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was used in the function set, which, for simplicity, was represented by “D”. It
is worth pointing out that no other element such as zero or one were used in
order to find the most parsimonious expressions to the functions presented in
Table 4.10 and, as you can easily check by drawing the expression trees, the
use of 1 would have made some of these solutions considerably more com-
pact as a total of five nodes are required to express 1 using just NAND gates
((
a
nand
a
) nand
a
).
The attractiveness of the NAND system (and the NOR system, for that
matter) resides not on its compactness (the classical Boolean system or the
Reed-Muller system are much more compact, for example) but rather on the
fact that just one kind of gate can be used to realize any kind of logical
function, which, on practical terms, is a great advantage. But designing par-
simonious solutions using just NAND (or NOR) gates is not as intuitive as
using the classical Boolean functions of ANDs, ORs, and NOTs and, in fact,
most NAND circuits are designed initially with ANDs, ORs, and NOTs and
only later converted into NAND circuits by applying the De Morgan's theo-
rems. But this is not the most efficient approach because it is possible to
design much more compact expressions by using the NAND gate directly as
your basic building block. And this is the reason why new techniques such as
gene expression programming might be useful for helping in the design of
even more compact and, consequently, faster digital circuits.
Nor Logic
The NOR system is, after the NAND system, the second most important
system in industry. There are, however, specialists that claim that this system
is better than the NAND system for a wide spectrum of applications, as there
are functions for which the NOR system produces more compact solutions
than the NAND system. But of course the reverse is also true and, overall,
both systems are most probably very similar. However, there are cases for
which the use of one system or the other might have drastic consequences in
terms of processing time. This is one of the reasons why having the right
tools to design and choose the most parsimonious configurations for a par-
ticular set of logical functions might bring large benefits.
The parsimonious solutions presented in Table 4.11 were created using
the same evolutionary strategies and the same chromosomal organizations
used in the previous sections, with the difference that now just the one func-
tion NOR was used in the function set, which, for simplicity, was repre-
sented by “R”. Again note that no other element such as zero or one were
