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function is the XOR function, which can also be called odd-2-parity function.
We will start our exploration of odd-parity functions from here and then go on
finding solutions to more complex odd-parity functions (up to odd-6-parity) by
trying to discover and explore patterns in their design.
So, suppose we were using the classical Boolean system of ANDs, ORs,
and NOTs, and we wanted to discover parsimonious solutions to a wide range
of odd-parity functions (say, up to the odd-11-parity function). Of course we
could try the direct approach and try to use gene expression programming to
discover a solution to each of these functions with just ANDs, ORs, and
NOTs in the function set, but we would have soon found out that these func-
tions are really hard to design and it would be almost impossible to find a
perfect solution to the higher order parity functions this way (Table 4.14). As
you can see, tremendous resources were already necessary to solve the odd-
4-parity function and higher order parity functions are even harder to design
Table 4.14
Parameters for the odd- n -parity problem without UDFs.
Odd-2
Odd-3
Odd-4
Number of runs
100
100
100
Number of generations
50
10000
100000
Population size
30
30
30
Number of fitness cases
4
8
16
Function set
A O N
A O N
A O N
Terminal set
a b
a b c
a b c d
Head length
7
10
20
Gene length
15
21
41
Number of genes
2
2
2
Linking function
A
A
A
Chromosome length
30
42
82
Mutation rate
0.044
0.044
0.044
Inversion rate
0.1
0.1
0.1
IS transposition rate
0.1
0.1
0.1
RIS transposition rate
0.1
0.1
0.1
One-point recombination rate
0.3
0.3
0.3
Two-point recombination rate
0.3
0.3
0.3
Gene recombination rate
0.3
0.3
0.3
Gene transposition rate
0.1
0.1
0.1
Fitness function
Eq. (3.8)
Eq. (3.8)
Eq. (3.8)
Success rate
100%
68%
3%
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