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Concepts and Principles of General Systems Theory

Prof. Dr. rer. nat. habil. Reinhold SchönefeldSchulung & Beratung für Software, Ilmenau

October 2009

Copyright © 2009 for all slides by Reinhold Schönefeld

Table of Contents

1. Short Introduction into Systems Engineering

2. What Precisely is a System?

3. Principles of Abstraction

3.1 What is Classification Abstraction?

3.2 What is Generalization Abstraction?

3.3 What is Composition Abstraction?

4. Essential Concepts of a System

4.1 Some Essentials to the Boundary

4.2 Essentials to the Concept of Interaction

4.3 Essentials to the Structure of a System

4.4 Additional Essentials to Holism and Emergence

4.5 Essentials to the Concept of Self-Regulation

4.6 Remarks to Emergence, Complexity, and Self-organization

1. Short Introduction into Systems Engineering

• Systems Engineering (SE) is concerned with the analysis and development of large complex artificial systems.

• SE is a generalization of the existing approaches to analyze and develop such large systems independently of the domain (Civil Engineering, Mechanical Engineering, Electrical Engineering, Software Engineering, Chemical Engineering, Bio Engineering, etc.).

• Theoretical foundations of SE are given by the General Systems Theory (GST).

• SE follows for analysis and building of large technical systems a few general steps (workflow) (pict. 1.2).

• Without precise requirements it is impossible to design and build any technical system.

• Engineering is principally different from the sciences (physics, chemistry, biology). Nature has built its systems and scientists are taking research on ready existing living and nonliving systems.

• Engineering has at first to specify the goals for an artifact and has than to realize the specification.

Abb. 1.2

Grobe Anforderungen des

Problems zusammenstellen

S

H

Exaktes Anforderungsdokument

aufstellen

Iterativer Entwurf und Aufbau

des System

Anforderungen

erfüllt?

nein

ja

nein

jaEvolutionäre

Änderungen?

Überführung in betriebliche

Nutzung Arbeitsfluss bei Analyse und Entwicklung

technischer Systeme

2. What precisely is a System?

• Examples: A human being, an animal, a tree, an engine, a computer program, a solar system, etc.

• System: A unit or an object, restricted from its environment by a boundary (borderline). This unit appears as a wholeness and interacts with its environment (pict. 2.1).

• Note: How the boundary is drawn, is subjectively.• The kind of interaction determines the type of system

(pict. 2.2)a) Open systemsb) Closed systems

c) Isolated systems

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Copyright © 2007 by Schenski

Systemgrenze

System und seine Wechselwirkung

System als Ganzes

Abb. 2.1

Wechselwirkung mit Umgebung

Umgebung

Umgebung

Anstoß

Reaktion

Offene, abgeschlossene und isolierte Systeme

Menge aller Systeme

abgeschlosseneSysteme

offeneSysteme

isolierteSysteme

Abb. 2.2

• Interaction: The environment effects permanently an open system by – Energy

– Matter

– Information

(cause) and the system reacts back to the environment in the same way (effect).

• In case of a closed system the system is only effected once and reacts as long as the resources hold.

• An isolated system is of theoretical interest.

• Examples:– Open: living systems, any engines, etc.

– Closed: clock, solar system, etc.

• An open system is embedded in its environment, it is part of the environment (pict. 2.3)

Einbettung und Wechselwirkungen zwischen offenem System und seiner Umgebung

Abb. 2.3

offenes System

Menge aller Wechsel-wirkungen

offenes System

Umgebung

Umgebung

• The open system is a whole by it self, but it is also a part of the environment.

• Sovereignty of control belongs to the environment.

• That causes a hierarchical structure for the interaction,(pict. 2.3). We distinguish two layers.

• The idea of forming a hierarchy could be recursivelycontinued by further layers, if the system consists of further parts and these parts contain again other parts etc. (pict. 2.4).

• The lowest layer contains always the parts which are indivisible.

• We call them elementary systems.

• Note: In physics we speak about elementary particles.

Hierarchie von Systemen

System

Eingaben Ausgaben

elementare Systeme

Ko

mp

ositio

n

De

kom

po

sition

Abb. 2.4

Hierarchieebene

System

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• Note: We have to recognize objective and subjective indivisibility (quarks and electrons e. g. in physics are objectively indivisible, atoms in chemistry are subjectively chosen elementary systems) (pict. 2.5).

• Out of this there is a chose able granularity to describe the elementary parts or objects of a system.

• Note: There are two views on a system– An outside view, between environment and system.

– An inside view, between the parts and the system as a whole.

• Outside view: To describe the interaction between environment and system, you don‘t have to regard the parts of the system. The system appears as a black box.

• In this case we abstract from the details, namely the parts.

• This kind of abstraction is the so called composition abstraction.

Hierarchie im subatomaren Bereich

Abb. 2.5

• Examples: a) Body of an animal or a human being.

b) Data abstraction with classes and objects in object oriented programming in software systems.

• Inside view: From this point of view we can find an other definition of a system. „A system is a restricted set of subsystems (parts, objects, elements, components, etc.) which are connected to an integrated whole to achieve a certain goal.“

• The system as a whole and its parts interact than in the same way as environment and system. This interaction is called the inner interaction (pict. 2.18 and 2.18a).

• This inner interaction keeps the system and the parts together. This kind of interaction is the „glue“ of the system. Remoteness!

Systemhierarchie mit zentralisierter Steuerung

System

Stimuli Reaktionen

elementare Systeme

Abb. 2.18

Hierarchieebene

System

Umgebung

äußere Wechselwirkung

innere Wechselwirkungen

Innere und äußere Wechselwirkungen in Systemen mit möglicher Selbstorganisation

Abb. 2.18a

elem. System

Umgebung

äußere WW

elem. System

elem. System

äußere WW

System aus vielen elementaren

Systemen

• Examples: H-Atom (hydrogen)

– Whole: H-Atom

– Parts: one proton and one electron

– Interaction: photon (invariant mass = 0) is the glue inside the H-Atom. The photon is the carrier of the electromagnetic force which keeps the parts together. Photons are also called force particle or light quanta.

• The „glue“ in natural systems exist in the form of the four fundamental forces

– Gravitation force (weakest force, far reaching, particles of interaction unknown so far, may be gravitons).

– Electromagnetic force (second strongest force, far reaching, photonsas interaction particles).

– Strong force (strongest natural force inside the nucleons, extremely short reaching, gluons as interaction particles).

– Weak force ( belongs to nucleons with radioactive decay, weak gauge bosons as interaction particles).

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• The „glue“ in information systems exist in the form of messages.

• Other examples of systems:

– Technical systems (car, aircraft, nuclear power station, computer, web, software, etc.)

– Physical systems (atom, molecule, pendulum, laser, weather on earth, universe, etc.)

– Living systems (cell, organ, plant, animal, etc.)

– Sociological systems (family, team work, group of interest, etc.)

– etc. (pict. 2.10)

Abb. 2.10

Allgemeines System

natürliches System künstliches System

unbelebtes

System

lebendes

System

Atom Planeten-

system

Tier Pflanze Pilz

technisches

System

ökonomisches

System

soziologisches

System

Speicherchip Unternehmen Partei

AutofirmaVolkswagen

Boskoop-

apfelbaum

. . . . . .

. . .

Konzepthierarchie für unterschiedliche Systemkonzepte

3. Principles of Abstraction (short overview)

• We distinguish 3 kinds of abstraction in GST:– Classification Abstraction– Generalization Abstraction

– Composition Abstraction

• Goal: Reduction of diversity and complexity of systems.• With all of these abstraction principles we can reduce a

set of several (possibly infinite) elements to a few elements or only one single element.

• Advantage: Large systems are better to get under „intellectual control“.

• The principles are based on sound mathematical foundations, but here we give only an simple overview.

3.1 What is Classification Abstraction?

• Let‘s take the set of chairs in this room. Than each chair is concrete thing (inventory number), we speak about an instance of the chair.

• We can abstract from the set of concrete chairs and speak about the chair in general or the class chair as a single thing.

• The only one chair in general represents the several equivalent concrete instances of chairs and reduces so the diversity.

• This consideration is very oversimplified (pict. 2.7). The classification abstraction is mathematically based by equivalence relations and equivalence classes.

Klassifizierungsabstraktion und ihre Umkehrung

Menge realer Gegenstände mit den selben Eigenschaften

ein abstraktes Konzept

(Repräsentant)

Klassifizierungsabstraktion

Instanziierung

Abb. 2.7

3.2 What is Generalization Abstraction?

• The class chair und say a class armchair can now be abstracted by a more general class seating place.

• By this we abstract from two or more classification abstractions (shortly called classes) to one single super class – seating place (pict. 2.11, also see 2.10).

• Note: The opposite of generalization is specialization.

• The relation between the super classes, classes and subclasses is called an „is-a relation“ (pic.2.12).

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Zusammenhang von Klassifizierungs- und Generalisierungsabstraktion als Schema

Abb. 2.11

Generalisierung

Spezialisierung

Klassifizierung

Instanziierung

Allgemeineres Konzept

(Oberklasse)

Konzept(Klasse)

Spezielleres Konzept

(Unterklasse)

Menge der konkreten

Gegenstände(Instanzen)

Unterschiedliche Darstellungen einer Generalisierungshierarchie

Mann

Mensch

Säugetier

Tier

symbolisiert die ist ein-Relation

Tier

Säugetier

Mensch

Mann

Abb. 2.12

3.3 What is Composition Abstraction?

• Each composition, beginning with the elementary systems, hides the inner details (parts) of the considered system to the next higher layer (pict. 2.18a).

• That leads to an composition abstraction up to the upper system layer.

• The environment is unable to observe which parts belong to the system of the most upper layer.

• The system appears to the environment as an integrated whole, also well known as a black box.

Systemhierarchie mit zentralisierter Steuerung

System

Stimuli Reaktionen

elementare Systeme

Abb. 2.18 a

Hierarchieebene

System

Umgebung

äußere Wechselwirkung

innere Wechselwirkungen

System

• In GST we speak about holism or holistic approach by looking on a system as a whole without concern to the parts.

• Advantage: To specify the outside interaction for a system with the environment, we don‘t have to regard the parts, what reduces diversity and complexity.

• Note: A composition hierarchy is different to a generalization hierarchy (pict. 2.15 and 2.15a).

• The relations between the layers are „has a-relations“.

Kompositionshierarchie eines PKW als Baumstruktur

PKW

Motor Kraftübertragung Fahrwerk Karosse Fahrzeugelektrik

Bremsanlage Feder. u. Dämpf. Lenkung Radaufhängung Räder

Abb. 2.15

hat ein-Relation

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Composition Hierarchy of a Car as Tree Structure

car

enginepower

transmissionchassis car body electricity

braking system suspen. / damp. steeringwheel

suspensionwheels

Pict. 2.15 a

has a-relation

4. Essential Concepts of a System

4.1 Some Essentials to the Boundary

• The boundary can be:

a) A closed cover, where at each point exist an interaction with the environment.

b) A set of discrete regions in which the interaction takes place with the environment.

• Examples:

a) Buoyant force or buoyancy (Archimedes 287-212 BC) (pict. 2.16)

b) Thermal bridges in a building

p2 > p1

Oberfläche des Körpers als Systemgrenze

Körper mit Masse m als System

Schwere-druck p1

Schwere-druck p2

Flüssigkeit als Umgebung

Gewichtskraft FG

Auftriebskraft FA

Auftrieb in einer Flüssigkeit

Abb. 2.16

• Interaction between environment and system is only possible via the boundary.

• In case of only discrete ranges of the boundary we often call these ranges interfaces of the system.

• Each interface has

– a statically aspect, that is the structure of the interface or the suitability for interactions passing through the interface.

– a dynamical aspect, that is the interaction by itself at the interface.

• Examples:

– Electric motor (pict. 2.17)

– Key and keyhole

– Cell membrane

– Graphical user interface (GUI) to a software system

– etc.

Mechanische Schnittstelle eines E-Motors

Der statische Aspekt der mechanischen Schnittstelle ist durch die Geometrie von Welle, Keil und Flansch gegeben.

Abb. 2.17

4.2 Essentials to the Concept of Interaction

• Systemic interaction is a generalization of the interaction of bodies, fields, and/or radiations in physics.

• In a system we distinguished up to now the interaction between

– Outside environment and system

– System and inside parts or subsystems

• A closer look shows that there is no difference between both. The system interacts in the same way with the subsystems as the environment with the system as a whole.

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• The sources (causes) of an interaction are located in the environment and they are systems apart or a continuum with distributed parameters (fields, fluids, gases).

• The trigger or activator of an interaction is of different physical nature according to the problem domain:

– Mechanics: forces, momentums, both push bodies

– Electrical Eng.: voltages are switched to electrical networks

– Software Eng.: messages are sent to objects

– Business: purchase orders come to an enterprise

– etc.

• In GST we generalize all these different triggers and speak about a stimulus to a system.

• We can see from the examples that the stimuli can be continuous or discrete in its values and in time (pict. 3.3-3.6):

– Analog stimulus (value and time are real numbers)

– Sampled stimulus (value real and time discrete with equal intervals)

– Event discrete stimulus (value and event time points are integer numbers)

– Digital stimulus (value discrete, time discrete with equal intervals)

• Note: An event in the environment becomes a stimulus to the system if it can pass the interface. Only than the system observes the stimulus.

• In electrical systems the stimuli are often called signals.

• In Information systems especially a stimulus is a message to the system. The message is carried by a physical carrier and has a certain semantics to the receiver – the system.

y n(k)

k=0 1 105 k

tnt

kk=0 1 2 3 4 5 6 7 8

y(t)

y n (k)

Abb. 3.3 Abb. 3.4

Abb. 3.5 Abb. 3.6

Quantisierungsstufe

n=0

1

2

3

4

5

n=0

1

2

3

4

5

Quantisierungsstufe

t1 t2 t3 t4 t5 t6 t7 t8

y(tn)

t0

• Each stimulus activates generally spoken a „movement“ in the system, we speak about a process in the system (pict. 2.19).

• The process is reflected inside the system by the changing state of the system.

• Today is known that some characteristic stimuli generate the state variables of the system.

• All state variables together form the system state.

• The cause of the state of any system is the impact by each stimulus to the system. There is a certain accumulation and superposition of the different stimuli to the system state.

• The state and its changing over the time expresses the dynamic behavior of the considered system.

• With other words the state of a system is of fundamental importance to GST.

Wechselwirkung zwischen Systemen S i und S j auf verschiedenen Systemebenen

SystemS j

SystemS i

Stimulus

Reaktion

Menge aller innerenWechselwirkungen

Prozess in S j

Systemebene i

Systemebene j

Zeitachse

Prozessdauer

tvor tnachAbb. 2.19

übt die Steuerhoheit aus

• Note: All phenomena of self-organization of a system are reflected in the movement of the state of that system (pict. 2.18b).

• Each research on self-organization needs absolutely the knowledge of the behavior of the system state.

• The state is an inner variable and usually not to be observed outside the system.

• What could be observed outside is the response of the system on a stimulus

– Mechanics: Bodies are moving, change their position in space with a certain velocity

– Electr.Eng.: In a network a current flows, in coils a magnetic field exist, charges are accumulated in a capacitor

– Software Eng.: Objects send back a return value to the message sending object

– Business: Enterprises deliver products or services back to the customer

• In GST we speak generally about reactions of a system.

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System

Stimuli Reaktionen

elementare Systeme

Abb. 2.18 b

Hierarchieebene

System

Umgebung

äußere Wechselwirkung

innere Wechselwirkungen

Azyklischer Graf von Wechselwirkungen

System

4.3 Essentials to the structure of a system

• Structure of a system: Which parts belong to the system and which do not.

• Structure is the static aspect of the connection of the parts.

• The interaction is the dynamic aspect of the connection of the parts.

• The static connection of parts is mathematically expressed by a relation (pict. 2.20).

• The structure of a system is formed by the parts and the set of all existing relations between the parts.

• The structure expresses the spatial organization of a system.

• The interaction or the behavior of a system expresses its temporal organization, often called process organization.

Binäre Relation

Menge A Menge BRelation R

Abb. 2.20

• Structure and behavior are mutually dependent. Interaction takes only place if there is a relation between the parts.

• The structure and the behavior of a system can be best demonstrated by a nested presentation (pict. 2.23).

• The system hides all its parts towards other systems in the environment. All parts are encapsulated.

• The interaction can only take place via the interface.

• Out of all of this we can take advantage of three facts:– The already known composition abstraction with hiding and

encapsulation.

– The system could be subdivided in a different number of parts at the next lower level under the condition that the parts fulfill a certain transactional closure.

– Each part hides again its parts and has an own behavior at its interface. If the inner parts are changed without disturbing the interface, than this changing is not observable at higher layers (principle of locality).

Rekursive Komposition von Teilsystemen in verschachtelter Darstellung

el. System0221

el. System0222

element.

System

012

element.

System

011

element. System 021

System 022

System 02System 01

System 0

element.

System

013

symbolisiert Schnittstelle

unerlaubter Zugriff

Abb. 2. 23

Schnittstelle von System 0

UmgebungSystem 1

System 11 System 12

el. System 111

Rekursive Komposition von Teilsystemen als Baumstruktur

el. System 113 el. System 121 System 122

el. System 1221 el. System 1222

el. System 112

Abb. 2. 24

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Proton und Neutron (räumliche Organisation)

+ 2/3

+ 2/3- 1/3

- 1/3

+ 2/3- 1/3

2 up-Quarks

2 down-Quarks

starke Ladung +2/3 starke Ladung -1/3

Proton Neutron

Abb. 2. 21

4.4 Additional Essentials to Holism and Emergence

• The essence of holism is the already known composition abstraction. Holism is characterized by the holistic view at a system, without knowledge about the parts of that system.

• Because of this it is possible to describe the structure of the environment and the system by a binary relation.

• The holistic view is also the reason for an easy description of the interaction between environment and system as the interaction between only two systems.

• However, holism is more as only composition abstraction.

• If a system contains parts, so these parts are depending each other after its integration into the system. Without the integration these parts are independent.

• The parts gain new qualities which are to be generated only by the integration.

• The system (the new whole) gains also new qualities which are not to be found in the parts.

• This coming out of the integration or composition of the parts is the so called emergence.

• Emergence and holism are tightly coupled. Some compositions show the effect of emergence. Some do not.

• Aristotle (383-322 BC):

– „Das Ganze ist mehr als die Summe seiner Teile“.

– „Whole proves to be mightier than the parts“.

• In modern GST the notion „holism“ was firstly mentioned by J. Smuts in 1926.

• Examples:– Molecules of water with the parts H and 0. Both are bound in H2O. Water

as a fluid has new other qualities as the gases H and O.

– Temperature und pressure of a gas are new other qualities and can‘t be found within a single molecule.

– An oscillator as an electrical circuit has new other qualities (e.g. resonance) that can‘t be found in the coil and the capacitor as the parts of the circuit.

• Murray Gell-Man: „You don‘t need something more to get more –that is what emergence means. Life can emerge from physics und chemistry, plus a lot of accidents. The human mind can arise from neurobiology, and a lot of accidents etc. It does not diminish the importance of these subjects, to know that they follow from more fundamental things plus accidents. It‘s a general rule!“

• P. W. Anderson: „This principle of emergence is as pervasive a philosophical foundation of the viewpoint of modern science as is reductionism.“

• How we could explain such emergent properties?

• I will give you a very simple sketch of an explanation.

• Emergence is most likely the result of nonlinear superposition during the interaction of the parts with the system as a whole.

• A linear superposition of the behavior of the system resulting from the behavior of the parts presupposes the additivity for the behavior of the parts (pict. 2.22 pp.).

• Additivity is only warranted in homogenous systems consisting of parts with similar behavior and very weakinteraction between the parts.

• Example: – Ideal gas in thermodynamics, e.g. Helium

– Composition of forces and velocities in mechanics

• Systems with emergent properties consist of a set of variousparts with strong interactions to each other.

u1 (t)

u2 (t)

.

.

.

un (t)

u(t) =

System

Ausgangs-größen

Eingangs-größen

y1 (t)

y2 (t)

.

.

.

ym (t)

y(t) =

Übertragungsfunktion T:

y(t) = T[u(t)]

u1(t) y1(t)

u2(t) y2(t)

un(t) ym(t)

.

.

.

.

.

.

Allgemeines System

Abb. 2. 22

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Systemeigenschaften

Übertragungsfunktion: y(t) = T [u(t)]

a) Additivität:

T [u1(t)+u2(t)] = T [u1(t)] + T [u2(t)]

b) Homogenität:

T [λ . u(t)] = λ . T [u(t)]

c) Superposition:

T [λ1 u1(t)+ λ2u2(t)] = λ1T [u1(t)] + λ2 T [u2(t)]

Systeme mit den Eigenschaften a) und b) sind lineare Systeme

Systemeigenschaften

Beispiel 1: (linear)

( )∫ ∫ ∫

∫

∞− ∞− ∞−

∞−

⋅+⋅=⋅+⋅=+⋅=⇒

+=

⋅=

t t t

t

yyduKduKduuKty

uutu

duKty

221122112211

2211

)(

)(:Ansatz

)()(

λλτλτλτλλ

λλ

ττ

Beispiel 2: (nichtlinear)

[ ][ ] [ ] [ ]

2211

22112221

212211

222

211

22211

2

22)(

)()(

yy

uuyyuuuuuuty

tuty

⋅+⋅≠⋅⋅+⋅+⋅=⋅⋅++=+=⇒

=

λλλλλλλλλλλλ

4.5 Essentials to the Concept of Self-Regulation

• If the output of a system is fed back to the input we have created a closed loop or a feedback system.

• Feedback is a fundamental strategy that allows the system to compensate unexpected disturbances (pict. 2.27).

• Automatic Control is an essential discipline of a GST.

• Positive feedback means the transition in the feedback loop is positive and greater than 1.

• Examples of positive feedback:– Acoustic feedback with a microphone and a speaker

– Chain reaction in an atomic reactor

– Inflation of the value of money

– etc.

• Negative feedback means the transition in the feedback loop is negative and less than 1.

• Examples of negative feedback:

– Regulation of the temperature at the heating

– Regulation of the route of an airplane by an auto pilot

– Regulation of the blood sugar level in the human body

– etc.

• A negative feedback stabilizes the output of the system. A positive deviation of the output leads to a negative deviation of the input and so to a correction of the output level.

Rückkopplung eines Systems

Eingaben Ausgaben

System

Zeitachse

Prozess

tvor tnach

System

Ü-Faktor

Rückkopplung

Eingaben Ausgaben

Abb. 2.28

Positive und negative Rückkopplung

Positive Rückkopplung zeigt divergentes Verhalten der

Ausgangsgröße

Negative Rückkopplung zeigt konvergentes Verhalten der

Ausgangsgröße

Abb. 2.29

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4.6 Remarks to Emergence, Complexity, and Self-organization

Emergence – some citations from Wikipedia

• “In philosophy, systems theory, and science, emergence is the way complex systems and patterns arise out of a multiplicity of relatively simple interactions. Emergence is central to the theories of integrative levels and of complex systems”.

• G. H. Lewes (1875): “Every resultant is either a sum or a difference of the co-operant forces; their sum, when their directions are the same -- their difference, when their directions are contrary. Further, every resultant is clearly traceable in its components, because these are homogeneous and commensurable.

It is otherwise with emergents, when, instead of adding measurable motion to measurable motion, or things of one kind to other individuals of their kind, there is a co-operation of things of unlike kinds. The emergent is unlike its components insofar as these are incommensurable, and it cannot be reduced to their sum or their difference.”

• Jeffrey Goldstein (1999): “Emergence is the arising of novel and coherent structures, patterns and properties during the process of self-organization in complex systems."

• Emergence is the formation of new internal structures and new external behaviors of a system because of the internal interaction of its parts. Thereby this new properties are not reducible to the properties of the single parts of the system.

Complexity – some citations from Wikipedia

• “Warren Weaver (1948) has posited that the complexity of a particular system is the degree of difficulty in predicting the properties of the system if the properties of the system’s parts are given. In Weaver's view, complexity comes in two forms: disorganized complexity, and organized complexity. Weaver’s paper has influenced contemporary thinking about complexity.”

• “A complex system is any system featuring a large number of interacting components, whose aggregate activity is non-linear and typically exhibits self-organization under selective pressures.”

• “Many definitions tend to postulate or assume that complexity expresses a condition of numerous elements in a system and numerous forms of relationships among the elements. At the same time, what is complex and what is simple is relative and changes with time.”

Self-organization – some citations from Wikipedia

• “Self-organization is a process of attraction and repulsion in which the internal organization of a system, normally an open system, increases in complexity without being guided or managed by an outside source. Self-organizing systemstypically (but not always) display emergent properties.”

• “Self-organization usually relies on four basic ingredients:– Positive feedback

– Negative feedback

– Balance of exploitation and exploration

– Multiple interactions”

• “Sometimes the notion of self-organization is conflated with that of the related concept of emergence. Properly defined, however, there may be instances of self-organization without emergence and emergence without self-organization, and it is clear from the literature that the phenomena are not the same.”

• “Self-organization is a process in which pattern at the global level of a system emerges solely from numerous interactions among the lower-level components of the system. More over, the rules specifying interactions among the system’s components are executed using only local information, without reference to the global pattern. The pattern is the emergent property of the system, rather than a property imposed on the system by an external ordering influence.

• Conclusion!

Map of Complexity Science (Wikipedia: Complexity)

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4.6.1 Software Object and Emergence

• Any software object (SO) is a composition of some program functions (PF) and some data objects (DO). Thereby the DO and the bodies of the PF are hidden to the environment of the SO (pict. 2.30).

• The SO is a composition abstraction or in Informatics known as data abstraction.

• Because this is a composition of things with different properties we should expect the emergence of new properties by the SO after composition.

• We’ll try to find the reason for the expected emergence by a deeply analysis of the ongoing composition.

• Let’s first look at the parts without the composition to the SO.

An Object composed by Functions and Data Objects

pict. 2.30

. . .

. . .

return object (RO)

to environment

argument object

(AO)

interface of the object

data objects

(DO)functions

(PF)

function head

function body

system

boundary or

capsule

message to an

object

Object (SO) as new integrated software unit

interaction with

environment

inner interaction

• A DO could be a simple type (integer, real, character, string, …) ore a composite type (array, unit, structure, …).

• A DO has a name and can have several values, which can only be effected by a PF or operation.

• A PF as we know, has a head and a body. After declaration we can use the PF by calling it, thereby passing some DO as parameters via the head to the PF.

• A PF has for the same input always the same output - the function value.

• A second way to pass a DO into a PF as a global variable (GV).

• Example: Side effect by a GV “y”f (x){ return x + y;}g (z){a = f (1); y = y + z; return a + f (1);}

• Both functions are using “y”. The PF g (z) has the side effect, that the value of y is changed and because f also depends on y, the calls of f(1) return different values, despite the parameter is equal.

• In structured programming the side effect as it is well known is an unwanted effect.

• The side effect disappears if we use y as a parameter:f (x, y) {return x + y;}g (z, y) {a = f (1,y); y’ = y + z; return a + f (1, y’);}

• Graphical metaphor of PF and DO

input

output

head body (implemented function)

program function (PF) data object (DO)

name and value

• Graphical display of the side effect

• PF f and g are interacting with y. f and g are coupled by y, causing the side effect.

• When forming an SO we are composing several PF and several DO to one single SO as a new whole (pict. 2.30). That means only the chosen PF can interact directly with the chosen DO.

• Other PF outside the SO don’t have a direct access to the inner DO. The SO is a new whole hiding its inner parts to the outside.

PF f

PF g

DO y as GV

• A DO in the domain of object oriented programming is an entity object with the most simple PF:

– create the object

– delete the object

– set the object to a certain value

– get the actual value of the object

• Each DO has an interface with these functions. The interaction between the PF of the SO and the DO is the inner interaction (pict. 2.30).

• Note: Any entity object or DO can only change its value by the set-function.

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• What are typical signs of emergence in an object (SO)?

1) Functions (PF) an data (DO) depend each other. Without composing, a function is independent of the data and vice versa.

2) PF and DO gain new properties by the integration to an SO. Only these PF can directly interact with the DO (inner interaction). Functions not integrated must not access directly the hidden DO.

3) All PF form with its heads (signatures) the interface of the SO to the environment. Other PF not integrated, can’t form such an interface.

4) The integrated DO carry the state of the SO. They span a state space of the SO. Not integrated DO can’t span a multidimensional state space.

5) The SO, the new whole, gains also new properties, which are not to be found in the PF and the DO. The SO has an identity, that means it is a real unity or instance.

6) The SO has an interaction with the environment at its interface, also called an input-output behavior. This behavior depends not only from the chosen PF and its arguments but also from the actual state of the SO in its state space.

7) The SO is a finite state machine.

8) The emergence of the properties of the SO could be explained by the nonlinear superposition of the DO inside the PF.

9) For this we should remember that we use three control structures to express the program flow (the algorithm) in a PF.

10) These control structures for sequential programs are:

− Sequence (also math. Composition)

− Alternation

− Iteration

11) Each DO and each argument object (AO) in a PF are operated in so called state functions (SF) (also called function nodes in the control structure):

− x := 3 (simplest SF as an assignment statement)

− z := x + 5

− Other expressions, statements, or functionsoperating upon the DO and AO

• Any SF on its part is included in the three control structures.

• Sequence of two SF g and h

Instead of the graphical rule write the mapping between I and O

whereg ; h = { (x, z) | (∃ u) (u ∈ g(x) ∧ z ∈ h(u))}

The PF of the sequence of two (or more) SF is the mathematical function composition of these SF:

[PF] = [g ; h] = [h] o [g] = [h (g (x))]

[ ] denotes the program function PF. “o” is the composition operator, also useable as “;” (fat semicolon).

g hx u z

g ; hI O

Note: With “;” the operands are exchanged. It is important to see that the superposition of the DO and AO takes place in the SF and than after this in the function composition h (g (x)).

• Alternation of two (or more) SF g and h

(graphical rule) (mapping between I and O)

where

if p then g else h fi = { (x, y) | (p(x) ∧ y ∈ g(x)) ∨ (¬ p(x) ∧ y ∈ h(x))}

The PF of the alternation is defined as follows:

[PF] = [if p then g else h fi] = ([p] = true =def [g] ∨ [p] = false =def [h]

gx

h

py

if p then g else h fix y

I O

T

F

The [PF] of the alternation of two SF g and h is in case p is true the PF of

[g] and in case p is false the PF of [h].Note: The PF of the alternation is a mixture of the two functions g and h, depending on p.

• Semi-Alternation

if p then g fi = { (x, y) | (p(x) ∧ y ∈ g(x)) ∨ (¬ p(x) ∧ y = x)}

[PF] = [if p then g fi] = ([p] = true =def [g] ∨ [p] = false =def [I])

where [I] is the PF of the Identity function.

gx

py if p then g fi

x y

I OT

F

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• Iteration of the SF g

The PF of an Iteration can only be calculated if the loop is terminating.The graphical rule for the loop

could be substituted by an other rule with the same mapping

[PF] = [while p do g od] = [if p then g ; while p do g od fi ]

g

xp

y

g

p

T

F

F

T

g

p

T

F

x yx y

I Owhile p do g od

The PF of the iteration is only defined recursively. As long as pis true each run through the loop is a function composition

(note the fat semicolon!) of g with itself.

• Conclusions!

• Importance of the functions!