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instance, in (a), a small building block (ADF
0
) involving arguments
a
and
b
,
and a larger one (ADF
1
) involving arguments
c
and
d
, were created and then
combined in different ways in the main program in order to design a perfect
solution to the odd-4-parity function. And as you can see, they are both called
twice from the main program. In (b), two relatively large building blocks,
each involving a different set of variables (ADF
0
is a function of
b
and
c
,
whereas ADF
1
is a function of
a
,
c
, and
d
) were discovered. Then these ADFs
are linked together in the main program to form a perfect solution. Note,
again, that each building block is expressed twice in the cell. And in (c), a
more verbose solution was created, involving two relatively large building
blocks, each using a different set of arguments (ADF
0
is a function of
a
,
b
,
and
d
, whereas ADF
1
is a function of
b
,
c
, and
d
). Then these ADFs are each
invoked from three different places in the main program and a perfect solu-
tion to the odd-4-parity function is also created with them.
6.3 Kepler's Third Law
Kepler's Third Law is an interesting problem to solve with the cellular system
of gene expression programming as it can be used to illustrate the advantages
of the flexible linking that takes place in the cells. Moreover, this scientific
law, considered by Langley at al. (1987) one of the most challenging to have
been rediscovered by a family of heuristic techniques for inducing scientific
laws from empirical data, was also rediscovered by genetic programming
using populations of 500 individuals for 51 generations (Koza 1992). We will
see here that gene expression programming can discover Kepler's Third Law
using population sizes of just 10 individuals evolving for 50 generations.
Kepler's Third Law states that
P
2
=
cD
3
, where
P
is the orbital period of a
planet and
D
is the average distance from the Sun. When
P
is expressed in
units of Earth's years and
D
is expressed in units of Earth's average distance
from the Sun,
c
= 1.
The fitness cases used to rediscover Kepler's Third Law are the astro-
nomical data for the six planets presented in Table 6.4, where
D
is given in
units of Earth's semimajor axis of orbit and
P
is given in units of Earth's
years (Urey 1952). So, the goal here consists of finding a relationship de-
scribing the orbital period of a planet as a function of its average distance
from the Sun. For that we could define the fitness to be the number of out-
puts that are within 20% of the correct value and, therefore, evaluate the
fitness by equation (3.1), thus giving
f
max
= 6.
