For the multicellular system without random numerical constants, both the parameters and performance are shown in Table 5.

Table 5
Settings and performance for the sextic polynomial problem using a multicellular system encoding 1, 2, 3, and 4 ADFs.

As you can see, these multicellular systems with Automatically Defined Functions perform extremely well, even better than the multigenic system with static linking (see Table 2). And they are very interesting because they can also be used to solve problems with multiple outputs, where each cell is engaged in the identification of one class or output. Here, however, we are using a multicellular system to solve a problem with just one output, which means that all the cells are trying to find the same kind of solution and, therefore, for each individual, the fitness is determined by the best cell. Obviously, the greater the number of cells the higher the probability of evolving the perfect solution or cell. But there is one caveat though: one cannot go on increasing the number of cells indefinitely because it takes time and resources to express all of them. The use of three cells per individual seems a good compromise, and we are going to use just that in all the experiments of this section.

Let’s take a look at the structure of the first perfect solution found using the multicellular system encoding just one ADF (which cell is best is indicated after the coma):

As its expression shows, the best main program invokes the ADF encoded in the conventional gene five times. Note, however, that this perfect solution is far from parsimonious and could indeed be simplified to (ADF)².

It is also interesting to take a look at what the other cells are doing. For instance, Cell₀ encodes zero and Cell₁ encodes (ADF)⁴, both a far cry from the perfect solution.

Let’s also analyze the structure of a program with more than one ADF, the individual below with four ADFs and three cells (the best cell is indicated after the coma):

As its expression shows, the best main program invokes two different ADFs (ADF₀ and ADF₂), but since ADF₀ encodes zero, the best cell could be simplified to (ADF₂)², which is again a perfect solution built with just one kind of building block (a³-a). It is also worth noticing that two of the ADFs (ADF₀ and ADF₃) and one of the cells (Cell₀) encode zero, and the numerical constant 1 is also encoded by ADF₁; they are all good examples of the kind of neutral region that permeates all these solutions.

Let’s now analyze the behavior of the multicellular system when random numerical constants are also incorporated in the Automatically Defined Functions.

For that purpose a similar set of experiments were done, using also 1, 2, 3, and 4 ADFs (Table 6). And as expected, a considerable decrease in performance was observed comparatively to the performance obtained in the multicellular system without random numerical constants (see Table 5). Notwithstanding, ADFs with random numerical constants perform quite well despite the additional complexity, and they may prove valuable in problems where random numerical constants are crucial to the discovery of good solutions.

Table 6
Settings and performance for the sextic polynomial problem using a multicellular system encoding 1, 2, 3, and 4 ADFs with random numerical constants.

Let’s take a look at the structure of a perfect solution found using the multicellular system encoding just one ADF with random constants (the best cell is indicated after the coma):

As its expression shows, the main program encoded in Cell₁ is far from parsimonious, but it encodes nonetheless a perfect solution to the sextic polynomial (14). The only available ADF is called four times from the main program, but in essence it could have been called just twice as it can be simplified to (ADF)².

Let’s now analyze the structure of a program with more than one ADF, the individual below with four ADFs and three cells (the best cell is indicated after the coma):

As its expression shows, the best main program invokes two different ADFs (ADF₀ and ADF₂), but the calls to ADF₂ cancel themselves out, and the main program is reduced to (ADF₀)², which, of course is a perfect solution to the problem at hand.