Exercise 7.30 (Complete Bi-Decomposition of the Incompletely Specified Function h3q(x), h3r(x)). Check first for each pair of variables whether for the incompletely specified function with the mark functions h3q(x) and h3r(x) calculated in Task 14 of Exercise 7.29 exists a bi-decomposition with regard to the OR-, AND- and EXOR-operation. Extend the found elementary bi-decompositions to complete bi-decompositions. Select the best existing bi-decomposition and calculate the decomposition functions.

Practical tasks:

1 Load the TVL system e73dec4.sdt of Exercise 7.29. This TVL system includes the function h3q(x) as object number 34 and h3r(x) as object number 35.

2 Prepare a PRP that checks for each pair of the three variables based on

(7.99) in [18] whether an OR-bi-decomposition exists.

3 Prepare a PRP that checks for each pair of the three variables based on

(7.106) in [18] whether an AND-bi-decomposition exists.

4 Prepare a PRP that checks for each pair of the three variables based on

(7.112), (7.113), and (7.114) in [18] whether an EXOR-bi-decomposition exists.

5 Are there strong one-to-one OR-bi-decompositions?

6 Are there strong one-to-one AND-bi-decompositions?

7 Are there strong one-to-one EXOR-bi-decompositions?

8 Are there strong one-to-two OR-bi-decompositions?

9 Are there strong one-to-two AND-bi-decompositions?

10 Are there strong one-to-two EXOR-bi-decompositions?

11 Load the TVL system e73dec4.sdt of Exercise 7.29 in order to remove the intermediate results.

12 Calculate the function g4(x) of the existing EXOR-bi-decomposition based on (7.115) in [18] as object 15.
13 Calculate the mark function h4q(x) as object 36 and h4r(x) as object number 37 using the formulas (7.116) and (7.117) in [18] for the existing EXOR-bi-decomposition with regard to (x2, [x3, x4]). Show the Karnaugh-maps of the calculated mark functions.
14 Select the function h4(x3, x4) to be realized in the multilevel circuit and store it as object 15. Show the Karnaugh-map of the selected function h4(x3, x4).
15 Remove the intermediated TVL and store for later use the TVL system as e73dec5.sdt.