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Gene Expression Programming: Mathematical Modeling by an Artificial Intelligence

Finding solutions to odd-parity functions with ADFs

Automatically defined functions were introduced in section 2.2.3. As you recall, in this complex system, normal genes encode a particular ADF whereas homeotic genes control which genes/ADFs are expressed in each cell. We also learned in chapter 2 that the number of cells per organism (or, in other words, the number of homeotic genes per chromosome) is arbitrary and, as the fitness of the individual is determined by the fitness of the best cell, the greater the number of cells the higher the probability of evolving the right cell or solution. But there is one caveat though: we cannot go on increasing the number of cells indefinitely because it takes time and resources to express all of them. As shown in Table 4.22, a total of three cells per individual is a good compromise.

As you can see in Table 4.22, the performance of this approach, in terms of success rate, is nearly the same as in the previous ones. But remember that the system is much more complex and requires the fine tuning of many more parameters. One advantage of this system, though, is that the linking of the sub-ETs is much more flexible and no assumptions whatsoever are made concerning the structure of the evolved solutions.

Table 4.22
Parameters for the odd-n-parity problem using ADFs.

 Odd-2 Odd-3 Odd-4 Odd-5 Odd-6 Number of runs 100 100 100 100 100 Number of generations 50 50 50 100 200 Population size 30 10 30 30 30 Number of fitness cases 4 8 16 32 64 Function set A O N A O N X A O N X A O N X (A O N X)2 Terminal set a b a b c a b c d a b c d e a b c d e f Number of normal genes 3 3 3 3 3 Head length of normal genes 5 5 5 5 5 Linking functions A O N A O N X A O N X A O N X A O N X Number of homeotic genes 3 3 3 3 3 Head length of homeotic genes 3 5 5 5 5 Chromosome length 54 66 66 66 66 Mutation rate normal genes 0.044 0.044 0.044 0.044 0.044 One-point recombination rate 0.3 0.3 0.3 0.3 0.3 Two-point recombination rate 0.3 0.3 0.3 0.3 0.3 Gene recombination rate 0.1 0.1 0.1 0.1 0.1 Gene transposition rate 0.1 0.1 0.1 0.1 0.1 IS transposition rate 0.1 0.1 0.1 0.1 0.1 IS elements length 1,2,3 1,2,3 1,2,3 1,2,3 1,2,3 RIS transposition rate 0.1 0.1 0.1 0.1 0.1 RIS elements length 1,2,3 1,2,3 1,2,3 1,2,3 1,2,3 Success rate 100% 94% 95% 95% 96%

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