1. Tautology

2. Tautology Decision May be able to use unateness to simplify process Unate Function – one that has either the uncomplemented or complemented literals for each variable Function F is weakly unate with respect to the variable X i when there is a variable X i and at least one constant a  P i satisfying F(|X i = a)  F (X 1,.., X i,.., X n ) The SOP F is weakly unate with respect to the variable X i when in an array F there is a sub-array of cubes that depend on X i and in this sub-array all the values in a column are 0.

3. Weakly Unate SOP F Example c 3 and c 4 depend on variable X 1 first column of c 3 and c 4 are all 0. Therefore F is weakly unate with respect to the variable X 1 X 1 X 2 X 3 1111 – 1110 – 1110 c 1 1111 – 1101 – 1101 c 2 0110 – 0110 – 1101 c 3 0101 – 0111 – 1101 c 4 F =F =

4. Tautology Decision - Weakly Unate Simplification Theorems Theorem 9.6 Let an SOP F be weakly unate with respect to the variable X j. Among the cubes of F, let G be the set of cubes that do not depend on the variable X j. Then, G  1  F  1. Theorem 9.7 Let c 1 = X j S A and c 2 = X j S B where S A  S B = P j and S A  S B =  Then, F  1  F(|c 1 )  1 and F(|c 2 )  1.

5. Tautology Decision Algorithm 1.If F has a column with all 0’s, then F is not a tautology. 2.Let F = { c 1,c 2,...,c k }, where c i is a cube. If the sum of the number of minterms in all cubes c i is less the total number in the univeral, cube then F is not a tautology. 3.If there is a cube with all 1’s in F, then F is a tautology. 4.When we consider only the active columns in F, if they are all two-valued, and if the number of variables is less than 7, then decide the tautology of F by the truth table.

6. Tautology Decision Algorithm (continued) 5.When there is a weakly unate variable, simplify the problem by using Theorem 9.6 6.When F consists of more than one cube: F is a tautology iff F(|c 1 )  1 and F(|c 2 )  1 where c 1 = X j S A and c 2 = X j S, S A  S B = P j and S A  S B = .

7. Tautology Decision Examples: 1. X 3 variable has column with all 0’s, so not a tautology. 2. F does not depend on X 1. Let c 1 = (11- 110 - 1111) and c 2 = (11- 110 - 1111) By Thm 9.7, F is a tautology. G =G = 01 – 100 – 1100 11 – 111 – 0010 F= 11 – 111 – 1111 F 2 = F(|c 2 ) = 11 – 110 – 1110 11 – 110 – 0001 11 – 001 – 1111 F 1 = F(|c 1 ) = 11 – 111 – 1110 11 – 111 – 0001  1 11 – 111 – 1111  1 

8. Generation of Prime Implicants

9. Definitions: Prime Implicant - an implicant contained by no other implicant. A set of prime implicants for a function F is denoted by PI(F ) Strongly Unate - Let X be a variable that takes a value in P={0, 1, 2, …, p-1}. If there a total order (  ) on the values of variable X in function F, such that j  k ( j, k  P) implies F (| X = j )  F (| X = k ), then the function F is strongly unate with respect to X. If F is strongly unate with respect to all the variables, then the function F is strongly unate.

10. Generation of Prime Implicants Definitions: Strongly Unate – Next, assume that F is an SOP. If there is a total order (  ) among the values of variable X, and if j  k ( j, k  P), then each product term of the SOP F (| X = j ) is contained by all the product term of the SOP F (| X = k ). In this case the SOP F is strongly unate with respect to X. If F is strongly unate with respect to X i, then F is weakly unate with respect to X i.

11. Strongly Unate Example F(|X 1 = 0) = (1111 – 1001) F(|X 1 = 1) = F(|X 1 = 2) = F(|X 1 = 3) = F(|X 1 = 0) < F(|X 1 = 1) = F(|X 1 = 2) = F(|X 1 = 3) F is strongly unate with respect to X 1 and to X 2 1111 – 1001 0111 – 0111 0011 – 0110 0001 – 0101 F =F = 1111 – 1001 1111 – 0111 1111 – 1001 1111 – 0111 1111 – 0110 1111 – 1001 1111 – 0111 1111 – 0110 1111 – 0101 F(|X 2 = 0) = (1111 – 1111) F(|X 2 = 1) = F(|X 2 = 2) = F(|X 2 = 3) = F(|X 2 = 2) < F(|X 2 = 1) < F(|X 2 = 0) = F(|X 2 = 3) 0111 – 1111 0011 – 1111 0001 – 1111 0111 – 1111 0011 – 1111 1111 – 1111 0111 – 1111 0001 – 1111

12. Generation of Prime Implicants Generation of Prime Implicants Algorithm

13. Generation of Prime Implicants Example:

14. Generation of Prime Implicants Example:

15. Generation of Prime Implicants Example:

16. Sharp Operation

17. Sharp Operation: (#) Used to computer F  G, assume For 2-valued inputs and F = U, n -variable function generates  ( 3 n / n ) prime implicants, so sharp function time consuming. Disjoint Sharp Operation: ( # ) Used to compute F  G. Cubes are disjoint, n -variable function has at most 2 n cubes.

18. Sharp Operation

20. Example:

21. Sharp Operation Example:

22. Sharp Operation Example:

23. Sharp Operation Example:

24. Problems to think and to Solve 1.Sharp operation for MV logic in Cube Calculus. 2.Realization of MV circuits and optimization using Sharp. 3.Applications of MV Tautology. 4.Strongly Unspecified MV functions. 5.Generation of Prime Implicants 6.Unate MV functions.