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parsimonious expressions irrespective of the universal system one chooses to
work with, as the problems chosen for this study clearly illustrate.
Nand Logic
The NAND system is arguably the most important in industry as most digital
circuits are built using NAND gates. It could easily have been the NOR
system to get this privilege, but it just happened this way or perhaps the
NAND system is overall slightly more compact than the NOR system (com-
pare Tables 4.10 and 4.11). Indeed, on the two-input functions, these systems
exhibit the same compactness, both requiring a total of 84 nodes. But on the
three-input functions, the NAND system performs slightly better than the
NOR system, with a total of 150 nodes as opposed to 158.
The parsimonious solutions presented in Table 4.10 were discovered us-
ing the same evolutionary strategies and the same chromosomal organiza-
tions used in the previous section, with the difference that now only NAND
Table 4.10
Parsimonious solutions designed with the NAND (“D”) system.
Rule Name
Rule #
Rule Table
Parsimonious Solution
Size
AND
8
1000
DDDabab
7
OR
14
1110
DDDaabb
7
NAND
7
0111
Dab
3
NOR
1
0001
DDDDDDaaabbaa = DD1DDaabb
13
LT
2
0010
DDDDbDbabbb = DD1Dbab
11
GT
4
0100
DDDaDbDabbb = DD1aDab
11
LOE
11
1011
DaDab
5
GOE
13
1101
DDbaa
5
XOR
6
0110
DDDaDDbabab
11
NXOR
9
1001
DDDDDabaabb
11
MUX
172 10101100 DDDDbacab
9
IF
202 11001010 DDDabDcac
9
MAJ
232 11101000 DDDDbacDDabcc
13
MIN
23 00010111 DDDDDDDaaccacbb
15
EVEN
105 01101001 DDDDDDDDDacDDbbabbcDDacaacc 27
ODD
150 10010110 DDDDDDcabDDDDaDbDabDDacbcacbb
29
NLM39
39 00100111 DDDDDDcaabcbb
13
NLM27
27 00011011 DDDDDDcacbbac
13
NLM115
115 01110011 DDbaDcc
7
NLM103
103 01100111 DDDDDDcaDccbbab
15
