# The Complexity Caused by the Inadequacy of the Structure Model

It is known that the most characteristic properties of the system, the emergent properties, are determined by the structure of the system (see Section 2.3, Part I). Therefore, in the disclosure of the complexity of the behavior of the system it is very important to identify the features of the structure that determine this emergent property of the system. This is clearly manifested in determining the causes of specific types of behavior of systems — archetypes: the reasons for this lie in the specific combinations of feedback loops in the structure (see Section 3.4.2). This is an example of studying the system, that is, building a model of its structure. In cases when the working model of the controlled system includes its *structure model*, control difficulties arise if the model contains errors of any of the four types (because the structure is a certain combination of inputs and outputs of all elements of the system). The problem with this type of complexity is the adoption of measures to prevent errors in the structure model (see Section 2.1).

# The Complexity Caused by Incomplete Information in the Combined Operating Model of the Managed System

The working model of the system can be a combination of the composition and the structure models of the managed system. If the working model lacks any information necessary to achieve the goal, the system falls into the category of “complex”. This term has a specific conditional meaning: it does not mean the property of the object, but the relationship between the controlling subject and the controlled object, it is the *complexity of ignorance.* The same object can be “complex” for one subject and “simple” for another: they have models of this object of varying degrees of adequacy. (For many, the control of the TV is a simple matter, but its repair difficult.)

Complex systems of this type are managed by trial and error method (see Section 4.3, Part I). This algorithm extracts a new piece of information about the system from the result of each subsequent (“trial”) control step, adds it to the existing model, and then plans the next control action according to the improved model. Thus, the method with each step increases the adequacy of the model, reduces the complexity of the system, and increases management efficiency. A measure of this type of complexity can be the number of accomplished cycles of algorithm, which help achieve a satisfactory approximation of the goal.

# The Complexity Produced by Probabilistic Uncertainty

The transition from the above static models to dynamic ones leads to the introduction of another class of complexity — the complexity of the control of random processes. We emphasize the difference between this type of systems from chaotic systems, the management of which is reduced to “dance” with them, that is, to attempts to adapt to the peculiarities of a particular chaotic implementation — by identifying a pattern, the links between the events, and attempts to “fit” into their flow in our own interests. But in this case, we are talking about the possibility of using information not only about the observed realization of a random process but also about the whole ensemble of its realizations, and even about the possibility to influence the characteristics of the process itself.

Such possibilities appear if the randomness of the trajectories is not completely chaotic, but organized and limited by a certain framework, as is the case in a strange attractor, but not only in it. The “limitation” is that such a randomness is fully characterized by the probability distribution function over the possible realizations of a random phenomenon (event or process). Moreover, the constraints on the process include the conditions of stationarity (statistical stability of each realization) and ergodicity (statistical homogeneity of all realizations).

The complexity of working with such an object is caused by the *uncertainty* of predicting its behavior. The observed or controlled parameters of a random object can be chosen as appropriate characteristics of its distribution, for example, the parameters of the position or scale of the distribution and various measures of connection between variables (correlation, regression); as a convenient measure of uncertainty distribution, C. Shannon proposed *entropy.* Any desired numerical characteristic of a random object can be expressed by some *functional of the distribution function, *analytically calculated or statistically estimated value of which is used for the object. The possibility of controlling a random process, that is, a purposeful change of some of its characteristics, is associated with the possibility of changing its probability distribution. Here, the dual nature of the concept of *probability* as a measure of uncertainty of the outcome of a random experiment comes to the fore.

The probability of outcome is defined as a measure of its possibility to occur *under a given set of conditions.* This set of conditions includes the laws of the stochastic nature of the phenomena (which determine the *objective* component of probability);

however, the level of uncertainty (and hence the value of the probability) is also affected by the *degree of knowledge* by the subject about objective conditions, which gives rise to the *subjective* component of probability. (A good example of controlling a random object due to the subjective part of the probability is given by a sharper in card or dice games; improving the quality of the weather forecast is carried out by taking into account additional factors affecting the weather; etc.).

Different degrees of reliability of knowledge about probability distribution dictate the need to extract the necessary information from the same data set in different ways. Special algorithms of experimental data processing have been developed for different levels of a priori information about the distribution. Mathematical statistics consists of four sections:

- 1. classical (parametric) statistics based on the assumption that the distribution function is
*known*up to a finite number of parameters; - 2. nonparametric statistics, assuming that the observations are subject to distribution, the functional form of which is
*unknown*; - 3. robust statistics considering cases where the distribution function is
*known approximately:*the real function is located in some neighborhood of a given function; - 4. semi-parametric statistics, assuming that observations belong to a parametric family,
*with random parameters.*

In accordance with this, the “Rules of statistical safety” when working with random objects are developed. Methods have also been developed to incorporate any additional (collateral) information into statistical data analysis procedures.

# The Complexity Associated with “Vague” Uncertainty

Classification of difficulties will be incomplete if we do not mention another type of complexity that often occurs in human practice. The above types of difficulties are typical for the cognitive or transformative activity of an individual subject, including it under the conditions of uncertainty associated with the *probabilistic* nature of the object of activity. However, uncertainty in the working model can be not only *random.*

In human practice, very often there are situations where certain activities must be carried out by several persons, together and in concert. This means that each of them should have its own working model of the situation in which it plans its actions; however, to coordinate the actions of all participants, it is necessary that the different models contain the same information about the common situation. (The most ancient example of the failure of the collective project is given in the biblical parable about the failure of the construction of the tower of Babel because of the incompatibility of the languages of the builders.)

Therefore, models are built in the languages of the problem situation configurator (see Section 5.5, Part II). If the corresponding language is professional, that is, it describes the situation accurately for all participants in the collective work (e.g., the languages of professional mathematicians, engineers, or repairers), then there are no difficulties in the work. But often (especially in the management of social systems) the configurator includes spoken language, with its characteristic ambiguity of the meaning of words, especially evaluative words that express the gradation of quality in weak measuring scales. This leads to complications in the joint work of subjects by giving them different meanings of the same evaluative word.

Vagueness and uncertainty in the meaning of words in the natural language generate another specific type of complexity, which has a purely *subjective origin* (since the ability to evaluate anything has only the subject). Almost all disagreements and conflicts that complicate the joint activities of people arise from the vagueness of the classification terms in the natural language (“What is good and what is bad?”).

Mathematical tools to describe this type of complexity were proposed by L. Zadeh in the form of the theory of *fuzzy sets.* The basic idea of this theory was the introduction of the *membership function p _{cla}Jx),* a numerical measure for the degree of confidence of the subject in the membership of the estimated object

*x*to the fuzzy class of objects.

The value range of the membership function ranges from 0 (“not exactly belongs to”) to 1 (“surely belongs”). Unlike a precise classification, in a fuzzy classification, there are no clear boundaries between classes, the functions of belonging to them can overlap, and this object can simultaneously belong to several classes (with different degrees of confidence).

For example, the fuzzy classification of numbers into three classes of “small”, “medium”, and “large” from the viewpoint of one subject can be determined by the dotted membership functions in Figure 9.1, and from the viewpoint of another by a continuous one.

In their joint work, it is necessary to develop solutions that take into account the views of both participants. To carry out the required information processing, the rules for calculating the membership function of specific combinations of different judgments are defined. For example, the “or” operation (U), p'U^{2}p^{2}class(x)=max [p_{ldass} (x), fi2cia«(^{x})]; ^{for the} operation “and” (Л) — p^{1 2} _{class}(x)=min [p_{lclass} (x), p_{2class} (x)], etc.

FIGURE 9.1 Fuzzy classes of numbers.

Management decisions are formed with the help of fuzzy logic in the automated processing of vague initial information about the controlled system. It is worth noting that practitioners sometimes wrongly interpret the value of the membership function as probability. This is a gross error: the properties of the membership function and the probability distribution function differ both in meaning and in the technique of their mathematical transformations.