ARCHITECTURE OF MODEL PARAMETRIC SPACE: HIERARCHY IN SIMON’S ARCHITECTURE OF COMPLEXITY

1. ARCHITECTURE OF MODEL PARAMETRIC SPACE: HIERARCHY IN SIMON’S ARCHITECTURE OF COMPLEXITY Y.R. Valkman, A.Y. Rykhalsky The International Research and Training Center of Information Technologies and Systems Е-mail: yur@valkman.kiev.ua

2. Simplicityis a sign of beauty. • Simplicity is a sign of truth. • Simplicityis a sign of genius. • Simplicityis a sign of efficiency. • However, everything is not as simple as it seems. That’s what this report is devoted to. • This work represents a continuation of the studies results of which can be found, particularly, in[Yu.R.Valkman. Information Theories of Resarch Design of Complex Ware: Strructure of Model and Parametr Space//International Journal on Information Theories & Applications. Sofia. – FOI-COMMERCE - 1995 - Vol.3, - No.10. - pp. 3-12.]. • The general goal of these studies is to develop methods and ways of building knowledge bases for the purpose of modeling complex systems using the apparatus of model parametric (<М,Р>) space. • The authors discuss the issue of reflecting the structure of complexity of knowledge of designers and researchers modeled in <М,Р> space.

3. THIS REPORT WILL TOUCH UPON THE FOLLOWING PROBLEMS: 1. Rigid and soft systemic thinking 2. Complexity problems and open systems 3. Complex systems and agents 4. Simon’s structure of complexity 5. Definition of a <M,P>-space, its property and structure 6. Architecture of model parametric space

4. 1. Rigid and soft systemic thinking First attempts to develop methodologies and technologies of systemic thinking were undertaken as part of military elaborations during World War II. Operational research, systemic analysis, and systemic engineering all have appeared during that time. Checkland later called these systems a "rigid systemic thinking." Their distinctive features included rigidly set end goals the achievement of which the system was aimed at. Problems were determined in course of the work, and mathematical and operating models of their solution were developed.

5. New theory was born, called “soft systemic thinking".This theory aimed at the study of first of all “live", social systems. Soft systemic thinking is based on the assumption that it is impossible to determine simple, clear, and constant goals for social system equally understood by all. Main attention was given to the integration of different, sometimes contradicting views of the problems and their solution in organization, which is necessary to prepare and implement the changes. The process is built to ensure that the system will learn and self-organize.

6. It’s worth noting that later, Academician I.V. Arnold proposed to introduce soft mathematical models, and even before that, L. Zade introduced the concept of soft mathematics. Soft systemic thinking especially emphasizes the role of values, beliefs, and a general worldview. Its main goal is the study and description of culture and policy of organization to ensure that the process of changes is supported by all members of this organization. Perhaps one could talk about "soft structure“ which provides for possibility of dynamic replacement of the system’s components. These model structures are supported by means of model parametric space. They ensure flexibility of relevant structures of knowledge.

7. 2. Complexity problems and open systems A flaw of the systemic thinking which often comes to the surface is that when addressing the complexity problem, almost every computer guru presents his views as if he is a pioneer and no similar problems have ever appeared in other fields of knowledge. Meanwhile, the complexity problem is not a discovery, far from it. Complexity is a widely used category; different fields of fundamental science have developed own perceptions of complexity.

8. In the data transmission theory, complexity is measured by the total number of properties transmitted by object and received by observer. Inphysics, complexity is determined by the probability of system state vector. Inmathematics, they are talking about computing complexity of algorithms. Incognitive psychology, complexity of problem is evaluated from the angle of possibility of solving this problem by human. In addition to that, GENERAL THEORY OF SYSTEMSalso has its own perception of complexity.

9. Let’s list the qualities which complex systems have according to this theory. • One of the most important characteristics of a really complex system is unpredictability; • Relations between components of complex system are quite short. System element usually receives information from its nearest neighbors, which means that when traveling large distances it undergoes changes; • Relations are not linear; therefore, small perturbing impact may cause substantial effect, and vice versa: large perturbing impulse may turn out to be ineffective;

10. Relations between components may include feedback, both positive (which oscillate the system) and negative (damping it); • By definition, complex system is OPEN; depending on the nature of the system its boundaries must be permeable either for information or for energy; • Complex systems have history;moreover, small changes in the present may result in significant changes in the future; • A characteristic feature of complex systems is NESTING; say, economy as a system may consist of the enterprises it includes, which are systems in themselves; the enterprises consist of individual employees who are also systems, and so on.

11. 3. Complex systems and agents From our point of view, use of the agent ideology is an extremely prospective method of presenting complex structures in <М,Р>-space. There are the following basic ideas of using a multi-agent concept in the complex system modeling. 1.Complex systems include autonomous objectswhich interact with each other when performing certain their tasks. 2.The agents adapt themselves – they must be able to react to their environment and, possibly, change their behavior based on information received.

12. 3.Complex systems are also characterized by their appearing structures. APPEARING STRUCTUREis a logically linked scheme formed as a result of interaction between agents.Results of functioning of the appearing structure may be both positive and negative, which means that they have to be analyzed when developing agent-based systems. 4.Successful systems with appearing structures often exist on the verge of order and chaos. If any organism or organization are in order at all times or are always in the state of chaos, it’s a sign of destruction. Nevertheless, the interim state is necessary for an object to exist. 5.We have to learn from the nature.For billions of years it has been solving serious combinatory problems, so when creating agent-based systems it makes sense to consider parasitism, symbiosis, reproduction, genetics, mitosis, and natural selection.

13. When creating agent-based systems, one has to devote special attention to the APPEARANCE CONCEPT. On the one hand, appearance may occur without our intention or consent, which may be good or bad. Examples include: ant colonies, bee swarms, bird flocks, traffic jams,etc. Note: ants (or cars) change, but the structures – colonies (or traffic jams) – remain. On the other hand, being developers of the system of knowledge about the object of study, we can try to “project“ appearance of the structures of knowledge that we need. In other words, we can try to project agents with the behavior necessary for the required structures to appear. THESE ARE TWO SIDES OF THE SELF-ORGANIZING STRUCTURES.

14. 4. Simon’s structure of complexity The main peculiarity of complex physical, social, biological, or technical systems per se is the fact that they have a CLEARLY DEFINED HIERARCHICAL ORGANIZATION. The occurring structure of multiple parts "nested" inside each other allows to describe these systems from the point of view of different levels (or modules) of organization, which leads to important consequences for the strategies of their study. Since separate parts located inside these levels interact among each other stronger than between the levels, when describing complex systems we may to a certain degree abstract away from their complexity and concentrate on the description of mechanisms of just one or two neighboring levels. Simon calls these systems "nearly completely decomposable" or, to be short, "NEARLY DECOMPOSABLE" (ND) complex systems.

15. Everybody knows about Simon’s "PARABLE OF THE TWO WATCHMAKERS" which illustrates the usefulness of the ND principle. One of the watchmakers tries to assemble the watch outright from the tiniest details, which means that any serious malfunctioning of the watch makes him start all over again from the very beginning. The other watchmaker puts together intermediate modules, each of which has certain autonomy, first, and only after that he sets on assembling the whole watch. As a result, any problem sends him back to the certain already sufficiently advanced phase of work. Structurally, complex systems are not homogeneous. They represent "interrelated islands" of more or less stable formations (modules). It reflects both the principles of self-organization in synergetic systems and certain approaches to the chaos theory.

16. These general considerations lead Simon to the following two fundamental issues. The first of them deals with the parameters of evolution processes related not to Charles Darwin’s natural selection (or Adam Smith’s "invisible hand of market") but to organisms and organizations built from the myriad of relatively autonomous and stable "functional blocks". The second issue is the issue of applicability of logical and mathematical methods of describing complex systems and their behavior. As Simon noted in, "complexity of systems can easily exceed possibilities of their modeling using the most powerful computers, both present and future".

17. 5. Definition of a <M,P>-space, its property and structure DEFINITION 1. Model-parametrical space (<M,P>-space)we shall understand a set of all models, parameters, relations between them, describing property (designed and/or researched) product (system). From our point of view, the most suitable means for description and research of structure of the <М,P> space is the graph theory. Elements (nascent components) of <М,P> are: models (set М), parameters (set P) and relations between them, i.e. M = {Mj}, j €J, P = {Pi}, i € I, where sets of the I and J indexes are determined by objects, considered in each concrete case, determining them explicitly and implicitly as integrated structures.

18. We attribute to the objects various aggregates, nodes, functional subsystems, components of a designed complicated product, the product itself, and the system that describes product’s behavior and functioning in the external environment. Hence, in the most general case, the <М,P> space is determined on the set of the direct Catresian product of M and P, i.e. <М,P>Мx P. To objects we refer various, knots, functional subsystems, item, it as a whole and system circumscribing it a behavior and operation in an external medium. Thus in the most common case direct decart a product of sets М and P, i.e. <М,P>Мx P.

19. We will prescribe a different meaning to arcs-relations between the models and parameters, depending on context of the consideration. But, by default it is supposed, that the arrow, directed from the parameter to the model, means that this parameter in the given model is independent, and the arrow, directed from the model to the parameter, corresponds to the case of dependence of the parameter on the model. Note that the reasons adduced (especially, as to the statement 1 interpretation) justify an expediency of exclusion from the consideration of the "undirected models". In fact, any model is intended for simulation of some properties and/or characteristics of the product under projection, i.e. it always possesses the input and output parameters.

20. Neighborhoods in <М,P>-space Word "neighborhood" has in ordinary speech such sense, that many properties, in which the mathematical concept participates called of themes by the name, act as mathematical expression intuitively of clear properties. DEFINITION 2. As a neighborhood of a <М,P>-space of the 1-st order concerning a model Мjwe shall name a set of parameters Pi, immediately connected with a model Мj. To designate this neighborhood we shall be [Mj]1. To designate this neighborhood we shall be [Мj]1 DEFINITION 3. As the boundary of a neighborhood k-order of a model Мjwe shall name a set of all elements of a <М,P>-space connected with Мj by a way, length equal "k". To designate the boundary we shall be — [Mj]k

21. Representation of model-parametric approach

22. An example <М,P>-neighborhoods 1-st, 2-nd and 3-rd order concerning a model М3

23. <М,Р>-neighborhood of the Р1 parameter

24. Union of <М,P>-neighborhoods М1, Р1 и Р2

25. Intersection of neighborhoods of elements Р1 и Р2 P 2 P 1

26. <М,P> SPACE Examples of the graphic interpretations of "intersection" and "union" of knowledge

27. A case history of outcomes of operation of intersection <М,P>-neighborhoodsР2и Р9

30. 6. Architecture of model parametric space Any model represents a system. Complex system may be represented only by complex system. The complexity of this model reflects in the need to support many models the structure of relations of which reflects the relations between components of the modeled object. The adequacy and stability (quality) of each local model reflects the level of our knowledge about modeled aspect of a designed or studied complex object. Therefore, the <М,Р> space was initially built to support multi-model structures.

31. In this case, models may be represented in different forms and formats: • frames, • products, • via semantic networks, • cognitive models, • statistical polynomials, • differential equations, tables, • diagrams, • on verbal level, etc.

32. Architecture of <М,Р> space ensures that Simon’s structure of complexity is reflected in the computer environment. • The following levels of "knowledge" can be defined in the hierarchical structure of this space: • parameters, • models, • <М,Р> neighborhoods, • methods of calculating different integral and • aggregated parameters, • characteristics of new complex hardware, • appearances (projections) of complex • systems.

33. Diagram hierarchical structures of a <M,P>-space Аppearances (projections) of complex systems <М,Р> -neighborhoods Parameters Models

34. At the same time, certain <М,Р> neighborhoods may become part of other <М,Р> neighborhoods thus ensuring RECURSIVENESS. Levels of ND architecture of <М,Р> space are limited ONLY BY THE COMPLEXITY OF THE PROBLEM. Design and study of complex systems had always been the job of experts in various problem and application fields.

35. Their knowledge represented using different models is integrated in <М,Р> space. These models may come in form of generally accepted and tested laws. But <М,Р> space may also include the models that only undergo the testing. Therefore, this space is HETEROGENEOUS from this point of view as well.

36. It is worth noting that the appearances are built for the purpose of modeling structures of different units, components, and subsystems of complex objects and processes of their functioning and behavior of the object in general in the outside environment. In [Yu. R. Valkman. Model calculus in concurrent engineering of complex products.// Proc. IMACS Multiconference "Computational Engineering in Systems Applications" (CESA'96), Lille-France, July 9-12, 1996,- pp. 909-914.] describers ideology of building hierarchical structures in <М,Р> space. The appropriate methods are based on the use of a SPECIAL MODEL CALCULATION APPARATUS (algebra and logic of model texts and contexts).

37. We believe that the model located on the top level of hierarchy plays the role of context for the models lying below it. That’s how the hierarchical structure of model contexts is formed. For the purpose of this work, context means formal representation of all aspects of adequate interpretation of the appropriate models.

38. Hierarchical structure of contexts Let’s take a look at the structure (context inside context) of properties and characteristics of: ship(Р1 = A, . . . ): • navigational qualities (Р2 = B, . . .): ►propulsion, ►controllability, ►rocking (Р5 = E, . . .), rolling, pitching, . . . ; • hull (Р3 = C, . . .): ►geometry, ►durability, . . . • power unit (Р4 = D, . . .): ►capacity, ►weight and dimensional characteristics, . . .

39. Р1 = А … Р2 = В … Р3 = С … Р4= D … Р5= Е … Р1 = А … Р3 = С … Р2 = В … Р5 =Е … Р4 =D … - texts - contexts Reflection of I-graph of the product’s structure in the graphical image of "context inside context"

40. The figure above shows examples of GRAM(graphic analysis of mathematical models) system performance results. Ideology of hierarchical contexts was extensively used when developing this system (and Database Management System of DRAWING system in general). Let’s assume that we have a model with the following text: Р1= F1 (P2, P3, P4, Р5). This model is difficult to analyze using virtual images. Therefore, this model was simplified in certain phase of the study and four- and five-dimensional discretely-continuous ГО were reviewed. Р1 = f2 (P2, P3),ifP4 = (а1, а2, , а3, …)andP5 = const. Now, parameters P4andP5were “transferred“ to the context of Р1= F1 (P2, P3, P4, Р5)model. We call this ГО “four-dimensional discretely-continuous". Fig. shows “five-dimensional discretely-continuous ГО": Р1 = f3 (P2, P3),ifP4 = (а1, а2, а3, …,)and P5 = (в1, в2, , в3, …,). Note, that the employer decided not to go with further “increase“ of ГО dimensions, such as, for example, image of “matrix of matrixes".

41. The purpose of GRAMsubsystem is synthesis of graphical presentation of mathematical models. Using the GRAM, researcher and designer can analyze models in form of visual images.

42. Подсистема ГРАММ предназначена для синтеза графического представления математических моделей. С помощью ГРАММ исследователь и проектировщик анализируют модели в форме визуальных образов. Two-dimensional, continuous graphical image

43. Подсистема ГРАММ предназначена для синтеза графического представления математических моделей. С помощью ГРАММ исследователь и проектировщик анализируют модели в форме визуальных образов. Three-dimensional, continuously-discrete graphical image

44. Подсистема ГРАММ предназначена для синтеза графического представления математических моделей. С помощью ГРАММ исследователь и проектировщик анализируют модели в форме визуальных образов. Four-dimensional, continuously-discrete graphical image

45. Подсистема ГРАММ предназначена для синтеза графического представления математических моделей. С помощью ГРАММ исследователь и проектировщик анализируют модели в форме визуальных образов. Five-dimensional, continuously-discrete graphical image

46. Подсистема ГРАММ предназначена для синтеза графического представления математических моделей. С помощью ГРАММ исследователь и проектировщик анализируют модели в форме визуальных образов. Three-dimensional, continuous graphical image

47. Подсистема ГРАММ предназначена для синтеза графического представления математических моделей. С помощью ГРАММ исследователь и проектировщик анализируют модели в форме визуальных образов. Four-dimensional, discretely-continuous graphical image

48. Five-dimensional, discretely-continuous graphical image

49. Image texts and contexts Apparently, it is appropriate to compare the center of <М,Р>-neighborhood, or the entire neighborhood (but in that case in the <М,Р>-space) with the center of attention, and consider the other models and parameters acontextrelevant image. The center of <М,Р>-neighborhood, or the entire neighborhood (depending on the goals of creation or study of the image space) may play the role of image text. We can prove formally that in case of this approach, all four context properties will be fulfilled.

50. (Im) image located in the center of attention, with ("far"and "near") images forming its context.