Untangling Complex Systems, page 92
• Class 4: Localized structures, moving around and interacting with each other in very com-
plicated ways.
Another famous cellular automaton is that invented by the mathematician John Conway (Gardner
1970) and called the “Game of Life.” It is an infinite two-dimensional grid of square cells, each of
which is in one of two possible states: off or on, 0 or 1, white or black, dead or alive, unpopulated or
populated. Every cell has eight neighbors: four adjacent orthogonally and four adjacent diagonally
(see Figure 13.14). When we start from a certain configuration of the grid, its evolution over discrete time steps is determined by a set of “genetic laws” listed as follows.
• Birth: a dead cell, surrounded by three live cells, becomes alive at the next time step.
• Survival: a live cell, surrounded by two or three live cells, stays alive.
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t
t+1
t+2
t+3
t+4
FIGURE 13.14 Evolution of a glider in the “Game of Life,” reproduced by using “Mini-Life” model,
Complexity Explorer project, http://complexityexplorer.org.
• Loneliness: a live cell, surrounded by less than two live neighbors, dies.
• Overcrowding: a live cell or a dead cell, surrounded by more than three live neighbors,
dies or stays dead.
Conway formulated these “genetic laws” after a long period of careful experimentation to create
interesting life-like behaviors. One interesting pattern is the “glider” represented in five consecutive
time steps in Figure 13.14. It moves in a southeast direction indefinitely, as if it were a chemical wave.
Other intricate patterns have been discovered, such as the “glider gun,” able to generate an infi-
nite number of “gliders.” Conway demonstrated that by assembling “glider guns,” “gliders,” and
other structures, it is possible to carry out the fundamental logical operations: AND, OR, NOT.
Therefore, the “Game of Life” can simulate a Turing machine that is a universal computer. It can
run any program that can operate on a standard computer. However, it is very challenging to design
initial configurations that can do non-trivial computations.
The most important message we receive from the study of cellular automata is that a simple sys-
tem, ruled by simple laws, can give rise to complex phenomena. Therefore, cellular automata pro-
vide alternative models to interpret Complex Systems. Models based on cellular automata harness
the parallelism of natural systems and are appropriate in highly nonlinear regimes where growth
inhibition effects are relevant, and discrete threshold values emerge. The theory of cellular automata
has been used in many fields. For example, it has been applied with success for the description of
excitable media, such as the growth of dendritic crystals, snowflakes (Zhu and Hong 2001), the
Belousov-Zhabotinsky reaction, and turbulent fluids (Markus and Hess 1990). Moreover, it has been
used for the representation of many biological phenomena (Ermentrout and Edelstein-Keshet 1993),
such as the formation of patterns, the pigmentation of many mollusk shells (Kusch and Markus
1996), and the uptake and loss of gases by plants through stomatal apertures (Peak et al. 2004).
Finally, cellular automata have been implemented in different ways; for example, by reaction-
diffusion processes (Hiratsuka et al. 2001; Stone et al. 2008), nanometric quantum dots (Snider
et al. 1999; Blair and Lent 2003) and memristors (Itoh and Chua 2009).
13.3.1.7 Artificial Intelligence, Fuzzy Logic, and Robots
Human intelligence may be conceived as the emergent property of the human nervous system
(HNS). The nervous system is a complex network of billions of nerve cells organized in a structure
where we can distinguish three main elements:1 (1) the sensory system (SS), (2) the central nervous system (CNS), and the effectors’ system (ES) (Paxinos and Mai 2004).
1. The sensory system consists of many different specialized receptor cells, which either
catch the stimuli coming from the external environment or monitor the inner state of the
body. The receptor cells transduce the different kinds of stimuli in electrochemical signals
that are sent to the CNS. The electrochemical signals contain the sensory information, and
the sensory information may or may not lead to conscious awareness. If it does, it is called
sensation.
1 Traditionally, the human nervous system is presented as constituted by two parts: (a) the Central Nervous System (CNS) that is the brain and the spinal cord, and (b) the Peripheral Nervous System (PNS) made of the nerves carrying signals to and from the CNS.
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2. The central nervous system consists of the brain and the spinal cord. The spinal cord is
the highway of information. It lets the information coming from SS to reach the brain,
and the information sent by the brain to reach the ES. The principal and fundamental roles
of the brain are those of processing the information coming from the SS, memorizing it
permanently or temporarily, and taking decisions. The brain extracts meaning from the
world by mixing the different forms of sensory perception (Stein 2012).
3. Effectors’ system consists of glands (both exocrine and endocrine) and muscles. Both of
them receive instructions from the CNS.
It is interesting to compare the structure and working mechanisms of our nervous system with those
of a von Neumann electronic computer. We may say that the brain plays the roles of both central pro-
cessing unit and memory. The sensory system along with the effectors’ cells play like the information
exchanger of an electronic computer, and the spinal cord is the data and instructions bus. Apart from
these formal similarities, there are sharp differences between the human nervous system and the von
Neumann electronic computer. In fact, in any von Neumann computer the information is encoded
by electrical signals, whereas in any brain through electrochemical signals. Moreover, in any von
Neumann computer information is digital, whereas in HNS it is both digital and analog. In any von
Neumann computer, we distinguish hardware and software. In brains, this distinction is not evident.
The hardware of a computer is rigid, whereas any brain is flexible, and the synaptic connections
among neurons are plastic. In a von Neumann computer, the operations are executed synchronously,
whereas different regions of the brain work asynchronously. A brain is smaller and more efficient
than a computer because it can achieve five or six orders of magnitude more computation for each
joule of energy consumed (Ball 2012). The more significant efficiency of the brain is partly due to its
3D network architecture that is sharply different from the 2D grid-like structure of a von Neumann
computer. A supercomputer is much faster in doing mathematical computations, and it has a much
stronger memory for numerical data than any brain, even if we consider the brain of a mathematical
genius. This evidence does not mean that we have reached the “Singularity,” the point where machines
surpass human intelligence, as envisioned by Kurzweil (2005). In fact, human intelligence does not
reside solely on its computing rate and memory space, but on its ability to learn, understand scenarios,
think creatively, react to unexpected events, recognize variable patterns, and compute with words.
We, humans, can handle both accurate and vague information by computing with numbers and
words. We can take decisions in complex situations when there are many intertwined variables,
and accuracy and significance become two attributes of our statements, which are mutually exclu-
sive (see Figure 13.15), in agreement with the Principle of Incompatibility (Zadeh 1973).
Therefore, it is evident that it is worthwhile trying to deeply understand the foundations and running
mechanisms of the human intelligence and reproduce them artificially. The term “Artificial Intelligence”
was coined by the American computer scientist John McCarthy. It was used in the title of a workshop
that took place in the summer of 1956 at Dartmouth College in Hanover, New Hampshire (Moor 2006;
Russell and Norvig 1995). The research on Artificial Intelligence has had cycles of success, misplaced
optimism, and resulting cutbacks in enthusiasm and funding. Recently, it has received a renewed boost
by the research initiatives called “The Decade of the Mind” started in 2007 (Albus et al. 2007), and “The
Human Connectome Project” launched in 2009. A comprehensive mapping of the structural and func-
tional connections, across all scales, from the micro-scale of individual synaptic connections between
neurons to the macro-scale of brain regions and interregional pathways (Behrens and Sporn 2011) should
facilitate the imitation of the human mind. The success in the imitation of the human mind will have a
significant impact on science, medicine, economic growth, security and well-being.
For mimicking human intelligence, it is crucial to simulate the human power of computing with
words and implement not only binary but also analog logic. These two tasks can be accomplished,
at the same time, by turning to Fuzzy logic (Zadeh 2008). Fuzzy logic is based on the theory of
Fuzzy sets. As we learned in Chapter 12, any Fuzzy set (Zadeh 1965) breaks the Law of Excluded-
Middle. In fact, an item may belong to a Fuzzy set and its complement at the same time, with the
How to Untangle Complex Systems?
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Accuracy
Significance
To probe the
economic health of
No way! It is
a country
enough to
accurately, we
monitor four
should monitor the
parameters:
activities of all the
GDP, inflation,
families and
deficit and
companies ...
interest rate
FIGURE 13.15 An illustration of the Incompatibility Principle. According to it, accuracy and significance
are two mutually exclusive properties of the statements regarding Complex Systems.
same or different degrees of membership. The Law of Excluded-Middle is the foundation of the
binary logic. In binary logic any variable is partitioned into two classical sets after fixing a threshold
value: one set will include all the values below the threshold, whereas the other one will contain
those above. In case of a positive logic convention, all the values of the first set become the binary 0,
whereas those of the other set become the binary 1. The shape of a classical set is like that shown in
Figure 13.16a. The degree of membership function, for such a set, changes discontinuously from 0
1.2
1.0
0.8
μ 0.6
0.4
0.2
0.0
(a)
Variable X
1.0
1.0
0.8
0.8
μ 0.6
0.6
0.4
μ 0.4
0.2
0.2
0.0
0.0
(b)
Variable X
(c)
Variable X
1.0
1.0
0.8
0.8
μ 0.6
μ 0.6
0.4
0.4
0.2
0.2
0.0
0.0
(d)
Variable X
(e)
Variable X
FIGURE 13.16 Shapes of the membership functions ( μ) for a generic variable X. The case of a “classical”
set: plot (a) Examples of Fuzzy sets are shown in (b) (sigmoidal shape), (c) (triangular shape), (d) (trapezoidal
shape), and (e) (Gaussian shape) plots.
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1.0
1.0 Very low
Low
Medium
High
Very high
0.8
0.8
0.6
0.6
µ
µ
0.4
0.4
0.2
0.2
0.0
0.0
0
10
20
30
40
50
60
0
10
20
30
40
50
60
(a)
T (°C)
(b)
T (°C)
FIGURE 13.17 Graphical representation of the first two steps needed to build a FLS: granulation (a), and
graduation (b) of the variable temperature (T) expressed in degree Celsius and taken as an example.
(below the threshold) to 1 (above the threshold). On the other hand, Fuzzy sets can have different
shapes. They can be sigmoidal (b in Figure 13.16), triangular (c), trapezoidal (d), Gaussian (e), to cite a few. For a Fuzzy set, the degree of membership function changes smoothly from 0 to 1.
It derives that Fuzzy logic is an infinite-valued logic. Fuzzy logic is suitable to describe any com-
plex non-linear cause and effect relation after the construction of a proper Fuzzy Logic System (FLS).
The development of a FLS requires three fundamental steps. First, the granulation of every variable
in Fuzzy sets (see Figure 13.17a). In other words, all the possible values of a variable are partitioned in a certain number of Fuzzy sets. The number, position, and shape of the Fuzzy sets are context-dependent. The second step is the graduation of all the variables: each Fuzzy set is labeled by a
linguistic variable, often an adjective (see Figure 13.17b as an example for the variable temperature expressed in degree Celsius). These two steps imitate how our senses work (Gentili 2017). Physical and
chemical sensations are epitomized by words in natural language. For instance, human perceptions of
temperature are described by adjectives such as “freezing,” “cold,” “temperate,” “warm,” “hot,” and
so on. Visible radiations of different spectral compositions are described by words such as “green,”
“blue,” “yellow,” “red,” and other words that we call colors. Savors are classified as salt, sweet, bitter,
acid, umami. There are different words to describe the other sensations. These words of the natural
language are cognitive granules originated by perceptions. Perceptions are constrained, noisy, and
they generate clumps of Fuzzy-information (remember what we studied in Chapter 12).
The third final step required to build a FLS is the formulation of Fuzzy rules that describe the rela-
tions between input and output Fuzzy sets. Fuzzy rules are syllogistic statements of the type “IF…,
THEN…” The “IF…” part is called the antecedent and involves the linguistic labels chosen for the
input Fuzzy sets. The “THEN…” part is called the consequent and contains the linguistic labels
chosen for the output Fuzzy sets . When we have multiple inputs, these are connected through the
AND, OR, NOT operators (Mendel 1995). At the end of the three-step procedure, we have a FLS.
In a FLS we distinguish three elements: A Fuzzifier, a Fuzzy Inference Engine, and a Defuzzifier (see
graph a in Figure 13.18). A Fuzzifier is based on the partition of all the input variables in Fuzzy sets.
A Fuzzifier transforms the crisp values of the input variables in degrees of membership to the input
Fuzzy sets. The Fuzzy Inference Engine is based on the Fuzzy rules (see graph b in Figure 13.18).
The IF-THEN rules are patches (Kosko 1994) covering the nonlinear function that is represented as
the winding black trace of graph b in Figure 13.18. The higher the number of rules, the more accurate is the description. The Fuzzy rules are formulated by experts (according to the Mamdani’s method),
or they are determined by automatic learning mathematical techniques applied to a set of training
experimental data (according to the Sugeno’s method).2 The Fuzzy Inference Engine turns on all the
2 For more details about the methods of formulating Fuzzy rules, read the clear tutorial by Mendel (1995).
How to Untangle Complex Systems?
475
Defuzzifier
Y VH
IF VH,
Output
THEN VH
variable
IF H,
VH
H
THEN H
Y
Fuzzy inference
engine
IF M,
Crisp
H
tput variable
M
THEN M
input
M
Ou
If-then
IF L,
L
THEN L
rules
L
IF VL,
THEN VL
VL
VL
0
1
Fuzzifier
1
1
VL
L M H
VH
0
VL
L
M
H
VH
Input variable X
0
Input variable X
Crisp
(a)
input
(b)
FIGURE 13.18 Graph a, on the left, shows the three elements of a FLS built according to the Mamdani’s
method (Mendel 1995). It consists of a Fuzzifier (based on Fuzzy sets of the input variable), a Fuzzy Inference
