TautologyTautology

Tautology DecisionTautology DecisionMay 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 Generation of Prime Prime

ImplicantsImplicants

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 Sharp OperationOperation

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:

Problems to think and to SolveProblems 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.