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Tautology

Tautology. 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

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Tautology

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  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 Xiwhen there is a variable Xiand 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 Xiwhen in an array F there is a sub-array of cubes that depend on Xiand in this sub-array all the values in a column are 0.

  3. Weakly Unate SOP F Example • c3and 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 X1X2X3 1111 – 1110 – 1110 c1 1111 – 1101 – 1101 c2 0110 – 0110 – 1101 c3 0101 – 0111 – 1101 c4 F=

  4. Tautology Decision - Weakly Unate Simplification Theorems Theorem 9.6 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=XjSAandc2=XjS B where SA  SB = Pj and SA  SB =  Then, F 1 F(|c1)  1 and F(|c2)  1.

  5. Tautology Decision Algorithm • If F has a column with all 0’s, then Fis 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 Fis not a tautology. • If there is a cube with all 1’s in F, then Fis 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.

  6. 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: Fis a tautology iff • F(|c1)  1 and F(|c2)  1 where • c1=XjSAandc2=XjS, • SA  SB = Pj and SA  SB =  .

  7. Tautology Decision 01 – 100 – 1100 11 – 111 – 0010 • Examples: • X3variable has column with all 0’s, so not a tautology. • Fdoes not depend on X1. • Let c1= (11- 110 - 1111) andc2= (11- 110 - 1111) • By Thm 9.7, Fis a tautology. G= 11 – 110 – 1110 11 – 110 – 0001 11 – 001 – 1111 F= 11 – 111 – 1110 11 – 111 – 0001 11 – 111 – 1111  1 F1= F(|c1)=  11 – 111 – 1111  1 F2= F(|c2)=

  8. Generation of Prime Implicants

  9. 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.

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

  11. Strongly Unate Example 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) Fis strongly unate with respect toX1and toX2 1111 – 1001 0111 – 0111 0011 – 0110 0001 – 0101 0111 – 1111 0011 – 1111 0001 – 1111 1111 – 1001 1111 – 0111 1111 – 1001 1111 – 0111 1111 – 0110 0111 – 1111 0011 – 1111 1111 – 1111 0111 – 1111 0001 – 1111 1111 – 1001 1111 – 0111 1111 – 0110 1111 – 0101

  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 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.

  18. Sharp Operation

  19. Sharp Operation

  20. Sharp Operation Example:

  21. Sharp Operation Example:

  22. Sharp Operation Example:

  23. Sharp Operation Example:

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

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