These problems are somewhat related, but also different. The cited thread is about converting DNF to Reed-Muller normal form while the images ask for the construction of FDDs. FDDs and Reed-Muller normal forms are closely related, so having one helps a lot to get the other one. Still a bit work has to be done to do that.
The example solutions construct the FDDs by performing the Davio decomposition using the cofactors. That is not the worst you can do for such problems.
An alternative would be to work with the relationship between RMF and FDDs .Let's consider 2016-Feb-2e: The task is to construct a FDD for the formula x1⊕x2. This formula is already in Reed-Muller normal form, but still we have to generate the FDD. Having the Reed-Muller normal form helps, since we just have to sort the monomials into those having x2 and others:
x1⊕x2 = (x1)⊕x2⋀(1) = (0⊕x1⋀1)⊕x2⋀(1)
For the other problem, it helps a bit more, since we can first rewrite x1➞x2 to its Reed-Muller normal form from a DNF:
x1➞x2 = ¬x1⋁x2 = ¬x1⋀¬x2 ⋁ x1⋀x2 ⋁ ¬x1⋀x2
= ¬x1⋀¬x2 ⊕ x1⋀x2 ⊕ ¬x1⋀x2
= (1⊕x1)⋀(1⊕x2) ⊕ x1⋀x2 ⊕ (1⊕x1)⋀x2
= 1 ⊕ x1 ⊕ x2 ⊕ x1⋀x2 ⊕ x1⋀x2 ⊕ x2 ⊕ x1⋀x2
= 1 ⊕ x1 ⊕ x1⋀x2
Again, from the above Reed-Muller normal form, we can get the FDD by sorting the monomials:
= (1 ⊕ x1) ⊕ x2 ⋀ (x1)
= (1 ⊕ x1 ⋀ 1) ⊕ x2 ⋀ (0 ⊕ x1 ⋀ 1)
But as you see, that was more work than performing the Davio decomposition as given in the example solution. Still, you can solve these problems this way.
