ARCHITECTURE OF MODEL PARAMETRIC SPACE: HIERARCHY IN SIMONS ARCHITECTURE OF COMPLEXITY

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ARCHITECTURE OF MODEL PARAMETRIC SPACE HIERARCHY
IN SIMONS ARCHITECTURE OF COMPLEXITY
Y.R. Valkman, A.Y. Rykhalsky  The International
Research and Training Center of Information
Technologies and Systems ?-mail
yur_at_valkman.kiev.ua
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Simplicity is a sign of
beauty. Simplicity is a sign
of truth. Simplicity is a
sign of genius. Simplicity
is a sign of efficiency. However, everything is
not as simple as it seems. Thats 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 (lt?,?gt) space. The
authors discuss the issue of reflecting the
structure of complexity of knowledge of designers
and researchers modeled in lt?,?gt space.
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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. Simons structure of
complexity 5. Definition of a ltM,Pgt-space, its
property and structure 6. Architecture of model
parametric space
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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.
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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.
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Its 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 systems
components. These model structures are supported
by means of model parametric space. They ensure
flexibility of relevant structures of knowledge.
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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.
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In the data transmission theory, complexity is
measured by the total number of properties
transmitted by object and received by observer.
In physics, complexity is determined by the
probability of system state vector. In
mathematics, they are talking about computing
complexity of algorithms. In cognitive
psychology, complexity of problem is evaluated
from the angle of possibility of solving this
problem by human. In addition to that, GENERAL
THEORY OF SYSTEMS also has its own perception of
complexity.
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• Lets 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

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• 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.

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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 lt?,?gt-space. There are the
following basic ideas of using a multi-agent
concept in the complex system modeling. 1. Complex
systems include autonomous objects which
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.
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3. Complex systems are also characterized by
their appearing structures. APPEARING STRUCTURE
is 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, its 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.
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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.
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4. Simons 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.
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Everybody knows about Simons "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.
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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
Darwins natural selection (or Adam Smiths
"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".
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5. Definition of a ltM,Pgt-space, its property and
structure
DEFINITION 1. Model-parametrical space
(ltM,Pgt-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 lt?,Pgt space is the
graph theory. Elements (nascent components) of
lt?,Pgt are models (set ?), 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.
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We attribute to the objects various aggregates,
nodes, functional subsystems, components of a
designed complicated product, the product itself,
and the system that describes products behavior
and functioning in the external environment.
Hence, in the most general case, the lt?,Pgt space
is determined on the set of the direct Catresian
product of M and P, i.e. lt?,Pgt?? ?? 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 ? and P, i.e.
lt?,Pgt?? ?? x P.
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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.
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Neighborhoods in lt?,Pgt-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
lt?,Pgt-space of the 1-st order concerning a model
?j we shall name a set of parameters Pi,
immediately connected with a model ?j. To
designate this neighborhood we shall be Mj1.
To designate this neighborhood we shall be ?j1
DEFINITION 3. As the boundary of a
neighborhood k-order of a model ?j we shall
name a set of all elements of a lt?,Pgt-space
connected with ?j by a way, length equal "k".
To designate the boundary we shall be Mjk
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Representation of model-parametric approach
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An example lt?,Pgt-neighborhoods 1-st, 2-nd and
3-rd order concerning a model ?3
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lt?,?gt-neighborhood of the ?1 parameter
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Union of lt?,Pgt-neighborhoods ?1, ?1 ? ?2
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Intersection of neighborhoods of elements ?1 ? ?2
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lt?,Pgt SPACE Examples of the graphic
interpretations of "intersection" and "union" of
knowledge
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A case history of outcomes of operation of
intersection lt?,Pgt-neighborhoods ?2 ? ?9
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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 lt?,?gt space was
initially built to support multi-model
structures.
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• 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.

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• Architecture of lt?,?gt space ensures that
  Simons 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,
• lt?,?gt neighborhoods,
• methods of calculating different integral and
  
• aggregated parameters,
• characteristics of new complex hardware,
• appearances (projections) of complex
• systems.

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Diagram hierarchical structures of a ltM,Pgt-space
?ppearances (projections) of complex systems
lt?,?gt -neighborhoods
Parameters
Models
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At the same time, certain lt?,?gt
neighborhoods may become part of other lt?,?gt
neighborhoods thus ensuring RECURSIVENESS.
Levels of ND architecture of lt?,?gt 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.
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Their knowledge represented using different
models is integrated in lt?,?gt space. These
models may come in form of generally accepted and
tested laws. But lt?,?gt space may also include the
models that only undergo the testing.
Therefore, this space is HETEROGENEOUS from this
point of view as well.
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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
lt?,?gt space. The appropriate methods are
based on the use of a SPECIAL MODEL CALCULATION
APPARATUS (algebra and logic of model texts and
contexts).
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We believe that the model located on the
top level of hierarchy plays the role of context
for the models lying below it. Thats 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.
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Hierarchical structure of contexts
Lets take a look at the structure (context
inside context) of properties and characteristics
of ship (?1 A, . . . )
navigational qualities (?2 B, . . .)
? propulsion, ?
controllability, ? rocking (?5
E, . . .),
rolling,
pitching, . . .
hull (?3 C, . . .)
? geometry, ?
durability, . . . power unit (?4 D, .
. .) ? capacity,
? weight and dimensional characteristics, .
. .
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Reflection of I-graph of the products structure
in the graphical image of "context inside
context"
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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). Lets assume
that we have a model with the following text ?1
F1 (P2, P3, P4, ?5). 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 ?? were reviewed. ?1
f2 (P2, P3), if P4 (?1, ?2, , ?3, ) and P5
const. Now, parameters P4 and P5 were
transferred to the context of ?1 F1 (P2, P3,
P4, ?5) model. We call this ?? four-dimensional
discretely-continuous". Fig. shows
five-dimensional discretely-continuous ??" ?1
f3 (P2, P3), if P4 (?1, ?2, ?3, ,) and
P5
(?1, ?2, , ?3, ,). Note, that the employer
decided not to go with further increase of ??
dimensions, such as, for example, image of
matrix of matrixes".
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The purpose of GRAM subsystem is synthesis
of graphical presentation of mathematical
models. Using the GRAM, researcher and
designer can analyze models in form of visual
images.
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???????????? ????????????? ??????????????
???????. ? ??????? ????? ????????????? ?
????????????? ??????????? ?????? ? ?????
?????????? ???????.
Two-dimensional, continuous graphical image
43
?????????? ????? ????????????? ??? ???????
???????????? ????????????? ??????????????
???????. ? ??????? ????? ????????????? ?
????????????? ??????????? ?????? ? ?????
?????????? ???????.
Three-dimensional, continuously-discrete
graphical image
44
?????????? ????? ????????????? ??? ???????
???????????? ????????????? ??????????????
???????. ? ??????? ????? ????????????? ?
????????????? ??????????? ?????? ? ?????
?????????? ???????.
Four-dimensional, continuously-discrete graphical
image
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?????????? ????? ????????????? ??? ???????
???????????? ????????????? ??????????????
???????. ? ??????? ????? ????????????? ?
????????????? ??????????? ?????? ? ?????
?????????? ???????.
Five-dimensional, continuously-discrete graphical
image
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?????????? ????? ????????????? ??? ???????
???????????? ????????????? ??????????????
???????. ? ??????? ????? ????????????? ?
????????????? ??????????? ?????? ? ?????
?????????? ???????.
Three-dimensional, continuous graphical image
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?????????? ????? ????????????? ??? ???????
???????????? ????????????? ??????????????
???????. ? ??????? ????? ????????????? ?
????????????? ??????????? ?????? ? ?????
?????????? ???????.
Four-dimensional, discretely-continuous graphical
image
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Five-dimensional, discretely-continuous graphical
image
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Image texts and contexts
Apparently, it is appropriate to compare
the center of lt?,?gt-neighborhood, or the entire
neighborhood (but in that case in the
lt?,?gt-space) with the center of attention, and
consider the other models and parameters a
context relevant image. The center of
lt?,?gt-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.
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(Im) image located in the center of attention,
with ("far" and "near") images forming its
context.
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Text of (?t) model in a multilayer structure of
Ct1, Ct2, Ct3 contexts (structural presentation
of lt?,?gt-neighborhood)
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Operation of overlapping lt?,?gt-neighborhoods of
?1 ? ?2 images. Note, that we can combine model
texts only when overlapping of their contexts (at
any level) is not empty.
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Operation of combining texts of three images
?1??2??3. Note, that in case of these
combinations contexts of relevant models must
transform into the text of the generalized model,
thus implementing explication of context.
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• CONCLUSION
• 1. Model context is a multilayer
  hierarchical structure (a context inside
  context").
• 2. This structure is built based on the
  hierarchy of parameters characterizing modeled
  knowledge about the object or process.
• 3. Hierarchical structure serves as the
  basis for
• integration of models into expedient
  multi-model systems
• combined analysis of different models
  (describing the same object)
• substantiation of possibility of generalizing
  and aggregating models
• analysis of consistency of lt?,?gt-space and
  study of integrity, completeness, and balance of
  its structure
• substantiation of possibility of model
  calculation operations (algebra and logic of
  texts and contexts).

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Conclusion Therefore, lt?,?gt space
represents a balanced, interrelated,
non-contradictory, integrated system of models of
the created and/or studied complex object.
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In the Autonomic Computing (www-1.ibm.com/industri
es/goverment/doc/ content/binauto.pdf) IBMs
Perspective on the State of Information
Technology manifest published in 2001 Paul Horn
declared
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