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

ARCHITECTURE OF MODEL ARCHITECTURE OF MODEL PARAMETRIC SPACE:PARAMETRIC SPACE:

HIERARCHY IN SIMON’S HIERARCHY IN SIMON’S ARCHITECTURE OF ARCHITECTURE OF

COMPLEXITYCOMPLEXITYY.R. Valkman, A.Y. Rykhalsky

The International Research and Training Center of Information Technologies and Systems

Е-mail: [email protected]

SimplicitySimplicity is a sign of beautyis a sign of beauty.. Simplicity Simplicity is a sign of truthis a sign of truth. . SimplicitySimplicity is a sign of genius.is a sign of genius. SimplicitySimplicity is a sign of efficiencyis a sign of efficiency..

HoweverHowever, , everything is not as simple as it seemseverything is not as simple as it seems. . That’s what That’s what this report is devoted tothis report is devoted to..

This work represents a continuation of the studies results of This work represents a continuation of the studies results of

which can be found, particularly, inwhich can be found, particularly, in [[Yu.R.Valkman. Yu.R.Valkman. Information Theories of Resarch Design of Complex Ware: Strructure Information Theories of Resarch Design of Complex Ware: Strructure of Model and Parametr Space//International Journal on Information of Model and Parametr Space//International Journal on Information Theories & Applications. Sofia. – FOI-COMMERCE - 1995 - Vol.3, - Theories & Applications. Sofia. – FOI-COMMERCE - 1995 - Vol.3, -

No.10. - pp. 3-12.No.10. - pp. 3-12. ].]. The general goal of these studies is to develop methods and The general goal of these studies is to develop methods and

ways of building knowledge bases for the purpose of ways of building knowledge bases for the purpose of modeling complex systems using the apparatus of model modeling complex systems using the apparatus of model parametric (<М,Р>) space. parametric (<М,Р>) space.

The authors discuss the issue of reflecting the structure of The authors discuss the issue of reflecting the structure of complexity of knowledge of designers and researchers complexity of knowledge of designers and researchers modeled in <М,Р> space.modeled in <М,Р> space.

THIS REPORT WILL TOUCH UPON THE THIS REPORT WILL TOUCH UPON THE FOLLOWING PROBLEMS:FOLLOWING PROBLEMS:

1. 1. Rigid and soft systemic thinkingRigid and soft systemic thinking

2. 2. Complexity problems and open systemsComplexity problems and open systems

3. 3. Complex systems and agentsComplex systems and agents

4. 4. Simon’s structure of complexity Simon’s structure of complexity

5. 5. Definition of a <M,P>-space, its property and Definition of a <M,P>-space, its property and structurestructure

6. 6. Architecture of model parametric spaceArchitecture of model parametric space

1. 1. Rigid and soft systemic thinkingRigid and soft systemic thinking

First attempts to develop methodologies and First attempts to develop methodologies and technologies of systemic thinking were undertaken technologies of systemic thinking were undertaken as part of military elaborations during World War II. as part of military elaborations during World War II. Operational researchOperational research, , systemic analysis, and systemic analysis, and systemic engineering systemic engineering all have appeared during that all have appeared during that time. Checkland later called these systems a time. Checkland later called these systems a ""rigid rigid systemic thinkingsystemic thinking..""

Their distinctive features included Their distinctive features included rigidly set end goals the rigidly set end goals the achievement of which the system was aimed atachievement of which the system was aimed at. . Problems were determined in course of the work, and Problems were determined in course of the work, and mathematical and operating models of their solution mathematical and operating models of their solution were developedwere developed. .

New theory was born, called New theory was born, called “soft systemic thinking"“soft systemic thinking".. This theory aimed at the study of first of all This theory aimed at the study of first of all “live"“live",, social social systemssystems. . Soft systemic thinking is based on the Soft systemic thinking is based on the assumption that it is impossible to determine simple, assumption that it is impossible to determine simple, clear, and constant goals for social system equally clear, and constant goals for social system equally understood by allunderstood by all..

Main attention was given to the integration of Main attention was given to the integration of different, sometimes contradicting views of the different, sometimes contradicting views of the problems and their solution in organizationproblems and their solution in organization,, which is necessary to prepare and implement the which is necessary to prepare and implement the changeschanges. .

The process is built to ensure that the system will The process is built to ensure that the system will learn and self-organizelearn and self-organize..

It’s worth noting that later, Academician I.V. Arnold proposed It’s worth noting that later, Academician I.V. Arnold proposed to introduce to introduce soft mathematical modelssoft mathematical models, , and even before and even before that, L. Zade introduced the concept of that, L. Zade introduced the concept of soft mathematicssoft mathematics..

Soft systemic thinking especially emphasizes the role of Soft systemic thinking especially emphasizes the role of values, beliefs, and a general worldviewvalues, beliefs, and a general worldview. . Its main goal is Its main goal is the study and description of culture and policy of the study and description of culture and policy of organization to ensure that the process of changes is organization to ensure that the process of changes is supported by all members of this organizationsupported by all members of this organization..

Perhaps one could talk about Perhaps one could talk about ""soft structuresoft structure“ which “ which provides for possibility of dynamic replacement of the provides for possibility of dynamic replacement of the system’s componentssystem’s components. . These model structures are These model structures are

supported by means of model parametric spacesupported by means of model parametric space. . They They ensure flexibility of relevant structures of ensure flexibility of relevant structures of knowledgeknowledge..

2. 2. Complexity problems and open Complexity problems and open systemssystems

A flaw of the systemic thinking which often comes to A flaw of the systemic thinking which often comes to the surface is that when addressing the complexity the surface is that when addressing the complexity problem, almost every computer guru presents his problem, almost every computer guru presents his views as if he is a pioneer and no similar problems views as if he is a pioneer and no similar problems have ever appeared in other fields of knowledgehave ever appeared in other fields of knowledge. .

MeanwhileMeanwhile, , the complexity problem is not a discovery, the complexity problem is not a discovery, far from itfar from it. . Complexity is a widely used categoryComplexity is a widely used category; ; different fields of fundamental science have different fields of fundamental science have developed own perceptions of complexitydeveloped own perceptions of complexity. .

In the In the data transmission theorydata transmission theory, complexity is , complexity is measured by the total number of properties measured by the total number of properties transmitted by object and received by observertransmitted by object and received by observer. .

InIn physicsphysics, complexity is determined by the , complexity is determined by the probability of system state vectorprobability of system state vector. .

InIn mathematicsmathematics, they are talking about computing , they are talking about computing complexity of algorithmscomplexity of algorithms. .

InIn cognitive psychologycognitive psychology, complexity of problem is , complexity of problem is evaluated from the angle of possibility of solving evaluated from the angle of possibility of solving this problem by humanthis problem by human..

In addition to that, In addition to that, GENERAL THEORY OF SYSTEMSGENERAL THEORY OF SYSTEMS also has its own perception of complexity.also has its own perception of complexity.

Let’s list the qualities which complex systems have Let’s list the qualities which complex systems have according to this theoryaccording to this theory..1.1. One of the most important characteristics of a really One of the most important characteristics of a really

complex system is complex system is unpredictabilityunpredictability;;2.2. Relations between components of complex system are Relations between components of complex system are

quite shortquite short. . System element usually receives System element usually receives information from its nearest neighborsinformation from its nearest neighbors, , which means which means that when traveling large distances it undergoes that when traveling large distances it undergoes changeschanges;;

3.3. Relations are not linearRelations are not linear; ; thereforetherefore, , small perturbing small perturbing impact may cause substantial effect, and vice versa: impact may cause substantial effect, and vice versa:

large perturbing impulse may turn out to be ineffectivelarge perturbing impulse may turn out to be ineffective;;

4.4. Relations between components may include Relations between components may include feedbackfeedback, , both positive both positive ((which oscillate the which oscillate the systemsystem)) and negative and negative ((damping itdamping it); );

5.5. By definition, complex system is By definition, complex system is OPENOPEN; ; depending depending on the nature of the system its boundaries must be on the nature of the system its boundaries must be permeable either for information or for energypermeable either for information or for energy; ;

6.6. Complex systems Complex systems have historyhave history;; moreover, small moreover, small changes in the present may result in significant changes in the present may result in significant changes in the futurechanges in the future; ;

7.7. A characteristic feature of complex systems is A characteristic feature of complex systems is NESTINGNESTING; ; saysay, , economy as a system may consist economy as a system may consist of the enterprises it includesof the enterprises it includes, , which are systems in which are systems in themselves; the enterprises consist of individual themselves; the enterprises consist of individual employees who are also systemsemployees who are also systems, , and so onand so on..

3. 3. Complex systems and agentsComplex systems and agentsFrom our point of viewFrom our point of view, , use of the agent ideology is an use of the agent ideology is an extremely prospective method of presenting complex extremely prospective method of presenting complex structures instructures in <М,Р>- <М,Р>-spacespace. . There are the following basic There are the following basic ideas of using a multi-agent concept in the complex ideas of using a multi-agent concept in the complex system modelingsystem modeling..1.1. Complex systems include Complex systems include autonomous objectsautonomous objects which interact with each other when performing certain which interact with each other when performing certain their taskstheir tasks. . 2.2. The agents The agents adapt themselvesadapt themselves – – they must be able they must be able to react to their environment and, possiblyto react to their environment and, possibly, , change their change their behavior based on information receivedbehavior based on information received. .

3.3. Complex systems are also characterized by their Complex systems are also characterized by their appearing structuresappearing structures. . APPEARING STRUCTUREAPPEARING STRUCTURE is a logically is a logically linked scheme formed as a result of interaction between linked scheme formed as a result of interaction between agentsagents.. Results of functioning of the appearing structure may Results of functioning of the appearing structure may be both positive and negativebe both positive and negative, , which means that they have to which means that they have to be analyzed when developing agent-based systemsbe analyzed when developing agent-based systems. . 4.4. Successful systems with appearing structures often Successful systems with appearing structures often exist on the verge of exist on the verge of order and chaosorder and chaos. . If any organism or If any organism or organization are in order at all times or are always in the organization are in order at all times or are always in the state of chaos, it’s a sign of destructionstate of chaos, it’s a sign of destruction. . Nevertheless, the Nevertheless, the interim state is necessary for an object to existinterim state is necessary for an object to exist. . 5.5. We have to learn from the natureWe have to learn from the nature.. For billions of years For billions of years it has been solving serious combinatory problemsit has been solving serious combinatory problems,, so when so when creating agent-based systems it makes sense to consider creating agent-based systems it makes sense to consider parasitism, symbiosis, reproduction, genetics, mitosis, and parasitism, symbiosis, reproduction, genetics, mitosis, and natural selectionnatural selection. .

When creating agent-based systemsWhen creating agent-based systems, , one has to devote one has to devote special attention to the special attention to the APPEARANCE CONCEPTAPPEARANCE CONCEPT. . On the one handOn the one hand, , appearance may occur without our appearance may occur without our intention or consentintention or consent, , which may be good or badwhich may be good or bad. . Examples Examples includeinclude: : ant coloniesant colonies, , bee swarmsbee swarms, , bird flocksbird flocks,, traffic jams, traffic jams, etcetc. . Note: ants Note: ants ((or carsor cars) ) change, but the structures change, but the structures – – colonies colonies ((or traffic jamsor traffic jams) ) – remain– remain.. On the other handOn the other hand, , being developers of the system of being developers of the system of knowledge about the object of study, we can try to knowledge about the object of study, we can try to “project“ “project“ appearance of the structures of knowledge that we need. In appearance of the structures of knowledge that we need. In other words, we can try to project agents with the behavior other words, we can try to project agents with the behavior necessary for the required structures to appearnecessary for the required structures to appear. .

THESE ARE TWO SIDES OF THE SELF-THESE ARE TWO SIDES OF THE SELF-ORGANIZING STRUCTURESORGANIZING STRUCTURES..

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

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

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

5. 5. Definition of a <M,P>-space, its Definition of a <M,P>-space, its property and structureproperty and structure

DEFINITION DEFINITION 11.. Model-parametrical space Model-parametrical space ((<M,P>-<M,P>-space)space)we shall understand a set of all models, we shall understand a set of all models, parameters, relations between them, describing parameters, relations between them, describing property (designed and/or researched) product property (designed and/or researched) product (system).(system). From our point of view, the most suitable means for From our point of view, the most suitable means for description and research of structure of the description and research of structure of the <М,P> space<М,P> space is is the graph theory. Elements (nascent components) of <М,P> the graph theory. Elements (nascent components) of <М,P> are: models (set are: models (set ММ), parameters (set), parameters (set P P) and relations ) and relations

between them, i.e. between them, i.e. M = {MM = {Mjj}, j }, j €€ J, P = {PJ, P = {Pii}, i}, i € € I I,, where sets where sets of the of the II and and J J indexes are determined by objects, considered indexes are determined by objects, considered in each concrete case, determining them explicitly and in each concrete case, determining them explicitly and implicitly as integrated structures. implicitly as integrated structures.

We attribute to the objects various aggregates, nodes, We attribute to the objects various aggregates, nodes,

functional subsystems, components of a designed functional subsystems, components of a designed

complicated product, the product itself, and the system that complicated product, the product itself, and the system that

describes product’s behavior and functioning in the external describes product’s behavior and functioning in the external

environment. Hence, in the most general case, the <М,P> environment. Hence, in the most general case, the <М,P>

space is determined on the set of the direct Catresian space is determined on the set of the direct Catresian

product of M and P, i.e. product of M and P, i.e. <М,P><М,P> ММ x P.x P.

To objects we refer various, knots, functional subsystems, To objects we refer various, knots, functional subsystems,

item, it as a whole and system circumscribing it a behavior item, it as a whole and system circumscribing it a behavior

and operation in an external medium. Thus in the most and operation in an external medium. Thus in the most

common case direct decart a product of sets common case direct decart a product of sets М М and and PP, i.e. , i.e.

<М,P><М,P> ММ x Px P. .

We will prescribe a different meaning to arcs-relations We will prescribe a different meaning to arcs-relations between the models and parameters, depending on context between the models and parameters, depending on context of the consideration. But, by default it is supposed, that the of the consideration. But, by default it is supposed, that the arrow, directed from the parameter to the model, means that arrow, directed from the parameter to the model, means that this parameter in the given model is independent, and the this parameter in the given model is independent, and the arrow, directed from the model to the parameter, arrow, directed from the model to the parameter, corresponds to the case of dependence of the parameter on corresponds to the case of dependence of the parameter on the model.the model.

Note that the reasons adduced (especially, as to the Note that the reasons adduced (especially, as to the statement 1 interpretation) justify an expediency of exclusion statement 1 interpretation) justify an expediency of exclusion from the consideration of the "undirected models". In fact, from the consideration of the "undirected models". In fact, any model is intended for simulation of some properties any model is intended for simulation of some properties and/or characteristics of the product under projection, i.e. it and/or characteristics of the product under projection, i.e. it always possesses the input and output parameters. always possesses the input and output parameters.

Neighborhoods in <М,P>-spaceNeighborhoods in <М,P>-space

DEFINITION 2. As a neighborhood of a <М,P>-space of the 1-st order concerning a model Мj we shall name a set of parameters Pi, immediately connected with a model Мj.To designate this neighborhood we shall be [Mj]1.

DEFINITION 3. As the boundary of a neighborhood k-order of a model Мj we shall name a set of all elements of a <М,P>-space connected with Мj by a way, length equal "k".

To designate this neighborhood we shall be [Мj]1

To designate the boundary we shall be — [Mj]k

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.

Representation of model-parametric approach

An example <М,P>-

neighborhoods 1-st, 2-nd and 3-rd

order concerning a model

М3

M3

M6

P3

P6

P8

P9

M5

M2

M7

M8

M1

P12

P7

P13

2

P10

2

M4

P11

2

P4P1

P2

P5

Р1

<<М,РМ,Р>>-neighborhood of the Р1 parameter

M

P2P1

M1

P

M

MP

P

P

P

P

M

M

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

P 1

P 2

Intersection of neighborhoods of elements Р1 и Р2

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

A case history of outcomes of operation of intersection <М,P>-neighborhoods Р2 и Р9

6. 6. Architecture of model parametric Architecture of model parametric spacespace

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

In this case, models may be represented in In this case, models may be represented in different forms and formats: different forms and formats:

frames, frames, products, products, via semantic networks, via semantic networks, cognitive models, cognitive models, statistical polynomials, statistical polynomials, differential equations, tables, differential equations, tables, diagrams, diagrams, on verbal level, on verbal level, etc.etc.

Architecture of <М,Р> space ensures that Simon’s Architecture of <М,Р> space ensures that Simon’s structure of complexity is reflected in the computer structure of complexity is reflected in the computer environment. environment. The following levels of "knowledge" can be defined The following levels of "knowledge" can be defined in the hierarchical structure of this space: in the hierarchical structure of this space:

parameters, parameters, models, models, <М,Р> neighborhoods, <М,Р> neighborhoods, methods of calculating different integral and methods of calculating different integral and

aggregated parameters, aggregated parameters, characteristics of new complex hardware, characteristics of new complex hardware, appearances (projections) of complex appearances (projections) of complex systems.systems.

Diagram hierarchical structures of a <M,P>-space

ParametersParameters ModelsModels

<М,Р> <М,Р> --neighborhoneighborho

odsods

ААppearanppearances ces

(projectio(projections) of ns) of

complex complex systemssystems

At the same time, certain <М,Р> neighborhoods At the same time, certain <М,Р> neighborhoods

may become part of other <М,Р> neighborhoods may become part of other <М,Р> neighborhoods

thus ensuring thus ensuring RECURSIVENESSRECURSIVENESS. .

Levels of ND architecture of <М,Р> space are Levels of ND architecture of <М,Р> space are

limited limited ONLY BY THE COMPLEXITY OF THE ONLY BY THE COMPLEXITY OF THE

PROBLEM.PROBLEM.

Design and study of complex systems had always Design and study of complex systems had always

been the job of experts in been the job of experts in various problem and

application fields..

Their knowledge represented using different Their knowledge represented using different

models is integrated in <М,Р> space. models is integrated in <М,Р> space.

These models may come in form of These models may come in form of

generally accepted and tested lawsgenerally accepted and tested laws. But . But

<М,Р> space may also include the models that <М,Р> space may also include the models that

only undergo the testing. only undergo the testing.

Therefore, this space is Therefore, this space is

HETEROGENEOUSHETEROGENEOUS from this point of view as from this point of view as

well.well.

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 In [[Yu. R. Valkman. Model calculus in concurrent engineering of Yu. R. Valkman. Model calculus in concurrent engineering of complex products.// Proc. IMACS Multiconference "Computational complex products.// Proc. IMACS Multiconference "Computational Engineering in Systems Applications" (CESA'96), Lille-France, July 9-12, Engineering in Systems Applications" (CESA'96), Lille-France, July 9-12,

1996,- pp. 909-914.1996,- pp. 909-914.] describers ideology of building ] describers ideology of building hierarchical structures in <М,Р> space. hierarchical structures in <М,Р> space. The appropriate methods are based on the use of The appropriate methods are based on the use of a a SPECIAL MODEL CALCULATION APPARATUSSPECIAL MODEL CALCULATION APPARATUS ((algebra and logic of model texts and contextsalgebra and logic of model texts and contexts). ).

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

Hierarchical structure of contextsLet’s take a look at the structure Let’s take a look at the structure ((context inside contextcontext inside context) ) of of properties and characteristics ofproperties and characteristics of: : shipship ((РР1 1 = A, . . .= A, . . . ): ): • • navigational qualitiesnavigational qualities ( (РР22 = B, . . . = B, . . .): ): ►► propulsion, , ►► controllability, , ►► rocking ( (РР55 = E, . . .), = E, . . .), rollingrolling,, pitchingpitching,, . . . ;. . . ; • • hullhull ( (РР33 = C, . . . = C, . . .): ): ►► geometry, , ►► durability, . . ., . . . • • power unitpower unit ( (РР44 = D, . . . = D, . . .): ): ►► capacitycapacity, , ►► weight and dimensional characteristics, . . ., . . .

Reflection of IReflection of I--graph of the product’s structure in the graph of the product’s structure in the graphical image of graphical image of ""context inside contextcontext inside context""

Р1 = А…

Р4= D…

Р3 = С…Р2 = В…

Р5= Е…

- contexts- texts

Р1 = А …

Р2 = В …

Р5 =Е …

Р3 = С …

Р4 =D …

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), if P4 = (а1, а2, , а3, …) and P5 = const.Now, parameters P4 and P5 were “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), if P4 = (а1, а2, а3, …,) and

P5 = (в1, в2, , в3, …,). NoteNote, , that the employer decided not to go with further that the employer decided not to go with further “increase“ of “increase“ of ГОГО dimensions, such as, for example, image of dimensions, such as, for example, image of “matrix of matrixes"“matrix of matrixes"..

The purpose of The purpose of GRAMGRAM subsystem is synthesis of subsystem is synthesis of graphical presentation of graphical presentation of mathematical modelsmathematical models..

Using the Using the GRAM, GRAM, researcher and designer researcher and designer can analyze models in can analyze models in form of visual imagesform of visual images..

Подсистема Подсистема ГРАММ ГРАММ предназначена для предназначена для синтеза графического синтеза графического представления представления математических математических моделей.моделей.

С помощью С помощью ГРАММ ГРАММ исследователь и исследователь и проектировщик проектировщик анализируют модели в анализируют модели в форме визуальных форме визуальных образов.образов.

Two-dimensional, continuous graphical image



Three-dimensional, continuously-discrete graphical image



Four-dimensional, continuously-discrete graphical image



Five-dimensional, continuously-discrete graphical image



Three-dimensional, continuous graphical image



Four-dimensional, discretely-continuous graphical image

Five-dimensional, discretely-continuous graphical image

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 a context relevant 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.

Image texts and contextsImage texts and contexts

((ImIm)) image located in the center of attention image located in the center of attention, , with with ((""farfar"" and and ""nearnear"")) images forming its images forming its

contextcontext..

Im

Text of Text of ((ММtt)) model in a multilayer structure of model in a multilayer structure of CCtt11,,

CCtt22,, CCtt33 contexts contexts ((structural presentation of structural presentation of <М,Р<М,Р>>--

neighborhoodneighborhood))

Ct3

Ct2

Ct1

ММtt

Operation of overlapping Operation of overlapping <М,Р<М,Р>>--neighborhoods of neighborhoods of ММ11 М М22 images.images.

Note, that we can combine model texts only when Note, that we can combine model texts only when overlapping of their contexts overlapping of their contexts ((at any level) is not emptyat any level) is not empty..

М1

М2

Operation of combining texts of three images Operation of combining texts of three images ММ11ММ22ММ33.. NoteNote, , that in case of these combinations that in case of these combinations contextscontexts of relevant of relevant models must transform into the models must transform into the text of the generalized modeltext of the generalized model, ,

thus implementing explication of contextthus implementing explication of context..

М1

М2

М3

CONCLUSION 11.. Model context is a multilayer hierarchical structure Model context is a multilayer hierarchical structure ((a a ““context inside contextcontext inside context"). "). 2.2. This structure is built based on the hierarchy of This structure is built based on the hierarchy of parameters characterizing modeled knowledge about the parameters characterizing modeled knowledge about the object or processobject or process. . 3.3. Hierarchical structure serves as the basis forHierarchical structure serves as the basis for: :

integration of models into expedient multi-model integration of models into expedient multi-model systems;systems;

combined analysis of different models combined analysis of different models ((describing the same objectdescribing the same object));;

substantiation of possibility of generalizing and substantiation of possibility of generalizing and aggregating models;aggregating models;

analysis of consistency of analysis of consistency of <М,Р>-<М,Р>-space and space and study of integritystudy of integrity, , completeness, and balance of its completeness, and balance of its structure;structure;

substantiation of possibility of model calculation substantiation of possibility of model calculation operations operations ((algebra and logic of texts and contextsalgebra and logic of texts and contexts).).

ConclusionConclusion

Therefore, <М,Р> space Therefore, <М,Р> space represents a balanced, represents a balanced, interrelated, non-contradictory, interrelated, non-contradictory, integrated system of models of integrated system of models of the created and/or studied the created and/or studied complex object.complex object.

""Complexity obstructs Complexity obstructs

further developmentfurther development..

Overcoming complexity Overcoming complexity

barrier becomes one of barrier becomes one of

the IT industry’s most the IT industry’s most

important objectivesimportant objectives..

IT’S OUR NEW GREAT IT’S OUR NEW GREAT

GOAL"GOAL"

In the Autonomic Computing (www-1.ibm.com/industries/goverment/doc/ content/binauto.pdf) IBM’s Perspective on the State of Information Technology manifest published in 2001 Paul Horn declared:

THANK YOU THANK YOU FOR YOUR FOR YOUR

ATTENTIONATTENTION!!

THANK YOU THANK YOU FOR YOUR FOR YOUR

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

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