolize here x that is universal quantifier Lx implies Ex all lions are animals we symbolize in this way then X no man is perfect X Mx implies not Px this implies that no man is perfect then here also we use universal quantifier but in case of i propositions i categorical propositions some students are clever we use existential quantifier means some students are clever this is dot not Mx some students are clever then you see some philosophers are not scientists here also we use existential quantifier then some philosophers are not clever we use we symbolize in this way the four types of categorical propositions A, E, I, O so you see it means that given any X if X is a lion then X is mortal that the X is if it means that given any X if X is a lion then X is animal then you see again the next one given any X if X is a man then X is not perfect the third one it means that there is at least one X such that X is a student and X is clever and fourth one is that there is at least one X such that X is a philosopher and X is not scientist so this is all about the modern classification of propositions simple and compound singular and general now after going through this unit now there are some highlighting points I would like to give for you and that highlighting points you see proposition is the basic point of logical unit of logical thinking by proposition we mean any statement which must be either true and false a proposition is expressed in the form of a sentence but it is not identical with a sentence similarly the truth or falsity of a proposition is determined by the fact but a proposition is different from a fact only indicative sentences can be regarded as propositions because they can be just to be either true or false they are learners unlike traditional propositions unlike traditional logicians modern logicians do not accept that every proposition must be expressed in the same logical form that is subject copula and predicate according to modern logicians there are innumerable kinds of propositions there are some drawbacks of the traditional classification of proposition though traditional logicians try to express all type of propositions in the same logical form fundamentally they are different in their logical structures therefore there seems to arise some confusion in their classification of proposition in order to get rid of the defects and limitations of traditional classification of proposition modern logicians attempt to present their classification scientifically according to modern logicians there are three kinds of propositions simple compound and general a simple propositions contains only one statement on the other hand and compound proposition contains two or more simple propositions and general propositions are about classes now they are learners this is all about the summary of this unit modern classification of proposition simple compound and general in order to understand the unit in a very lucid manner you have to study some books and that books are you take logic in formal symbolic and inductive that is written by Sanda Sakramoti and another very important book introduction to logic written by I. M. Kopi, Kohen and Karl and Kohen another very important book modern elementary logic written by Susan Alstabing and very important one that is logic written by Ossockard and Ankush Sawant so I think they are learners you have understood the unit 6 modern classification of propositions simple compound and general thank you


The U.S. Army launched this robotic collaborative technology alliance in the early 2000s just as robots began entering the battlefield. With the global war on terrorism came IEDs, and to combat them, the U.S. military teamed with industry and academia to tackle what would become its greatest challenge, replacing teleoperated robots with autonomous systems that could turn robotic tools into robotic teammates. Robots depend on our robots being precise, being accurate, and being dexterous. Through focused research pursuits, the Army is advancing research to solve some of the most difficult problems facing allied forces. By collaborating with America's top academic institutions, the Combat Capabilities Development Command Army Research Laboratory is pushing the frontiers of scientific research to help robots autonomously perceive, plan, and interact. The military and the Army context means that you can't know the environment ahead of time. It's going to change. Much of today's advancements in robotics, autonomy, and artificial intelligence is credited to the foundational research pursued by America's military and its university and industry partners. Strategic investments in defense-led foundational research has resulted in advanced science and four critical areas of ground combat robotics that affect the way U.S. warfighters see, think, move, and team. Academia is all about solving the most challenging problems out there. This collaboration is about developing the algorithms that look at the problems in a basic shape and form, find the algorithms that solve those problems, and then transitioning those algorithms into the technologies that are much more mature. And this is where the Army and collaboration with industry really helps, because researchers on the Army side can actually take those algorithms and actually build the technologies that can solve the problems using the algorithms that were developed on Academia. Here at MIT, we've tried really hard to provide value to the Army Research Lab through the various programs and by working with the researchers. We developed new forms of planning algorithms. We developed new forms of algorithms that allow robots to team naturally with human partners. So for instance, I and my research lab with my students and my postdocs, we've developed a number of systems that allow a robotic wheeled vehicle or a robotic mobile manipulator to actually take instructions from a human operator or a human partner, carry out complex temporal extended tasks over long distances in order to meet the needs of the warfighter. And so we're developing new kinds of algorithms, we're developing new models, and we're showing how these systems can perform well, can perform at army tempo, at op tempo, without the training data that most existing systems assume. I think I have been involved in the RCT program for six years. I think this is a very rewarding appearance for me because it allows me to interact with people, especially talented scientists and engineers from the different places. So my work focuses on a certain component which is critical in autonomous systems, but the interaction and collaboration with other experts allows me to have a more global understanding of the entire autonomous systems and I also learn from them. Being in a collaborative team I think really helps because one person cannot do all these things, right? And you also get to learn from people who are actually experts in these different areas. And I think that's one of the major reasons why I think collaboration is very useful. I have neither faculty, I am neither a student, so I am a person that works in between the faculty and the students. I have had the opportunity to see these students on the academic part in the classroom with faculty and then developing their approaches in the lab and then taking them to the field to work on real problems on the field with actual robots and that is an invaluable experience. These very same students also have the opportunity to interact with, for example, soldiers in the field or with colleagues from the ARL that have very different opinions and insights into what is relevant in the field of robotics. It was a great opportunity to test robots and environments that you don't usually get to use. As a university we're constrained to drive around in our hallways or around on campus and that's a very limited set of environments that don't reflect the challenges of the real world and Campbell June gave us an opportunity to go out and try more difficult things and see where the state of the art fails and where it succeeds. The RCTA project has been invaluable in helping us take some of the research that we're doing and partner with researchers and engineers in the Army and put it out in the real world to see how it performs. Some of the lessons learned are very harsh. The algorithms that I thought I had created that would work really well out in the real world ended up not performing well at all when subjected to the harshness of the real world. So I think what's really valuable is in having problems that are real, having problems that are meaningful, having clarity in metrics and partnering with people who have the same vision that we have of getting robots into the real world solving real problems. The collaboration with RCTA or in general with other researchers really highlights different perspectives of looking into the same problems and also by working with RCTA I get to see different types of problems that I can't really think of on my own when I'm researching in this perfectly safe environment. The students benefit from exposure to real world problems. They benefit from exposure to problems they wouldn't necessarily see in other domains and other disciplines. They benefit from the rigor that is brought to these programs. They benefit from being able to collaborate with other researchers and they also benefit technically from being able to advance the state of the art. The undergraduate and graduate students in my lab and in my collaborators labs, the opportunity to work on the highest priority problems of national defense has represented opportunity to get exposure to problems they wouldn't necessarily see elsewhere. Knowing how to get rid of a lot of the assumptions that exist in a lot of commercial systems and put them into systems for the military context is hugely valuable for the students. I got to visit the Army Research Lab and hear a lot more about what was going on and have like ongoing dialogue with the researchers where they would pose problems and questions that I would then go off and they would shape my research direction. So I was also funded under the RCTA during my faculty career. We're able to educate an entire new generation of PhD students, master students, undergrads, young faculty to think about what it is to solve problems out in the real world. So I think that we're sort of sowing the seeds of trees that will grow and really be the future of robotics for the United States. The Army Research Lab has contributed substantially to the next generation of engineers in industry and academia. A lot of industrial results are really dependent on the PhDs that are growing inside the collaboration with the Army Research Lab. I believe for students it's really beneficial to not just work on their small isolated projects but really see the bigger picture, working on interdisciplinary teams and I think that's also of course going to be really crucial later on when they go into the workforce and that is also of course exactly the kind of lessons they learn through participation in programs like the RCTA. Many of my students have gone on to academic careers, many of them have gone to found their own companies, many of them have gone to work in large defense contractors as well. This is all tremendous benefit to them and hopefully benefit to the nation as well and they're working on the next generation of problems. I just actually graduated this year and I will be joining Google DeepMind. This is a group that actually does like artificial intelligence research so I'll be doing basically AI research. I'm going to essentially continue my PhD work forward so still again working with manipulation, working on like perception and trying to sort of get robots into unstructured environments. I came to the University of Pennsylvania as a master's student but during my time on the RCTA I was accepted into the University of Pennsylvania as a PhD program in large part due to my work on the program. I feel very fortunate to have had the opportunity to work with the great people Army Research Lab, people like Dr. Stuart Young, people like Dr. Ethan Stump, Dr. John Fink and many others. A lot of the people at the Army Research Lab represent the best and the brightest of what our nation has to offer. They bring tremendous intellectual heft and rigor to the problems that we're tackling and they represent a real commitment to keeping the nation safe in a variety of ways. I think the Army Research Lab has gone unrecognized a lot of the time for its contributions in developing basic technology and transitioning that technology out into the world. Through this program there have been a lot of folks who graduated have gone to all the top companies that are building robotic systems. From my lab I have placed students at Uber, Apple, Amazon Robotics and a number of those companies and they're bringing in those kind of knowledge and expertise from the programs like the RCTA into the industrial world. The RCTA has given me better appreciation for the challenges of field robotics, moving robotics out of like a research lab and into the real world. My father was in the Army so I definitely knew there were talented people in the Army, talented engineers in the Army but what I didn't know was that there were talented researchers. I suppose I was sort of vaguely aware that the Army must be researching the future of war but I thought most of the hands-on engineering research would be going on in defense contractors' labs and so this collaboration with the Army Research Lab really highlighted that no that's not the case and that a lot of the most interesting research is being directly supported by the Army and by civilians working with the Army. I think the RCTA is a tremendously exciting project and it was a privilege to be a part of it. I learned a tremendous amount and I hope the Army funds similar things in the future and that I can be part of it.


In order to file for a patent, you must be the inventor. Now, if you find an existing product and you modify it and you apply it in a unique method, you might be considered the inventor and you can proceed with your own patent. You can also improve existing products and if your improvement is seen as being new, useful and it's not an obvious variation of what already exists, you could qualify for a patent. Some examples of obvious improvements that will not qualify changes in size or changes in color or shape. Of course, if there's some obstacle that you're able to overcome by changing the size, for example, making the hearing aids significantly smaller or if changing the shape or color allows you to overcome an obstacle, then these improvements could be seen as being non-obvious and you still might qualify.


Do you recall last episode where at the start I said there's one rule that we can't talk about the challenge wheel from episode 5? We can throw that rule out the window. I think I redeemed myself in episode 6. What's good boys? Hey, welcome back. We are on the hottest of hot streaks. Do you smell it? I don't know if it was the dirt jersey, if it was the shade jersey. I don't know what it was, but we're on a four win streak, about four and two. The Wheel of 2k, going into the seventh episode. If you take a peek at the cheat sheet, you'll know now that the worst case scenario is a subs truth or drink. Where I record a truth or drink and you guys choose every single question, we're hot right now.   Three signed jersey giveaway, five signed jersey giveaway, playing 2k high, henna tattoo, subs DM 10 people. It's all still available and I'm feeling good. Team wise, we picked up clay and we have yet to use Taukuzma or Mike Dunleavy. New addition to the team. Excited to see him. Yao Ming, still incredible. And you know what? I kind of like LeBron, but I feel like he gets outboarded a lot at the power forward. I feel like my best case starting lineup would be, I get Dark Matter Jimmy once I keep upgrading him. I move LeBron to the three and then we get a super legit power forward, like someone very big. I don't know if David Robinson can also play at the four, but I'd like someone like him. I really like rebounding on people. I think what I'd like on today's episode, I can't really replace Ja, that's my team captain, but I was going to say like a Dark Matter point guard. All right, well, you know what? The wheel's going to tell us what we get. The power of the dirt jersey remains. Let's see what we got on this one. Jackpot, but six, seven plus. Wait. Triple threat, steal a player. Okay. This could be insanely good. It could be so trash. Triple threat online. All I'm going to do is go into one triple threat game. I can steal one player from my opponent. Just have like invincible shack. I hope it's not team-based matchmaking because my team sucked that I just queued up with.  Dark Matter, Dr. J. Signature, Luca. Dark Matter, Yolkich.  Let's go. We're getting a Dark Matter for sure. I think I want the Luca, right? How expensive is Signature series Luca? Oh my God. He's point guard, shooting guard. Oh yeah, he's going in for Clay for sure and Clay's moving to the bench. I know there was Dr. J. and Luca. I'd rather have Donchich. Damn, someone bid it up to $5.95. Please don't bid again. Let's go.  All right, we're getting Luca and 333,000 MT back. Good to see it. All right, so we'll bring Luca in. Jaw is still an amazing point guard, so we're going to keep him in. Luca, dark guy, we've got two Dark Matters in this starting lineup now. Luca got to be spectacular. Unlimited contracts, love to see that. And badges, let's see who's got stats wise.   Draw file, close shot, everything's spectacular on him. Even his driving dunk is still an 85. Burner defense, 95 is way higher than I think it should be, but I'll take it. 99 speed with ball, 95 speed. 42 gold, 25 Hall of Fame. All right, so Clay, Kuzma, Rodman, Cat, Border, Kirillinko. Yeah, it's got to be. I feel so bad having Kirillinko this deep. We'll see if Kuzma's better. I might put Kirillinko back in for Kuzma, but I love Dennis Rodman. I love his defensive effort. I love his rebounding. I'm not taking Dennis Rodman out. Cat's the only pure center, and Kuzma's Galaxy Opal, so I kind of want to keep him. If I lose Dark Matter Luka right now because I can't win, I'm going to be livid. Well, it's okay, challenge wheel clutched up last time. Yellow Ming, Lebron James, Jimmy Butler, Lucadon should jam around with an interesting squad. And it's, what a challenge. Squatty points with our newest player. Just so happens to be a Dark Matter Lucadon, which I think I can do it. Cade Cunningham, Jalen Brown, all George, Cat, and you know what? It's not an all Dark Matter squad.  Why do they let people do this?   Oh, no way. Got her on board, and he's got a chrome ball.  Look at his jerseys, too. This is cancer. Oh my God. All right, whatever. He's breaking already. It's low key hard to tell how open I am. I'm going to do a post spin. It's going to be gnarly. Oh, the double.   What just happened? All right, so we need 20 with this Luca, yeah? Let's get two right here.  Can't imagine I miss free throws with this Luca. And 96. Take those. We take those.  Is this Paul George on Yao?   A? Yao with the rip.  Damn, Paul George is about to be heating up. But Jimmy can have three? Yes sir. I'm telling you bro, it's legitimate. Why do I get a three point shooting foul? Every single game. What even was that one? Whatever, take your points. Good God. Computer Jalen Brown was kind of botting me, but you know what? Luca's going to the line again. That's four for Luca. And I need 20 with him to complete our challenge. Ooh, Luca with a clean rip and he gets fouled. Oh yep, down to Clay, pick and roll, beauty. And there's Kuzma's first ever buckets. Ooh, good to. All right, Kuzma's screen and roll, I'ma pass that. Hey, another one for Kuzma.  This is what you get for off balling, bro. I'ma off ball you right back. Kuzma, one more time, Kuzma, let's go. Kuzma's got six straight. All right, I think we made a right call with Kuzma right now. I'll fight with that shot.  She can turn over? No. Good. 17 to 15 hanging on to a small two-point lead right now. He gets possession in the second quarter. I think I got 10. Oh, great defense, no shot that's going in. Oh, oh, Kira, Linko is open open. Oh, thank God for yelling. This man does not know how to guard y'all down there. 12 points to the paint to his two. I play in the paint. It's how I play Touquet. And ever since I got y'all, things have been looking very up for us.  I'd get the double with y'all. Yes, sir, he still tries to shoot it. What's he doing? All right, let's get more buckets with Luca. I only got four right now. Okay, that works for me. Michael Porter does not get a lot of... Michael Porter rarely comes in and he chucks up a brick. It was not a good release though, so I accept a small amount of responsibility.  Oh, just kidding. That was great defense. All right, all right, all right. Ooh, he drills one. Easy bucket. Good, ooh! Ref, what are you smoking? That was textbook perfect defense. Oh, whoa, whoa, whoa, whoa. Get it, right there. Dude, that's the best file he could have done at a wide open three. Oh, go, Luca, all the way. I'm gonna stand in the paint with y'all and I'm gonna get points. Sick of being poverty. Okay. I'm not getting Luca points.   Oh my God, you're small. Who are you?   Somebody's not guarded at all. Y'all with the defensive recovery. Good drop step, Luca. All right, we take those. No shot, dark matter, Luca just missed that. Absolutely nobody on him. And he just put that in on y'all. Kuzma, bad release. That was just a bad shot. It was a bad shot by me. Oh, and now I'm gonna lag right here, dude. And we just got a shooting foul.  Beauty. All right, it's a five point game. It'll be a lot worse. Y'all's in, let's pull him. He's tired. Let's pull Jimmy for Rodman. Let's play some defense. Good recovery defense from LeBron. Jimmy's gonna get five right there. Let's go, what a play. Gray play, Jimmy Butler. Somebody's not guarding somebody, but a stroke of luck goes our way until LeBron tips the ball directly out of bounds.  Who's on who? We got a board. Do I know how, no, but Luca hit the leaner, Luca. Let's go, Luca. Damn, let's get cooked. Hey, was that LeBron? Let's go, the signature dog too. We're up by two now. Hell of a run right now. Look at the defense. Luca, we need one from you. Yes, sir. Luca's got nine. Good close out. No shot, you're making that. Oh, yes, sir. Jop up fake. Oh, I thought that'd get him. Oh, but Bron from deep, Bron from deep. Easy shot. Well, how is that red? Goodie. No, you didn't.  All right, your luck's gonna wear out again. Once his luck wears out, I'll get my run back. We'll win the game. This is so wack right now. We have to get a stop here. It'll be so bad if I can't get a stop. This is bad. Who are you passing to?    And he drills that. Oh, this is horrible.  Hits another one right there. Carl at 30 towns. I'm about to quick shell you and just use anybody else. Let's go, y'all. Let's go. Tell me a little early. That's a three ball. Everybody clamped, great D. No, you didn't.  Oh my God.  And he's gonna get a wide open three because of it. Ooh, it's pretty good release. I was gonna green that.  What is this shit? How is he doing this? One more. Beauty. I'm gonna miss that. My whole team is just completely collapsing. We're scoring quickly. That's an N-1. It's gonna be single digits if y'all can hit. I mean, Robin's in. I've got the defensive lineup in right now. Yeah, LeBron's in too. This is my defensive lineup. There's y'all. No. No, no, no, no, no, no. That was supposed to be past the y'all.  Good D. What a pump. Wait a minute. 10 point ball game. There's no two minute warning basketball. Why was I just looking for a two minute warning? Good D? Good D. No. Oh no, this clock is bad. Give him a bailout on a seven point game? Okay. We had to stop too. Can't stop y'all right now. Quick score for y'all, which is huge.  Damn, what have been a five point ball?  Can't even see anything, even if I wanted to. You gotta love that. I just gotta take a bad shot. Wait a minute.  That was why I luck his game this season.  Definitely my unluckiest game of the season, dude.  Yeah, I was drilling him really.  I don't even want a personal foul anymore. Team comparison, let's see it. So we shot a better field goal percentage than him. A better field goal percentage than him. That is so annoying. He made three more threes. He shot 2% better. He went to the line 10 more times. And I get that two of those right at the end. The other eight, it was 13 to five. So the other eight, I had more points in the paint, more second chance points. He out rebounded me.  Moves us, I don't keep Luca now. You know what? I'm not even that mad about not keeping Luca. He did not play that good. It was a good game. I'm glad it was a well contested game. We haven't had a good game in a while, but wow, that sucked. But okay, so now Luca is going out entirely. And Don Levy comes back in, right? Wow, I did all that for the Luca. Now I just gotta sell him. All right, boys. I might have to bring out the final jersey for the next game. We fall to four and three. Five episodes left. I'll see you boys in the next one. Peace.