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The Combinatory System Theory
A New Approach for Understanding
and Controlling Collective Phenomena
University of Pavia
- Typology of Systems
Piero Mella, Chair of Business Economics
Faculty of Economics, University of Pavia, ITALY - Email: [email protected]
This course
My aim is twofold:
1. to present a particular class of Complex Systems which
I have defined as
Combinatory Systems.
Due to the simplicity of their structure and functioning
logic, I have also provocatively proposed to name these
systems as
Simplex Systems.
2. to illustrate, in particular – with the aid of simple
combinatory automata – phenomena as intriguing as
they are emblematic of the action of the synergetic
principles in social collectivities.
Piero Mella - 2005
1 – Typology of Systems
The programme
 Section 1 – Introduction: typology of systems
 Section 2 – Observing collectivities through the
Combinatory Systems view
 Section 3 – Models and typology of Combinatory
 Section 4 – Examples and Business applications
 Section 5 – The heuristic power of the
Combinatory System Theory
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Defining systems
von Bertalanffy, The general system theory, p.80
 I feel it necessary to present a general conceptual framework
of the different types of systems.
 I would first point out that Ludwig von Bertalanffy, the
“father” of the General System Theory, defined the system
as a group
 of interacting elements
 that we observe as a whole,
 that obeys the non-summation principle.
 That is, at a macro level, the macro process and the output of
the system, as a unit, are not equivalent to the sum of the
micro processes and outputs of its constituent elements.
 Ergo: the characteristics of the system, as a whole,
are emerging and new.
Piero Mella - 2005
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Models based on systems
 von Bertalanffy clearly stated that
… in one way or the other we are forced, in all areas of
knowledge, to deal with complexity, with “wholes”, with
“systems”. And this implies a basic reorientation of
scientific thinking. (GST).
 The message that von Bertalanffy wished to give us in his
General Systems Theory is clear and powerful:
we must develop a system thinking;
our ability to observe, to understand and to explain our
Universe will improve the more we are able to build
models of reality in system terms.
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Synthetical and analytical models of
 We can build synthetical and analytical models of a system.
From an exogenous point of view, a system can be
observed as a unity (macro) strongly characterized by its
own states, with an autonomous significance.
The synthetical models show the macro processes
of the system as a whole acting within its environment.
From an endogenous point of view, a system is
conceived of as a structure of interconnected elements
which develop a network of micro processes.
The analytical models show the structural map of
its elements and their micro processes.
Piero Mella - 2005
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Structural Map
and Macro Processes
Boundary of
the system
Synthetical model
Piero Mella - 2005
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Synthetical Models of Systems
 The simplest way to consider dynamic (dynamical) systems is
the synthetical or exogenous point of view, which leads to
models which are typically mathematical.
 These kind of models:
interpret these systems as black boxes,
and describe them through a system of difference or
differential equations,
which express the relations among the variations of
inputs and those of outputs,
considering the latter as the consequence of variations in
the internal state variables.
Piero Mella - 2005
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Systems with feedback loop
 A more usual block model introducing the feedback loop is
the following.
state transition
h(I, S)
output transition
g(I, S)
Melay Systems.
f(O, I)
For more general models you can see the evergreen book:
Sandquist (1985), Introduction to System science, Prentice-Hall
Piero Mella - 2005
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Analytical models
 The analytical approach is typically logical.
 It considers systems as white boxes and tries to understand:
the nature of the internal elements constituting the
the logical rules that specify how these elements
interact to produce both the micro behaviours of the
elements and the macro behaviour of the system as a
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Organized and
non-organized systems
 With regard to the nature of the internal elements
constituting the structure, we can distinguish between:
organized systems or structured systems: these are
characterized by distinct and specialized organs
connected by stable relations conceived of as the
organization of the system1;
non-organized systems or social collectivities: these
are composed of similar elements, or agents, without
being necessarily interconnected in a stable structure
(network, web or tree structure).
1 In the sense of Maturana & Varela (1980), Autopoiesis and
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The concept of organization
 “The relations that define a machine as a unit and determine
the dynamics of the interactions and transformations that it
can bear as a unit, represent the organization of the
 The effective relations that occur among the components that
integrate a specific machine in a given space represent its
 See: Maturana & Varela, Autopoiesis, p. 129.
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 An organ is a structural component characterized by the
following four features:
a precise spatial and temporal placement (topology),
a specialized function that both specifies the admissible
input and output that the component can show and
delimits the interactions with the other elements,
a specific functionality, regarding the contribution to the
entire structure,
a set of functioning standards that depend on the nature
of the system.
Piero Mella - 2005
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Organized system
Structure of organs
Structural elements or
The organs as
differentiated elements
which form a whole: the
The stable relations of
function, functionality
and topology that give
meaning to the elements
independently of their
System unit
Specific elements which,
through the
organizational relations,
form a lasting structure
(analytic vision).
Lasting structure of
organized elements that
are observed as a unit that
presents emerging
characteristics (synthetic
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Typology of organized systems
 Within organized systems, considering the nature of the
organs, we can further distinguish between:
Mechanisms, if the structure is composed of physical
elements and organs;
Organisms, if the structure is composed of biological
elements and organs;
Organizations, if the structure is composed of individual
autonomous agents forming organs.
Piero Mella - 2005
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Organizations as
multi-agent, multi-layer systems
Individual element
1st-level organ
2nd-level organ
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Holonic Systems
 Organized Systems may be composed of semi-autonomous
subsystems conceived of as holons.
 Holons are “Janus-faced entities which display both the
independent properties of wholes and the dependent properties
of parts”.
 Holons are subsystems that can be conceived of as:
autonomous systems, if observed in isolation,
wholes including their parts, if considered from a lower level,
parts composing a greater system, if considered from a higher
 A Holarchy is defined as a hierarchically organized structure of
holons obeying the
whole/part relationship.
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Holons: the original quotes
The term holon was coined by Arthur Koestler (1967) in his The Ghost in the Machine (Arkana,
London) and has been widely analysed by Ken Wilber (1995), Sex, Ecology, Spirituality: The Spirit
of Evolution, Shambhala Publications
 “Parts and wholes in an absolute sense do not exist in the
domain of life ... The organism is to be regarded as a multileveled hierarchy of semi-autonomous sub-wholes, branching
into sub-wholes of a lower order, and so on. Sub-wholes on
any level of the hierarchy are referred to as holons.
Biological holons are self-regulating open systems which
display both the autonomous properties of wholes and the
dependent properties of parts. This dichotomy is present on
every level of every type of hierarchic organization, and is
referred to as the Janus Effect ... The concept of holon is
intended to reconcile the atomistic and holistic approaches.”.
(Koestler, 1967, Appendix I.1).
“The world is not composed of atoms or symbols or cells or
concepts. It is composed of holons.”. (Wilber, 2001: 21).
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Model of Holarchy
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Collectivities as non-organized systems
A definition
 Non-organized systems are collectivities of individual agents
forming an autonomously observable whole.
 We can define a collectivity as
a plurality of similar elements or agents,
 which are unorganized: that is, not specialized according
to function, functionality, functioning and topology,
 and produce an analogous micro behaviour over time,
 or similar micro effects,
 but, considered together, are capable of developing a
macro behaviour– and/or macro effects – which can be
attributed to the collectivity as a whole.
 For this reason these systems can be succinctly denominated
Agent-Based Systems.
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Non-Organized systems:
collectivity of analogous elements
Herd of elephants
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A gaggle of greylag Geese
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Flock of Birds
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School of fish
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A cluster of bathers on a beach
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Can can dancers
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Typology of collectivities
 Collectivities can be
observable if the agents act simultaneously (for example,
swarms, flocks, crowds, spectators at a stadium, students
in a classroom, persons that are talking in a crowded
room, dancers doing the Can Can),
imaginable if the agents act at different times or in
different places (for example, trailer-trucks traveling a
stretch of highway in a month, the noble families of Pavia
who erected the 100 towers in the span of two centuries,
a group of scientists who dedicate themselves to a branch
of research, the consumers of a particular product during
its entire life-cycle, stockbrokers working on a certain day
in world or European stock markets).
Piero Mella - 2005
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Complex and Combinatory Systems
 We can further divide non-organized systems into:
complex (adaptive) systems: the agents normally and
prevalently interact according to local rules that establish
how the micro behaviour of an agent derives from, or
conditions, that of its neighbours;
combinatory systems: these represent a particular class
of complex systems whose macro behaviour derives from
the combination – appropriately specified – of the
analogous micro behaviours (or effects) of its similar
agents (hence the name Combinatory System),
following macro and micro rules.
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Some References
For Agent Based Systems and Complex Adaptive Systems see:
 Axelrod
(1997), The Complexity of Cooperation, Princeton
University Press
 Gell-Mann
(1995), Complexity, Vol. 1, no.5 ©, at:
 Goldspink
(2000), Modelling social systems as complex: Towards a
social simulation meta-model Journal of Artificial Societies and
Social Simulation vol. 3, no. 2, 31, at:
 Holland
(1995), Hidden Order: How Adaptation Builds Complexity,
Perseus Books, Cambridge, Massachusetts
Piero Mella - 2005
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Some References
I have conceptualized the Combinatory System Theory in:
Mella (1997), Dai sistemi al pensiero sistemico, Franco Angeli
Mella (1999), Razionalità e libertà nel comportamento collettivo,
Franco Angeli
Mella (2000), Combinatory System Theory:
Piero Mella - 2005
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System Theory
For more …
[Many simulation models are shown]
Remarks, suggestions and criticisms
are welcome!
My web page and e-mail:
[email protected]
Piero Mella - 2005
1 – Typology of Systems
Piero Mella - 2005
1 – Typology of Systems