Untangling Complex Systems, page 96
Crisp
Fuzzy
logic
logic
logic
e
Sigmoid
Smooth
input-output
input-output
Decoherenc
relationships
relationships
Single atoms
Collection of
or molecules
atoms or molecules
FIGURE 13.25 The kinds of logic that can be processed by using either single molecules or collection of
atoms and molecules. (From Gentili, P.L., ChemPhysChem., 12, 739–745, 2011a.)
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tput
tput
Ou
Ou
(a)
Input
(b)
Input
tputOu
tput
tput
Input
Ou
Ou
(c)
Input
(d)
Input
FIGURE 13.26 Examples of linear (a), smooth (b and c) and sigmoid (d) non-linear output-input functions.
In the inset of graph D, a two-step sigmoid function is shown.
collected even by human eyes, they easily bridge the gap between the molecular and our macro-
scopic world. The relationships establishing between inputs and outputs have many shapes: they can
be either linear or non-linear, and when they are non-linear, they can be either smooth or sigmoid
(see Figure 13.26). Whenever the input-output relationship is sigmoid, having a steep slope around the inflection point, it is suitable to process crisp logic. In particular, if the input-output function is
characterized by just one step, it is proper to implement binary logic. For this purpose, it is neces-
sary to establish a threshold value and a logic convention for every input and output variable. The
variables can assume merely high or low values that become digital 1 or 0, respectively, in the posi-
tive logic convention, whereas the negative logic convention reverses this relationship. On the other
hand, if the input-output function is characterized by three or k > 3 steps, it is suitable to process
three- or multi-valued logic, respectively. Whenever the input-output relationship is smooth, it is
not adequate for crisp logic, but it is proper to implement infinite-valued logic, for instance, Fuzzy
logic.6 For this purpose, the entire domain of each variable referred to as the universe of discourse is divided into different Fuzzy sets whose shape and position define their membership functions
(Gentili 2011b).
This method is not the only way for processing Fuzzy logic by molecules. Another approach
considers the absorption bands of a compound as Fuzzy sets (Gentili et al. 2016), as we learned in
Chapter 12 for the interpretation of the human color vision. When polychromatic radiation interacts with a compound, it belongs to more than one spectral Fuzzy set, with the same or different degrees
of membership. The ensemble of degrees of membership of the radiation to the absorption bands of
a compound is Fuzzy information encoded at the molecular level. The consequent photo-induced
reactions process the collected Fuzzy information.
6 This distinction in the use of sigmoid and smooth input-output functions holds in current electronic circuits, as well.
In fact, when the electrical signals vary steeply, in sigmoid manner, they are used to process Boolean logic. On the other hand, the best strategy to implement Fuzzy logic is through analog electronic circuits that are based upon signals varying smoothly.
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Even a third way of implementing Fuzzy logic by using molecules has been proposed, so far.
It exploits compounds that exist as ensembles of many conformers (Gentili 2014b). The physical and
chemical properties of such ensembles are context-dependent, like the properties of Fuzzy sets and
the meaning of words in natural language. A set of conformers becomes a “Molecular Fuzzy set,”
and at the same time a “Word of the chemical language.” A chemical reaction becomes an event of
Fuzzy information processing, which is ruled by inter- and intra-molecular forces. The Fuzziness
of a chemical compound becomes particularly important when we consider macromolecules, where
we can have many conformational structures. For example, we already know that many proteins
in their native, functional state, cannot be described adequately by a single conformation (Tompa
and Fuxreiter 2007). Structural disorder becomes relevant in eukaryotic proteomes and correlates
with important functions. Cellular processes become computational events of Fuzzy information.
This feature should be considered when the method of logic modeling is used to describe molecular
and gene networks in synthetic biology and the development of virtual organisms (Le Novère 2015;
Macia and Sole 2014; Bavireddi et al. 2013).
A buzzing question in this research field is: how will the chemical computers look like? All the
reasonable architectures thought so far, can be partitioned in two principal families: one family
based on “interfacial hardware” and the other based on “wetware” (Gentili 2011c).
In the case of “interfacial hardware,” the computations are carried out by molecules anchored to
the surface of a solid phase. The computing molecules can be organic semiconductors, proteins or
even DNA, to cite just a few examples. Organic semiconductors can behave like silicon-based semi-
conducting devices. Information is encoded through electric signals exchanged with the outside
world through electrodes (Joachim et al. 2000). Proteins can compute through their ability to recog-
nize a specific type of molecule and switching its state by making or breaking a precisely selected
chemical bond. DNA derives its computing power from the hybridization reaction, i.e., the ability
of nucleotides to bind together using Watson-Crick pairing. There is a code of “stick” or “don’t
stick,” with DNA chains binding to form regions of double-stranded molecules or remaining free as
regions of single-stranded DNA.
In the case of “wetware,” soups of suitable chemicals process information through reactions,
coupled or not with diffusion processes. These soups can work inside a test tube wherein com-
putations are performed through perturbations coming from the outside world. Inside the test
tube, we may think of putting a “Molecular Turing machine.” The latter might be constituted by
(I) a polymer that, as the tape of the Turing machine, has sites that can be modified chemically
and reversibly, and (II) a polymer-manipulating catalyst, anchored to the macromolecular tape,
which, as the head of the Turing machine, modifies the sites of the tape, controlled by external
stimuli (such as light, electrons or acid/base chemicals) (Varghese et al. 2015). Alternatively,
the chemical soups can operate in microfluidic systems structurally related to the pattern of
the current electronic microchips. The microfluidic channels are the wires conveying chemical
information, while logic operations are processed inside reaction chambers. In these systems,
computation can also be pursued through chemical waves propagating across excitable chemical
media. An alternative approach consists in devising open systems wherein properly combined
chemical reactions mimic the signaling network of cells. Finally, it might be worthwhile trying
to mimic a nervous system by implementing a network of distinct processing units, similar to
neural networks. In a network of neurons, information is processed in two different kinds of
spatial coordinates: a horizontal one and a vertical one. Along the vertical coordinate, informa-
tion is processed hierarchically, from the molecular level (especially through conformations),
to the mesoscopic one (through reaction-diffusion processes), up to the macroscopic one by
the appearance of ordered structures playing the role of communication channels between the
microscopic and the macroscopic worlds. Along the horizontal coordinate, information is pro-
cessed and conveyed due to the chemical waves interconnecting spatially distant processing
units. With a such complex computing system, it will be possible to devise a chemical computer
more similar to the brain rather than to the current electronic computers.
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13.3.2.4 The “Ultimate Laptop”
The physicist Seth Lloyd (2000) has calculated the physical limits of computation for an “ulti-
mate laptop” operating at the limits of speed and memory space allowed by the laws of phys-
ics. According to the time-energy Heisenberg uncertainty principle extended by the Margolus and
Levitin theorem, a quantum system in a state with average energy E takes time at least ∆ t = π /2 E
to evolve to an orthogonal state. If we assume that our “ultimate laptop” has a mass of 1 kg, it has
an average energy of E = mc 2 = 8 9874
.
×1016J. Therefore, its maximum speed of computation is
5 4258 1050
.
×
operations per second. For the ultimate laptop, the rate grows by increasing its mass.
Conventional laptops operate well below the ultimate laptop. In fact, in a conventional laptop, most
of the energy is locked up in the mass of the materials, leaving only a tiny portion for performing
calculations. Moreover, in a conventional laptop, many electrons are used to encode just one bit.
From the physical perspective, such a computer operates in a highly redundant fashion. However,
redundancy is required to reduce errors. A way of limiting redundancy is to compute with single
subatomic particles.
The maximum amount of information, which the ultimate laptop can process and store, is
determined by the total number of distinct physical states that are accessible. This number is,
of course, related to the information entropy of the system. The calculation of the total entropy of
1 kg of matter contained in 1 L of volume would require complete knowledge of the dynamics
of the elementary particles, quantum gravity, and so on. We cannot access all this information.
However, its entropy can be estimated by assuming that the volume occupied by the laptop is a
collection of modes of elementary particles with total energy E. It results (Lloyd 2000) that the
amount of information that can be stored by the ultimate laptop is ∼1031 bits. This amount is much
larger than the ∼1010 bits stored in a current laptop. This discrepancy is because conventional
laptops use many degrees of freedom to store just one bit, assuring stability and controllability.
If the computation to be performed is highly parallelizable or requires many bits of memory,
the volume of the computer should be large, and the energy available should be spread out evenly
among the different parts of the computer. On the other hand, if the computation is highly serial
and requires fewer bits of memory, the energy should be concentrated in a smaller volume. Ideally,
the laptop can be compressed up to the black-hole limit (a black-hole of 1 kg has a “Schwarzschild
radius” of 1027m). Then, the computation becomes fully serial. A black-hole is suitable for comput-
ing because, according to the quantum mechanical picture, it is not entirely black. In fact, a black
hole emits the so-called Hawking radiation that contains information about how it has been formed:
what goes in does come out but in an altered form.
13.4 LAST CONCLUSIVE THOUGHTS AND PERSPECTIVES
We are at the end of this fascinating journey of discovering Complexity. We have learned that
Complex Systems are (I) networks with many nodes (II) in out-of-equilibrium conditions,
(III) which exhibit emergent properties. Complexity is “disorganized” (to cite a term used by
Warren Weaver (1948) in his farsighted paper titled “Science and Complexity”) when the networks
are random. The probability theory and statistical mechanics are powerful tools for the description
of the random networks. Complexity is “organized” when the networks are either regular, small-
word, scale-free, modular or hierarchical. Their behavior can be investigated by analytical math-
ematical methods, such as that used by Marcus Covert and his colleagues at Stanford University
for Mycoplasma cell’s overall functions (Karr et al. 2012). However, if in a system of “Organized
Complexity” there is an enormous number of factors that are all interrelated into an organic whole,
an analytical approach becomes computationally intractable. In this latter case, we collect Big
Data. But, then, the overflow of data must be transformed into information, knowledge, and finally
wisdom (according to the DIKW pyramid). For this purpose, it is necessary to develop a model.
Into the model, we are not expected to incorporate everything we know about a Complex System.
In fact, we could originate a model that is too complicated to be understood. One can make a
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complicated model do anything one likes by fiddling with the parameters (suffice to think about
the remarkable predictive power of Artificial Neural Networks). A model that predicts everything is
not necessarily useful for understanding the phenomenon it refers to. A model should be an inten-
tional symbolic simplification of a Complex System. The features of the model depend on which
level we want to describe the behavior of a Complex System. As properly stated by the British
statistician George Box (1919–2013), “All models are wrong, but some models are useful.” The
perfect model is not the model that best represents the world around us but, instead, is a model that
in some ways exaggerates the aspects of the world we are most interested in and can help us win the
challenges we are facing. For instance, can we predict when and where earthquakes occur? How
much do human activities affect the climate? How can we save the biodiversity of our ecosystems?
Which are the best strategies to guarantee a sustainable economic growth everywhere in the world?
How can we guarantee social justice? Can we defeat cancer? Is it possible to slow down the aging
of our bodies? And so on.
A useful methodology to build models for Complex Systems is represented in Figure 13.5.
It assumes that any Complex System computes. Therefore, we must discern the inputs, outputs, and
the computations that the system performs. Then, we need to develop algorithms that might carry
out those computations. A suitable algorithm will become a predictive tool and a decision support
system to try to win the Complexity Challenges related to that particular Complex System. Finally,
we may contrive a mechanism for implementing the algorithm. If such a mechanism exists, it will
contribute to the development of technology.
By constructing good models of Complex Systems, we should store the knowledge necessary to
formulate a new theory and promote the expected third “gateway event” in the humankind journey
to discovering the secrets of nature (remind Chapter 1). In fact, this new theory will probably apply equally well to the cell, the immune system, the brain, the ecosystem, human societies, the economy,
and perhaps even the climate and the geology of our planet. Most likely, this new theory will have
information as the pivotal variable. Information will be included not only in terms of quantity but also
in terms of quality. The concepts of granulation and graduation of the variables, which are corner-
stones of the theory of Fuzzy logic, could give clues on how to rationalize the quality of information.
To succeed in our effort of understanding Complex Systems and winning the Complexity
Challenges, it is vital to support the formation of interdisciplinary research teams and promote
the diffusion of interdisciplinary academic degrees focused on Complexity. In the interdisciplin-
ary academic degrees in Complexity (see Figure 13.27), we should prepare the new generations of
“Philo-physicists,” i.e., scientists having the following attributes:
• Wonder for the beauty of nature.
• Curiosity for the unknown.
• Open-mindedness, multidisciplinary interests, and knowledge. It is fundamental to have
polymath minds who approach problems by making analogies among ideas and concepts
belonging to different disciplines.
• Dedication, patience, and perseverance in doing experiments and collecting data. In other
words, patience in querying nature and catching its answers, by using the scientific method
based on the rigorous laws of rational logic.
• High standards of personal honesty and love of truth when interpreting the answers of
nature to our queries.
• Critical thinking, creativity, independence, and resourcefulness to think “out of the box”
and formulate new ideas and theories.
• Ingenuity to contribute to the development of technology that exerts always a positive feed-
back action on the scientific enquiry.
• Resilience to recover from defeats and criticism.
• Awareness and excitement for having the possibility of contributing to the development of
science and the improvement of human psycho-physical well-being.
