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212 CHAPTER 5 / FUNCTION MINIMIZATION

where use has been made of the XOR form of Eqs. (3.31), and the XOR DeMorgan identities
given in Eqs. (3.27) from which, for example, there results (C + D) 0 C = CD 0 C in
g\. Introducing these coefficients into Eq. (5.27) and simplifying by using the XOR form
of Eqs. (3.30) gives the minimum result

Z AB = C 0 D 0 B 0 BCD 0 ACD 0 ABCD

= B 0 C 0 D 0 ACD 0 ABCD, (5.28)

which is a three-level function with a gate/input tally of 6/15.
Repeating the same procedure for Fig. 5.4c and bond set {A, C}, the function Z is recast
into the CRMT form

ZAC = (AC)/ 0 0 (AC)/, 0 (AC)/2 0 (AQ/ 3
= (AC)(B 0 D) 0 (AQ(B + D) 0 (AC)fi 0 (AQ(B 0 D)
)g 2A® g3AC, (5.29)

where the g coefficients become

go = B 0 D
gi=(£ + Z))0fl0D = .eZ)0fl0D=:flD
g2 = B ®B@D = D
D®BD = BD.

Then by introducing these g coefficients into Eq. (5.29) and simplifying with the XOR form
of Eqs. (3.30), there results the exact minimum

Z AC = B 0 D 0 BCD 0 CD 0 AD 0 ABCD
(5.30)

which is a three-level function with a gate/input tally of 6/14 excluding possible inverters.
The results for ZAB and ZAC are typical of the results for the remaining four two- variable
bond sets {C, D}, {B, D}, {A, D}, and {B, C}. All yield gate/input tallies of 6/14 or 6/15 and
are three-level functions. Thus, in this case, the choice of bond set does not significantly
affect the outcome, but the effort required in achieving an exact minimum may vary with
the choice of the bond set. No attempt is made to use single- or three- variable bond sets for
function Z.
A comparison is now made between the CRMT minimization method and other ap-
proaches to the minimization of function Z. Beginning with the canonical R-M approach
and from Fig. 5.4a, the / coefficients easily seen to be

/I = /2 = /4 = /6 = /7 = /8 = /9 = /10 = /1 1 = /14 = /15 = 1 and
/O = h = f5 = f\2 = /13 = 0.