Development of Chemical Systems to Process Fuzzy Logic at the Molecular Level
Pier Luigi Gentili
Department of Chemistry, Perugia University, Via Elce di sotto 8, Perugia, Italy
Keywords: Molecular processors, Qubit, Fuzzy sets, Wavefunctions, Fuzzy logic systems, Logistic function.
Abstract: Current Information Technology is pursuing a revolution in the design of computing machines: it is trying
to pass from macroscopic processors miniaturized through top-down approaches, to microscopic processors
made of single molecules assembled through bottom-up approaches. When computations are carried out by
single atoms and molecules, quantum logic can be processed. It is difficult to devise a quantum computer
due to the decoherent effects exerted by the surrounding environment. However, it is still possible to work
out with molecules, by abandoning the lure of quantum logic and processing classical logic. Single
molecules make binary computations, whereas ensembles of molecules can be used to implement either
Boolean logic gates or Fuzzy inference engines. The behaviours of two chemical compounds after photo-
excitation are described as examples of quantum systems whereby Fuzzy logic can be processed by
exploiting the decoherent effects exerted by the surrounding microenvironment.
Information Technology is trying to develop systems
capable of processing larger amounts of information
at increasingly high speed and lower power, volume
and price. Current computers are based upon
semiconductor technology and electrical signals.
Their computational power has been growing
exponentially. The pace of their improvement is
epitomized in the empirical Moore’s law, stating that
the number of transistors per chip doubles every
eighteen months (Jurvetson, 2004). Moore’s law has
been obeyed, almost precisely, in the last forty years,
by virtue of the continuous progress in the
miniaturization of computer’s processors.
In current computing machines based upon
classical physics, both Boolean and Fuzzy logic can
be processed. Binary information is recorded in
macroscopic two level systems: i.e., when there is no
electrical current flowing through a wire, it
represents a logical “0”, whereas when there is some
current flowing through, it represents a logical “1”.
These two states form a bit of information. All
Boolean computations are based on logical
manipulation of bits through logic gates acting on
wires representing these bits. On the other hand, the
most effective implementations of Fuzzy logic have
been achieved by the use of analog electronic
circuits that are based on continuously variable
electrical signals. Boolean binary logic has the
peculiarity of manipulating only statements that are
true or false, reducible to strings of zeros and ones.
However, quite often, the available data and
knowledge suffer a certain degree of uncertainty and
imprecision, especially when they are based on
subjective linguistic statements. In all these cases, it
is still possible to process information by
abandoning hard computing, based on binary logic
and crisp systems, and adopting soft computing,
based on Fuzzy logic, neural nets and probabilistic
reasoning (Zadeh, 1994). Fuzzy logic is likely to
play an increasingly important role in the conception
and design of systems whose machine intelligence
quotient is much higher than that of systems
designed by conventional methods, since it affords
to deal with certain and uncertain information,
objective and subjective knowledge.
Until now, the miniaturization of computer’s
elements has been pursued by the top-down
approach through photolithography and related
techniques. The race towards always smaller
dimensions is now approaching some fundamental
limits, because processors are being made of a few
atoms. Fundamental technological problems arise,
such as current leakage and heat dissipation.
Therefore, an alternative strategy, named as bottom-
Gentili P..
MOLECULAR FUZZY INFERENCE ENGINES - Development of Chemical Systems to Process Fuzzy Logic at the Molecular Level.
DOI: 10.5220/0003125102050210
In Proceedings of the 3rd International Conference on Agents and Artificial Intelligence (ICAART-2011), pages 205-210
ISBN: 978-989-8425-40-9
2011 SCITEPRESS (Science and Technology Publications, Lda.)
up approach (Feynman, 1960), has been put forward
over the last few years. It is based on the idea of
building a computer with its underlying hardware
based on single molecules, self-assembled
supramolecular entities and/or chemical reaction
networks coupled with diffusion processes. The
development of molecular logic gates will allow not
only electrical but also other physical and chemical
inputs and outputs to be exploited. The purpose is to
find out molecular systems whereby not only
Boolean but also Fuzzy logic can be processed.
Working with single atoms and molecules entails to
deal with the laws of the quantum-mechanics,
therefore quantum computation can also be carried
out. The elementary unit of quantum information is
the qubit or quantum bit (Schumacher, 1995). A
qubit is a quantum system that has two accessible
states, labelled
0 and 1 , and it can exist as
superposition of them; in other words, a qubit (
Ψ )
is a linear combination of
0 and 1 :
wherein a and b are complex numbers, and a
normalization convention
ba is generally
adopted. A computer based upon qubits promises to
be immensely powerful because it can be in
multiple states at once. For instance, if it consists of
n unmeasured qubits, it can be in an arbitrary
superposition of up to 2
different states
simultaneously, differently from a classical
computer that can only be in one of the 2
states at
any one time (Bennet, 2000). The superposition can
involve also the quantum states of physically
separated particles, if they are entangled (Plenio,
The main difficulty in building a quantum
computer comes from the fact that quantum states
must constantly contend with insidious interactions
with their environment triggering loss of coherence.
The superposition state of a qubit, for example
defined in equation (1), collapses by decoherence,
into a single state,
0 or 1 , with probability a
, respectively.
Whenever decoherence effects are unavoidable,
the lure of quantum information vanishes. However,
it is, anyway, possible to compute with molecules by
processing crisp logic. Since a qubit,
Ψ , can
collapse into one of two available states,
0 or 1 ,
it seems obvious that just Boolean logic can be
implemented at the molecular level.
The ability of making computation by molecules
resides in their structures and their reactivity (i.e.
affinity). The order, the way the atoms of a molecule
are linked, and their spatial distribution rule the
intra- and inter-molecular interaction capabilities of
the molecule itself, defining its potentiality of
storing, processing and conveying information.
Any molecule or supramolecular assembly that
can exist in two states of different chemical or
physical properties, may be regarded as a potential
logic gate if there exist physical or chemical stimuli
that can change reversibly the state of the system.
Computing with molecules allows multiple inputs
and outputs to be used: not only electrical but also
chemical, optical, mechanical, thermal, magnetic
and other physical ones. The nature of logic gates
that can be implemented depends on the response of
chemical compounds to the physical or chemical
When computations are performed through an
ensemble of a huge number of molecules, the
collective response of the chemical system is
continuous on a macroscopic level, although only
discrete processes of Boolean character are involved
at the molecular level. Whenever the macroscopic
input-output relation has sigmoid shape, it has
digital character and is suited to process binary
logic. For this purpose, it is necessary to establish a
threshold value and a logic convention for every
input and output variable. The variables can assume
simply high or low values that become digital 1 or 0,
respectively, in the positive logic convention,
whereas the negative logic convention reverses this
relationship. On the other hand, whenever the output
variable varies smoothly as response of the
continuous variation of the input, their relation has
analog character and can be exploited to process
Fuzzy logic. 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
Different technological solutions have been put
forward for the implementation of chemical
computers. They can be grouped in two sets: one
that can be defined as based upon “interfacial
hardware” and the other that is based upon the so-
called “wetware”. In the case of “interfacial
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
hardware”, the computations are carried out by a
single or an ensemble of molecules anchored to the
surface of a solid phase (see Figure 1 A).
Figure 1: Technological implementations of chemical
computing: (A) interfacial hardware; wetware (B) in a test
tube, (C) through a microfluidic system, and through a (D)
type-cell system wherein “r” stands for receptor and “e”
for effector.
In the case of molecular computing based upon
“wetware”, soups of suitable chemicals process
information through reactions, coupled or not with
diffusion processes. These soups can work inside a
test tube (Figure 1 B) wherein computations are
performed through perturbations coming from the
outside world. Alternatively, the chemical soups can
operate in microfluidic systems (see Figure 1 C)
structurally related with the pattern of the current
electronic microchips: the microfluidic channels are
the wires distributing the information, while logic
operations are processed in the reaction chambers. A
refined way of implementing chemical logic gates
entails emulating the complex molecular signalling
circuits that are active inside a living cell (see Figure
1 D). These circuits consist of (i) receptor units,
sensing the inputs coming from the outside, (ii)
processors, made of reaction-diffusion processes,
and (iii) effectors, unveiling the results of
So far many chemical systems have been proposed
as digital logic gates (Szaciłowski, 2008), whereas a
few have been found suited to implement Fuzzy
inference engines. An example of the latter (Gentili,
2007b) is offered by aromatic carbonyl and nitrogen-
heterocyclic compounds (see Figure 2), exhibiting
Proximity effects in their photophysics (Gentili,
Figure 2: (a) 6(5H)-Phenathridinone, an example of
aromatic carbonyl compound, and (b) phenanthridine, an
example of nitrogen-heterocyclic compound.
The excitation of these compounds by UV-
visible radiation of right frequency triggers the
formation of a quantum state,
, that is the
superposition of two pure electronic excited states,
as indicated in equation (4):
The two wavefunctions,
, are
relative to the electronic (
,*) and (n,*) states,
primarily due to the C=O and C=N chemical groups.
qubit has usually a short lifetime, of the
order of nanoseconds; therefore, it is not suited to
implement quantum computation.
collapses to one of its states,
. If it
collapses to
, the molecule can emit light,
whereas if it collapses to
, the molecule does
not fluoresce at all, since it relaxes thermally to the
electronic ground state bypassing the (
,*) state.
The probability of getting
is equal to a
whereas that of getting
is equal to b
. The
coefficients, a and b, depend on the vibronic
coupling between the two close-lying (
,*) and
*) states (Siebrand, 1980). In planar aromatic
molecules, such as those of Figure 2, the mode that
couples the (
,*) and (n,*) states is a low
frequency out-of-plane bending mode since n and
orbitals are symmetric and antisymmetric with
MOLECULAR FUZZY INFERENCE ENGINES - Development of Chemical Systems to Process Fuzzy Logic at the
Molecular Level
respect to reflection through the molecular plane
(Lim, 1986). The wider the energy gap between the
,*) and (n,*) states, the weaker the coupling
between them. When the two electronic excited
states couple weakly, the a coefficient of equation
(2) assumes large values. That means the probability
) that
collapses to
is high. If a
large, the observable fluorescence quantum yield,
measured for an ensemble of molecules, results
It is possible to control the extent of the coupling
between the (
,*) and (n,*) states, and hence the
fluorescence quantum yield (
) of an aromatic
carbonyl and nitrogen-heterocyclic compound, by
some environmental conditions, such as the
temperature and the solvent. In fact, high
temperature (T) implies large thermal energy
available to the molecular vibrational motions (in
particular to the low frequency out-of-plane bending
mode, cited above), and hence strong coupling.
Moreover, if the lowest excited state occurs to be
,*) in character, the energy gap between the
,*) and (n,*) states may increase in going from
aprotic to protic solvents, since the (n,
*) state blue
shifts, whereas the (
,*) state red shifts, under the
influence of hydrogen bonding. In other words, by
choosing solvents with strong hydrogen bonding
donation ability (HBD), it is possible to weaken the
coupling between the two excited states.
An example of the dependence of
on T and
HBD ability of solvent is shown in Figure 3 for
6(5H)-Phenanthridinone. From the 3-D plot of
Figure 3, it is evident that
varies smoothly with T
and HBD ability of the solvent, therefore their
relation is suited to process Fuzzy logic. Fuzzy
Logic Systems (FLS), based upon the photophysics
of 6(5H)-Phenanthridinone, can be implemented.
This is due to the fact that the quantum phenomenon
of superposition, underlying the Proximity effect of
6(5H)-Phenanthridinone, has characteristics
common to the algebra of Fuzzy sets. In fact, the
, produced by absorption of an UV
photon, can be conceived as a Fuzzy variable
divided in two Fuzzy sets:
. ,
which is a superposition of
belongs to both Fuzzy sets, at the same time. The
degree of membership of
to the
set is a
, whereas the degree of membership to the
Fuzzy set is b
. The degree of
membership of
rules the fluorescence
quantum yield for an ensemble of molecules. The
values of the degrees of membership of
to the
two Fuzzy sets can be modulated through external
macroscopic parameters, such as T and HBD ability
of solvent.
Figure 3: Dependence of the fluorescence quantum yield
) of 6(5H)-Phenathridinone on temperature (T) and
Hydrogen Bonding Donation (HBD) power of the solvent.
It ensues that FLS can be built by means of this
class of compounds with T and HBD ability of the
solvent as inputs,
as output and ultraviolet (UV)
radiation as power supply. Through the Mamdani’s
or Sugeno’s methods, the input and output variables
are fuzzified, i.e. they are partitioned in Fuzzy sets,
defining their related membership functions (μ) and
assigning linguistic variables to each Fuzzy set. IF-
THEN statements, wherein the multiple antecedents
are connected through the AND operator, are fixed
as rules. Each Fuzzy rule is interpreted as a Fuzzy
implication. Since the antecedent parts of the rules
are connected through the AND operator and the
cornerstone of scientific modelling, i.e. the cause
and effect relation, has to be respected, the
membership functions of the rules (
),( kj
) are
defined only by the minimum (equation 3) and the
product (equation 4) t-norms:
As the way of determining
),( kj
is fixed, it is
necessary to specify how to combine the IF-THEN
rules. Generally, they are combined through the t-
conorm operator, i.e. the Fuzzy union. The last
element of a FLS is the defuzzifier. A criterion for
its choice can be based on the attempt of optimising
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
the prediction capabilities that the built FLS exhibit
towards the Proximity Effect phenomenon of 6(5H)-
Another example of molecular system whereby
Fuzzy logic can be processed is offered by
tryptophan (Gentili, 2008a). When tryptophan
absorbs an UV photon, it passes from the electronic
ground state (S
) to the first excited state (S
). In S
tryptophan is unstable. It decays in a few
nanoseconds by following different paths (see
Figure 4): it can emit light (fluorescence); it can
relax by dissipating the electronic energy in heat
(thermal relaxation); it can chemically transform by
electron or proton transfer reactions and finally, in
the presence of an effective fluorescence quencher,
such as flindersine, it can transfer its energy in
excess (Gentili, 2008b). These different relaxation
pathways are in kinetic competition: the faster the
route, the higher the probability of occurring and
hence its quantum yield. It is possible to influence
the speed of some of these processes and hence the
fluorescence quantum yield (
) through external
physical and chemical inputs, such as the
temperature (T) and the content of the quencher
flindersine. At low temperature and in the absence of
is high. By increasing T,
slightly since it becomes easier the thermal activated
reaction path, i.e. the energy barrier, E
, is more
easily overcome (see Figure 4).
Figure 4: Relaxation dynamics of tryptophan after photo-
is also reduced by adding the quencher
flindersine. Since
varies smoothly with T and the
moles of flindersine, it is possible to exploit the
photobehaviour of tryptophan to implement Fuzzy
Logic Systems (Gentili, 2008a). The temperature
and the extent of flindersine act as inputs, UV
photons as power supply and
of tryptophan as
output. In ways similar to those explained above for
6(5H)-Phenanthridinone, the macroscopic variables
involved are fuzzified; Fuzzy rules, wherein the
multiple antecedents are connected through the
AND operator, are defined and Fuzzy Inference
Engines area started up, based upon the cornerstone
of scientific modelling, i.e. the cause and effect
There are also chemical reactions that allow
Fuzzy logic to be processed. An example is the
biochemical reaction network controlling the
glycolysis/gluconeogenesis functions (Arkin, 1994).
Here, fructose-6-phosphate (F6P) is interconverted
between its two bisphosphate forms by specific
kinases and phosphatases. The enzymes in this
kinetic mechanism are under the allosteric control of
many of the chemical signals of cellular energy
status such as cyclic-adenosine-monophosphate
(cAMP) and citrate. The dependence of the
concentration of F6P on those of cAMP and citrate,
gives rise to a 3D surface showing a not abrupt
transition from low to high values, such as that of
Figure 3. The profile of the 3D surface has a smooth
hyperbolic shape and not a steep sigmoidal response:
it is suited to process Fuzzy logic.
Another example of a Fuzzy chemical reaction is
DNA hybridisation wherein two single-stranded
DNA molecules (oligonucleotides) bind to form a
double stranded DNA duplex. At room temperature,
the hybridisation reaction is not a two-state, all or
none process, but it is inherently Fuzzy because it is
a continuum of outcomes (Deaton, 2001). The pairs
of oligonucleotides formed inside a test tube cannot
be divided into distinct sets of hybridised and
unhybridised species, but each molecule would have
a degree of membership in both.
The best implementations of Fuzzy Logic
Systems are human senses, that have to be mimicked
by Information Technology in order to reach high
intelligent quotients in artificial intelligence. Sight,
hearing, taste, smell and touch are inherently fuzzy.
They fuzzify the crisp inputs coming from the
outside and send the information to the human brain,
that is a Fuzzy inference engine, capable of facing
up problems based on subjective or imprecise
knowledge. Senses are based on a discrete number
of perceiving cells acting as Fuzzy sets (Gentili,
2009). For example, in the case of colour perception,
we have three types of cones, whereby we
distinguish colours: one cone absorbing mainly the
blue portion of the visible spectrum, another
absorbing mainly the green and the third principally
sensitive to the red. Their absorption spectra in the
MOLECULAR FUZZY INFERENCE ENGINES - Development of Chemical Systems to Process Fuzzy Logic at the
Molecular Level
visible, can be conceived as Fuzzy sets, having
Gaussian shape: one centred at 437 nm, the other
centred at 533 nm and the third centred at 564 nm.
When a radiation, having wavelengths included in
the visible, hits the retina of our eyes, it activates the
three cones in a specific proportion, i.e. it will have
specific values of membership functions in three
Fuzzy sets. Each combination for the values of three
membership functions will be transduced into the
perception of a specific colour inside our brain.
Computers of the future will probably consist of
molecular processors. Atoms and molecules process
quantum logic. However, the insidious actions of the
environment trigger detrimental decoherent effects
on the qubits. If decoherent phenomena acting on the
qubits are unavoidable, it is still possible to compute
with molecules by abandoning the lure of quantum
logic and processing classical logic. When
computations are carried out by single molecules,
only Boolean logic gates can be implemented. When
computations are performed through a huge
collection of molecules, both Boolean and Fuzzy
logic can be processed. If the input-output relations
are abrupt and they have sigmoidal shape, they are
suited to implement binary logic gates. If, on the
other hand, the output varies smoothly with the
inputs, their relation become suited to implement
Fuzzy inference engines. Photo-responses of
molecules such as 6(5H)-Phenanthridinone and
tryptophan can be used to realize Fuzzy Logic
Systems wherein the multiple antecedents are
connected through the AND operator. New chemical
systems have to be found to process complete Fuzzy
Inference engines. They will be implemented by
following either the strategy of the “interfacial
hardware” or that of “wetware”. The possibility of
processing Fuzzy logic at the molecular level will
allow high quotients to be within reach of the
Artificial Intelligence.
This research was funded by the Ministero per
l’Università e la Ricerca Scientifica e Tecnologica
(Rome, Italy) and the University of Perugia
(PRIN2008, 20088NTBKR).
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