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Table 4.9
Parsimonious solutions designed with the universal system {AND, OR, NOT}.
Rule Name
Rule #
Rule Table
Parsimonious Solution
Size
AND
8
1000
Aab
3
OR
14
1110
Oab
3
NAND
7
0111
NAab
4
NOR
1
0001
NOab
4
LT
2
0010
ANba
4
GT
4
0100
AaNb
4
LOE
11
1011
ONba
4
GOE
13
1101
OaNb
4
XOR
6
0110
AONabAab
8
NXOR
9
1001
OANabOab
8
MUX
172
10101100
OAAacNba
8
IF
202
11001010
OAAabNca
8
MAJ
232
11101000
AOOaAbcbc
9
MIN
23
00010111
NAOOaAbcbc
10
EVEN
105
01101001
AOOAOANabNcONcOabAabab
22
ODD
150
10010110
OAAONAcaOAabbcOOAbacac
22
NLM39
39
00100111
ONAONcacb
9
NLM27
27
00011011
NAOOaNbcc
9
NLM115
115
01110011
NAObNca
7
NLM103
103
01100111
AONObANcbca
11
The comparison of the classical Boolean system with the MUX system is
also interesting (see Table 4.13). The MUX system is considerably more
compact with a total of just 82 nodes for the 10 functions of three arguments
as opposed to 115 nodes required for the classical Boolean system; indeed,
even after discounting the MUX and IF functions, the MUX system requires
a total of 74 nodes to realize all the remaining functions, whereas the classi-
cal Boolean system requires a total of 99 nodes. For the functions of two
arguments, though, the Boolean system is more compact with a total of 46
nodes, whereas the MUX system requires a total of 52 nodes.
It is worth emphasizing, however, that the goal of this study does not con-
sist in discussing the compactness of the different logical systems (although
for the functions of two arguments the discussion is as good as any as the
entire set of relevant functions was used), but rather in showing that gene
expression programming can be successfully used to discover the most
