Re: Boolean Algebra / Karnaugh Maps with N > 2 (Higher Dimensions)


David T. Ashley wrote:

However, it also occurs frequently that an integer range (x, 0-255, say) is
"paneled" into smaller pieces of significance (0-10, 11-67, and 68-255,
say), and this can lead to truth tables that are a mixture of these
"paneled" ranges and Boolean values. The simplest example would be a table
with 3 columns (corresponding to the three ranges above), and two rows (y,
one for F, one for T).

In "reducing" a 3 x 2 table like this, one could easily end up with a
Boolean-valued function like:

Uhuh (looks up, down, and around to see if a trick question is about to
drop on my head or run me over).... maybe I missed something really
significant, but you apparently answered your own question implicitly
with the little piece of C you gave. What is (V > 10) if not a Boolean
result?

Instead of thinking about this as a single Boolean input A and a binary
variable V that can lie in one of three possible "panels" of interest,
you should think about this as a single Boolean input (from your
original problem/circuit), and two additional Boolean inputs, which
I'll call C and D.

C = 1 iff (v is an element of {0..10})
D = 1 iff (v is an element of {11..67})
[Note there is an implicit third state where C=D=0, i.e. v > 67].

You now have a vanilla three-input K-map with inputs V, C and D. Once
you reduce this to a Boolean expression, you can substitute the
appropriate numeric comparisons:

C = (V <= 10) (implies D')
C' = (V > 10) (implies D=X)
C'D = (V > 10 && V < 68)
D' = (V < 11 || V > 67)
C'D' = (V >= 68)
(CD) = impossible

(Hey, I never thought I would use a K-map again, I guess that wasn't
time wasted at college after all).

.

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