Simplification of boolean function

Question

I arrived in a boolean function and I wonder if there is a simpler form or if mine is right.

The function is:

I arrived at the following reply:

I’m wondering if this is the simplest solution I could come up with...

Answer 1

Let’s set up the truth table of f:

w x y z f
0 0 0 0 1
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 1
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 0
1 0 0 1 0
1 0 1 0 1
1 0 1 1 0
1 1 0 0 0
1 1 0 1 0
1 1 1 0 1
1 1 1 1 0

Let’s reorder the table by placing the x in the first column (and reorder the rows so that the set xwyz is ordered from 0000 until 1111):

x w y z f
0 0 0 0 1
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 0
1 0 0 0 1
1 0 0 1 0
1 0 1 0 1
1 0 1 1 1
1 1 0 0 0
1 1 0 1 0
1 1 1 0 1
1 1 1 1 0

The first half of the table is equal to the second. That is to say, x is irrelevant. Here’s how the remaining table looks:

w y z f
0 0 0 1
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 0
1 1 0 1
1 1 1 0

There are several possible expressions that express this truth-table, and they all necessarily depend on w, y and z (there is no other irrelevant variable).

Other possible truth table ordering are these:

y w z f
0 0 0 1
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 0

z w y f
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 0

Among the possible ways to express the content of these tables, we have:

1. (NOT w AND NOT z) OR (y AND NOT w) OR (y AND z) - your solution.
2. (NOT w AND NOT z) OR (y AND (NOT z OR NOT w)) - Lucas Percisi’s solution.
3. (NOT w AND NOT y AND NOT z) OR (NOT w AND y AND NOT z) OR (NOT w AND y AND z) OR (w AND y AND NOT z) - list of lines with 1 in truth tables.
4. (NOT w AND NOT y AND NOT z) OR (NOT w AND y) OR (w AND y AND NOT z) - simplification of 3.
5. (NOT z AND (w <-> y)) OR (NOT w AND y) - simplification of 4.
6. IF w THEN (y AND NOT z) ELSE (y OR NOT z) - using w as a test in IF.
7. IF y THEN (w NAND z) ELSE (w NOR z) - using y as a test in IF.
8. IF z THEN (NOT w AND y) ELSE (NOT w OR y) - using z as a test in IF.
9. (y AND NOT z) OR (w AND (y <-> z)) - separating the cases where y and z are different from those where they are equal.
10. (NOT w AND y) OR (NOT z AND (w <-> y)) - separating the cases where w and y are different from those where they are equal.
11. (NOT w AND NOT z) OR (y AND (w XOR z)) - separating the cases where w and z are different from those where they are equal.

In my personal opinion, solution 7 is the simplest, but you may disagree. There is no way to simplify much more than these alternatives that are there.

Answer 2

I arrived in another reply:

F(x,y,w,z) = (!w * ! z) + (y * (!z + ! w));