I came up with a Boolean function and would like to know if there is a simpler form or if mine is right.
The function is:
Icameupwiththefollowinganswer:
I'm in doubt if this is the simplest solution I could get ...
Boolean function simplification
1
asked by anonymous 04.11.2017 / 16:59
2 answers
2
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 putting the x in the first column (and reorder the rows so that the xwyz set is sorted from 0000 to 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 equals the second. That is, x is irrelevant. Here's how the remaining table is:
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 all of them necessarily depend on w , y and z (there is no longer any irrelevant variable).
Other possible truth table sorts 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 contents of these tables, we have:
(NOT w AND NOT z) OR (y AND NOT w) OR (y AND z) - your solution. (NOT w AND NOT z) OR (y AND (NOT z OR NOT w)) - Lucas Percisi solution. (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) - listing the lines with 1 in the truth tables. (NOT w AND NOT y AND NOT z) OR (NOT w AND y) OR (w AND y AND NOT z) -
simplification of the 3. (NOT z AND (w <-> y)) OR (NOT w AND y) - simplification of 4. IF w THEN (y AND NOT z) ELSE (y OR NOT z) - using w as test in IF . IF y THEN (w NAND z) ELSE (w NOR z) - using y as test in IF . IF z THEN (NOT w AND y) ELSE (NOT w OR y) - using z as test in IF . (y AND NOT z) OR (w AND (y <-> z)) - separating cases where y and z are different from those in which they are equal. (NOT w AND y) OR (NOT z AND (w <-> y)) - separating cases where w and y are different from those in which they are equal. (NOT w AND NOT z) OR (y AND (w XOR z)) - separating cases where w and z are different from those in which 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 the alternatives that are there.
05.11.2017 / 02:03
0
I came up with another answer:
(x, y, w, z) = (! w *! z) + (y * (! z +! w));
04.11.2017 / 21:00