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Tautology Decision

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 Decision

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  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 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.

  2. 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=

  3. 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.

  4. 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.

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

  6. 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)=

  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 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

  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:

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