Tautology Decision

May be able to use unateness to simplify process

Unate Function – one that has either the uncomplementedor 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.

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 X2 X3

1111 – 1110 – 1110 c1

1111 – 1101 – 1101 c2

0110 – 0110 – 1101 c3

0101 – 0111 – 1101 c4

F =

Tautology Decision - Weakly Unate Simplification Theorems

Theorem 9.6Let 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 = XjS

A and c2 = XjS

B

where SA SB = Pj and SA SB =

Then, F 1 F(|c1) 1 and F(|c2) 1.

Tautology Decision Algorithm

1. If F has a column with all 0’s, then F is not a tautology.

2. 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.

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.

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(|c1) 1 and F(|c2) 1 where

c1 = XjS

A and c2 = XjS

,

SA SB = Pj and SA SB = .

Tautology Decision

Examples:

1. X3 variable has column with all 0’s, so not a tautology.

2.

F does not depend on X1.

Let c1= (11- 110 - 1111) and c2= (11- 110 - 1111)

By Thm 9.7, F is a tautology.

G =01 – 100 – 110011 – 111 – 0010

F=

11 – 111 – 1111F2= F(|c2) =

11 – 110 – 111011 – 110 – 000111 – 001 – 1111

F1= F(|c1) = 11 – 111 – 111011 – 111 – 0001

1

11 – 111 – 1111 1

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.

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.

Strongly Unate Example

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

1111 – 10010111 – 01110011 – 01100001 – 0101

F =

1111 – 10011111 – 0111

1111 – 10011111 – 01111111 – 0110 1111 – 10011111 – 01111111 – 01101111 – 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)

0111 – 11110011 – 11110001 – 1111

0111 – 11110011 – 1111

1111 – 11110111 – 11110001 – 1111

Generation of Prime Implicants

Generation of Prime Implicants Algorithm

Generation of Prime ImplicantsExample:

Generation of Prime ImplicantsExample:

Generation of Prime ImplicantsExample:

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.

Sharp Operation

Sharp Operation

Sharp OperationExample:

Sharp OperationExample:

Sharp OperationExample:

Sharp OperationExample: