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4.6 ENTERED VARIABLE K-MAP MINIMIZATION 163

BD -v r- ABC
BC / / \BC \ I ^-f- 1
00 01 11/ 10 / /\\ QQ\ 01 ' 11/1 0
0 $ D </> + D <^D * 0 \ D prf 0
*
0 1 3 2 * 0 1 3 2
1 1 0 0 *R Jl -> l) 0 0
^ + D 5 7 6 4 5 7 6
^ 4
/ 'SOP
n\ /
0 0
(a)
1 1
v RP.
B+D
(* D ) D */
0 1 3 2
A 1 1 1 fo ~^]
4 5 ' 7 6
*
I /
C \
\ - -
^-A+B
(c)
FIGURE 4.33
(a) First-order compression plot and submaps for the function / in Eq. (4.45). (b) Minimum SOP
cover and (c) minimum POS cover.

into the third-order K-map in Fig. 4.33a, a first-order compression with a Map Key of 2.
Here, the subfimctions are presented in their simplest form yet preserving all canonical
information. In Figs. 4.33b and 4.33c are shown the minimum SOP and POS covers for this
function, which produce the expressions

(4.46)
fpos = (A + B+ D}(B + D)(A + fi),

both of which have a gate/input tally of 4/10. In extracting the minimum expressions of
Eqs. (4.46), the loop-out protocol is first applied to the entered variable D and then applied
to the 1's or O's.
Some observations are necessary with regard to Fig. 4.33 and Eqs. (4.46). First, these
expressions are logically equivalent but are not algebraically equal. The reason is that the
don't cares 0 4 and 0 7 in cells 2 and 3 are used differently for the fsop and fpos- For
example, (0 7 + D) SOp = 1 for </> 7 = 1 but (</> 7 + D) Pos = D, since, in this case, 0 7 = 0.
Second, the extraction process involved some techniques in dealing with </>'s that have
not been discussed heretofore. These techniques are set off for reference purposes by the
following: