## 1 Introduction  
To fully describe the operation of a digital combinational design, a truth table showing the required output state exhibiting every possible combination of input states is sufficient. However, digital sequential circuits present a much more complex problem. In the case of these systems, a full description of their operation is much more complicated, and a truth table is not enough. To fully describe the operation of a sequential system, a state table or other state diagram representation is necessary. Therefore, there are significantly fewer works concerning the automatic design of sequential systems. These works usually use descriptions of a given system in the form of a state table or other representation of transition graphs to describe possible changes in system states. This paper takes a completely different approach, in which the description of the sequential system results from the study of the reaction to external signals from a given unknown system, which is treated as a black box. This approach may be useful when we want to recreate the internal structure of a given, unknown system or when we want to obtain a system based only on information about the reactions of this system to signals from outside. It should be noted that in the automatic design of sequential systems, typical functional tests created to test such systems are not as effective as might be expected. Therefore, in the case of the automatic creation of sequential systems, one should create a sequence that describes system operations unlike those used for testing systems; it should be focused on system creation. Generally, there are complete functional tests that allow for 100% determination of the correct operation of the sequential system, but there are no functional tests that allow for a 100% confirmation of the proper operation of the createdsystem. This is due to the fact that for each test predicting states for \(n\) clock ticks for arbitrary large \(n\), it may turn out that the system after these \(n\) clock ticks will behave contrary to our expectations. This could be caused by the resulting system having more possible states than expected. The paper presents problems that arise when creating descriptive training sequences of such situations and the reasons why they may appear. During the research, a Multivalued Cartesian Genetic Programming (M-CGP), which was used to design sequential systems, was created, as well as a system based on this model. Generally, in M-CGP, it was assumed that when decoding the genotype, it is possible to additionally divide the types of functions performed by the nodes into any number of groups. This division is used so that the nodes performing the functions of each of these groups can perform calculations based on the results obtained by this group or by other established groups. For this to be the case, the results obtained by the functions of each group can be remembered in separate memory places. Thanks to this approach, M-CGP allows the designing of common systems for different types of elements, i.e., gates as well as flip-flops. It allows the designing of a system based on the tables that describe the system's responses. With such a description of the system, and without using the mechanisms proposed in M-CGP, the automatic design of sequential systems, which are treated as a whole using CGP, would simply not be possible. Therefore, no studies using CGP for designing the sequential systems that are based on tables, which describe system response, have been conducted because it simply could not be completed. Thus, what we have here is the qualitative superiority of M-CGP over CGP, namely M-CGP, which allows for designing sequential systems in situations where the basic CGP is of no use. All the examples of sequential systems presented in this work were created using M-CGP, which is based on the tables that describe the system's responses.