1. 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 Xi when there is a variable Xi and at least one constant a  Pi satisfying F(|Xi = a)  F (X1 , , Xi , , Xn )
The SOP F is weakly unate with respect to the variable Xi when in an array F there is a sub-array of cubes that depend on Xi and in this sub-array all the values in a column are 0.

2. Weakly Unate SOP F Example
c3 and c4 depend on variable X1
first column of c3 and c4 are all 0.
Therefore F is weakly unate with respect to the variable X1
X1 X X3
1111 – 1110 – c1
1111 – 1101 – c2
0110 – 0110 – c3
0101 – 0111 – c4
F =

3. Tautology Decision - Weakly Unate Simplification Theorems
Let an SOP F be weakly unate with respect to the variable Xj. Among the cubes of F, let G be the set of cubes that do not depend on the variable Xj. Then, G  1  F  1.
Theorem 9.7
Let c1 = XjSA and c2 = XjS B
where SA  SB = Pj and SA  SB = 
Then, F  1  F(|c1)  1 and F(|c2)  1.

4. Tautology Decision Algorithm
If F has a column with all 0’s, then F is not a tautology.
Let F = {c1,c2 , ,ck}, where ci is a cube. If the sum of the number of minterms in all cubes ci is less the total number in the univeral, cube then F is not a tautology.
If there is a cube with all 1’s in F, then F is a tautology.
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.

5. Tautology Decision Algorithm (continued)
When there is a weakly unate variable, simplify the problem by using Theorem 9.6
When F consists of more than one cube: F is a tautology iff
F(|c1)  1 and F(|c2)  1 where
c1 = XjSA and c2 = XjS ,
SA  SB = Pj and SA  SB =  .

6. Tautology Decision Examples: G =
01 – 100 – 1100
11 – 111 – 0010
Examples:
X3 variable has column with all 0’s, so not a tautology.
F does not depend on X1.
Let c1= ( ) and c2= ( )
By Thm 9.7, F is a tautology.
G =
11 – 110 – 1110
11 – 110 – 0001
11 – 001 – 1111 F=
11 – 111 – 1110
11 – 111 – 0001
11 – 111 – 1111
F1= F(|c1) =
 1 
11 – 111 – 1111
F2= F(|c2) =
 1

7. Generation of Prime Implicants
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.

8. 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 Xi, then F is weakly unate with respect to Xi.

9. Strongly Unate Example
1111 – 1001
0111 – 0111
0011 – 0110
0001 – 0101
F(|X2= 0) = (1111 – 1111)
F(|X2= 1) =
F(|X2= 2) =
F(|X2= 3) =
F(|X2= 2) < F(|X2= 1) < F(|X2= 0)
= F(|X2= 3)
F =
F(|X1= 0) = (1111 – 1001)
F(|X1= 1) =
F(|X1= 2) =
F(|X1= 3) =
F(|X1= 0) < F(|X1= 1) = F(|X1= 2)
= F(|X1= 3)
F is strongly unate with respect to X1 and to X2
0111 – 1111
0011 – 1111
0001 – 1111
1111 – 1001
1111 – 0111
1111 – 1001
1111 – 0111
1111 – 0110
0111 – 1111
0011 – 1111
1111 – 1001
1111 – 0111
1111 – 0110
1111 – 0101
1111 – 1111
0111 – 1111
0001 – 1111

10. Generation of Prime Implicants
Generation of Prime Implicants Algorithm

11. Generation of Prime Implicants
Example:

12. Generation of Prime Implicants
Example:

13. Generation of Prime Implicants
Example:

14. Sharp Operation
Sharp Operation: (#) Used to computer F  G, assume For 2-valued inputs and F = U, n-variable function generates  (3n / 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 2n cubes.

15. Sharp Operation

16. Sharp Operation

17. Sharp Operation
Example:

18. Sharp Operation
Example:

19. Sharp Operation
Example:

20. Sharp Operation
Example: