Information and Entropy in Cybernetic Systems

Heikki Hyotyniemi
Cybernetics Group
Helsinki University of Technology, Control Engineering Laboratory
P.O. Box 5500, FIN-02015 HUT, Finland
heikki.hyotyniemi@hut.fi
http://www.control.hut.fi/hyotyniemi

Abstract. It has been shown that the cybernetic approaches can eciently be used for analysis and design of complex networked systems.
Still, the earlier discussions were bound to the actual application domain
at hand. This paper gives more intuition in what truly takes place in a
cybernetic system from another point of view. Information theory, and
specially the concept of entropy, oer a yet more general perspective to
such analyses.

Introduction

There are many approaches to address the problems of real-life complex systems,
each of them concentrating on some specic issues and more or less ignoring the
others. The approach that is introduced in [19], or neocybernetics, is a new
theoretical framework that diers from the existing ones, compensating for the
shortcomings:
In complexity theory one studies structurally complex nonlinear functions as
independent entities; now the emphasis is on structurally simple large-scale
systems where complexity is caused by high-dimensionality and system-wide
interactions.
In system theory (or general system theory) discussions also embrace the
whole system, but they are limited to abstract, holistic studies; now, on the
other hand, concrete down-to-earth analyses give substance and semantics
to the discussions.
In control theory, also being a branch of system theory, studies are similarly
oriented on concrete rather than abstract systems; however, as compared to
the current approach, there the emphasis is solely on centralized rather than
distributed control structures.
In traditional cybernetics (being rediscovered various times under dierent
names like autopoiesis and synergetics) the decentralized structures also communicate with each other; however, the studies are stuck with the mechanisms, and the issues of emergence are not addressed. The assumption now is
that without explicit emphasis on emergence it is impossible to understand
the overall behaviors the actual essence of cybernetic systems.

In neocybernetics, the emphasis is on the emergent pattern rather than on the

individual interaction processes that nally lead to that emergent pattern (see
[19]). This emergent pattern is the (hypothetical) dynamic equilibrium after all
transients in the system have decayed. Stabilization of the system, or nding
the balance among tensions, is the ultimate goal of the self-regulatory structures that are based on interactions and (negative) feedbacks within the system.
Indeed, these balances (or higher-order balances) are the most characteristic feature that is common to very dierent cybernetic systems. The balances can be
characterized as minima of some cost criteria; imbalance means tensions or free
energy still remaining in the system.
The discussions in [19] where this kind of concepts were introduced were
rather heuristic and high-spirited. To keep things in control, a strong conceptual
structure is needed; such framework will be studied in this paper.
Wittgenstein observed that a language is needed for expressing ideas: Without strong concepts and grammar structures not everything can be said, and
discussions remain vague ands descriptive. There are dierences between languages; whereas human languages are practical in everyday life, it is mathematics that is the natural language of nature. Mathematics is pure syntax, but when
the syntactic structures and manipulations are applied to semantic entities, new
interpretations can be made ...
What kind of conceptual structures can be constructed when the cybernetic concepts are thought of applying the language of mathematics?
When deriving the cybernetic models, it turned out that down-to-earth analyses brought necessary substance and semantics to discussions. And, again, it
turns out that when the nonidealities of natural domains are actively taken into
account, new intuitions can be gained. Counterintuitively, applying more and
more specic toolboxes, more and more general conclusions can be drawn. Earlier, the power of system theory as a source of conceptual tools was praised
now the role of control engineering is emphasized as a source of intuition. It turns
out that without understanding the ideas of model-based control and ltering of
signals the behaviors in cybernetic systems cannot truly be mastered.
This paper discusses the relations between the information theoretic concepts
and cybernetic models; and, truly, it seems that some deep paradoxes can be
seen from fresh points of view. For example, why does it seem that the complex
systems seem to become more and more complex? Why do there exist systems
that seem to defy the arrow of entropy? What are the mechanisms for complexity
cumulation?
In the Middle Ages, it was thought that there needs not exist common principles in nature: The sublunar and translunar phenomena, for example, followed
their own laws. The breakthrough in physics came when it was observed that
the planetary motions were also governed by earthly, not by divine principles.
Similarly, today it seems that the systems are divided into two classes: Normally,
the systems follow the thermodynamic principles; but then there exist systems
where these principles seem not to hold. It seems that in the framework of cybernetic systems, these dilemmas can be studied in a coherent way. To put it

Fig. 1. Cybernetic systems:

Framework for systems
ghting against the ow of
entropy

informally: The stronger the ow of entropy is, the more probably there also
exist countercurrents, or whirls, in that ow (see Fig. 1). It turns out that
cybernetic systems are systems where the arrow of entropy is inverted.

From physical constraints ...

Before concentrating on the highly abstract analyses concerning information

and entropy, it needs to be noted that there is always a physical nonideal system
beyond the abstractions. Here, such nonidealities are rst briey studied and,
counterintuitively, it turns out that these concrete analyses give the concrete
means to carry out the highly abstract and general discussions.
2.1

Modeling of complex systems

The engineering-like approach to understanding any real-life problems is modeling, constructing simplied, abstracted representations capturing the relevant
system properties in a compressed form. The key question here is whether a
complex system can be simplied without missing its essence: If the whole is
more than parts, reductionistic methodologies collapse.
This intuition seems to be the mainstream attitude among complex systems
theorists meaning that there is scepticism against the potential of traditional
modeling and mathematics in general. For example, in [13] the starting point
is that whereas simple systems are modellable, complex systems by denition
defy modeling attempts: The possibility of complexity, in an intuitive sense,
only arises when a system acts in unexpected ways; that is, in ways that do not
match the predictions of models. Another formulation for this thought is given
in [18], where it is claimed that all descriptions of complex behavior necessarily
are still more complex than the original behavior: The simplest representation of
a system is the system itself, and no models can be constructed. Such pessimism
has resulted in predictions concerning End of Science [5].
However, the end of science has been prophesized many times in the past:
Always when old approaches have been exhausted, there has been scepticism
before new paradigms are found. The huge successes of the scientic paradigm,
and the ever regenerative power of mathematical tools, motivates the Pallas
Athene Hypothesis, in the spirit of the Gaia Hypothesis: Whereas Gaia supports
the life processes on the Earth, Pallas Athene supports scientic work. Indeed,

more than being a matter of fact, this is a confession of faith. But this optimistic
attitude pays back: There is still plenty to study, and it is the Old Science that
suces when studying complex systems.
To start with, Wolframs arguments [18] are indeed undeniable: If the modeling framework is too powerful, the mathematical analysis framework collapses.
Even if studying more and more complex systems, the model structures also
must not become more complex beyond a certain limit. How can this incompatibility be resolved? When the belief of not-yet-exhausted power of mathematical
modeling is adopted as a starting point, it is easier to redirect analysis eorts.
The only way to avoid the deadlock is to take another route: When the approach
starting from chaos theoretical problem setting turns out to lead nowhere, the
opposite way to proceed is to study emergent behaviors. Rather than looking
at the process, consisting of nonlinear iterations, one can look at the pattern,
the nal outcome of the iterations when the steady state has been reached. One
does not need to follow the individual behaviors if one knows the goal where
the system is going towards; in mathematical terms, this nal goal can be characterized in terms of cost criteria that are being minimized (maximized) in an
optimization process. The dynamic processes are caused by tensions in the underlying pattern that has not yet converged. Whereas the formulations based
on cost criteria are often mathematical, with no physical relevance, in the case
of cybernetic models the cost criteria have direct interpretations, implementing
a connection between abstract and concrete, between local agents and global
systems.
Perhaps because of the marvellous successes of computer technology, the
process view of thinking about systems dominates today. For example, in the
contemporary agent frameworks, the agents are software constructs. The problem here is that the topological structure beyond the algorithms is too vague
to oer any added value beyond the explicitly implemented functionalities. It is
important to observe that the algorithmic approach is not the only alternative,
at least not in all cases.
These discussions can be extended also to more abstract cases. There are
many domains where the shift from immediately-observable proceses to fundamentally-existent patterns has not yet taken place, and where it is dicult to
see what this transition would mean in the rst place. It is typical that modeling
is rst implemented by mimicking the observed surface-level behaviors or intuitions: For example, the self-organizing Kohonen SOM algorithm was rst dened
as a procedural algorithm; only later, it was observed that this algorithm is just
a way (gradient descent algtorithm) for minimizing a special energy function,
and new possibilities for further analysis and algorithm development opened up
[6]. It is easy to see developments in retrospect but, for example, there may
be possibilities of seeing also the processes of developmental biology, or in biological evolution, in the cybernetic pattern perspective. Such issues will be later
elaborated on.

2.2

Role of individuals

When the cybernetic models were derived (see [19]), the discussions were carried
out in a purely information theoretic setting. It was assumed that information
can be transferred in a causally unidirectional manner, without output aecting the input coming from outside the cybernetic system. However, when the
information carriers are populations, for example, and information transfer takes
place in the form of lower level prey being eaten by higher-level predators, this
idealized view necessarily collapses. There is no such thing as information without some physical carrier, and in some environments this fact becomes specially
acute.
The cybernetic models were derived by closing all loops inside the system;
now the nal loops are closed, coupling the system output with the system input.
First, study an ecosystem where the species on a specic trophic level compete for resources on the previous level. As presented in [19], the cybernetic
model becomes
dx
= A x + B u.
dt

(1)

Here, x is the vector of relative population activities (not simply the number
of individuals, or biomass; how the actual biomass is distributed, and what are
the losses, is not concentrated on here), and u is the vector of resources coming
from outside. The matrix A contains the interaction factors among the samelevel actors, revealing the patterns of competition, and B contains the forage
proles, revealing how the predators exploit the resources. The model can be
interpreted so that some kind of life force, or elan vital, emanating from the
environment and cumulating in the populations. Whereas this dissipative ow
never ceases, a cybernetic equilibrium among populations is found when the
opposing forces are in balance.
The model (1) is generic, and not very much can be said about it in general
terms. Assuming that the system is truly cybernetic (see [7]), there holds
xuT },
A = E{
xx
T }, and B = E{

(2)

where and are adaptation parameters, and x denotes the steady-state values
of x corresponding to the almost constant u:
x
= T u = E{
xx
T }1 E{
xuT } u.

(3)

Assuming that the dynamics of x is truly much faster than that of u, and if the
behavior of u is smooth, one can substitute x
with x in the formulas.
Whereas x = T u reveals the population balance, that is, the distribution
among population activities in steady state as a whole, in reality there is each
day a new competition among the individual actors for the available resources.
To make the rather abstract model (1) more concrete, this everyday struggle
will now be modeled. It is these everyday actions that also aect the available
resources, thus constituting the feedback structure between the system and its
environment.

Whereas there is the elan vital coming from the environment, the inverse
eect in the feedback loop caused by the individuals exhausting the resources
could be called elan letal. An individual tries to take the resources it needs,
pushing competitors away. As it is the individuals that are the agents for the
inverse action, rst study a single individual and its eect in the environment.
Dene the vector v that reveals the eective activity of a population; the
contribution of a single individual i to the whole grid of population activities
can be expressed as


dv
= A v + 1i .
(4)
d i
The vector of elan letal being caused by the single individual is here dened as
T
1i = ( 0 0 1 0 0 ) .
     
i1

(5)

ni

It needs to be recognized here that there can be numerical parameters adjusting

the overall behavior; it can still be assumed that qualitatively the eects remain
essentially intact. The individuals belonging to a certain population aect other
populations as revealed by the competition matrix A. The time axes dier in
dierent submodels the dynamics of a single individual is fast (rate of change
in variables being typically of order from hours to days), the dynamics of a
population is slower (from days to weeks), and the dynamics of the environment
is still slower (from weeks to months). The speeds of individual dynamics can
be emulated in dierent ways; here it is assumed that all integrators operate on
dierent time scales.
Again, as seen in the wider perspective, when elan letal cumulates, a cybernetic equilibrium is eventually found, and the eective activities reach a balance:
(
v )i = E{xxT }1 1i ,

(6)

so that the eect of an individual on the resources can be expressed as

(U )i = B T v = E{xuT }T E{xxT }1 1i ,

(7)

where the forage proles in B are utilized. Here the sign is explicitly included
to emphasize that the resources are exhausted (however, the consumption of
some resource type can also be negative, meaning that this resource is saved
this is caused by the screening eects among individuals).
Because of system linearity, the eects of one individual can be freely scaled;
the overall eect of all individuals in the ecosystem can be calculated as a
weighted sum:
u =

n

i=1

xi (U )i =

n


i=1
T 1

= E{xu } E{xx }
T T

E{xuT }T E{xxT }1 xi 1i

x.

(8)

Population
level

Food Forward

Adaptation

Resource

u^

duin

A
Level of
individuals

Feed Back

Fig. 2. Action (upper part) and reaction (lower part) automatically implemented by a cybernetic system. The structure looks strangely symmetric
remember Heraclitus: The way up and the way down are the same

This is how much of the resources are being used by the whole system, revealing
the rate of exhaustion. Assuming that the pool of resources is large, it will not
be exhausted immediately; instead, a dynamic structure is instantiated:
du
= u + u.
d

(9)

where u(t) is the natural rate of growth in the resources; the time scale, or
how is related to real time t, is dependent of the system. The overall system
structure can be expressed as shown in Fig. 2. Assuming that the inner dynamics
are much faster, the outermost loop is always stable (phase shift being only 90
degrees).
The mathematical formulation of the feedback structure (8) looks strangely
familiar. Indeed, it equals the multilinear least-squares regression model between
x and u, that is, whatever x is, the best (linear) mapping from it to output (in
terms of minimized error variance) is obtained by the MLR formula
u
= E{xuT }T E{xxT }1 x.

(10)

If spans the principal subspace, as has been shown in [19], mapping from
the corresponding x to output applying MLR implements principal component
regression (see [8]). It has been observed that PCR is an ecient way to reach
robust regression structures, ltering out noise and exploiting the correlations
in data optimally.
Finally, note that if the natural reproduction uin in (9) is constant, and
if the resource variations have been appropriately captured by the system, a
balance is found where u
compensates the growth in u; the system remains in

balance where u
equals u. If there is variation in the growth rates, the principal
component regression from input back to input still does a good job trying to
eliminate the variation; and it can be assumed that this principle applies
at least approximately also to other cybernetic domains. This issue deserves
closer analysis.
2.3

Models and control

The principal subspace representation of the input data is, indeed, a rather
sophisticated model of the environment. This fact needs to be emphasized:
The mirror image that is constructed by a cybernetic system (see [19]) is
not just any storage of observation data; it is an optimized model of the
properties of the environment (as measured in terms of covariations).
Why does nature construct such sophisticated models? The deep insight necessary here is oered by control theory. There it is well known that the best control
result, or the best elimination of external disturbances can be implemented if
a model of those disturbances is available. Now the key observation concerning
the nature of cybernetic systems can be expressed as follows:
The interactions implemented by a (truly) cybernetic system are not
whatever feedback structures; it constitutes an optimized model based
control trying to eliminate the disturbances coming as input from the
outside environment.
Traditionally in cybernetic (synergetic, autopoietic, etc.) studies, it is assumed
that feedback alone is enough to implement the polished behaviors observed in
complex dynamic systems. However, it is not so that whatever negative feedback
would do a good job; indeed, the whole eld of control theory studies this subject
of constructing good feedback strategies. What is more, there are also semantic
consequences: It is the whole conceptual universe of established theories and
tools that become available. For example, applying the intuitions from control
engineering, one can further elaborate on the true essence of cybernetic systems:
The controls implemented by a (truly) cybernetic system are not whatever control structures; it constitutes an optimized adaptive control continually changing its behavior according to the changes in the environment.
Adaptive controllers are capable of automatically adjusting their behaviors based
on the properties of the observed environment. Because of their promises, adaptive structures have been studied actively in control theory [1]. However, adaptive
controllers have not become very popular in practice it has been recognized
that there emerge behaviors that are not benecial and are dicult to master.
The diculties are fundamentally caused by loss of excitation in the system.
Adaptation of behaviors can only be carried out if there exists some kind of a
model of the environment, identied on the basis of the available information as

delivered by the measured signals. Adaptive control is fundamentally based on

model adaptation. Problems emerge when this adaptive model is connected in
the feedback loop to implement model-based control. Whenever some behavioral
pattern in the signals is detected, or when an appropriate model for it is found,
it can be exploited and the pattern can be compensated; after this elimination,
the behavioral pattern no more exists and the model becomes obsolete, so that
a new model needs to be found. But when the previous obsolete model is forgotten, information is lost, and the original patterns of disturbance can again freely
pop up. It is not so that this model adaptation would be a renement process; it
is a more or less cyclic process between states of bad behavior and good identication versus good behavior and bad identication, making the overall system
behavior non-stationary and very complicated.
This adaptive control view opens new horizons to analysis of behaviors in
cybernetic systems. Again, as in technical control systems, the structure of the
excitation whatever is the phenosphere changes, as the originally most
relevant input disturbances are compensated and less evident ones remain to
be discovered. In this sense, as the cybernetic control system has evolved so
much that it has enough momentum for essentially aecting its environment,
eliminating the relevant input variation, the system adaptation starts getting
astray, concentrating on the less relevant patterns. The nal outcome can be
that the model becomes ruined altogether and the boundary to chaos is crossed:
If for some reason the original disturbance pattern is evoked, in the streamlined
system there are no more mechanisms for tackling with it. Only after a turbulent
period, the necessary structures are reconstructed but this reconstruction of
the control structures can be based on very dierent underlying subsystems and
agents after the collapse.
As an indication of what kind of analyses will be carried out later in tis paper,
let us apply the above technical studies to non-technical cases. It is tempting to
extend these considerations also to very complex domains, like human societies:
For example, after the Pax Romana was reached, the army was no more so
essential, and not so much resources was put to it; nally, after centuries of
peaceful status quo, the barbarians managed to conquer the empire. It seems
that this is inevitable in all too developed systems; one can try to maintain the
old good structures, but as the Soviet example reveals, the tensions just become
stronger and stronger until the structures necessarily collapse. And it needs to
be recognized that this process of decay is comprehensive: The loss of excitation
is experienced by all subsystems that independently optimize themselves. It is
hopeless to ght against the natural tendencies. It is not so that concentrating on
some specic detail and xing it the inevitable could be avoided (for example,
putting more resources to military expenses does not necessarily help if the
soldiers are unmotivated because of the missing threat). How could the collapse
be avoided, then? In practical control systems it has been observed that the
adaptive controls must not be too good, one should not implement the best
possible control, allowing strong enough disturbances excite the systems or
extra noise can explicitly be added in the system. It seems that robustness and

optimality in a system are contradictory goals. Again drawing bold conclusions:

In the case of a culture, this means that new ideas, etc., need to continually
enrich the society. A continuous turmoil is better than the complete collapse
after a stagnation!
The above leaps from simple, concrete, low-level systems to societal systems
consisting of humans were huge and heuristic. Too wild speculations are useless.
The eld of complexity theory is a notorious example of systems where the hole
is larger than its parts, resulting in empty words with no substance. However,
the framework of cybernetic systems oers concrete concepts and tools to motivate and functionalize the intuitions. This claim will be elaborated on in the
rest of this paper.
A good control has to be based on a good model of the system behavior,
capturing the essence of the domain. Above, in the concrete ecological case, the
feedback that was based on the latent PCA-type variables implements a control
of resources that is optimal in a simple environment; however, as more and
more complex systems are being discussed, when it is not only the second order
statistical properties of data that are needed to capture the system dependency
structures, more and more sophisticated models are needed for implementing
good feedback. The control theoretical intuitions still apply when analysing such
more complex systems. Indeed, in real life the models can become extremely
complicated, and the agents implementing the control strategies, trying to drive
the system towards balance, can be equally complicated. This applies also to
cybernetic systems consisting of humans as agents: These actors have more or
less thorough understanding (model) of the environment. Depending on this
understanding, they have a vision of how a non-tension situation could be
reached, and the actions are implemented accordingly ...
However, it needs to be admitted here that the necessary conceptual tools
that are needed to turn intuitions into veriable/falsiable theories are not yet
available. To extend the intuition to new, less structured domains, beyond simple
population models studied above, more powerful ideas need to be employed.
When modeling complex systems (biological, cognitive, etc.) the details are so
overwhelmingly complex that the overall picture cannot be seen when looking at
the individual processes alone. The number of underlying structural alternatives
is essentially higher than the number of visible patterns, so that there exists
an innite number of alternative structures to choose from when explaining
behaviors. To capture the most appropriate model structure, one needs strong
modeling principles stronger than what exist today. It turns out that to reach
this one has to answer not only the how questions, but also the why questions.
Traditionally, such teleological problem settings have been thought to belong to
metaphysics, not to natural sciences. So, what is the essence of Logos, or the
underlying elan vital in cybernetic systems, large and small?

... Towards universal principles ...

It is also a non-trivial control system that is being implemented by the cybernetic

system. How can this intuition be exploited?
3.1

About entropy and order

The concept of entropy is among the most fundamental ones in nature, and when
searching for universal laws governing cybernetic systems, these issues need to
be addressed.
Applying the thermodynamic interpretation (as dened by Clausius), entropy
reveals the extent to which the energy in a closed system is available to do work
(as dened in a somewhat sloppy manner). The lower the entropy level is, the
more there is free energy. In a closed system, entropy level cannot decrease; it
remains constant only if all processes within the system are reversible. However,
because the natural processes typically are irreversible, entropy in the system increases, so that energy becomes inert. Even though the total amount of energy
remains constant, according to the rst law of thermodynamics, it becomes less
useful, according to the second law of thermodynamics. Ultimately, the system
ends in a thermodynamic balance, or heat death, where there is no more free
energy available.
The cybernetic systems, as dened in [19], are also characterized by balances:
First, the determination of x
is based on nding the dynamic equilibrium as
determined by the system model (1). Second, the matrices A and B, as dened
in (2), are also dynamic equilibria as determined by the statistical properties in
u (see [8]). Indeed, in a cybernetic system there are balances at each level
and, in this sense, the convergence towards a steady-state model is completely
in line with the second law of thermodynamics.
However, the above observation is not yet intuitively sucient, and some
more analysis is needed. There are many denitions for the concept of entropy,
and it seems that the corresponding intuitions are to some extent contradictory,
or at least obscure.
In statistical mechanics (by Boltzmann and Gibbs), and analogously in information theory (by Shannon), entropy is related to probability: More probable
states (observations) reect higher entropy than less probable ones. In a sense,
entropy is the opposite of information less probable observations contain more
information about the system state. In such discussions, the second law of thermodynamics, or the increase in entropy, is reected so that systems tend to
become less ordered, and information becomes wasted. This probability-bound
interpretation of entropy is intuitively appealing, but it seems to result in paradoxes: For example, a symmetrical pattern is intuitively more ordered, containing more information, and consequently having lower entropy than a completely
random pattern on the other hand, symmetric pattern can be seen to contain less information than a random pattern, because the redundancies caused
by the symmetricity can be utilized to represent the patterns more eciently,
so that the entropy level should be now higher now. Indeed, as discussed in

[11], the algorithmic entropy is higher in a symmetric pattern than in a nonsymmetric one. To confuse concepts concerning order and symmetry even more,
or, rather, to reveal the inconsistencies in our intuitions, think of the following
claim: A totally unordered system can be said to be extremely symmetric as the
components cannot be distinguished from each other.
However, here it is assumed, according to the original intuition, that orderliness is a manifestation of low entropy. The key point here is that the simplicity
of symmetric patterns, or ordered patterns in general (loss of information in
them), is just an illusion: The missing information of the pattern is buried in
our pattern recognition capability. If the same data is to be presented without
the supporting underlying mental machinery, or specialized interpretation and
analysis tools, there is no handicap the redundancy cannot be exploited, and
no compression of data can be reached. In general, a higher-level representation
makes it possible to abstract the domain area data; in other words, a model is
the key to a compressed representation.
Similarly, in cybernetic systems one seems to be facing paradoxes, or fundamental inconsistencies that are not only of semantic origin. How is it possible
that in some systems being equally subject to real-life constraints and laws
of nature the arrow on entropy seems to be inverted, so that rather than getting disordered, new order emerges in them? For example, in cognitive systems,
in social systems, and in living systems in general, more and more complicated
structures are introduced in the course of evolution. Of course, there are no outright contradictions here (the subsystems getting ordered are open systems, the
overall entropy in the universe all the time increasing), but why do the systems
not select the easy way, exhausting energy for simply increasing entropy? Why
are there such countercurrents in the ow of entropy?
As in the case of symmetric patterns above, it is a higher-level structure
representing the lower-level data that looks more ordered and smart, as interpretated by our perception machinery. The PCA-based cybernetic system
distinguishes between random noise and correlated variation in the data, thus
compressing information so that it contains less noise. As this correlated variation in the environment is interpreted as information, the cybernetic system
seems to act like a Maxwell Demon, distinguishing between two containers
of information and noise, compressing information and pumping negative entropy into the emerging structures (see Fig. 3).
Applying the discussions in Sec. 2, the mystery of cumulating complexity can
be resolved: It is the control system intuition that is needed to solve the arrow
of entropy paradox. Even though it seems that entropy level goes down in some
subsystems, when one looks at the structure among the systems more closely,
the contradictions vanish: The system of decaying entropy is the supersystem,
or control system, driving the subsystem more eciently towards increased entropy, or local heat death, as characterized by the stable balance (see Fig.
4). The minor decrease in entropy in the supersystem is compensated by the
major increase of entropy in the subsystem, so that the second law of thermodynamics is obeyed and, indeed, this entropy is now pursued more eciently

High probability
High entropy

Low probability
Low entropy

Outside the model

Inside the model

Fig. 3. Schematic illustration of how information, or the units of covariation,

are condensed in a cybernetic system

than otherwise would be possible. If the data properties are stationary, the cost
of the constant higher-level structure becomes negligible as compared to evercumulating entropy on the lower level. More explicitly it can be claimed that
the second law of thermodynamics is the motivation for emergence of order in
complex enough environments (see Sec. 3.2).
Perhaps the most important consequence of the new interpretation of the
cybernetic systems is that reductionistic approaches become possible: Traditionally, the only systemically consistent level of studying entropy-decaying systems
was the holistic level, the whole Earth as one entity, now each subsystem as
studied alone is also thermodynamically consistent.
The new view of cybernetic systems as pursuing balance is fundamentally
dierent from traditional intuitions. It has been assumed that interesting complex systems are at the edge of chaos. For example, when studying the processes
of life, the mainstream view is expressed by Ilya Prigogine: Life is as far as possible from balance, whereas death means nal balance. Erwin Schr
odinger [14]
phrased this as What an organism feeds upon is negative entropy; it continues
to suck orderliness from its environment. Also in cybernetic systems, static balance means death but a living system is characterized by (thermo)dynamic
balances. Now the roles are essentially inverted: Whereas a living thing is traditionally assumed to play an active role, now it just has to adapt to its environment; it is the environment that pumps disorder into the system, and life
processes try to restore balance. It is not imbalance but the homeostasis, as
explained by Bernard and Cannon, that is the essence of life processes; this balance is not only an emergent phenomenon but the very kernel of the relevant
functionalities (see [19]). It is not about minimization of entropy, but, on the
contrary, it is explicit maximization of overall entropy in a cybernetic system
this entropy increase is just channelled in a smart way!
Heraclitus claimed that the Logos running the world is re. Perhaps a better
characterization of cybernetic processes is that the universal mind running them
is a re extinguisher incoming excitation is being attenuated. It seems that,
again, the Eastern tradition is perhaps deeper than the Western is: The underlying vitality principle beyond the Chinese philosophy and medicine is based on
balancing and ordering; see Fig. 5. On the other hand, in Indian philosophy

Traditional view

Flow of entropy

New view

Flow of entropy

Fig. 4. The roles of subsystems and supersystems need another look (see text)

many principles (stationarity, desire and consequent suering, etc.) also reect
the cybernetic ideas.
To summarize the above discussion, one can say that cybernetic (sub)systems
constitute a framework for connecting successive system levels in the same framework, so that the emergence of structures within the overall entropic model
can be explained. Structure (model) on the higher level means higher entropy
on the lower level. It is the available resources that are the driving force keeping
the system running, and maintaining the dissipative, irreversible processes in
a cybernetic system; however, this underlying machinery is another thing from
the actual essence determining the cybernetic functions. It is not so that the
available resources would simply constitute the supply of free energy in the
thermodynamic sense, so that the amount of resources would directly stand for
amount of neg-entropy; the opposite of entropy, or incoming information, is
revealed in the form of variations in resources. This is the same thing as in
thermodynamics where free energy is buried in temperature dierences rather
than in temperatures themselves. As Gregory Bateson intuitively puts it [3]:
Information consists of dierences that make a dierence. Whatever is the interpretation of the resource vector, whatever are the physical dimensions of the
input, the driving force in the cybernetic information cumulation is variations
coming from outside; the system does what it does trying to eliminate these variations. The Shannons formula just denes a static entropy measure, connecting
information theory to thermodynamic domain in a formal way; it may be that
the deepest interpretation and the most fruitful framework for cybernetic studies comes from a combination of these two elds in a more fundamental way.
Perhaps one could here speak of information theoretic thermodynamics. This
is not only jargon: It turns out that the above discussions can also be put in
practice.
3.2

Principle of least dierence

Traditionally, the second law of thermodynamics is thought of as being a universal, more or less metaphorical principle. The existence of systems with inverted,

Fig. 5. Chinese symbol for the ordering

principle, also denoting air or vapour

entropy-decaying nature has made it dicult to motivate explicit utilization of

this principle in practice: It seems that the entropy principle cannot be applied
in a reductionistic way for analysis of concrete large-scale systems.
Now, according to the above discussions, the entropy in a subsystem always
increases when seen from the higher-level system. In a cybernetic system, entropy increases in a consistent manner, there is balance pursuit at all levels,
completely in line with the second law where thermodynamical balance is the
ultimate goal. Because of this consistency, any subsystem at any level can be
studied separately, and also holistic systems can be analyzed in a reductionistic
manner. In this sense there is no more dierence between dierent kinds of complex systems: Living systems and non-living ones, for example, can be modeled
in the same framework. Whereas the rst law of thermodynamics (energy principle) oers powerful tools for deriving static models, it seems that the second
law (entropy principle), being a fundamentally ux-based concept, oers generic
tools for deriving dynamic models.
The entropy level, or, rather, changes in levels, can be applied as the measure
for tensions in a cybernetic system. To make the very abstract entropy principle
applicable in practical modeling tasks, one rst has to dene what free energy
is in general terms; that is, one cannot rely on the particle-level considerations,
but a system-level characterization is needed. The starting point here is that
as it is the balance, or match with environment, that is being pursued in a cybernetic system, free energy originates from mismatch between the system and
its environment. In this sense, one can speak of principle of least dierence,
extending the principles of least action or minimum energy, as originally proposed by Maupertuis, and later extended by Euler, Lagrange, and Hamilton. In
concrete terms, one can dene a cost criterion J, or the energy function, as being
the inverse of the tness:
1
T
(11)
J(x, u) = (u x) W (u x) .
2
Here, u stands for the state of the environment, whereas x is the internal state
of the system; it needs to be recognized that the state variables (elements of the
vectors u and x) operate in dierent domains, and in between, a mapping matrix

 is needed. Matrix W makes it possible to weigh data components in dierent

ways. Typically the dimension of x is lower than that of u, so that no complete
match can be reached. Now, free energy in the system state x can be dened as
the dierence between J(x, u) and its minimum value J(
x, u) in minimum,
free energy is exhausted.
The vector u can not only represent the real observed state of the environment it can also stand for some hypothetical state. As presented in the next
section, the same cybernetics-motivated modeling principles can be applied also
for the iterative technical product development processes (and also in economical optimization processes); assuming that, for example, the goal is to reduce
some quantity (weight, elapsed time, etc., depending on the device), in the goal
state this quantity should be zero, and this deviation then constitutes the free
energy, giving raise to technical development. As long as zero-delay, zero-cost,
etc., has not been reached, the system will evolve. Of course, this optimum state
is never reached, and the nice results concerning the cybernetic balances cannot
be directly exploited. However, there are clear patterns here, too even though
the developments are random and sporadic, on average the developments take
place in the direction of maximum gain, and typically the optimum state is being
approached in a more or less consistent manner (typically following the exponent
curve).
Mathematical formulations sound rather trivial, and it is dicult to see how
any more complicated problems could be formulated in this vector-based framework. However, it needs to be rocognized that also the most complicated articial
intelligence methodologies have traditionally been formulated applying problem
spaces and goal states. Reasoning and planning tasks, for example, can be
formulated so that a goal state is being searched for. Even though the problem
space in AI cases is not necessarily a mathematically compact vector space, it
needs to be recognized that the structural complexity can be substituted with
dimensional complexity that is, introducing enough new variables, the search
problem in the high-dimensional space becomes simpler (remember the idea of
support vector machines). Indeed, if correct variables are selected, so that local minima can be avoided, also reasoning problems become pattern recognition
tasks in the high-dimensional space.
Entropy is also always being maximized in a cybernetic system. But powerful
principles not only explain behaviors they can also help to predict behaviors
beyond the range of the original model. Indeed, the principle of least dierence
can be strenghtened: It can be assumed that a system tries to maximize entropy as fast as possible. Ideas concerning maximum (generalized) entropy have
been studied a lot, but mainly for determining static optima; only in limited
application domains such ideas have been applied for dynamic analyses. In the
eld of autokatakinetics, as proposed in [15], the idea of maximum entropy
production is intuitively proposed. Now the concrete, simple formulations make
it possible to reach practical results and powerful modeling tools.
What is, then, the fastest route to maximum entropy, characterized by a
dynamic balance with hidden tensions? It needs to be recognized that there

is no centralized master mind in nature. Nature does not know the global
optimum, or where to go to reach it. The optimization strategies that nature
implements are decentralized, distributed to very local agents that only see their
local environments, and do not know the big picture. Generally, the direction
of fastest local decay in cost criterion is revealed by the (negative) gradient
and, indeed, at least in simple environments, the processes proceed so that higher
densities, concentrations, temperatures, etc., are discharged towards lower ones.
To emulate such locally consistent behaviors one can rst write the gradient for
the criterion (11):
dJ
(x(t)) = T W x(t) T W u.
dx

(12)

Now the continuous-time version of the steepest descent gradient algorithm can
be written in the state-space form:
dx
(t) = A x(t) + B u,
dt

(13)

There are physical restrictions that determine what is this rate of adaptation,
and these factors are collected in . Combining (12) and (13), the matrices A
and B are dened as
A = T W , and B = T W.

(14)

To apply the simple dynamic model, the global criterion J is not explicitly
needed; to justify the linear dynamic model, it is just assumed that it exists. On
the other hand, if the criterion is known, the entropy principle oers a practical
way to determine dynamic models also for complex systems.
A cybernetic system also carries out pattern matching against its environment, nding a balance between the outer and inner states. At least in special
cases the above hypotheses are justied: In a truly cybernetically optimized system, as in a Hebbian/anti-Hebbian neuron system, denes the basis vectors of
the principal subspace of the input u, and W is the input data covariance matrix;
this assures that the system obeys not only the rst-order balance but also the
second-order balance (see [19]). It turns out that the quadratic formulation of
the cost criterion can also be motivated in the Hebbian/anti-Hebbian framework
but, from the point of view of universal applicability of the maximum entropy
principle, can this intuition be generalized; why should this formulation apply
to other cybernetic systems in other domains?
The key notion here is that of a diusion process: In diusion systems the
behavior is characterized by an explicit search for balance, or maximization of
entropy. The tensions causing interactions come from the free energy that is
manifested in the form of imbalances in concentrations, temperatures, or other
distributed quantities. The diusion process is internally balanced, negative feedbacks being automatically built up. Dierent kinds of diusion processes are
typically linearly dependent of the dierences in the system and this linearity
can be interpreted in terms of quadratic cost criteria in the form (11). If nding

ui+2
Ai+1

xi+1
u^

Bi+1

ui+1
Ai

i+2

xi
vi+1

T
Bi+1

u^

Bi

ui

i+1

Ai+1

vi

Bi

u^ i

Ai

Fig. 6. Connection of trophic levels

the balance in a cybernetic system can be seen as a generalized diusion process, the same framework (13) can always be utilized. When looking at a (truly)
cybernetic process in a static perspective, it can be said to constitute a higherorder balance system; on the other hand, when seen in a dynamic perspective,
it is a system characterized by higher-order distributed diusion.
As discussed in [16], the exponential model is very plausible according to
theoretical analyses and practical experiences, at least when studying population dynamics. The problems arise when the constraints have to be taken into
account, resulting in very non-universal and intuitively non-appealing model
structures. Luckily enough, in cybernetic models integrate the constraints in
the same simple model structure. The model x = Ax + Bu with real-valued
eigenvalues in A is a general form of interacting (perhaps cascaded) diusion
processes with exponential behavior.
Another point worth mentioning here is that, again, one should take into
account all feedback loops: It can well be so that the developments are not, after
all, so straightforward as the diusion intuition might suggest. For example,
after a structrural experiment, carried out by some of the system agents, it
is not only the internal parameters of the system that get adjusted as the new
balance is restored, but perhaps also the external parameters. In retrospect, some
ecological/economical selections may look as the best choice merely because
those decisions determined the parallel universe that was realized. It is not
necessarily so that the system is an image of its environment; a strong enough
system can make its environment reect its own structure!
3.3

Escaping from a domain to another

Where does the free energy in nature, or variation in resources, originate from?
The energy producing processes in the Sun are relatively stationary, providing

technical
systems

social
systems

ecological
systems

economical
systems

memetic
systems

biological
systems

physical
systems

Fig. 7. Schematic illustration of cybernetic systems extinguishing the re

a rather constant ow of energy, no variations. However, the orbiting of planets

causing day and night, summer and winter, introduce the rst variations; after
that, there are the locally unstable (chaotic) processes further creating environmental variation. First there was chaos: Chaos truly creates information
as unanticipated uctuations are generated in such processes information that
can subsequently be exploited by other systems (for analysis of information, see
Sec. 4). For example, the fast enough uid ows based on the Navier-Stokes equation introduce turbulence and nontrivial spatio-temporal distribution in energy.
These physical processes are still sub-cybernetic they remain stable because
of the inherent nonlinearities, not because of the actual feedback structures. Or,
even though the interactions can be interpreted as feedbacks, these feedbacks
are physical, bound to concrete location, whereas the feedbacks in actual cybernetic systems are more like information theoretical. For example, planetary
motion is not cybernetic; Newtonian action and counteraction is not sucient
to implement negative feedbacks in the cybernetic sense. What is more, there is
no structural adaptation or evolution in these systems; qualitatively nothing different emerges as there exist no learning mechanisms (sand dunes, for example,
even though virtually being manifestations of new order, have no negative feedback eect on the wind). However, as claimed in [4] the principle of maximum
entropy production can be applicable also to such sub-cybernetic processes (the
convection patterns of the Benard process being used as an example).
This incoming variation is then processed by the actual cybernetic systems,
trying to exhaust it. The complex dynamics, in the form of chaotic-like oscillations, is not an inherent property of the cybernetic model structure; the dynamics
arises as a reaction to the external excitation. When dierent kinds of cybernetic systems are combined, qualitatively new levels of model complexity can be
reached, resulting in more and more smart-looking systems. For example, when
a cognitive system is integrated in a biological system, as is the case with individual humans, much more ecient exploitation of the environment is reached.
Similarly, as a single cognitive system is integrated in a population system, as
in the case of an intelligent organization where individual humans are appro-

priately networked, the resulting system can outperform the capabilities of the
individuals. In any case, it is the entropy principle that still applies, all systems
trying to eliminate the incoming variation (see Fig. 7). Thus, one could perhaps
distinguish between dierent kinds of cybernetic systems: There is a continuum
from physical systems through natural (or normal) cybernetic systems to constructivistic systems, and further to technical systems, depending of how much
of the emergent structure is determined directly by the environment; in physical
systems, there are no degrees of freedom, whereas in constructivistic systems
the emergent structure is practically free of the environmental constraints. In
technical systems it is not only the system that is (more or less) articially constructed, as in constructivistic systems, but also the optimum state where the
system tries to balance itself is imaginary (see later). Indeed, in economical systems the imaginary goal (maximum amount of money, or minimum cost) is
visible in its most explicit form.
Entropy is universal, but it is not centralized: It separately governs the behaviors of the smallest systems as well as cosmic ones. There are no centralized processing or control units in nature; the only information available is
the information delivered by the system itself, and the only energy available
for self-organization is the energy engaged in the tensions of the system. The
entropy-pursuit machinery is thus localized, characterized by distributed negative feedback mechanisms. Ultimate functionalism cannot be reached in such a
distributed system of systems. This means that even though the model-based
control can optimally attenuate variations in some subsystem, the optimality is
lost in wider perspective. Indeed, this loss of optimality is a very fundamental
phenomenon: Paradoxically, it turns out that locally optimized entropy maximization, or elimination of information, results in maximal preservation of overall information. This can be explained as follows: A good controller is aware of
what is happening in the system being controlled, or what is the state u of the
outer environment; this information that is used for controlling the environment
is captured in the internal state of the controller x. This x containing the essence
of the environment is available for yet higher level systems to be exploited as
input resource! And, further, this relaying of information from former systems
to latter ones can be repeated; there can exist an arbitrary number of successive levels in the chain of cybernetic systems, and the information is maximally
transferred up to the top layer (see Fig. 6).
After the long slow evolution of natural systems, resulting in more and more
complex structures, developments became much faster when a new machinery
was once introduced: The human, and specially his/her cognitive capabilities offer a general-purpose control platform for implementing dierent kinds of higherlevel systems. In this case the model is in another domain (phenosphere) as
compared to the system being modeled. Nature relies on completely distributed
strategies when implementing control systems, and, as was observed, this results
in extreme complexity and illogicality; the human is needed to implement a more
streamlined system, where the non-optimalities are ripped o. From the ecologi-

cal point of view, this simplication and loss of diversity in the form of increased
functionality and consistency of course has to be seen as impoverishment.
In human-constructed systems, the balances need not be ght-or-perish
equilibria, they can also be negotiated. The human societies can be based on
ideas by John Forbes Nash, not only by Adam Smith; the welfare society can,
after all, be a good idea if the faith of the underlying dynamics can be
appropriately foreseen in advance.
It is not only optimization that is carried out by humans completely
new resources of free energy are also released by humans. In concrete terms,
new variables are introduced in the resource vector by human activity. In a
sense, the human has the role of a catalyst: New resources become available
because of human activity, and processes that otherwise would never take place
are activated. For example, the oil reservoirs would never have been exhausted
if the human culture had not done that. Thus, after all, exhaustion of natural
resources and destruction of the environment is inevitable and predestinated by
the entropy law. And the success of the human can be measured in terms of
increased entropy, or the rate of consumption!
It needs to be recognized that in more complex domains the solutions to
the optimization problems, or models of how the free energy in the environment
can be used, are by no means unique. For example, if there is money available
in the market, there are many dierent ways of exhausting it. According to
the selected strategy, a process can be instantiated to produce it; this process
is further divided in subtasks with simpler goals, employing human sta for
running the subprocesses, keeping the subcontrols in balance. The original push
(economical pressure) is thus divided into investments, constituting a delicate
structure. Within a selected (sub)structure, the implementation is more or less
unique, but there are always many ways to select the structure.
In all systems, the bottleneck is caused by the scarcity of information and
understanding. When something is better understood, or when a better model
exists, more or less immediately after that (or, at least, when the new balance
after the transient has been reached) it is exploited and new feedback structures
are constructed. In this sense, one is always on the edge between the known and
the unknown. This is easy to see in technical and economical systems, where the
whole closed loop between the observations and actions is explicitly optimized;
understanding is exploited in a straightforward way and more streamlined systems are implemented immediately when it is justied in terms of economical
etc. considerations. However, also in politics the control of the society is implemented in a rather straightforward way by applying legislation, according to
estimates and assumptions about future that are based on more or less accurate models of the society dynamics. The current state of the societal system is
measured in terms of statistics, and also by opinion polls.
Today, the models can be constructed proactively rather than reactively: The
company hierarchy can be designed for some production task, and the customer
demand is created only afterwards. In complex domains there are no straightforward patterns to be matched in the environment, or, more accurately, the space

of available variables is not xed beforehand. In such cases, evolution is not a

random but a goal-directed process; this evolution is much faster than in biological, etc., environments because it is based on human intermediating agents:
Best strategies are searched for in a goal-directed manner, and spreading of new
strategies can be instantaneous.
A human with such freely reprogrammable modes of behavior is a powerful
agent for implementing dierent kinds of cybernetic systems. The human mind
is a platform for societies, economical and scientic systems alike. It is not only
the hardwired animal instincts but also the needs and desires that aect the
human behaviors. Because of the exibility of the human mind, the rules of the
game common to all agents need to be imposed consistently upon everybody;
otherwise, if the agents do not behave in a coherent way, there is no orchestration,
and no cybernetic functionalities emerge in the overall system. In short, this
coordination is implemented in the form of some kind of moral imperatives or
codes, either religious, or philosophical/political. Indeed, it seems that religiosity
of humans had an evolutionary advantage when organized societies and cultures
were to emerge; otherwise, uncoordinated anarchy would have prevailed. Instead
of rigid moral codes, nowadays the moral control is based on much more
exible and ecient means: Todays humans are controlled to a large extent by
fashions, and specially by money. It is money that oers a universal measure
for quantifying very dierent kinds of things in a transparent way, making the
control actions by the system very fast and unambiguous; in concrete terms,
money makes things commensurable, so that the above proposed vector-form
coding of complex phenomena is justiable however ethically objectionable
such rude evaluating and assessing might feel like. However, it is with human
agents as it is with ants travelling on their paths: There is the mainstream
ow, but there also exist rebellious dissidents, introducing necessary variation
in behaviors so that there is possibility of changes in behaviors.
It is in the human nature that one searches for new frontiers to discover
and conquer, to understand and exploit or, to model and control. Putting
it poetically, one can admit that God gave man dominion over the world
but there is catch: Man rst has to detect and identify that world. The human
builds models, but, simultaneously, he is also a model of his environment; man
is an image of God but this God is Gaia. And it is the lure of money that
implements the whispering of the modern secular conscience.
Grasping a thing means that a model capturing the relevant phenomena
has been successfully constructed. Arthur Schopenhauer already observed that
human understanding and intellect are there just for biological reasons, or rational thinking is a tool for fullling the physical needs. But Schopenhauer claimed
that art is a key to escape the rat race: Aesthetic experience is, according to him,
free of any concrete utilitarian purposes. However, following the above discussions, it turns out that also the aesthetic experience can be explained in the same
framework of better fullling ones needs: It is an ecient means of constructing
more versatile models of the reality. Remember that there often exist analogues
among systems in dierent domains, that is, there can exist common patterns

among very dierent systems; or, more accurately, the same model structures
can be applicable in dierent domains. The more general the modeling principles
are, the more probable it is that there are similarities. For example, learning to
see symmetries in concrete patterns can help to see corresponding structures
also in more abstract domains, so that such pre-created model structures can be
reused for reinterpreting, that is, for nding creative new associations. In this
sense, art (or, actually, practically any eld of special expertise) can truly help
in seeing nature in new ways, and constructing more ecient (subconscious)
models for other systems.
3.4

Basics of constructivism

The claim here is that, really, it is the entropy principle, as implemented in cybernetic control structures, that governs also complicated cybernetic systems,
like all human behavior and activity. The purpose of all human information
gathering, for example, is gaining knowledge and understanding; understanding
is the route to exploitation of deposits of variation that exists in the natural
resources. The feedback from understanding to explotation is seen as a control
loop. What makes this often dicult to recognize is the fact that the implementations of cybernetic control can be so distributed, and the application domains
are typically so non-mathematical.
In cybernetic systems it is populations of agents that determine the system
behavior; in ecosystems, etc., the agents themselves are a part of the system, operating in the same domain, whereas in more complex environments, the agents
operate in some other domain. Typically this means that there are more degrees
of freedom available, and the laws governing the adaptation are not so stringent.
In such constructivistic systems dierent kinds of approaches are necessary; however, it can still be claimed that the operations still maximize overall entropy
production.
Technical product development processes are typical constructivistic cybernetic systems, where the dynamics is caused by tensions determined by external
constraints and the technological drive. How does development of computer
technology, for example, boost entropy in the overall system, then? Word processors, typical software products, are used to construct descriptions of complex
domains; these models are (more or less balanced!) views of the domain eld.
More sophisticated word processors make this model construction process faster;
and the new hardware and software tools make it possible to share this model,
and compare its virtues with competing models. Specially, when doing scientic research, the Internet technology has made the paper production process
much faster as the information availability has increased. Computer technology
also boosts the delivery of new models (ideas and theories) from one domain
(cognitive system) to another (scientic community). A scientic paradigm is
determined as a balanced interplay between such theories, being a cybernetic
combination trying to explain (model) the subject domain. Straight after such
modeling is satisfactory, technical applications are introduced, where the developed model is applied for controlling, or manipulation of the domain eld.

In this sense, development of the computer technology nally result in better

mastering of the environment, in practice meaning that the available resources
become more eciently exploited, so that the free energy decays faster.
Quite concretely, the programming languages are tools for easier construction
of models of complex domains; the better the tool, and the more polished the
program is, the better match with the reality can be reached. Simulation tools, on
the other hand, are tools for transforming the information in static descriptions
into functionally more applicable form. The computer is really a universal machine, oering a general-purpose platform for implementing any kinds of models
and for functionalizing them, to be applied in dierent ways for nally somehow
aecting the environment.
The key point here is that there is a huge number of subprocesses intertwined
in a higher-level cybernetic domain; indeed, there seems to exist a fractal structure of cybernetic systems involved. It is not only the original system; it is also
the information availability, message transfer speed, quality in terms of accuracy,
etc., that are typically under the process of ever-continuing development. This
fractality of parameters cannot easily be formalized, because all substructures
can further be split in subparts. Indeed, whenever a subsystem has become appropriately identied, it becomes a subject of further development. Typically
in technical environments, this development means making the system faster,
smaller, cheaper, etc. How does the agent doing the development work know
that there are some cosmic consequences, about entropy, or control of environmental variations? The agents in ecolocigal systems do not know anything about
their control function, and neither do the humans know anything about the big
picture the overall behavior is again an emergent phenomenon. It is as it is
with simpler cybernetic systems: If the agents just follow the common principles (go towards resources, avoid competition), the higher-level functionalities
automatically emerge, and, as a system, the adopted strategy has evolutionary
advantage. However, as compared to ecosystems, etc., now there are no immediately evident resources or competitors, there is no immediate benet or threat,
and these simple principles are no more applicable as such. The goals in a constructivistic system are dened in terms of a not-yet-existing environment; more
powerful hardwiring of the agents is needed.
In short, the new constructivistic imperative that is needed to explain the
development in complex cybernetic systems can perhaps best be paraphrased as
citius, altius, fortius. The urge to make things in some sense better is reected
in the technological drive, and in human ambition in general, whatever is the
branch of activity. As motivated later, in constructivistic systems this principle is
just another formulation for good modeling of not existing but possible systems.
And, again, because of its success among the alternative strategies, this has also
the evolutionary advantage. In concrete terms, what are the properties that are
needed so that the agent the human could fulll its task?
How systems become constructed? When escaping from a simgle domain to another, the agent needs to be smart enough, and, additionally, to
try something before there is any evident use of it, the agent needs to be

curious that is, the agent needs to be simultaneously homo sapiens and
homo ludens.
How systems get adapted? To make the systems adapt when there is no
immediate need for that, a driving force is needed; this constructivistic imperative is not any explicit rule, it is an implicit tendency that is manifested
in greediness and ambition.
It has been claimed that one of the main dierences when humans are compared
to animals is that humans can think of the future, they can plan, and they
can imagine what the environment could be rather than merely adapting to
the current circumstances. Human behavior is proactive rather than reactive.
Humans can visualize the optimum state and the route to that state.
And, of course, yet another key feature in humans from the point of view of
acting as agents in constructing cybernetic systems is their social nature: Because of the monkeying, imitating others behavioral patterns, is so natural to
humans, new ways of behaving can easily be instantiated and new kinds of systems are possible. Even though free will is said to be one of the main things that
characterizes us as human beings, it is the absence of free will that characterizes
societies. This can also be expressed as (mental) laziness: Avoiding diculties,
going where it is easiest, not against opposing forces but following them, leads
to a dynamic balance also in abstract systems. However, it needs to be recognized that always when behaviors become too homogeneous and predictable,
there is room for local opportunistic optimizations, perhaps resulting in parasitic
strategies.
When does the group-think, or the group being dumber than the individuals,
change into system intelligence, where the society is cleverer than the individuals? Perhaps the key here is according to the above intuitions adaptation
and balance: Only after the sustem reaches the steady-state, internal tensions
compensating each other, the system level structures emerge.
3.5

Return to Natural Philosophy

Neocybernetics is not just another scientic paradigm; no, it goes beyond that,
shaking the very foundations of science. Traditionally, it is thought that it is philosophy (logic) that is the basis for mathematics, and, further, mathematics is
the basis for technical research (engineering disciplines). Now, in the framework
of cybernetic systems, this thinking can be inverted, as shown in Fig. 8: Mathematics (linear algebra) oers the syntax (language) and engineering (control
practices) oers the semantics (interpretation) for philosophical considerations
(metaphysics). Put in another way, empirism precedes rationalism, giving substance to das Ding an Sich. This claim is further elaborated on below.
To start with, it is the same with neocybernetics as it is with other scientic
disciplines: One has to admit that models are always false. The essence of the
real world cannot be captured, and the models should never be mixed with reality. It was the start of modern science when one started doing physics based on

Traditional view

Cybernetic view
Philosophy
(metaphysics)

Philosophy
(logic)
Mathematics
(linear theory)
Engineering
(control)

Mathematics
(syntax)
Engineering
(semantics)

Fig. 8. Relations among theories and metatheories

empirism, only trying to explain the observations, rather than metaphysics, trying to explain the underlying reasons for those observations. However, things are
dierent when one is constructing higher-order models, or models for models.
Now it can be claimed that models are essentially true. What does this mean?
One has to remember that the model construction of cybernetic systems also
applies to the cognitive domain: The mental system constitutes a mirror image
of the environment as determined by the observations. No matter what the
underlying realm truly is like beneath the observations, the mental machinery
constructs a model of it. If the same modelling principles are copied in the
computer, there will be a fundamental correspondence among the data structures
as constructed by the computer, and the mental representations as constructed
by the brain in the same environment. This is an extension of the Kantian
revolution: Perceptions are not observed but constructed as the observations
are matched against what there already exists. This makes it possible to reach
intersubjectivity of representations, technical or natural: The world models can
be essentially the same, not only between humans but also between humans and
computers. This makes it perhaps possible to reach Articial Intelligence in the
deep, not only in the shallow sense. Clever data processing becomes possible:
The computer can carry out the data preprocessing in a complex environment,
and the constructed data structures can be interpreted naturally in terms of
corresponding mental representations.
But this intersubjectivity is not all there is; indeed, one can reach interobjectivity. If nature itself tries to construct models for eliminating free energy in the
system, as presented above, the human trying to model these cybernetic systems
can touch not only the shadows of the behaviour (in the Platonian sense), but the
actual essence these models can be fundamentally the same. This means that if
some naturally evolved cybernetic system (an ecological system, for example) is
modeled by a human applying the appropriate principles, this model has a deep
correspondence with the system itself; what is more, in environments that are
in a transient (an economical system, for example), the cybernetic models can
predict what the nal system would look like after the stationary state perhaps
is reached. In this sense, the new models can perhaps give insight in the true
essence of complex systems and in the hidden tensions in such systems. This ob-

servation has also cosmic consequences: Whatever are the systems on the other
planets like, assuming that those systems are similarly based on local agents and
evolution processes towards better exploitation of the environmental resources,
they must obey the same universal principles. The celestial ecosystems, or social
systems, etc., most probably do not essentially dier from the earthly ones
of course the details dier (like surface patterns), but the principles remain the
same.
Universe constructs models and, after all, models are used for simulation.
It is not only Douglas Adams who claimed (in his book The Hitch Hikers
Guide to the Galaxy) that the Earth itself is a huge computer carrying out
(distributed) simulation: Edward Fredkin proposed in early 1980s a new theory
of physics based on the idea that the Universe was comprised ultimately of
software.
There are many philosophical issues that can be attacked in the cybernetic framework from the fresh point of view. It may need to be admitted
that metaphysics cannot be addressed in the current framework, physics being based on spatial interactions among particles, but one can perhaps speak of
metabiology, metaecology, etc. What do these metatheories stand for? There are
fundamental questions concerning modeling issues that have not seriously been
questioned before: For example, the Pallas Athene Hypothesis mentioned above
(or, perhaps more accurately, Antero Vipunen Hypothesis) is not just an unsubstantiated claim but there is some deeper essence there. Indeed, this hypothesis
can be expressed in a stronger form: It seems that system complexity and analyzability go hand in hand: If Nature has been able to construct sophisticated
model structures, why not us? The clain here also is that cybernetic systems
can always be modeled, one just needs to nd the appropriate model structure.
Perhaps a new era of positivism is ahead? And, to mention another deeply
philosophical principle: Ockhams razor is routinely being applied in modeling
(simplest explanation is the most appropriate), but the motivation for this idea
is typically merely pragmatic. In the framework of optimized cybernetic systems,
the models being based on principal subspaces, etc., extreme compactness truly
is the nal faith of the fully evolved systems.
But this optimality in cybernetic systems applies only when the system is
seen in the local perspective. Indeed, the discussions above can be summarized so
that it is not, after all, some intelligent designer that is responsible for all natural
diversity rather, looking at the immense inconsistence, one could speak of a
hardworking idiot: The left hand does not know what the right hand is doing.
The resulting system of systems is neither systematic nor systemic. However,
it may be so that nalism will have a renaissance: As explained in connection
with entropy, there exist goals in natural systems and such systems are not
constrained to biological or ecological domains.
Most human endeavors can also be interpreted as manifestations of the same
cybernetic principles. One specially interesting group of cybernetic domains of
human activity is that of scientic research. Scientic theories are again models
of the environment, whatever is the branch, no matter if it is natural or human

sciences. The more complex the domain eld is, the more there are degrees of
freedom, and the less the available data constrains the possible solutions. As
the hierarchy of complexity evolves, it is dicult to evaluate the priority among
candidate explanations. The latter layers are dictated more by the prior layers
than the actual environment being explained; in complex enough cybernetic
domains the system starts to create its own meanings not bound to the outer
realm. This is not dependent of the eld of study, this is more like a property of
too evolved science, where there are more theories than evidence (for example,
take cosmology and its wormholes, parallel universes, etc., being manifestations
of ironic science). The cybernetic view towards doing science makes it perhaps
easier to reach reconciliation between the two cultures within sciences [17]; in
these postmodern times, as there is more and more pressure towards new results,
the scientic explanations are similarly constructivistic in natural sciences as
they are in human sciences. Also natural scientists and engineers should be
humble. It is often claimed that science proceeds positively towards higher levels
of perfection following its own internal laws and these laws should be only
determined by objective criteria of truthfulness. However, it is the human that
is always integrated in the loop of doing science this means that it is not only
the match against evidence that alone determines the vitality of a paradigm. The
humans determine what is hot and what is not. The shifts between Kuhnian
paradigms are not so clear-cut, and it seems that also science is on the edge of
chaos; the term ironic science has been coined [5]. Within a cybernetic system
there are subsystems; indeed, science is a fractal structure of cybernetic systems
such embedded systems are studied in what follows.

... And back to concrete domains

The above discussion is not merely semantic jargon; down-to-earth analyses are
possible. To elaborate on the abstractions, the concept of information turns out
to be useful. Information can be studied in mathematical terms, and it can be
used as a link between the abstract and concrete ideas.
4.1

About information

Regardless of the underlying mechanisms (interacting agents, or explicit constraints), a cybernetic system is characterized by dynamic balances. When seen
from outside, and when the phenomena have been quantied appropriately, the
system implements principal component analysis, or, actually, principal subspace analysis of the incoming data. In either case, the model captures the
(co)variation in the data in the most ecient and compact way. Simultaneously,
the PCA model is capable of structuring and reproducing the variation: The
compression of high-dimensional multivariate data is based on the distribution
of variation. Now, if this variation in data is called information, there are ecient
means of mathematically processing and analyzing that information.

Information is here a purely syntactic concept, being measurable in strictly

mechanistic means. At this level there is no need for any interpretation or semantics, and information should not be mixed with knowledge.
The above denition of information is only technical, but it is also intuitively useful and justiable. First, there is the pragmatic support: Because of
the mathematical benets, error square criteria have traditionally been applied
in modeling quadratic criteria result in linear optimization methods. Variance
also enables commensurability of signals in dierent dimensions: Variances can
directly be added together, giving a single scalar measure (assuming uncorrelatedness). Physically, the variation is easily interpretable because it stands for
signal power. But perhaps the most important motivation for variance-oriented
approaches is that it gives a possibility to capture semantics in some limited
sense: Assuming that it is contextual semantics one is interested in, the covariation among variables carries a scent of the essence of the system. This interpretation of meaning sounds trivial, but the mathematically solid basis makes
it possible to process and reprocess the data, and when enough of it has been
carried out, something qualitatively new can emerge. And when anthropocentric
interpretations are added, results can look smart. It needs to be emphasized here
that semantics is not only related to the cognitive system but to all cybernetic
domains.
It is this information that is now identied with the free energy that was
discussed above: Information carries the correct connotations in this context. It
needs to be emphasized that it is not the level or absolute value of the signal
that is of relevance, but its deviations from the long-term average. If the levels
of all incoming resources remain constant, or even if they vary in an identical
cycle reecting the same underlying variable, there is little free energy available:
Constant ow of information results in heat death, or trivialization of the
system, because only one principal component is needed to represent all the
variation, so that a single population/species can exhaust all resources in a
centralized manner; this means that diversity will be lost. Complex variation of
resources, on the other hand, results in nontrivial distribution among actors.
Just as in traditional control systems, also in cybernetic systems there is
the physical and the metaphysical level: There are the process ows, and
there is the ow of information. Contrary to the Prigoginian views, it is abstract
information rather than the concrete energy that ows through in the dissipative
processes.
Even though variation in signals has always been applied for modeling purposes, now there is a fundamental shift in thinking: Traditionally, the fresh variation is seen as noise that is being minimized now, on the other hand, variation
is explicitly taken as welcome phenomenon. Indeed, measurements always tell
about system properties but also about the observation process, and very different results are found when the point of view is changed. This claim is best
claried when one looks at the cost criteria that are being minimized in the
modeling processes: Again, study the general criterion (11), where u is the input data and x is its reconstruction applying the model. The weighting matrix

Genetic
activity

B1
A1

Nucleus
Cell
Environment

Genetic state
= enzyme/transcription levels
Chemical
activity

B2
A2

Metabolic state
= chemical levels/flows

Fig. 9. Levels
in a cellular
system: Genetics
and
metabolics

W determines the mutual relevances among data components, thus (partially)

determining how the system sees the world. When doing traditional maximum
likelihood matching, for example, one has
W = E{uuT }1 ,

(15)

meaning that variation in those directions where there is most variation is suppressed. In the cybernetic case one has
W = E{uuT }

(16)

instead: It is evident that the more there is variation in some direction, the
more that variation is weighted in matching. Deviations are now interpreted as
valuable resource.
4.2

Emergence of structures

Cybernetic system becomes a mirror image of its environment, the representations being optimized in the local-level interaction processes. All agents experience the same environment; how is it possible that the emergent systems have
self-organizing structures where dierent agents have varying roles? To have
more insight, let us study two very dierent examples.
In Fig. 9, a schematic illustration of the proceses in a living cell are presented
(cf. [19]). There are two very dierent kinds of subsystems: The rst metabolic
subsystem is based on chemical balance reactions, where the balance is being
found as being constrained by the incoming chemical ows through the cell wall.
The set of available chemical reactions is essentially dictated by the enzymes
that are produced as determined by the active genes. The second level, or the
genetic subsystem, is based on very dierent kinds of elementary processes, like

The Company
years - months

The Branch
decades - years

Bosses
months - days

Electric network
seconds - milliseconds
Workers
days - hours

Higher-level controls
hours - seconds

Fig. 10. A cybernetic economic system is constructed of various subsystems

that are equally cybernetic

message-RNA production, transfer, and decoding, and it is practically impossible to model this dynamics in a mathematically compact form; the situation
becomes still more complicated as there are chains of gene activations, and the
interactions esentially form a deeply interconnected network. The transcription
factors essentially dictating the gene activities are products of some other gene
activity. However, again, when one concentrates only on the resulting balance
after the transients have decayed, things become much simpler, and it can be
modeled as a linear system around the equilibria. In both subsystems, the enzyme
/ metabolite levels are integrals of the genetic / chemical activities, respectively,
and their local dynamics around the equilibria can be modeled applying the
basic cybernetic model. Enzymes being produced in the genetic subsystem are
catalysts, not being exhausted in the subsequent metabolic processes, so that the
link betveen the subsystems is, in principle, unidirectional; however, as there is
also metabolic feedback from the outside the nucleus into it, the hierarchy is
not complete rather, the structure is cyclic. In any case, it is a reasonable
abstraction to distinguish between the two levels: this is due to the dierent time
scales between the subsystems. Chemical balances are reached much faster than
the gene activation balances.
In Fig. 10, on the other hand, illustrates the many-level structure of a typical company: On the highest level, there is the outside society determining the
operating environment, and the lower levels represent the organizational arrangements within the company. It is assumed here that the company being studied
is an electric company the lowest-level subsystems are again outside the
company, their behaviors being determined again by the environment, whereas
the company actions try to keep these outside systems in some intended balance.
Again, each level of subsystems operates in dierent time scales (as illustrated in
the gure). The higher-level subsystem seems to determine the set points for

fs
xs
System s

fs
xs

System s

Fig. 11. Where the feature

extraction can be implemented

the lower-level subsystems. The same structure can be detected also in industrial automation systems: on the lowest level, there are the natural (chemical)
processes being stabilized by the lowest-level controls; the next level of balances
is determined by the regulatory controls, trying to reach the reference values;
the yet higher level in the cascade structure of controllers is determined by the
production optimization. When seen as a single system, the dynamics is sti,
containing very slow and very fast modes; the system can be seen as a combination of algebraic and dynamic constraints. Such stiness problems vanish when
dierent levels are studied separately.
These examples show how levels and (fractal) hierarchies are encountered
and manifested in real life, the subsystems being more or less independent. Each
of the subsystems tries to answer to all demands and needs, resulting in local
balances but the balances in dierent levels are tightly linked together. How
to functionalize the notion of hierarchies in a cybernetic model? Where does the
seemingly universal multi-level structure of balances emerge from? It seems to
be natural to distinguish between levels, but how can this be implemented in a
mathematically solid framework?
4.3

Frequency-domain hierarchies

Whatever the environment is really like, it is visible to the agents only in the
experienced observation data. It is not the objective environment that is being
experienced by all agents in an absolutely homogeneous way; the agents world
views are subjective, depending on how the signals are seen and how they are
interpreted. In other words, this can be expressed in the form: How the features
are extracted from the data? Within a single subsystem the feature extraction
principles remain invariant (see Fig. 11).
In mathematical terms, feature extraction can be simplest implemented (that
is, applying strictly linear methodology) by appropriate weighting of the signals.
When this weighting is carried out in frequency domain, the time scales can
naturally be taken into account and it was these time scales that seemed
to be the main dierence between subsystems. So, let us introduce a new func-

lf

ent

nm

ro
nvi

m
ste

itse

Sy

ten

got
For

le
Vei

w
gs

gs

ms

Fig. 12. How information distribution can be analyzed applying control engineering intuitions (see text)

tional structure in the cybernetic model: Assume that the signals coming in the
subsystem s are ltered through the low-pass lter
dus
= s us + s uin ,
dt

(17)

so that the Laplace-transformed input signal is

U (s) = Fs (s) Uin (s),

(18)

where the transfer function is

s
Fs (s) =
s + s

(19)

with time constant s = 1/s . This means that beyond the angular frequency
= s the visible signals start decaying; the subsystem essentially cannot see
behaviors that take place at higher frequencies (they are veiled, as shown in
Fig. 12). The information content (power in terms of variation) in the ltered
signals can be expressed in terms of the power spectrum:
U () = Hs () Uin (),

(20)

where the power spectrum corresponding to (17) is

Hs () =

2s
.
2 + 2s

(21)

Note that more precise cut-o behaviors can be implemented applying higherorder lters; for example, one can dene
Fs
(s) = Fs (s)Fs (s) =

2s
.
s2 + 2s s + 2s

(22)

It is interesting to recognize that whatever is the rate of decay, the curve is

always piecewise linear on the log/log scale, that is, when both frequencies and
signal powers are presented as logarithmic quntities. Remember the scale-free

structures: Also in fractal systems, the dependencies between variables often

follow the same functional form. Perhaps this view oers yet another intuitive
explanation why such power law behaviors are so common in nature?
Information below the cut-o frequency is seen by the system with no attenuation, but that information does not cumulate in the subsystem: Signals pass
directly through, even though they can be changed. Only eects that cumulate
in the covariance structures remain in the system. This gives us a new interpretation of the covariance matrix estimate adaptation formulation applied in
the cybernetic models: Information is being ltered through a linear rst-order
lter. Information below the cut-o frequency = s represents the essence
of the subsystem (see Fig. 12), so that
s xT }
dE{x
s
s xTs } + s xs xTs ,
= s E{x
dt

(23)

s uTs }
dE{x
s uTs } + s xs uTs ,
= s E{x
dt

(24)

and

where
(or, actually, log
log ). If there are subsystems in still lower
frequency regions, the corresponding lower-frequency informaton is mostly captured by those dedicated subsystems; on the other hand, if there exists information in the signals at higher frequencies than , there will probably be another
subsystem. As seen by this intermediate subsystem, the exterior ambient information is abstracted, and seen only as constant data determining the more or
less xed environment: Too fast signals are ltered so that only the mean value
remains to be seen, and too slow signals are being faithfully followed. The subsystems outside the information horizon cannot be explicitly controlled by the
current subsystem: It must be assumed (in the spirit of cybernetics) that the
higher-frequency systems succesfully balance themselves only reference values
can be delivered by slower (higher-level) systems. The intermediate system is,
on the other hand, subject to the reference values coming from the yet higherlevel systems; as long as there is dynamics in the slow signal range, the outer
environment changes, and there is need for continuing adaptation. To repeat
now data represents the resource, level in data represents the environment,
and variation in data represents information; and models of that information
represent memory.
In traditional cybernetic studies (Bateson, Maturana, etc.) the interactions
and feedbacks in the system are very concrete sequential chains of eects or processes. Now the time scales, or frequency ranges, are selected so that the actions
at one level become seemingly instantaneous and simultaneous. Rather than
studying dynamic phenomena, one studies static patterns. Earlier this static nature of the patterns was explained so that it is balances that are being studied;
now, however, this simplicity is reached thanks to the powerful mathematical
machinery. The stationary statistical phenomena can compactly be manipulated
in frequency domain, where the individual signal realizations, their initial values, etc., are abstracted away. In principle, stationarity of signals is assumed,

but because it is not assumed that the system has reached the nal balance, the
analyses can be made more versatile. Luckily enough, because of the selected
information interpretation, information being identied with signal energy, the
essential signal properties are so easily transferred from time domain into frequency domain.
In the proposed framework of modeling cybernetic systems (to reach emergent models), one must always abstract the actual time axis away, concentrating
on statistical patterns rather than dynamic processes. In evolution, for example,
the basic unit is one generation, so that the time constants must typically be
of the order of hundreds of years and statistical learning necessarily takes
thousands of years! There are phenomena taking place also faster; indeed, the
interaction structure within the system is very much dependent of the selected
frequency range, dierent time scales oering very dierent views into the phenomena taking place in the system. The physicists dream concerning the Theory
of Everything, or the theory explaining everything in terms of elementary particles, or, in this case, in terms of elementary cybernetic entities, is futile. A good
model studies only the most appropriate phenomena visible at a specic level, so
that when seen from dierent distances there are dierent models. No universal
model exists; however, when studying cybernetic systems the model structure
may still remain the same.
It is natural that phenomena at higher frequency ranges are observed more
often than phenomena at lower frequencies. This means that when starting
from scratch there is statistically relevant information available rst only
concerning the phenomena with fastest dynamics, and only this information
is modeled originally. As time passes, also slower phenomena can be mapped
by successive subsystems. In this sense, in complex enough environments (like
ecosystems) it is natural that the structure of cybernetic systems is in a constant turmoil, because there always exist lower frequency regions where there
is non-exhausted information. The newer systems typically utilize earlier ones,
and hierarchies emerge also without explicit ltering of signals, because of the
temporal evolution, and because the stationary state has not yet been reached.
There are also technical reasons motivating the introduction of the ltering
scheme (17). Many constructivistic systems operate in discrete time that
is, information (statistics, etc.) are collected at certain time points, and it is
assumed that behaviors between the sampling instants remains approximately
the same. Now there is yet another intuition from control engineering available.
Theory tells us that the information sampling rate in the system determines the
Nyquist frequency: The maximum frequency in the signals must not exceed half
of the sampling frequency, otherwise the high frequency components get aliased
on the lower frequencies, creating phantom behaviors. Too high frequencies
have to be ltered away. Yet another related problem emerges when the sampled
measurements are used for control: Faster and faster controls make the closedloop system time constant shorter, thus increasing the high-frequency contents
perhaps beyond the sustainable level. Finally this is the problem that is faced in
quartal capitalism, where the company performance is to be optimized without

longer-term horizon, short-term benets being maximized, trying to be faster

than what is the natural dynamics in the business eld; similarly, this problem
applies also in politics, where the approval and popularity has to be earned in
a shortsighted manner, following momentary fashions and public opinions. And
even in scientic work where the production rate is being monitored, it may be
that the overall quality suers: Experts have no time to evaluate the innovations,
and no conclusions are ever drawn. Trying to go beyond the range of available
information means going beyond the edge of chaos.
But it seems that this is a completely natural and unavoidable tendency in
constructivistic cybernetic systems that are subject to all-embracing optimization activities. Remember that in Sec. 2 the control actions tried to minimize
the resource deviations; this is only one view of eliminating free energy from
the input in an adaptive system. Time integrals of variations are reduced either through making the signal amplitudes lower, or through minimizing the
time span of the deviations. Whereas only the rst alternative (making controls
more accurate) is available in natural systems, in constructivistic systems also
the second one is available (making controls faster). Both approaches eliminate
information from the lower-level system, so that they both are routes to chaos.
In an ever-evolving cybernetic environment, the closed loop behavior becomes gradually more and more pathological: As the controls become faster, the
inner loops are nally no more essentially faster than the outer ones are, and the
dynamics become mixed, ruining the guaranteed stability properties of the individual loops. What is the (imaginary) nal outcome from this process, assuming
that such optimization continues innitely? Next, it is shown that the system
can reach a qualitatively new level, where it is again very dierent theoretical
tools that are needed.

4.4

Relations to optimal control

Optimality and robustness are often contradictory goals: Faster control means
higher sensitivity to disturbances, and this means worsened stability properties
and lower robustness against noise. Is this unavoidable, or are there ways to
circumvent this? Indeed, there is some counterintuitive intuition available here.
Assume that the time constants within the cybernetic system are essentially
in the same range, that is, the feedbacks have become so fast and powerful
that the signals have no time to naturally converge but the whole feedforward/feedback structure constitutes a single dynamic system. This is not all;
assume that the feedbacks are so powerful that the environment can be completely controlled, that is, u(t) can be altered at will, so that eectively one has
u = u. The vector x(t) is the state of the cybernetic system, starting from
some non-optimal state x0 , and the goal is to bring it to optimum state (for
simplicity, assume that this optimum state is x = 0). Because of the internal
system dynamics, governed by the formula (1), the state cannot be immediately
altered; the faster one wants to make the eects, the larger are the controls that

are needed. One faces an optimization problem that can be formulated in the
following form:
 t1

T
J=
(25)
x (t)Qx(t) + uT (t)Ru(t) dt + xT (t1 )Sx(t1 ).
t0

Here, matrices Q, R, and S are positive (semi)denite; for simplicity, Q and R

can be selected to be identity matrices. As shown in [10], the optimum behavior
of the controlled system can be expressed in the form
 dx
1 T
B v
dt = Ax BR
(26)
dv
T
1
x,
dt = A v Q
assuming that the state can be directly measured (that is, matrix D in [10]
equals identity). Because A has all its eigenvalues in the non-positive halfplane, it is stable but simultaneously AT = A must be unstable, there is
positive feedback, meaning that there is inherent instability buried in the system
structure! However, as the LQ theory also dictates the boundary values as

x(t0 ) = x0
(27)
v(t1 ) = Sx(t1 ),
it turns out that the solution is meaningful; because of the boundary value
problem formulation, the solution of the problem in this form involves iterative
simulation-based approaches. The symmetricity of the structures in Fig. 2 (and
the symmetricity of A) resembles the structure in (26); indeed, this becomes still
clearer when one rewrites (26) so that dynamics of v is represented in inverse
time:
 dx
1 T
B v
dt = Ax BR
(28)
dv
dt = AT v + Q1 x.
This can be expressed as shown in Fig. 13; the inverted integral denotes the
backward nature in the feedback branch. There is a qualitative transition here,
because the causality structure has been inverted. Are there natural systems
where such optimal control strategy would have been implemented? Of course
not uncausal structures going backward in time are nonphysical, and unstable
systems when going forward in time are equally unphysical. But in articial systems, like in constructivistic cybernetic systems, such uncausal/unstable models
could easily be implemented?
The optimality formulation can be simplied by relaxing the end time; the
stationary innite-time control problem can be expressed in terms of

T

x (t)Qx(t) + uT (t)Ru(t) dt.

(29)
J =
t0

To implement the corresponding control one has to solve the steady-state Riccati
equation
0 = Q P BR1 B T P + AT P + P A.

(30)

System
state

-1

R-1

Conjugate
state

Fig. 13. Implementation of the optimally controlled system (see text)

When this is solved for P , it turns out that that the boundary-value problem
can be transformed into an initial value problem:

x(t0 ) = x0
(31)
v(t0 ) = P x(t0 ),
so that the nal, time-invariant state control law (skipping dynamics in the
feedback branch) becomes
u(t) = R1 B T P x(t).

(32)

It needs to be recognized that the system model (in terms of matrices A and
B) by no means needs to be optimal, as long as they describe the signal dependencies appropriately. Depending of the model, x is the interpretation of the
corresponding world state, and within this framework, the presented structures
still optimize the controls: The interpreted system state x0 can be brought to
optimum (zero state) applying the assumed available inputs to the system. What
is interesting here is that information on both trophic levels is simultaneously
manipulated and minimized; signal squares (variances) being appropriately suppressed not only in the lower-level system as is traditionally the case. Perhaps in
the constructivistic systems one can get from the hardworking idiot metaphor
back to intelligent design?
4.5

Agents vs. populations

Because of the principal component nature of the information manipulation in

a cybernetic system, information is maximally relayed from u to x; this is what
theory says. However, this information preservation is not complete: As the dimension of x is assumedly lower than that of u, typically variation (information)

in some direction in the data space is ignored. But it turns out that cybernetic
systems are truly marvellous: Nature can do better, exceeding the theoretical
constraints; in principle, all information in data can be exploited by the system.
The vector x has been interpreted (in ecological, etc., applications) so that
xi stands for the size of the population representing the population i. The key
to understanding the above claim is to recognize that whereas the number of
variables within the vector x, or the number of distinct populations is less than
the dimension of u, the total number of individuals within the populations still
by far exceeds the degrees of freedom in u. Indeed, this information/entropy
analysis gives us tools for understanding the diversity of individuals within a
population, and why such nonhomogeneity has evolutionary advantage and thus
never disappears even in stationary environmental conditions.
Assume that the dimension of u is m and that of x is n. After the n most
signicant principal components have been captured in the subspace spanned by
the populations characterized by the variables xi , the remaining m n dimensions in data (assuming that the data is non-redundant) are visible within each
population, and subpopulations with diering properties can emerge within the
main populations, the intra-population behaviors following the same principles
that also govern the inter-population behaviors. Remember that typically there
are no clear knees in the eigenvalues when covariance matrices are analyzed;
this means that it is dicult to say what is the natural dimension of the data.
Nature has the same problem when it implicitly operates on that same data,
dimension selection meaning selecting a specic number of variables, or populations, in the model this means that there is no such thing as the absolutely
correct number of populations (or species) to be chosen. There is fractal continuity of properties within an ecosystem, and the distinction between populations
and subpopulations becomes blurred; however, because of the biological reasons
and ecological denitions, clear boundaries between species exist. As a matter of
fact, another factor increasing diversity within an ecosystem is the observation
that, when looking at the formulae in Sec. 2, it seems that the higher the populations of identical agents are (xi ) the lower is the available activity level (vi )
of the individuals; in this sense, it is clear that subpopulations try to dierentiate themselves in terms of total number of individuals, it is better to have
two (or more) nonidentical subpopulations than only one, because this reduces
competition.
The current framework makes it possible to analyze the property distributions further, and it is also possible to make predictions. For example, it can
be assumed that in the nal state, the total number of subpopulations (within
each population separately) equals the number of remaining degrees of freedom
in data; and the spectrum of properties in dierent species is the same. This
means that within the populations there are latent substructures if the environmental conditions suddenly change, so that some less signicant principal
components suddenly become more prominent, there already exists capability
of reacting to such changes in the ecosystem, and the adaptation to the new
conditions can be remarkably fast.

4.6

Views of thew world

The very concrete view of the emergent (sub)system structure being dictated
by the available information deserves to be studied still closer. The cybernetic
system always optimizes its representation of the visible information; or, as the
system is living in its own narrow world, it thinks it is behaving optimally. This
optimality is determined in terms of the properties of the measurement data, dictated by the statistical signal properties, so that the analysis of the nal system
properties can be reduced to analysis of statistical signal properties. There are
many possibilities to utilizing this observation. In concrete terms, parallel systems forage dierent information when their feature extraction strategies dier
from each other.
Information in signals cannot be increased at will, but it can be reduced appropriately. A good example was presented above, where dynamic lters where
applied to suppress certain frequency bands. The approach presented in Sec. 4.3
was specially interesting, because it was through applying linear dynamic structures that the features were dened, so that the there are again powerful tools
for analysis of the closed-loop system available. It needs to be commented here
that it is typically assumed that linear structures are uninteresting when representing complex phenomena, because several linear layers do not increase the
expressional power of the system; also, in a cybernetic model one linear layer
already exhausts available correlations-related information. However, multiple
layers still can be interesting in the cybernetic case: Because it is also nature
that constructs non-optimal redundancy, one layer on top of another explaining
the same variation, the layer-like linear models can reect the cybernetic realm.
But truly unanticipated phenomena in the signals can be revealed when nonlinearities are employed. There are many alternatives to how features can be
extracted, or how some information can be ignored, applying nonlinear functions; for example, there are examples in [8] where sparse coding of data is
reached when cut function is applied, eliminating negative signal values altogether. Another possibility is to implement topology among the structures using
nonlinearities: For example, localaized sensor networks were studied in [19] by
explicitly cutting away the far-apart agent connections. This kind of locationbased representations are typical in nature, where local populations that are
located far from each other, for example, do not interact.
The structural nonlinearities can also be utilized to extend the dynamic structure in cybernetic models. Normally it is static mappings from u to x that are
being modeled here, so that there is assumed to be no connection between, say,
u(k) and u(k + 1). In nature, the successive inputs are, however, typically not
independent of each other: There exist correlations between resources in successive years, for example. And assuming that predators are longer-living than
prey, they see longer periods of time, and it is the total amount of prey over the
whole life span that determines the population rather than the situation during
one single year. Indeed, as presented in [2], this kind of dynamic states can be
captured in terms of time series samples. If one augments one single signal by including also the past values in the data vector, one can start modeling dynamics,

y
y = xx

y = tansig(xx)
-2

-1

xx

-1

-2

Fig. 14. Nonlinear cumulation of information

so that static principal component analysis changes into subspace identication

(see [12]):

ui (t)
ui (t 1)

(33)
u
i (t) =
.
..

.
ui (t d)
It has been assumed that information is interpreted directly in terms of variations
(that is, the atoms of information are coded in the form E{u2i }, for example)
or covariations (in the form E{ui uj }). However, the nonlinearities can also be
applied here. Whereas the storages for information were in the linear case the
covariance matrices, now one can dene, for example,
(xuT )}
dE{f
(xuT )} + f (xuT ),
= E{f
(34)
dt
the nonlinearities in the matrices being calculated elementwise. This means that
not all information has the same weight. For example, in Fig. 14 a nonlinearity is
presented where the high ends have relatively lower emphasis, the nonlinearity
being mathematically dened as
f (xi uj ) =

2
1.
1 + exp(2xi uj )

(35)

The results after model convergence are shown in Fig. 15. The data in the experiments was rst whitened, that is, E{uuT } = I, so that second-order statistical
properties in the data vanish; whereas the standard cybernetic model will not
have any preferences what comes to directions in the data space, the model
with the nonlinear covariance calculations seems to have consistent behavior.
Indeed, when data is concentrated at a certain distance from the center, it has
proportionally higher weight, and the principal component axes will be tilted
accordingly. Without any concrete proofs, it is claimed that rather than implementing principal subspace analysis (PSA), actually the model carries out
independent subspace analysis (ICA or ISA) here (see [9]).

4
3
3

2
1

-1

-1

-1
-2
-2

-2

-3
-3

-2

-1

-4

-3
-5

-4

-3

-2

-1

-4

-3

-2

-1

Fig. 15. Resulting behaviors of nonlinearized cybernetic models: Latent variable axes turn towards independent components

This implementation of cybernetic ICA is conceptually so simple that it

may oer fresh intuitions for further developments of higher-order statistical
analyses. Can it be assumed that the cybernetic combination of feedforward and
feedback structures always results in lower-level properties being inherited on
the higher level, that is, properties dened in terms of scalar functions f being
generalized into high-dimensional functionalities?

Conclusion

Cybernetics is interdisciplinary, the same principles being applicable in very

dierent disciplines. And this is not all; the interdisciplinary nature extends not
only over the spectrum of models for natural systems, but also over the spectrum
of modeling paradigms and philosophies. It is not only analysis but also synthesis
of systems that can be understood within the cybernetic framework; and it is not
only humans that are faced with the cybernetic complexity, but also Nature itself
is. Modeling of natural systems is not only describing behaviors, but models
when implemented in an appropriate way can also tell something fundamental
about the underlying system. The humans subjective world is model of the
environment but also the objective world consists of models. It seems that
building mechanical brains in the sense of deep AI may sometime be possible:
Man can understand the mechanisms of nature, and these mechanisms can also
be explicated and implemented in technical environments (see Fig. 16).
It may be that the above discussions go too deep in mathematical nuances.
However, the beauty of Nature is buried in the details. And now there exist
powerful frameworks for keeping the details in control multivariate statistics, linear system theory, and specially control engineering are the basics of
tomorrows intellectual avant garde. Philosophy and logics is the basis of mathematics, and mathematics is the basis of applied mathematics and engineering;
however, it can be claimed that applied mathematics gives substance to future
philosophies. Without concrete down-to-earth application domains, and without
the mathematical tools and control engineering intuitions, the abstract claims
about entropy etc. would have no substance.
The diversity of nature can be explained in the cybernetic framework, as
soon as the kernel of variation exists. The role of the divine designer can be
reduced to the question: Why is there something instead of nothing?

Fig. 16. In some dissipative systems (ideal mixers) adding energy results
in higher entropy, whereas
in some other (idea mixers) structures emerge and
entropy goes down

References
1. K.J.
Astr
om and B. Wittenmark: Adaptive Control. AddisonWesley, Reading, MA
(1989).
2. K.J.
Astr
om and B. Wittenmark: Computer-Controlled Systems Theory and
Design. Prentice Hall, Englewood Clis, NJ (2nd edition 1990).
3. G. Bateson: Steps to an Ecology of Mind. Paladin Books (1973).
4. A. Kleidon, R.D. Lorenz (eds.): Non-Equilibrium Thermodynamics and the Production of Entropy: Life, Earth, and Beyond. SpringerVerlag, Berlin (2004).
5. J. Horgan: The End of Science: Facing the Limits of Science in the Twilight of the
Scientific Age. Broadway Books (1996).
6. H. Hy
otyniemi: Self-Organizing Artificial Neural Networks in Dynamic Systems
Modeling and Control. Helsinki University of Technology, Control Engineering Laboratory, Report 97 (November 1994).
7. H. Hy
otyniemi: Cybernetics Towards a Unied Model? Submitted to the Finnish
Artificial Intelligence Conference (STeP04), Vantaa, Finland (September 2004).
8. H. Hy
otyniemi: Hebbian and Anti-Hebbian Learning: System Theoretic Approach.
Submitted to Neural Networks (2004).
9. A. Hyv
arinen, J. Karhunen, and E. Oja: Independent Component Analysis. John
Wiley & Sons, New York, NY (2001).
10. H. Kwakernaak and R. Sivan: Linear Optimal Control Systems. Wiley (1972).
11. S.-K. Lin: Correlation of Entropy with Similarity and Symmetry. Journal of Chemical Information and Computer Sciences, Vol. 36, pp. 367376 (1996).
12. P. van Overschee and B. de Moor: Subspace Identification for Linear Systems:
Theory Implementation Applications. Kluwer Academic Publischers, Boston,
MA (1996).
13. R. Rosen: Essays on Life Itself. Columbia University Press (1998).
14. E. Schr
odinger: What Is Life? Macmillan (1947).
15. R. Swenson, R.: Autocatakinetics, Evolution, and the Law of Maximum Entropy
Production: A Principled Foundation Toward the Study of Human Ecology. Advances in Human Ecology, Vol. 6, pp. 146 (1997).
16. P. Turchin: Complex Population Dynamics: A Theoretical/Empirical Synthesis.
Princeton University Press (2003).
17. E.O. Wilson: Consilience: The Unity of Knowledge. Abacus (1999).
18. S. Wolfram: A New Kind of Science. Wolfram Media (2002).
19. Additional material on cybernetics will be available in public domain in near future
at http://www.control.hut.fi/cybernetics.