## 5 Results and Discussion
### The Fundamental Problem with Sequential Systems  
Another example of the obtained system implementing the same Table 2 with the same set of functions but for the parameter \(number\ columnsCGP=50\) is the following genotype: *([1]3 0 0), *([2]0 1 0), *([3]0 1 2), ([4]1 0 3), *([5]1 3 1), *([6]3 0 1), *([7]1 2 5), *([8]2 3 5), *([9]0 3 7), *([10]2 1 7), *([11]3 3 6), *([12]0 7 7), *([13]1 5 1), *([14]0 13 3), *([15]0 14 10), ([16]0 13 5), *([17]0 15 2), *([18] 10 17 3), *([19]1 12 13), *([20]0 6 15), *([21]3 6 19), *([22]2 0 13), *([23]1 22 14), *([24]1 1 12), *([25]0 7 6), *([26]0 19 23), *([27]0 3 12), *([28]0 26 21), *([29]3 21 19), *([30]1 11 0), *([31]1 27 29), *([32]0 24 18), *([33]3 20 9), *([34]0 18 6), *([35]0 16 31), *([36]1 19 15), *([37]3 19 3), *([38]0 26 8), *([39]3 3 27), *([40]1 9 38), *([41]2 33 33), *([42]2 35 1), *([43]1 21 24), *([44]0 34 43), *([45]1 21 6), *([46]3 2 30), *([47]0 44 40), *([48]0 36 37), *([49]2 20 27), *([50]1 42 42), 47 26 14 22 representing the system shown in Figure 5.  
As can be seen in Figure 5, if we do not use optimization due to the number of elements, the \"imagination\" scope of genetic programming when obtaining sequential systems is enormous. The system shown in the drawing certainly meets the precisely presented table describing the system. However, it may add elements that do not contradict this table. For example, a circuit that, after the 17 lines in Table 2 shown, will operate in such a way that after each subsequent rising edge on clk, the code of the next digit of the PI number expansion will appear on the four outputs of the circuit, which does not contradict the presented table. Of course, it is unlikely that such a system will arise by chance, but it is possible. It is also possible and more likely that the system will behave in accordance with the table, and after the initial 17 lines described in Table 2, an arbitrary sequence of values will appear at the outputs, inconsistent with what we would expect. To present the problem  
Figure 4: The correct synchronous counter circuit obtained based on Table 2, in which a combination of JK 05 and JK 06 flip-flops was used instead of an AND gate. Figure 5: Synchronous counter system obtained on the basis of Table 2 for the number of CGP columns that are equal to 50.  
at hand using the simplest example, let us try to create a trivial circuit that will implement a table describing an AND gate. Of course, an AND gate is a combinational circuit, but sequential circuits may contain combinational elements in many places. Let us assume that we will be able to use both gates and JK flip-flops to build it. For the constructed system, the table describing the operation of the system will take the form of Table 3.  
For the parameter \(number\_columnsCGP=8\) and the set of functions left unchanged, the most frequently obtained result shows the genotype:  
*([2]1 0 1), ([3]2 1 1), ([4]3 2 0), ([5]0 3 3), ([6]1 1 4), ([7] 1 1 5), ([8]2 5 3), ([9]2 2 3), 2 representing the system shown in Figure 6.  
The obtained result is consistent with what we expected. However, for the same parameters \(number\_columnsCGP\), the same set of functions, and the same table, data such as, e.g., genotype was also obtained:  
*([2]0 0 1), ([3]2 2 0), ([4]0 2 1), *([5]0 2 1), ([6]0 5 0), ([7] 0 6 1), ([8]2 1 0), ([9]0 0 8), 5 representing the system shown in Figure 7.  
This system works according to the table describing the system, but it is not consistent with what we expected. If, after the sequence of signals described in the table, we set the state of the INPUT0 and INPUT1 inputs to 1 and give a rising edge to clk, the output state will assume the value 0. This is inconsistent with the operation of the AND gate. Another genotype obtained with the same parameters and the same table describes an even more interesting system:  
Figure 6: The circuit designed to implement an AND gate obtained on the basis of Table 3.