[MUSIC] Let's see how this Algorithm works in the context of triangles. We will use it to generate concept of tons, clause attribute sets. So, the first thing we do is we call first quarter and first quarter gives us the lactical first closed attribute set. It's the closure of the empty set. Well, in this formal context, the empty set is closed. So, our first closure is the empty set and then we run next closure with empty set as input. And next closure, we hope, will give us the lactical next closed set after the empty set. So, A in our algorithm is empty. And next closure goes through all attributes from e to a and tries to add them to the set A. So we start with e and we compute, we form the union of the set A, attribute m. In this case we get a set consistent of one attribute, "e", and then we compute it's conclusion. This is going to be the set "B" in our algorithm. Okay so, what objects have attribute "e"? It's T2 And T7. And they don't have anything else in common. So the closure of e is the set e. And the smallest new element in the set is e. So, indeed, in this case, the set a is e less than the set b, the first element in which a and b differ is e. So this is our next close Set e, we're on Now with e as input and next closure again starts with the largest attribute, and tries to modify the set A. Well, the largest attribute is e, but e is already here. So when the attribute e is part of A, the algorithm removes it. And so our A becomes the empty set. The next attribute is d of the colossal of d in this formal context is d itself. And because of the object T5 which has d and no other attribute, so B equals d, and the smallest new element is d. So this is our next closed set, d. At the next round of next closure, A equals. D and we add attribute e. Well, the closer of the e is the sum of all attributes because there's no object here that has both d and e. On the smallest new element inside M is a. So this with the first attribute in which the set capital A, and the set capital B differ. Here we have d, here we have a, b, c, d, e and the first item between which they differ is a. So a is greater than e, so we can say that B is e less than A, so we ignore this iteration of the algorithm. And next closure gets us to the next attribute. Well, the next attribute is d. So, it removes d from A and we have the empty set here. The next attribute is c. The closure of c is c. And the smallest near element is c again. So this is our next closed set. C. The algorithm continues with C, as A. The largest element is e. The of closure CE is M because we have no objects that have both c and a. And the smallest near element is a, so we ignore this, And we go to the next attribute. The next attribute is d. The closure of c, d Is M because again there is no, there is no object that has c and d together and we ignore it again and c is removed from a and and then the next that would be The closure of b is {b}. The smallest new element, relative to the empty set, is b. And so we've computed the next closed set, which is b. Then, the largest element is e. The closure of b, e is b, e, because of the the object T2, which has precisely these two attributes, b and e. The smallest new element is e, so, we've computed the next close set B-E, then we remove E from the B and the next attribute is D. So we compare the closure of b,d. And b,d is closed. Right, b,d, is closed here. And the smallest new element is d. So this is our next closed set, b, d. Now, a equals to b,d, and we'll add the new element, the largest element, e. But there's no object that has all the three attributes b,d,e. So the closure of b,d,e is m. The smallest new element is a and a is greater than e, so we don't need this close set yet. Go to the next attribute, the next attribute is d so, it's remote from our set a which becomes b. And the next attribute is c, so the closure of b, c is b, c itself. The smallest new element is c. Everything is fine. So b, c, Is our next closed set. Then we add e. But the closure b, c, e is M. So we ignore it, then we add d. But the closure of b, c, d is again M, so the smallest new element is A and we ignore the A. Then we remove C and we remove B and we are left with an empty set. And the largest element that remains is A. Okay, so because of A in this formal context is ABC because there is only one Object T4 that has attribute a and we also has b and c. So the closure of a is a, b, c and here the smallest new element is a relative to the empty sun, it's a. Okay, so this is our next close set. B, c and we continue with it as a, as capital a. So the next, the largest attribute is e. The closure of a, b, c, e. Is M, and the smallest new element here is d, which is less than e, so we ignore it. And instead we go to the next attribute, which is d. Now the closure of {a, b, c, d} is M again. But now the smallest new element is d, relative to a, b, c. So, this is the right time for M to be generated. A, b, c is indeed deal less than M. So this is our next close set and actually this is all closet because with computer theoretically large closet. It has all the attributes. The algorithm stops. It might look as if we simply go through all subsets of M and compute closure for each of them or like we do in the main algorithm, but this is not true, this is not the case. For instance,we I considered the closure of CDE, once we recognized that CDE is not good, we proceeded immediately with set B. And in large form contexts, where we have lots of closed elements and lots of subsets, this can really make things more efficient. [SOUND] [MUSIC]