1 Henry Selvaraj Henry Selvaraj Henry Selvaraj Henry

1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Definition: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Let. Henry B Selvaraj; Y = {y 1, . . . , ym} and p, q M. Minterms p, q Henry Selvaraj; Henry Selvaraj; CON(B) and. Selvaraj; only if y(p) ~ y(q) for. Selvaraj; every. Henry y Selvaraj; B, where Henry Selvaraj; if Henry Selvaraj; Henry a Selvaraj; 1 ~ a 2 Henry Selvaraj; if. Henry Selvaraj; means that. Henry a 1 and a 2 Henry are equal defined. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; If Henry every. Selvaraj; pair of minterms in a. Selvaraj; set is. Henry consistent then. Selvaraj; the Henry Selvaraj; Henry Selvaraj; Henry which Selvaraj; Henry Selvaraj; set constitutes a consistence class. The classes are not the Henry Selvaraj; Henry Selvaraj; subsets any other consistent Maximum Henry Selvaraj; of Henry Selvaraj; Henryclass Selvaraj; are Henrycalled Selvaraj; Henry Selvaraj; Henry Selvaraj; Consistent Classes (MCC). Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Basic notions contd. . . Henry Selvaraj

2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; F Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; F Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Basic notions contd. . . The same notation as defined earlier for the inputs can also be used for the output consistence class, i. e. P (B), where the index F differentiates the IND and CON relations. When B = Y, the CON relation is denoted as P and is called the output partition of the function F. Henry Selvaraj

3 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; 1 Henry 1 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 2 2 1 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; Henry Selvaraj; Henry 2 Selvaraj; 1 Henry 2 Selvaraj; Henry 1 Selvaraj; Henry 2 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Basic notions contd. . . Definition: A specification table is said to be consistent if and only if for every pair of vectors r = (x , y ) and r = (x , y ), the equality x = x implies y ~ y. It means that for every pair of minterms v and v , if v = v then F(v ) ~ F(v ). A function expresses the functional dependency between its inputs X and outputs Y only when the specification table T is consistent. Henry Selvaraj

4 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

5 Henry Selvaraj; Henry Selvaraj; Using the Henry notations already tuple T: Henry Selvaraj; Henrydefined Selvaraj; for Henry Selvaraj; M Henry Selvaraj; Henry Selvaraj; = {1, . . . , 10}, Henry Selvaraj; Henry Selvaraj; = {x. Selvaraj; 1, . . . , x 8}, Henry Selvaraj; Henry Selvaraj; XHenry Selvaraj; YHenry Selvaraj; Henry Selvaraj; = {y. Selvaraj; 1, . . . , y 4}. Henry Selvaraj; Henry Selvaraj; Example for BSelvaraj; and Selvaraj; B 2 = {x. Henry be written as 1 = {x 1}Henry 2, x 3} can Henry Selvaraj; IND Henryrelations Selvaraj; Henry Selvaraj; P(B Henry Selvaraj; Henry Selvaraj; 1) = {{6, 7, 10}, {1, 2, 3, 4, 5, 8, 9}} Henry Selvaraj; Henry Selvaraj; (Abstract classes of equivalence (blocks of Selvaraj; a partition) will Henry Selvaraj; relation Henry Selvaraj; henceforth be separated by semicolons and. Selvaraj; not by. Henry parenthesis. ) Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; P(B 2) = {1, 2, 3, 4; 5; 9; 6, 7, 8, 10}. Henry Selvaraj; Henry Selvaraj; Examples of Maximum Consistent Classes (MCCs) for. Selvaraj; the Henry Selvaraj; Henrycalculated Selvaraj; Henry Selvaraj; Henry Selvaraj; given CON relations are. Henry as follows: Henry Selvaraj; Henry Selvaraj; = {1, 2, 4, 6, 7, 10; 3, 4, 5, 6, 8, 9} F(y 1) Selvaraj; Henry Selvaraj; PHenry Selvaraj; Henry Selvaraj; Henry Selvaraj; F = {1, 6, 7; 3, 5; 2, 4, 7; 6, 9; 4, 8, 9; 7, 10}. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

6 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; In Boolean algebra, a cube is expressed as a product of Henry Selvaraj; Henry Selvaraj; variables Henry Selvaraj; Henry binary as Henry given. Selvaraj; by the mapping WB: Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; ki = 1, Henry Selvaraj; i, if. Selvaraj; Henry Selvaraj; x Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; WB(k Henry Selvaraj; x Henry Selvaraj; i) = i if ki = 0, Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; ki = *, Henry Selvaraj; Henry Selvaraj; 1 Henryif. Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; where ki is a component of the cube K = (k 1, . . . , kn). Cubes Henry Selvaraj

7 Henry Selvaraj; Henry Selvaraj; Product the. Selvaraj; cubes. Henry C and D Henry denoted as Henry C Selvaraj; D (or C Selvaraj; D) Henry Selvaraj; of Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; is the cube E with components e as follows: i Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 0 Henry Selvaraj; Henry if ci Selvaraj; and di. Henry are respectively equal. Henry to 0, 0; 0, *; *, 0; Selvaraj; Henry Selvaraj; Henry Selvaraj; if ci Selvaraj; and di. Henry are respectively equal. Henry to 1, 1; 1, *; *, 1; Selvaraj; Henry Selvaraj; 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; if c. Selvaraj; respectively equal to Selvaraj; 0, 1; 1, 0; i and d. Henry i are. Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; *Henry Selvaraj; Henry Selvaraj; if c and d are respectively equal to *, *; i i Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Selvaraj; Henry Selvaraj; function Henry Selvaraj; The concept of a. Henry cube extends the Henry Boolean Henry Selvaraj; Henry Selvaraj; specification to include the case where the table T = Henry Selvaraj; Henry Selvaraj; Henrywhere Selvaraj; K Henry Selvaraj; (K, A, X, Y), is the set Henry of cubes and Vx. Selvaraj; = {0, 1, *}. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

8 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Now, the specification table can be redefined as T = (K, A, X, Y), where K is the set of cubes and it is said to be compatible if and only if for every pair of C, D K, C and D are compatible (C ~ D) and their respective output vectors are consistent i. e. F(C) ~ F(D). For example, if for the cubes C = (0*00) and D = (*100) the respective output vectors are (110) and (011), then they are inconsistent. On the other hand, if the respective output vectors are (-10) and (01 -) then the cubes are consistent. Henry Selvaraj

9 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; The. Selvaraj; collection of subsets K/(COM) as Henry well. Selvaraj; as K/CON are Henry Selvaraj; Henryand Selvaraj; Henry Selvaraj; Henry called r‑partitions on. Henry K and are Henry denoted as. Henry COM(X') and Selvaraj; Henry Selvaraj; Henry Selvaraj; CON(Y') Henry Selvaraj; respectively. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; R‑partitions can be used as a tool to classify cubes of a Henry Selvaraj; Henry Selvaraj; function Henry Selvaraj; Henrythat Selvaraj; Boolean description. It. Henry can Selvaraj; be shown the. Henry r‑partition Henry Selvaraj; Henry Selvaraj; concept is a generalization of the ideas of partitioning a set into Henry Selvaraj; Henry Selvaraj; Selvaraj; Henry Selvaraj; consistent classes and. Henry partitioning a Selvaraj; set into. Henry compatible classes. Henry Selvaraj; Henry Selvaraj; Therefore, partitioning a set S into consistent classes of certain Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry relation RHenry and Selvaraj; partitioning a set Henry into Selvaraj; compatible classes of Selvaraj; a Henry Selvaraj; Henry Selvaraj; certain relation T can be treated as special cases of r‑partitioning. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; are Henry Selvaraj; All Selvaraj; other symbols and. Henry operations of Selvaraj; partition algebra Henry Selvaraj; Henry Selvaraj; applicable to r‑partitions also. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

10 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 2 Henry Selvaraj; Henry Selvaraj; Henry 1 Selvaraj; Henry Selvaraj; 2 Henry Selvaraj; 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry 2 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; 2 Henry Selvaraj; Henry 1 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Henry Selvaraj; 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 2 Henry Selvaraj; 1 Henry Selvaraj; 2 Henry Selvaraj; Henry 1 Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 2 Henry Selvaraj; Henry Selvaraj; 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; It is said that the r‑partition is not greater than ( ) if every block of the r‑partition is included in at least one block of the r‑partition . The product of the r‑partitions and is a r‑partition , the blocks of which are the nonempty products of the blocks of and . The division of r‑partitions and is the r‑partition of blocks of , where every block B is the sum of the minimal number of blocks of . Henry Selvaraj

11 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 3 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 3 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; For = {0, 2, 5, 8; 1, 3, 4, 6, 7, 9} = {0, 4, 6, 7, 8; 1, 9; 2, 4, 5, 6; 3, 9} = {0, 5, 6, 7, 9; 1, 2, 3, 4, 8} = {0, 5; 2, 8; 6, 7, 9; 1, 3, 4} = {(0, 8), (2, 5); (1, 9), (3, 9), (4, 6, 7)} Henry Selvaraj

12 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; An example: Henry Selvaraj

13 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 3 Henry 4 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 5 Henry Selvaraj; 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 2 5 Henry Selvaraj; Henry Selvaraj; 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; B=x x x It is also not precise to say that cube C is compatible with either cube C or cube C. According to definition 3. 5, cubes of the set C = {1, . . . , 9} can be classified as: 1, 2; 1, 5; 3, 4; 6, 7; 8, 9. This is evidently not correct because the cube 0** contains the minterm 000 as well, and this minterm is different from both C and C. Therefore the cube C has to belong to an another class also and the classes ought to be as follows: 1; 1, 2; 1, 5; 3, 4; 6, 7; 8, 9. Henry Selvaraj

14 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; of Henry Selvaraj; Let C(Li. Henry ) denote the product all. Selvaraj; the cubes belonging to Henry Selvaraj; Henry Selvaraj; compatible of P(B) defined over. Henry a set. Selvaraj; of cubes Henry Selvaraj; Henry. LSelvaraj; Henry. C. Selvaraj; i class. Henry Selvaraj; Henryis Selvaraj; Henry Selvaraj; ASelvaraj; r‑partition of (B) a family of subsets of the. Henry set Selvaraj; C Henry Selvaraj; Henry Selvaraj; such that: (B) = P(B) C Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; where c Henry CSelvaraj; if and. Henry only. Selvaraj; if there are two Henry blocks Li of. Henry P(B) Henry Selvaraj; Selvaraj; Henrycubes Selvaraj; C(L Henry) Selvaraj; containing the cube. Henry c and the Henry sum Selvaraj; of all the i is Henry Selvaraj; Henry Selvaraj; different from the cube c. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; In other words, for every c C all possible Henry Selvaraj; Henry Selvaraj; combinations of minterms vc are assumed: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Modified definition: Henry Selvaraj

15 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; i Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; i Henry Selvaraj; Henry Selvaraj; Henryi. Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; The r‑partition generated by a set B is nothing but the product of r‑partitions generated by the variables x B: (B) = (x ) i: x B Henry Selvaraj

16 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry 1 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 3 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry 4 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; 3 4 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 3 4 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; For the table from the example, = {1, 2, 5; 3, 4, 6, 7, 8, 9} = {1, 3, 6, 8; 2, 4, 5, 7, 8, 9} = {1, 2, 3, 4; 1, 5, 6, 7, 8, 9} = {1, 5, 6, 7; 1, 2, 3, 4, 5, 8, 9} Therefore, for the set B = {x , x }, (B) = = {1; 1, 2; 1, 5; 3, 4; 6, 7; 8, 9}. Henry Selvaraj

17 Parallel Decomposition Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Parallel decomposition relies on. Henry dividing outputs into Selvaraj; Henry Selvaraj; the Henry Selvaraj; Henrygroups Selvaraj; Henry Selvaraj; two. Selvaraj; separate so that each. Henry group of outputs depend on Henry Selvaraj; Henry Selvaraj; lesser or the same number of variables as the given function F: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; X= X g X h Selvaraj; Henry Selvaraj; Henry X X Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Xg. Henry Selvaraj; Henry Xh Selvaraj; F Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Y G H Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Yg Henry Selvaraj; Henry Yh Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Y Henry Selvaraj; Henry =Y g Y h Selvaraj; Henry Selvaraj

18 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Parallel Decomposition Henry Selvaraj

19 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; X= Xg Xh Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry X 9 Selvaraj; Henry Selvaraj; Henry Selvaraj; X 9 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Xg. Henry Selvaraj; Henry Xh Selvaraj; 5 5 Henry Selvaraj; Henry Selvaraj; F Henry Selvaraj; Henry Selvaraj; G H Henry Selvaraj; Henry Selvaraj; Y Henry Selvaraj; 6 Henry Selvaraj; Yg Henry Selvaraj; Henry Yh Selvaraj; Henry Selvaraj; Henry 3 Selvaraj; Henry Selvaraj; 3 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Y = Selvaraj; Yg Y h Selvaraj; Henry Selvaraj; Henry Selvaraj; Parallel Decomposition Henry Selvaraj

20 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Truth table Yh Selvaraj; Henry Truth table Yg. Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

21 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Serial Decomposition: Example Henry Selvaraj

22 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Serial decomposition relies on partitioning the input variables in such a way as to obtain a two-level functional dependency F = H(A, G(B C)). Henry Selvaraj

23 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; X Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; A Henry Selvaraj; X Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; B C Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; G F Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Y Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; H Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Y Henry Selvaraj; Henry Selvaraj; Serial Decomposition Henry Selvaraj

24 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Non-disjoint Serial Decomposition models Henry Selvaraj

25 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; G Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; G Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; G Henry. FSelvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Necessary and sufficient condition for the existence of a serial decomposition There exists a serial decomposition of F if and only if there exists a partition , such that: (a) P(B C), and (b) P(A) P. Henry Selvaraj

26 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; GHenry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry 2 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Partition in theorem represents function G. In particular, the number of blocks in , , determines the number of output variables of G, log . Henry Selvaraj

27 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Example: Henry Selvaraj

28 Henry Selvaraj; Henry Selvaraj; Let. Henry A = {x 1}, B= {x 2, x 3, x 0}, = . Selvaraj; Henry Selvaraj; C Henry Selvaraj; Henry Selvaraj; GSelvaraj; = , . . , Henry = (1, 4, 7, 9; 1, 2, 3, 9, 11; 0, 1, 2, 7, 10, 11; 5, 6, 8). Henry Selvaraj; Verify Henry Selvaraj; Henry if serial. Henry decomposition exists. Selvaraj; for the Henry given Selvaraj; bound set B. Selvaraj; Henry Selvaraj; Henry Selvaraj; P(A) Henry=Selvaraj; Henry Selvaraj; (0, 3, 5, 7, 9, 10; 0, 1, 2, 4, 6, 8, 11) Henry Selvaraj; Henry Selvaraj; P(B) = P 0(B), . . , P 0, 1, 7; 5, 6; Henry 0, 1, 2; Selvaraj; 1, 4, 7, 9; Henry 10, 11; Selvaraj; 1, 2, 3, 9, 11) 6(B) = (6, 8; Henry Selvaraj; 3, 4, 5, 6, 8, 9; 0, 1, 2, 7, 10, 11; 1, 3, 4, 5)Henry Selvaraj; P Henry Selvaraj; F = (7, 9; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 7, 9; Selvaraj; 1, 2, 11; 3, 9; 0, 7, 10; 0, 1, 2, 11; 5; 6, 8) Henry Selvaraj; G = (1, 4; Henry Selvaraj; P(A)· Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; P(A) Henry·Selvaraj; Henry Selvaraj; Hence, G PFHenry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; function Henry Selvaraj; Thus, according to Theorem, F is decomposable as Henry Selvaraj; F Henry Selvaraj; Henry Selvaraj; = H(x 1, G(x 2, x 3, x 0)), where G is a two-output function. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

29 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; G Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry m Selvaraj; Henry Selvaraj; 1 Selvaraj; 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; The main task of decomposition is to verify if a subset of input variables for function G which, when applied as a subfunction to function H will generate the final function F, i. e. to find a P(B), such that there exists P(B) which satisfies condition a) and b) in theorem. To solve this problem, consider a subset of input variables B, and an m‑block partition P(B) = (B ; . . . ; B ) generated by this subset. Henry Selvaraj

30 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; j Henry Selvaraj; Henryi Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; i j j Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry F Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; A relation of compatibility of partition blocks is used to verify whether or not partition P(B) is suitable for serial decomposition. Two blocks B , B P(B) are compatible if and only if partition obtained from partition P(B) by merging blocks B and B into a single block B satisfies conditions in the decomposition theorem, i. e. if and only if P(A) P Henry Selvaraj

31 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry i 1 Selvaraj; i 2 Henryin. Selvaraj; Henry Selvaraj; ij Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Compatible class A subset of n partition blocks, B = {B , . . . , B }, where B P(B), is a class of compatible blocks for partition P(B) if and only if all blocks in B are pairwise compatible. Henry Selvaraj

32 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Maximum Compatible Class A compatible class is called Maximum Compatible Class (MCC) if and only if it cannot be properly covered by any other compatible class. Henry Selvaraj

33 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

34 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; 2 4 Henry Selvaraj; Henry 3 Selvaraj; 5 Henry Selvaraj; Henry Selvaraj; F Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Let B = {x , x } and A = {x , x }. Then, P = (3, 10, 14, 15; 12; 3, 16, 12, 14; 5, 7, 8, 13; 1, 8, 9, 14; 4, 8, 11, 12; 2, 6, 8, 12, 14), P(A) = (2, 13, 14; 5, 11; 1, 3, 7, 8, 9, 15; 4, 6, 10, 12) and P(B) = (14, 15; 2, 3; 9, 11, 12; 6; 13; 1; 8, 10; 4, 5, 7) Henry Selvaraj

35 Henry Selvaraj; Henry Selvaraj; Henryif. Selvaraj; Henry Selvaraj; Let. Selvaraj; us check B 0 = (14, 15) and. Henry B 2 =Selvaraj; (9, 11, 12) are compatible. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; We. Selvaraj; have: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; P 02(B) = (9, 11, 12, 14, 15; 2, 3; 6; 13; 1; 8, 10; 4, 5, 7) Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; P(A) P 02 Henry (B) = (2; 13; 14; 5; 11; 1; Selvaraj; 3; 7; 8; Henry 9, 15; 4; 6; Henry 10; Selvaraj; 12), Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; does. Selvaraj; not satisfy P(A)Henry PSelvaraj; PF, Selvaraj; Henry Selvaraj; Henry 02(B) Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; B 0 and B are not compatible. Henry Selvaraj; 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

36 Henry Selvaraj; Henry Selvaraj; Selvaraj; Henry Selvaraj; For B and B 1 Henry = {2, 3}, we obtain 0 = {14, 15} Henry Selvaraj; Henry Selvaraj; P (B) = (14, 15, 2, 3; 9, 11, 12; 6; 13; 1; 8, 10; 4, 5, 7) Henry Selvaraj; 01 Henry Selvaraj; Henry Selvaraj; Henry and Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; P(A) P 01(B) = Henry (2, 14; Selvaraj; 13; 5; 11; 1; 3, 15; 7; 8; 9; 4; 6; 10; 12) PF. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Thus, B 0 and B 1 are. Henry compatible. In a similar way we check the Henry Selvaraj; Henry Selvaraj; compatibility of Selvaraj; each pair of blocks P(B)Selvaraj; finding. Henry the following relation Henry Selvaraj; in. Henry Selvaraj; Henry. COM Selvaraj; Henry, B Selvaraj; Henry Selvaraj; = {(B 0 1), (B 0, B 3), (B 1, B 3), (B 2, B 4), (B 2, B 5), (B 3, B 4), Henry Selvaraj; Henry Selvaraj; (B 3, B 5), (B 4, BHenry 4, B 5 Selvaraj; 6), (BSelvaraj; 4, B 7), (B 5, B 6)}. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

37 Finding the MCCs Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Let Sj be Henry the set containing all the. Henry blocks Bi for which Bj Henry and BSelvaraj; i are Henry Selvaraj; Henry Selvaraj; compatible. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry a) Selvaraj; A compatible list (CC‑list) is initiated with one. Henry set containing the. Selvaraj; first Henry Selvaraj; Henry Selvaraj; block. Henry Selvaraj; Henry Selvaraj; b)Selvaraj; Starting. Henry from. Selvaraj; B 1 to B by one, the. Selvaraj; set Sj is. Henry formed and. Henry tested. Henry Selvaraj; m, one Henry Selvaraj; of Henry Selvaraj; c) Selvaraj; If Sj is Henry an empty set, Henry a new class. Henry consisting one. Selvaraj; block is added to Henry Selvaraj; Henry Selvaraj; the CC‑list before moving to the next S. (Since block j is in conflict with Henry Selvaraj; Henry Selvaraj; blocks B 1 Henry to Bj-1 Selvaraj; it is placed in a single element Henry Selvaraj; set) Henry Selvaraj; Henry with Selvaraj; Henry. CC Selvaraj; Henry Selvaraj; d)Selvaraj; If Sj is not. Selvaraj; empty, Henry its Selvaraj; intersection every of the current Henry Selvaraj; Henry Selvaraj; CC‑list, Sj CC is calculated. If the intersection is empty, the sets are Henry Selvaraj; Henry Selvaraj; not. Selvaraj; changed otherwise a new. Selvaraj; class is. Henry created by adding to the intersection Henry Selvaraj; Henry Selvaraj; an. Selvaraj; one-element set Bj. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

38 Henry Selvaraj; Henry Selvaraj; Hence, that the blocks of. Selvaraj; PG are. Henry denoted Henry Selvaraj; assuming Henry Selvaraj; B Henry Selvaraj; 1, . . . , B 6, Henry Selvaraj; Henry Selvaraj; we. Selvaraj; obtain the. Selvaraj; Sj sets and corresponding MCCs (for. Henry the Selvaraj; sake Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; of. Selvaraj; simplicity we write SSelvaraj; j sets and C-lists by indexes of Bj) Henry Selvaraj; Henry Selvaraj; Henry S 0 Selvaraj; = Henry Selvaraj; CC 0 Henry = 0 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; S 1 Selvaraj; = {0}Henry Selvaraj; CC 1 Henry = {0, 1} Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; S 2 Selvaraj; = Henry Selvaraj; CC 2 Henry = {0, 1}, {2} Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; S = {0, 1, 2} CC = {0, 1, 3}, {2, 3} Henry 3 Selvaraj; Henry Selvaraj; 3 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; S 4 Selvaraj; = {2, 3} CC 4 Henry = {0, 1, 3}, {2, 3, 4} Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; S 5 Selvaraj; = {2, 3, 4} CC 5 Henry = {0, 1, 3}, {2, 3, 4, 5} Henry Selvaraj; Henry Selvaraj; S 6 Selvaraj; = {4, 5} CC 6 Henry = {0, 1, 3}, {2, 3, 4, 5}, Henry Selvaraj; {4, 5, 6} Henry Selvaraj; Henry Selvaraj; S 7 Selvaraj; = {4} Henry Selvaraj; CC 7 Henry = {0, 1, 3}, {2, 3, 4, 5}, {4, 7} Henry Selvaraj; {4, 5, 6}, Henry Selvaraj; Henry Selvaraj

39 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry 4 Selvaraj; 7 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 4 Selvaraj; 5 Henry 6 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 2 Selvaraj; 3 Henry 4 Selvaraj; 5 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 0 1 3 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; As the last C-list represents Maximum Compatible Classes we conclude: MCC 0 = {B MCC 1 = {B MCC 2 = {B MCC 3 = {B Henry Selvaraj , B , B , } , B }, , B }.