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C. FERREIRA

In A. Abraham, B. de Baets, M. Köppen, and B. Nickolay, eds., Applied Soft Computing Technologies: The Challenge of Complexity, pages 517-536, Springer-Verlag, 2006.


Designing Neural Networks Using Gene Expression Programming

Neural network for the 6-multiplexer
 

The 6-bit multiplexer is a complex Boolean function of six activities. Its rule table is shown in Table 3.

Table 3
Lookup table for the 6-multiplexer. The output bits are given in lexicographic order starting with 000000 and finishing with 111111.

For this problem, and in order to simplify the analysis, a rather compact chromosomal organization was chosen and the “Q” function was not included in the function set. Thus, F = {3U, 3D, 3T}, where “U” represents a function with connectivity one; T = {a, b, c, d, e, f}, representing the six arguments to the 6-multiplexer function; and W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, each taking values from the interval [-2, 2].

For the experiment summarized in the first column of Table 4, single-gene chromosomes were chosen so that the simulation of the 6-multiplexer function, a four modular function, went totally unbiased. One of the most parsimonious solutions designed by GEP-nets is shown in Figure 6.

Table 4
Parameters for the 6-multiplexer problem.

   Unigenic System Multigenic System
Number of runs 100 100
Number of generations 2000 2000
Population size 50 50
Number of fitness cases 64 (Table 3) 64 (Table 3)
Function set 3U 3D 3T 3U 3D 3T
Terminal set a b c d e f a b c d e f
Linking function -- O
Weights array length 10 10
Weights range [-2, 2] [-2, 2]
Head length 17 5
Number of genes 1 4
Chromosome length 103 124
Mutation rate 0.044 0.044
Intragenic two-point recombination rate 0.6 0.6
Gene recombination rate -- 0.1
Gene transposition rate -- 0.1
IS transposition rate 0.1 0.1
IS elements length 1,2,3 1,2,3
RIS transposition rate 0.1 0.1
RIS elements length 1,2,3 1,2,3
Weights mutation rate 0.002 0.002
Dw-specific transposition rate 0.1 0.1
Dw-specific IS elements length 2,3,5 2,3,5
Success rate 4% 6%


Obviously, we could explore the multigenic nature of GEP chromosomes and evolve multigenic neural networks. The solutions found are, however, structurally more constrained as we have to choose some kind of linking function (Ferreira 2001) to link the sub-neural nets encoded by each gene. For this problem, the Boolean function OR was chosen to link the sub-NNs. (If the mixing of OR with “U”, “D”, and “T” functions is confusing, think of OR as a function with connectivity two with a threshold and weights all equal to 1, and you have a neural net for the OR function.)


Figure 6. A perfect solution to the 6-multiplexer function discovered with GEP-nets. a) Its chromosome and corresponding array of weights. b) The fully expressed neural network encoded in the chromosome.

In the experiment summarized in the second column of Table 4, four genes posttranslationally linked by OR were used. The first solution found in this experiment is shown in Figure 7. Note that some weights in genes 1 and 2 have identical values, and that the same happens for genes 3 and 4. This most probably means that these genes share a common ancestor.


Figure 7. A perfect solution to the 6-multiplexer problem encoded in a four-genic chromosome. a) Its chromosome with each gene shown separately. W1-W4 are the arrays containing the weights of each gene. b) The sub-neural networks codified by each gene. In this perfect solution, the sub-NNs are linked by OR.

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