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system requires a total of 158 nodes. This again shows that for each set of
logical functions it is possible to find out which of the universal systems is
more appropriate to design the digital circuits.
Reed-Muller Logic
The Reed-Muller system {AND, XOR, NOT} has strong adepts in the in-
dustry as there is a wide variety of applications where the use of this system
is responsible for dramatic improvements in terms of processing time. Curi-
ously enough, we will see that, for the functions used in this study, this sys-
tem surpasses all others in terms of compactness.
The Reed-Muller system with ANDs, XORs, and NOTs is very similar to
the classical Boolean system with ANDs, ORs, and NOTs in the sense that
they both use the same kind and number of basic building blocks: two func-
tions of two arguments and one function of one argument. And for gene
expression programming, the algorithm we are going to use to design the
most parsimonious representations, this is almost all that matters as this means
that similar patterns will be followed to evolve the shape and size of all the
candidate solutions. Hence, in this study, we are going to use exactly the
same kind of chromosomal architectures and evolutionary strategies used in
the classical Boolean system study.
The parsimonious solutions designed by gene expression programming
using the Reed-Muller functions {AND, XOR, NOT} are shown in Table
4.12. And as you can see, the Reed-Muller system surpasses all the other
systems in terms of compactness (compare especially with the classical
Boolean system in Table 4.9 and the MUX system in Table 4.13). Indeed, for
the functions of two arguments, of all the systems analyzed here, the Reed-
Muller system is the system with better performance, requiring a total of 43
nodes, whereas the second best, the classical Boolean system (see Table 4.9),
requires a total of 46 nodes. The MUX system is also fairly compact on the
two-input functions, requiring a total of 52 nodes (see Table 4.13), thus more
compact than the NAND and NOR systems, both requiring a total of 84
nodes (see Tables 4.10 and 4.11).
On the functions of three arguments, the results obtained with the Reed-
Muller system are also interesting and, even after discounting the even- and
odd-parity functions, this system still remains more compact than the classi-
cal Boolean system, requiring a total of 69 nodes to realize the remaining
eight functions of three arguments, whereas the classical Boolean system
needs an extra two nodes.
