While studying or modeling the behavior of systems of interacting components, we describe their state in terms of states of components' properties, as well as of the system as a whole. The behavior of these systems is usually considered as a sequence of changes of states of their properties over time, and is explained by the existence of dependence of each following state of the systems on their state preceding the change. Such systems are called dynamical systems.

Properties of some dynamical systems are measurable. Their states represent the magnitude of properties and therefore can be denoted by numbers and deemed as quantitative. An obvious example of these systems is our solar system. The state of this system is characterized by such properties of the sun and the planets as their mass, gravitational force, and many more. Accordingly, systems, all properties of which chosen for studying or modeling their behavior are perceived as quantitative, and changes of the properties can be computed by reproducing in the model the dependence of the next state of the properties on their current state, may be classified and referred to as the Quantitative Dynamical Systems.

Meanwhile, the properties of many systems may often be perceived as qualitative. Considering systems, we can attribute qualitative assessments to their properties and describe the state of the systems in the form of affirmative propositions. So, for example, one may say: “My car is red” or “This window is only slightly open”. These descriptions, in the general case, contain: name of the component, optional name of the property, and mandatory predicate part, which serves as the qualitative assessment of the state of the property of the component. Thus, the systems, whose properties chosen for studying or modeling their behavior, are qualitative, and the change of assessments of states of properties can be explained by the existence of dependence of their following state on the previous, may be classified and referred to as the Qualitative Dynamical Systems.

The objective of this page is to consider the nature of the Qualitative Dynamical Systems and the characteristics of their behavior, as well as to propose a way of specifying the behavior of qualitative dynamical systems that can be used for creating models of behavior of these systems.

A note. All illustrations, shown in the figures of the sections of this page, are fragments of the snapshots taken from the screen of a computer, presenting different models of qualitative dynamical systems. They are created using the modeling formalism proposed by the Kaleidoscope project. Therefore, for details on the content of the illustrations, see page "Multicolored Logical Net Modeling Formalism."

Notion of the Qualitative Dynamical System and Feasibility of Their Mathematical Modeling

Although the expressive power and flexibility of natural languages make it possible to describe any system in terms of its qualitative characteristics, not all the systems perceived in that way may become suitable for studying their behavior by developing their mathematical models just because states of their properties were given qualitative assessments.

For the idea to describe states and changes of the states of dynamical systems in the form of their qualitative assessments to be practically meaningful, that is, could be studied, or designed, by creating their dynamical models, these systems must not only be qualitative and dynamical. They also should, in their essence, be discrete, since the qualitative assessments attributed to them are discrete, and their behavior has to be reproducible in the dynamical models created for the systems. In other words, they just have to be discrete and modelable.

Accordingly, the study of feasibility of modeling behavior of the qualitative dynamical systems, conducted by the Kaleidoscope Project, has resulted in (1) defining the notion of the qualitative dynamical system, given in the form of the listing of all characteristics of nature and behavior, that should be inherent in all the discrete and modellable systems, that can be considered as the qualitative dynamical systems, and, thus, constitute the class of that systems, on the one hand, and (2) developing the qualitative dynamical systems behavior mathematical modeling technique that enables creation of mathematical models for any system of the defined class, on the other hand.

A note. Since, in practice, all systems described on this and the other pages are strictly discrete, dynamical, and modellable, continuous repetition of the terms “discrete”, “dynamical”, and “modellable” may seem a bit redundant. Therefore, further on, they will be mentioned only in the titles of the sections and in cases when it is necessary to emphasize the fact that the qualitative system has one of these classifiers. Otherwise, they will often be omitted. Respectively, phrases like “qualitative dynamical system(s)” will also often be written merely as the "qualitative system(s)".

The rest of the sections describe: characteristics of the qualitative system, and the content of specifications of behavior of qualitative systems that can be used at the time of developing their models, suggested by the Kaleidoscope Project.

Characteristics of Nature and the Behavior of the Qualitative Dynamical Systems

The nature of something is what it has become in the process of its creation, and what is practically impossible to change later. Therefore, different systems, regardless of whether they are living organisms or man-made devices, may exhibit distinct behavior. The character of this behavior is a manifestation of their nature, and thus, cannot be changed either. This section lists those characteristics of behavior, inherent in systems, that make it possible to classify them as qualitative systems and develop models of these systems utilizing the qualitative dynamical systems behavior mathematical modeling technique.

Qualitative Systems are Driven by Their Own Behavior. According to the notion of the qualitative dynamical system, defined by the Kaleidoscope Project, all systems considered qualitative have their purpose and goals to achieve. Therefore:

Qualitative Systems May Interact With the External World. The qualitative dynamical systems, just like many other systems, may affect external systems or experience the influence of everything that is going on around them. But, unlike the systems driven exclusively by external input, they do not have a special “input channel” through which the influences could control their behavior directly. External influences on the qualitative systems (if they are a part of the organization of behavior of the systems) may only make modifications in the systems, perceived as changes of their properties. For the observer "living in the world of that system", influences are supposed to look like spontaneous changes of the state of the properties of the system, which are absolutely indistinguishable from the changes occurring in the result of the internal processes in the system. Accordingly:

Qualitative Systems are Discrete Systems. Studying processes of changes occurring in various systems, developing over a certain time comparable to the time of development of processes in all the systems around us, we perceive these processes as behavior of systems and always deal with two fundamentally different mechanisms causing changes in the systems and thus determining the character of their behavior. These are the processes developing as a result of continuous changes in systems caused by continuously acting forces of nature and processes formed of discrete changes produced by actions of limited duration. So.

Number of States of Properties of a qualitative system. Depending on the nature of a particular discrete system, the number of possible states that the properties of the system can take may be in the range from just a few or several, to quite many different states. Meanwhile, as of the notion of the qualitative system, the number of states that the properties of these systems may take is always reasonably limited.

Asynchronicity and Fragmentary Changes of States of Properties. Being the systems of interacting components, whose behavior is naturally a consequence of the activity of individual components, their interaction with each other, with external systems, and with the environment. Therefore:

Perception of Behavior of Qualitative Dynamical
Systems

While reasoning about behavior at the everyday level, we primarily pay attention to the actions that someone or something performs with respect to surrounding systems or the environment. And that is understandable. As the actions are a way of interaction between people, machines, and people and machines, they are always important. However, when studying the behavior of natural systems or phenomena, or the devices created by us, we, in order to gain an understanding of the character of their behavior, tend to reveal the mechanisms that drive the behavior of the systems. Therefore, in that study, we are primarily interested not so much in learning the character of the actions of the systems (after all, often the same result may be attained by using different actions), as in identifying the causes that directly or indirectly induce activation of actions, and in those changes in the systems which these actions produce.

Situations as Causes of Changes of States of Properties

Considering the causes of changes occurring in the discrete qualitative systems of interacting components from the point of view of describing character of their behavior, one may explain behavior of these systems as: 1) emergence in the systems internal situations, resulted from activity of components of the systems and actions of the environment, and 2) responses of the systems to the emergence of important for the systems situations. Accordingly, when describing behavior of the qualitative systems, we pay special attention to the situations, responses of the systems to the situations, and changes produced as part of the responses. They are detailed as follows.

(o) Situation in a qualitative system (in contrast to the external situations that may arise around us) is simply a particular combination of states of a group of properties of the system, characterizing important circumstances, present in the world of the system. Such combinations are formed as a consequence of changes in the state of the properties of the system caused by the activity of the system itself, and/or the direct actions on the states of the properties of the system by other systems, Emergence of situation in a system generally means accumulation of important changes in the state of the system that require it to respond to the formed situation by producing certain modifications either in the system itself, or in its environment, or both.

(o) Response is a certain way with which a certain system reacts to the situation arising, by activating certain actions aimed at producing required modifications corresponding to the situation, in the system itself and/or in its environment.

(o) Change is the result of acting, activated as a part of the response of the system. It is observed as the appearance of a new state of property of the system, produced by the system as the result of its response to the situation.

In a formal way, the change of a single state of a single property of a system carried out as the response to a single situation can be considered as the Property State Dependence Relation (PSDR) and be schematically written as a chain of the constituents of the response. It can be viewed as the abstract piece of the process of behavior of a system:

“Change <= Response <= Situation”

or shorter as /C-R-S/. This expression states that the occurrence of a particular situation S, in a system, is the cause of its response R to the situation S, which, in turn, is the cause of a change C in the state of a particular property of the system corresponding to the occurrence of the situation S. This chain of the relations is intentionally written as directed from right to left to evoke association of the scheme of dependence relations with the process of computation of state of a property, that might be presumably implemented in the form of mathematical expression of a model of a system. (For details of modeling of behavior of qualitative systems, see next page: “Qualitative Dynamical Systems Mathematical Modeling Technique”.

Behavior as Correspondences of States of Properties to Situations

The Described Property State Dependence Relation schema is a convenient abstract way of representing the idea of the existing dependence relation of an abstract state of a property on the emergence of an abstract situation. But it may have limited application when it could be necessary to consider how the particular responses of a system to the occurrence of particular situations could be used to construct a schema larger than /C-R-S/, describing the behavior of a system as a composition of a few or several particular responses to particular situations. To describe the connections of these particular responses in a description of the organization of the system's behavior, the elements of the description should not be just abstract ideas of dependency. They should have all the concrete elements participating in the response, allowing the description to explicitly identify each property by its name and each state as a certain constant used to represent a particular state of the property.

The practically useful way of describing the concrete dependence relations of the state of properties on the situation, enhancing the abstract idea of that relation described as the Property State Dependence Relation schema, is the method suggested by the Kaleidoscope project, which is developed for creating specifications of the behavior of qualitative systems. This method consists of representing the dependence relations of the state of a property on a situation in the form of a Correspondence. The Correspondences serve as the real constructive elements describing the behavior of concrete qualitative systems. When used for describing the behavior of a system, they are supposed to be created for every concrete dependence relation of the state of a property of a system on each concrete situation causing change in the state of the property. And each concrete Correspondence is defined to hold the content of the concrete dependence relation describing a concrete state of a concrete property on a concrete situation, and hence is defined as a pair of the dependent and independent elements of the relation. The first, dependent element of the relation in this pair is the property and the constant representing the concrete state, called "proposed state of the property", which the property takes as corresponding to the situation, called "expected situation of the relation", described in the second, independent element of the pair. Whereas, the independent element of the relation is a list whose elements are the property included in the description of the group of properties, characterizing the expected situation, and the constant representing the expected state of the property in the expected situation.

After all the explanations of the meaning of Correspondence, its concise definition may be given as:

This definition assumes that all states of the properties are denoted in the way chosen for the representation of states of the properties of the system.

An example of the implementation of a single Correspondence and the properties associated with it is presented in Figure 1.

Figure 1. The Correspondence.

Figure 1 shows a single correspondence of the dependence of the state of Property P1 on the states of Properties P2, P3, and P4, characterizing a single situation. It depicts properties as large circles whose states are denoted by different colors filling the circles. Correspondences are represented as four arcs connecting the Property-circles with a small gray circle representing the “fact” of the existence of the situation expected by the Correspondence. Three Property-circles: P2, P3, and P4 represent properties, one of the combinations of states of which should eventually represent the expected situation. The property circle P1 represents the dependent property of the Correspondence. The fact-circle is not part of the Correspondence, but is convenient for its graphical representation as a point at which arcs, exiting from the properties P2, P3, and P4, converge, and from which the arc, directed to the dependent property P1, exits. Arcs exiting from the properties P2, P3, and P4 are marked with colored dots denoting expected states of their input properties. Arc directed from the “fact” to the property P1 is marked with an arrowhead, the color of which determines the proposed state that the property P1 should take in the event of the occurrence of the expected situation specified by the colors of the dots located on the arcs exiting from the properties P2, P3, and P4.

However, unlike Figure 1, not so many properties of the modeled system can change their state only once and then remain in this state for the rest of the system's life. The state of most of the properties of a dynamical system is a consequence of its responses to various situations. And this assumes that, in general, the structure of the dependence relations of the state of properties on the states of groups of properties characterizing situations should be significantly more complex.

Everything complex is always a combination of something simpler

Even without analyzing characteristics of behavior of qualitative systems, but merely turning to our own experience and recalling everything we know about situations and reactions to them, it is possible to say that when thinking of the organization of behavior of the qualitative systems as of consisting of standard elements, first thing that springs in the maid is the two natural way of connecting the Correspondences making them work together. And for that, it is necessary to realize just two things:

(1) Each situation may produce changes in the state of not only a single property; it may also act simultaneously upon several properties by causing them all to take either the same or different states, depending on the purpose and needs of a particular modeled system.

(2) The state of each property of the system may depend not only on a single situation, but on several different situations. Thus, in case of the occurrence of such situations, dependent property may take either the same or different states proposed by different correspondences of states of properties to different situations, which are explicitly specified in the Correspondences.

An example of a fragment of a model of a qualitative system, illustrating both ways of composition of Correspondences, is presented in Figure 2.

Figure 2. Composition of six Correspondences.

Similar to Figure 1, Figure 2 represents properties as large circles, facts as small circles, and Correspondences as marked arcs directed from the properties to the fact and from the fact to the dependent property. But unlike Figure 1, which shows only one Correspondence and its associated properties, Figure 2 shows property P1 as dependent on three situations represented by the groups of property (P2, P3), (P4), and (P5, P6, P7). But the property P1, in this figure, is not only dependent. There are three properties, P8, P9, and P10, depending on its state. So, three different states of property P1 represent three different situations and therefore are described as having three Correspondences. Expected situations in these Correspondences are represented as arcs marked with blue, green, and red dots, directed from P1 to three facts and as arcs from those facts, marked as magenta, blue, and green arrowheads directed to properties P8, P9, and P10.

Thus, based on the behavior described so far and in accordance to its vision developed in the Kaleidoscope project, the mechanism of behavior of qualitative systems is determined: a) by the Correspondences that explicitly describe dependence relations of states of properties on the emergence or disappearance of situations, and, b) by, not explicitly described, but implicitly present, in the state of the systems, dependence of formation of situations represented by combinations of the state of properties, change of which occurs as a result of changes in the state of properties, included in the description of these situations, when the changes of the properties characterizing situations were produced earlier as the result of response of the systems to the situations that has been formed earlier.

Behavior as Step-by-Step Process of Transitions of State of System

Taking into account all that has been described above, the behavior of qualitative systems can be viewed as a process consisting of a sequence of steps, the content of which is always the same. Each step responds to the situations formed at the previous step by performing modifications in the system corresponding to the situations being processed, results of which are observed as the fragmentary changes of the state of the properties of the system. And since the states of some properties can be included in the descriptions of the expected situations, the consequence of the changes in the state of such properties can cause changes in the situations presented in the state of the system.

Depending on the current state of the system at each step, changes in the groups of properties characterizing situations subjected to modifications can cause:

(a) emergence of situations, completion of formation of which was expected as a change of the state of one, more, or of all states of the groups of properties characterizing situations; and these changes happened during the execution of the current step;

(b) update of states of groups of properties characterizing situations which are not yet entirely formed with one or more, but not the last, replacements of the state of properties characterizing situations, which make the groups proceed in the formation of the situations;

(c) destruction of existing situations represented by combinations of state of groups of properties, that were formed in the past, but the result of changes of the state of properties of the groups, produced by the system during the step, destroyed the combination of the state of groups of properties, since the properties, states of which have been changed, are included in the descriptions of the situation, so that new combinations of the state of properties no longer represent existing situations.

Possible cases when the execution of the current step of the process has ended, but none of the combinations of the state of the groups of properties characterizing the expected situation has been formed, and therefore, the next step of the process cannot begin, is an entirely anticipated form of termination of the process. Such a stop of the process is simply an indicator that everything that should be done by the system as its response to the occurrence of a single or a chain of situations has been done, and that thus the period of active behavior of the system has been completed. But such completion does not mean that the system has stopped acting at all. The activity of its components and other systems with which the system under consideration can interact continues. So, sooner or later a moment should come when either in the result of the ongoing processes in the components of the system, or under the influence of the surrounding systems or phenomena, someone or several properties will spontaneously change their state and these changes will create one or several new situations that will revive the active behavior of the system.

As follows, the description of the behavior of the qualitative system as a sequence of steps, two described factors determine the organization of behavior of the qualitative system. They are: the way of connection of the Correspondences and the step-by-step mechanism of application of Correspondence forming the process of behavior of the systems, as the dynamically organized behavior of systems of varying complexity.

The description of various forms of organizing the processes of behavior of the systems is described on the pages presenting approaches to different representations of models of the system, studied and proposed by the Kaleidoscope project.

So, What Is the Qualitative Dynamical System
After All?

Summarizing, all that has been described and discussed so far about the perception of systems as qualitative and about the characteristics which the qualitative systems should have, one would specify the qualitative systems as:

The example presented in Figure 3 illustrates a complete model of a qualitative dynamical system named “Accumulated Situations”. Similar to the fragments presented in Figure 1 and Figure 2, this model was developed based on the qualitative systems behavior modeling technique and utilizing the development tool called “Qualitative Dynamical Systems Modeling & Simulation Environment”, both provided by the Kaleidoscope Project.

Figure 3. Model of qualitative dynamical system:
“Accumulated Situations”.

For details on the presentation in Figure 3, see the section: Explanation of Modeling Formalism Used in Figures, right below.

Qualitative Systems Behavior Specification

The necessity to document our knowledge on the behavior of qualitative systems, gained in the process of their study or design, as well as to subsequently create models of their behavior, presupposes having a method for describing and storing all accumulated information in a form convenient for its updating, editing, and use. The proposed method of specifying this information should also be formal enough so that work with the document could be performed with the help of utility computer programs.

Accordingly, the structure and the content of the data characterizing behavior of a qualitative system, proposed in this section, assumes that the Qualitative System Behavior Specification, or shorter QSBSpec, should be created as consisting of two sections.

Unique Identifiers and Their Text Interpretations

Although the description of qualitative states of system properties expressed in the form of statements made in natural language is convenient for understanding the essence of the states of properties, it becomes inconvenient and cumbersome when describing states of dependent properties on the states of independent ones. After all, most properties may be both dependent and independent, which means that they are included in the descriptions of various Correspondences, and thus must be presented in the specification several times.

Therefore, to make the specification more compact, the task of the first section, entitled “Unique Identifiers and Their Text Interpretations”, is: a) creating a file elements of which defined as pairs: unique identifier of a name of a property and textual representation of the name of the property, and a file elements of which defined as pairs: unique identifier of a state of a property and textual representation of the state of the property; and c) using, in the second part of specification, unique identifiers instead of names of properties and states of properties, that otherwise should be extracted from the propositions.

For the sake of clarity, further on, the format of the unique identifiers of names and states of properties will be defined as: a lowercase letter “p” followed by an integer number that serves as index of the text in the file of correspondences of these identifiers to the names of the properties, or a lowercase letter “c” followed by an integer number that serves as index of the text in the file of correspondences of these identifiers to the states of properties.

Qualitative System Behavior Specification: Structure and Content

The second section, entitled "Contents of the Quality Property Dependency Specifications", is the actual description of the system's behavior. It defines the Qualitative System Behavior Specification, or QSBSpec, as a listing of the Qualitative Property Dependence Specifications, or shorter QPDSpecs. The content of each QPDSpec describes the dependence of the state of the property of the QPDSpec on all situations that occur, causing a change in the state of this property. So, the content of the QPDSpec is, in essence, based on the content of those Correspondences, definition of which includes the proposed state of the property of the QPDSpec, and the combination of states of independent properties characterizing the expected situation, occurrence of which causes a change in the state of the QPDSpec property.

The general structure of the QSBSpec, written using notation proposed by the first section, is presented in Figure 4.

      };

Figure 4. Structure of Qualitative System Behavior Specification.

This figure presents QSBSpec as a list of QPDSpecs, and defines each of QPDSpecs as a pair: a unique identifier of the dependent property p(i) of the specification QPDSpecs, and the second as the list of Correspondences, denoted by symbol C. Each j-th Correspondence of the list C is, in turn, again a pair. The first element of this pair is a unique identifier of the proposed state of the property of the QPDSpecs ps(i,j), while the second is the description of the expected situation specified in the form of the list S. And the elements of S are also pairs. They are written as a unique identifier of the independent property p(i,j,k) and a unique identifier of the expected state of the independent property es(i,j,k) of the Correspondences.

The way of description of the QPDSpecs utilizes the notation proposed by the Kaleidoscope project. In accordance with this notation, presentation of the lists in Figure 4 is provided with the help of constructs: Lⁿ_i=0,  C^cn(i)_j=0,  and S^sn(i,j)_k=0 that are the project’s proprietary approach to denoting list iterators of elements of the specification. These elements are written as bold uppercase characters L, C, and S, whose indices i, j, and k change from 0 to n, 0 to cn(i), and 0 to sn(i,j), respectively. Thus.

Construct Lⁿ_i=0 represents a list iterator through all QPDSpecs, index i of which changes from 0 to n, where n is the total number of QPDSpecs in the QSBSpec.

Construct C^cn(i)_j=0, is a list iterator through all Correspondences, index j of which changes from 0 to cn(i), where cn(i) is the number of Correspondences in the i-th QPDSpec. And finally.

Construct S^sn(i,j)_k=0 stands for the list iterator through all the pairs: [variable/value] describing the expected situation, index k of which changes from 0 to sn(i,j), where sn(i,j) is the number of pairs characterizing situation defined in j-th Correspondence of i-th PSDSpec.

Elements p_i and p_i,j,k, in the presented QSBSpec are the unique identifiers of the properties of the system, and elements es^t_i,j,k and ps^t+1_i,j, are the unique identifiers of the states of the properties, where the es^t_i,j,k stands for expected state of the property p_i,j,k at discrete time t, while the ps^t+1_i,j stands for proposed state that the property p_i should take at discrete time t+1.

Adaptation of the QSBSpec to its application for creating models of qualitative systems is described on the next page, "Qualitative Dynamical Systems Mathematical Modeling Technique", in the section: Qualitative Model Behavior Specification.

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Everything described on this page could be considered as just an interesting reflection, if it would not turn out that dependence of states of some qualitative properties of the qualitative systems on the states of some others, independent qualitative properties, can be computed in the same way as we can compute unknown estimates of truth or falsity of some propositions depending on the known estimates of truth or falsity of some other propositions.

Next page: “Qualitative Dynamical Systems Mathematical Modeling Technique”, presents results of the research on the feasibility of defining models of qualitative dynamical systems in the form of systems of algebraic equations.