Acta Polytechnica Hungarica Vol. 11, No. 4, 2014 [600455]

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 79 – Fuzzy Membership, Possibility, Probability and
Negation in Biometrics
Nicolaie Popescu-Bodorin †, Valentina E. Balas ‡
† University of S@E Europe Lumina, Colentina 64B, 02 1187, Bucharest, Romania.
‡ ‘Aurel Vlaicu’ University, Bd. Revolutiei 77, 3101 30, Arad, Romania.
† [anonimizat], ‡ [anonimizat]
Abstract: This paper proposes a new formalization o f the classical probability-possibility
relation, which is further confirmed as a much comp lex, but natural provability –
reachability – possibility – probability – fuzzy me mbership – integrability interconnection.
Searching for the right context in which this relat ion can be consistently expressed for the
particular case of experimentally obtained iris rec ognition results brought us to a natural
(canonic) and universal fuzzification procedure ava ilable for an entire class of continuous
distributions, to a confluence point of statistics, classical logic, modal logic, fuzzy logic,
system theory, measure theory and topology. The app lications – initially intended for iris
recognition scenarios – can be easily extrapolated anywhere else where there is a need of
expressing the relation possibility – probability – fuzzy membership without weakening the
σ-additivity condition within the definition of prob ability, condition that is considered
here as the actual principle of possibility-probabi lity consistency.
Keywords: σ-additivity, principle of possibility-probability c onsistency, provability,
reachability, possibility, probability, fuzzy membe rship, Riemann integrability, negation,
consistent experimental frameworks, Turing test, ir is recognition, biometrics.
1 Introduction
Implementing biometric identification systems means advancing from human
intuition to artificial but formally correctly biom etric decisions. To be specific,
when a human agent analyzes a pair of two good qual ity iris biometric samples
(two iris images) – for example, it is easy for him to decide if the pair is a genuine
or an imposter one, hence it is simple for him to h ave the intuition that designing a
reliable artificial agent able to recognize genuine against imposter pairs should be
possible. A first guess is that, as a decision syst em, the biometric system should be
a binary one. Theoretically, in ideal conditions, i t should be able to map all its
legal inputs (pairs of iris templates) onto a set o f two concepts and linguistic
labels ‘ imposter’ and ‘ genuine’ whose extensions should be disjoint, since in a
logically consistent iris recognition system and al so in our reasoning, no imposter

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 80 – pair is a genuine one and vice versa. Unfortunately , the field of iris recognition is
full of counter@examples to this ideal situation, s ome of them as old as the domain
of iris recognition itself [3], [4] and others, ver y recent indeed [6]@[10], [24], [25],
some of them in more direct connection with what fo llows to be presented in this
paper [2], [12]@[17], [19]@[23]. In all of these co unter@examples it happens that the
linguistic labels ‘ imposter’ and ‘ genuine’ are, in fact, represented as two
overlapping fuzzy sets of recognition scores, the o verlapping being itself a fuzzy
boundary in between the first two mentioned fuzzy s ets. Therefore, a second guess
is that, as a decision system, the biometric system outputs three possible values:
‘imposter’ , ‘ uncertain’ and ‘ genuine’ . A step further is analyzing the causes for
which some imposter similarity scores are too big a nd some genuine similarity
scores are too small, in an attempt to better isola te from each other the extreme
values imposter and genuine . The result is a 5@valued recognition function tha t
splits the input space (of iris template pairs) in five fuzzy preimages of five labels
imposter , hyena (degraded imposter), uncertain , goat (degraded genuine) and
genuine , all of them represented as fuzzy sets of recognit ion scores. All in all, the
non@ideal conditions occurred from the stage of ima ge acquisition up to the stages
of image processing and matching fuzzify the protot ype binary crisp recognition
function that all biometric systems are normally ex pected to have (in ideal
conditions) up to a binary, ternary, quaternary or even quinary fuzzy recognition
function. Therefore, the initial intention of desig ning an identification system must
be weakened when necessary to designing verificatio n systems. However, when
the aim is to design a logically consistent recogni tion system within the limits of
consistent biometry [22], the difference between ve rification and identification
vanishes.
This framework of iris recognition (and biometric r ecognition at large) is the
context in which we talk here about fuzzy membershi p, probability and negation
while searching for appropriate ways of expressing (precisiating, [31]) facts, rules
and phenomena of iris recognition in a computationa l manner such that to
maximize the cointension [31] between the real worl d of iris recognition and its
computational model. By adopting a Turing perspecti ve [27], we classify such a
task as a process of human intelligence and its com putational model as an artificial
agent whose degree of intelligence can be determine d through the test that today
bears his name (Turing test, [27]). In other words, ideally, artificial intelligence is
a way of representing processes of human intelligen ce as computational (artificial)
software agents with as much cointension as possibl e, the degree of cointension
being verifiable in principle [1] through a Turing test [27]. The results of all major
iris recognition experiments (such as [4], [8], [10 ], [13], [19], [20], [22], [24]) are
in fact partial Turing tests of iris recognition in which only the software agent is
interrogated. Completing these partial tests can be done easily by attaching to
them the corresponding iris recognition results obt ained while interrogating
qualified human agents using the same iris image da tabases. Surprisingly,
especially when using iris image databases containi ng good quality images, the
histogram of biometric decisions given by the human agents are indeed very

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 81 – different by those summarizing the biometric decisi ons given by the artificial
agents (the iris recognition systems). The huge dis crepancy between the two types
of iris recognition results was for the first time observed and understood in [21]
and further commented and used in [22] as an argume nt for searching a better
methodology for non@stationary machine@precisiation of iris recognition. Here we
will insist now on the lack of cointension between the human@precisiated and
machine@precisiated iris recognition and on the lac k of instruments for quantifying
it. Let us start with a tentative of quantifying th e degree of separation between the
distributions of imposter and genuine similarity sc ores. In the classical statistical
approach of iris recognition proposed by Daugman, t he decidability index,

2 22| |'
GIGId
σσµµ
+= , (1)
is such a measure of separation (where Iµ, Gµ, 2
Iσ and 2
Gσare the means and the
variances of imposter and genuine score distributio ns respectivelly), values of 4,
8, and 14 being already reported in the literature [3], [4]. Hence, the separation
between the distribution of imposter and genuine si milarity scores is in this case
precisiated numerically in such numeric values exem plified above. However, the
histograms summarizing the biometric decisions give n by qualified human agents
are crisp 0@1 (binary) histograms for which the sam e decidability index takes the
value of positive infinity. How relevant are 4, 8, or 14 against the infinity? How
much cointension is in this case in between the hum an@precisiated and machine@
precisiated iris recognition? We could not say othe rwise than not too much at all,
and this is what motivates our paper, which is furt her organized as follows:
1.1 Outline
Section 2 presents a newly proposed formalization o f the classical probability@
possibility relation, whereas the sections 2.1 and 2.2 introduce the notions of
consistent experimental setups and frameworks exemp lified in section 2.3. Section
3 presents and analyzes some cases of imperfect exp erimental frameworks
(especially cases of iris recognition, see section 3.1) study that leads to the finding
of a new possibility – probability – fuzzy membersh ip relation for Gaussian
distributed random numbers and also for other conti nuously distributed random
numbers, finding that points out that weakening the σ−additivity condition is not
necessarily required for establishing a consistent possibility@probability@fuzzy
membership relation. On the contrary, the σ−additivity condition is the bridge
that ensures this relation and therefore statistics and fuzzy logic could share a
common side@by@side evolution – fact that is commen ted in section 3.4. The
sections 3.5 and 3.6 are dedicated to exemplifying some issues that negation has
when it comes to deal with fuzzy membership assignm ents. At last, the 4 th section
deals with two types of negation in the context of implemented biometric systems
and is followed by conclusions.

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 82 – 2 A new formalization of the classical probability-
possibility relation.
Let us consider a data spring, i.e. an input@output relation R of a theoretical system
ST actually implemented as S P, a data spring that throws an uniformly distribute d
number y in the discrete set of fuzzy values Y=[0:1 :255]/255={y k=k/255 |
k∈Z256 } as a response to the excitation x ∈X. By intentionally confounding the
output y with the state of system S, X is the input space and Y is the state@output
space. Given the fact that the data spring is a uni formly distributed number in Y,
all y ∈Y are observable outputs/states and the event y ∈Y is not just possible, but
certain. The possibility that y ∈Y originates in the nature of data spring R,
whereas for any R taken such that R(X)=Y, the proba bility of y ∈Y is unitary. In
such cases, while the system S P is functioning, the actual outcome y=R(x) cannot
enter in the output/state space through its empty s ubset, i.e. the event R(x) ∉Y is
impossible, whereas its probability is zero. In suc h cases, even if we consider the
possibility is a matter of degree, the (maximally) impossible event is not
improbable (as Zadeh said in [30]), but actually it is not probable at all (i.e. it is
0−probable). On the other hand, the maximally possibl e event is the certain event
whose probability is unitary. In what follows here, all 0@probable events are
impossible, all p@probable events with 1≥p>0 are po ssible and all 1@probable
events are not just possible, but certain. The poss ibility of an event is not fuzzy,
but binary: all possible or impossible events have their possibility coefficients
equal to 1 or 0, respectively. Hence probability fo llows, originates in and
expresses possibility (probable events are not impo ssible) and possibility is causal
to probability. This state of facts is already expr essed in probability theory as a
precise law, namely the σ−additivity axiom within the definition of probabili ty,
which can be also viewed as a principle of consiste nt possibility@to@probability
translation, i.e. as an instance of the so called possibility/probability consistency
principle introduced by Zadeh in [30],
∑
==n
kkkp
1πγ , with p and πdenoting probability and possibility values, (2)
where all possibility values are taken unitary. In our modeling, a state or an output
is observable if and only if it is possible, i.e. t he possibility and observability are
interchangeable (synonyms). In terms of formal lang uages, there is a formal
grammar that describes the systems S T and S P able of producing that specific
state/output also. In a consistent experimental set up, all possible events should be
observed and, as a consequence, their statistics ca n be made, probability being a
nuance, a refinement of possibility (among all poss ible events, some are more
probable than others), a finer precisiation / quant ization of possibility in a numeric
space after the knowledge resulting from a certain experiment is gathered as
statistics data. Hence, the main differences betwee n the classical possibility@
probability relation that we are bounded to here an d the model proposed by Zadeh
in [30] are the following:

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 83 –
Possibility is encoded binary. Possibility is a mat ter of degree.
Possibility and probability values
satisfy the σ−additivity condition,
and therefore, they are cointensive
and fully consistent to each other –
i.e. in formula (2), γand possibility
values are all unitary. Possibility and probability values are
not necessarily cointensive, nor fully
consistent to each other, since their
degree of consistency (2) marks a
weakening of the
σ−additivity condition.
For any consistent experimental setup ‘E’, any even t ‘e’ and any p ∈(0,1]:
impossible(e) ↔ 0@probable(e, E) impossible(e) → improbable(e)
possible(e) ↔ p@probable(e, E) probable(e) → possible(e)
(if contrapositive principle still stands)
certain(e) ↔ 1@probable(e, E) formula (2)
Let us comment now on these matters. First of all, the relation between
impossibility and 0@probability,
impossible(e) ↔ 0@probable(e, E), (3)
tells that in any consistent experimental setup ‘E’ (situation further denoted
as ξ∈E , where the space of consistent experimental setups ξ follows to be
defined on the run, by natural restrictions that ap pear during formalization), an
impossible event cannot be observed as an outcome, or otherwise, the
experimental setup is not consistent (situation fur ther denoted as ξ∉E ). This
mechanism can be used to endow any computational ar tificial agent with the
capacity of predicting (having an expectation and a prior knowledge on) the future
outcomes of an experiment that it follows to witnes s, observe and understand, and
also with the capacity of knowing who is responsibl e when these outcomes do not
meet its expectation. However, formula (3) is a sim plified instance of a more
complex one that belongs in a second@order formal l anguage describing the
systems S T and S P (in what follows, t and f are used as true and fal se):
impossible(e) ↔ {(ξ∈∀E )[ t → 0@probable(e, E) ]} (4)
telling that an event ‘e’ is impossible if and only if, in any consistent experimental
setup ‘E’, its probability is null. Its dual by con traposition principle is:
possible(e) ↔ { (ξ∈∃E )[ 0@probable(e, E) → f] } (5)
telling that an event ‘e’ is possible if and only i f there is a consistent experimental
setup in which the assertion that ‘e’ is 0@probable in ‘E’ is false. Formula (5) is
further equivalent to:
possible(e) ↔ { (ξ∈∃E )( ] 1 , 0 ( p∈∃ )[ t → p@probable(e, E)] } (6)
telling us that an event ‘e’ is possible if and onl y if there is a consistent
experimental setup in which the assertion that the probability of ‘e’ is not null in
‘E’ is true.
Secondly, the relation between possibility and p@pr obability (when p ∈(0,1]),

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 84 – possible(e) ↔ p@probable(e, E), (7)
is also a simplified instance of a more complex for mula:
possible(e) ↔ { (ξ∈∀E )( ] 1 , 0 ( p∈∃ )[ t → p@probable(e, E) ] }, (8)
that tells that an event ‘e’ is possible if and onl y if, in any consistent experimental
setup ‘E’, (is true that) its probability is not nu ll. Formula (8) is actually a
strengthening of (6) by universal quantification of E in ξ, and therefore, a
strengthening of (4). Besides, the relations (8) an d (5) could not be simultaneously
true outside ξ, hence ξ is necessarily defined as follows:
2.1 Consistent experimental setups
Definition 1 The space of consistent experimental setups ξ for the system S T
whose observable state/output space Y is entirely c overed by its input@output
(X@R@Y) relation (i.e. R(X)=Y) is a space of experi ments (on implemented
systems S P) in which:
(i) any two members are interchangeable, i.e.:
t → {[(ξ∈∃E )(P(e, E))] → [(ξ∈∀E )(P(e, E))]}, (9)
where P(e, E) is a property of ‘E’ relative to a gi ven event ‘e’.
(ii) the following rewriting rule holds true:
{( ] 1 , 0 ( p∈∃ )[ t → p@probable(e, E) ]} ↔ [0@probable(e, E) → f] (10)
(iii) there is a negation operator ‘n’ defined such that n 2=1 (in terms of string
functions) and:
n{( ξ∈∀E )[t →0@probable(e,E)]} ↔{(ξ∈∃E )[0@probable(e, E) →f]}, (11)
(iv) the following rewriting rule for complementary even ts e and e holds
true:
(ξ∈∀E )( ] 1 , 0 [ p∈∀ ){[p@probable(e,E)] ↔[(1@p)@probable( e,E)]} (12)

(8) ↔(4)
/barb2up
(8) ↔(5) AND (5) ↔(4)*
/barb2up
(8) ↔(6) AND (10)
/barb2up
(9)
(where * is provable by (11) and contraposition pri nciple) (13)

In these conditions, a formal proof that (8) ↔(4) is presented here as formula
(13), where the equivalence relation * within the f ormula (13) is provable by (11)

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 85 – and contraposition principle. The condition (iv) wi thin Definition 1 establishes the
natural meaning of both labels impossible and certain by correspondence with the
trivial two@elements Boolean algebra defined by the empty set and Y, on the one
hand, and with the two extreme values (0 and 1) tha t the cumulative of probability
takes for the empty (impossible) and total (certain ) events, respectively, on the
other.

In the third place, the relation between certain an d 1@probable events:
certain(e) ↔ 1@probable(e, E), (14)
is also a simplified instance of a more complex for mula:
certain(e) ↔ {( ξ∈∀E )[ t → 1@probable(e, E) ]}, (15)
affirming that the event ‘e’ is certain, if and onl y if, it is 1@probable in any
consistent experimental setup ‘E’. Given the rewrit ing rule (iv) stated in
Definition 1, formula (15) is further equivalent to :
impossible( e) ↔ {( ξ∈∀E )[ t → 0@probable( e, E) ]}, (16)
that further is an instance of formula (4). By summ arizing this section up to this
point, an important remark is that the three ways o f describing the classical
possibility@probability relation for the impossible , possible and certain events,
namely the formulae (4), (8) and (15), or their sim plified forms (3), (7) and (14)
and also the axioms within the definition of probab ility are not independent, but
intimately interconnected as three images of the sa me thing, namely the concept
denoted above as ξ @ the space of all consistent experimental setups ξ for the
system S T whose observable state/output space Y is entirely covered by its input@
output relation. ξ is a formal, logical, computational and physical c oncept, a
coherent non@contradictory framework of expressing the natural relation between
possibility, probability and negation for all obser vable states/outputs of a physical
system S P given as an implementation of S T. By contrast, now the reader knows
what could mean to weaken any of the three axioms w ithin the probability
definition while attempting to define a possibility @probability relation different
from the canonic natural one that exists by default in ξ.
A second remark is that investigating the possibili ty@probability relation in ξ
shows how many things confirm each other and group together coherently in a
consistent and computational knowledge ensemble.
A third remark is that all formulae from above that contain the symbols t and f @
i.e. (4)@(6), (8)@(12), (15) and (16) @ are written in a formal logical dialect that
extends the cognitive dialect introduced in [17], w hich on its turn is derived from
the sound and complete formal theory of Computation al Cognitive Binary Logic
(CCBL) introduced in [15] and implemented online in [18]. However, the marker
(!:) signalizing an assertion in cognitive dialect is omitted in the above formulae,
because here in this article there is no chance of confusing queries and assertions.
Hence, our discourse presented here on the possibil ity@probability relation is
strictly a logical and computational one.

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 86 – 2.2 Consistent experimental frameworks
An artificial computational agent produces its stat es/outputs accordingly to a
sound formal theory in which the states/outputs are formally and logically
provable in terms of a generative grammar. Hence, i n ξ, a more complex relation
holds between possibility, probability, observabili ty and provability: observable
events (states/outputs) of S T and S P are possible, probable and provable, any time
when the input@state/output relation is theoretical ly known as a provable formula
of a sound theory describing S T and practically implemented in S P without errors.
This motivates the following definition:

Definition 2 A Consistent Experimental Framework (CEF) is a form al
ensemble (X,R,Y,S T,TS,ξ) in which:
(i) ξ is the space of consistent experimental setups for the system S T;
(ii) the observable state/output space Y of the system S T is entirely covered by
the input@output (X@R@Y) relation (i.e. R(X)=Y);
(iii) input@state/output relation (X@R@Y) is know explici tly as a provable formula
of a sound formal theory T S describing S T, on the one hand, and implemented
without errors on S P, on the other.

In this context, since T S is a sound formal theory, it cannot prove contradi ctions.
Therefore, contradictions are impossible / unreacha ble / unprovable events in T S.
Since the formal theory of T S actually describes the functioning of S T and S P, the
practice on S P is not able to deliver events that are theoretical ly impossible /
unreachable / unprovable (in S T), and therefore, the practice on S P (further denoted
PS) cannot deliver counter@examples for S T and T S. In other words, there is a
unitary cointension between S T and S P, between the theory T S and the practice P S
on the theoretical and actual systems S T and S P. In general, the lack of cointension
between practice (experiments) and theory could be expressed in experimental
results that contradict the theoretical model of th e system in some aspects. On the
contrary, a unitary cointension between practice an d theory ensures that in any
consistent experimental setup it is impossible to o btain experimental results that
contradict the expectations motivated by the sound theory T S. There are only two
possible gates that could allow inconsistency (cont radiction) within an
experimental framework, namely T S is not sound or P S is outside ξ. In addition,
there are three ways in which contradiction could b e expressed in an experimental
framework: inside T S, inside P S, and between theoretically predicted events in T S
and the actual events taking place in P S, i.e. between expectations and the actual
experimental outcomes.
2.3 Examples of consistent experimental frameworks
The extension of the concept introduced in Definiti on 2 from above is not the
empty set: propositional logic is the formal theory that describes any logical
circuit both as theoretical design S T and as implemented system S P. Given a

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 87 – sequence of excitations in the input space, the cor responding outputs are formally
provable in propositional logic (hence, accordingly to the formal theory of the
system they are expected values) and practically ob servable. For a given input@
output relation, any two consistent experimental fr ameworks differ only by
equivalent (interchangeable) logical circuits that support that input@output relation.
Given an input@output relation and a trajectory (x, y) t in X×Y, output statistics is
invariant along all consistent experimental framewo rks that support that input@
output relation, and therefore, the possibility@pro bability relation is invariant.

This example is also important for establishing two things, once for good: logical
circuits are the most basic intelligent agents (whe re intelligence means Turing
defined intelligence, i.e. the artificial intellige nce provable by Turing tests) and
secondly, in a Turing test, it is mandatory that th e human agent to be qualified
(besides being informed). Obviously, there is only one chance for the human agent
to predict correctly all outputs of a logical circu it, namely to know propositional
logic (besides knowing the circuit design). Otherwi se, the results of a Turing test
on a logical circuit are not relevant at all. On th e other hand, the human or
artificial agent that organizes the Turing test mus t be able to recognize intelligence
regardless if is human intelligence or artificial i ntelligence. The true output is
either certain and 1@probable or possible and p@pro bable (with p ∈(0,1)) or
impossible and 0@probable for any logical circuit i mplementing a tautology / a
contextual truth / a contradiction, respectively.

A second example of consistent experimental framewo rks can be built for any
digital circuit in general, by analogy with the fir st example from above. We make
this remark because, ultimately, a biometric system can be viewed today as a
complex digital circuit.
3 Imperfect experimental frameworks
ξand CEF are introduced exactly for ensuring that wh at is theoretically possible /
probable / certain is also practically possible / p robable / certain, respectively. On
the other hand, ξ and CEF allow us to study weakened models, when th e
weakening is made otherwise than changing the axiom s within the definition of
probability, for example. We do not have any clue t hat weakening probability
axioms could prove to be maximally productive, beca use even in imperfect
experimental frameworks the probability theory cont inues to function despite the
adequacy of our beliefs and intentions. In other wo rds, when a system evolves on
and within measurable sets/spaces, we can ignore th e probability theory if we
prefer, but we cannot abolish it, simply because it is engraved/embedded within
the structure of the space itself. Besides, we will illustrate further, how probability
distribution functions (PDFs) and cumulative distri bution functions (CDFs)
appearing in biometrics (or anywhere else) can gene rate fuzzy membership
functions easily, in more than a single way. Hence, adopting a fuzzy approach in a

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 88 – given matter is not necessary equivalent with contr adicting probability theory (by
weakening some of its axioms). The major difference between our approach and
[30] is that in ours, the classical possibility@pro bability relation (described above
in terms of ξ and CEF) stays unchanged while the experimental fr ameworks that
we work with @ despite being allowed to be imperfec t @ are not allowed to decay
up to the abolition of probability theory.

A first degree of imperfection illustrated here is when accurately implementing S T
is practically impossible or technically and econom ically unfeasible. For example,
when N is big enough, a preferable alternative for implementing N@dimensional
dynamical systems with predefined input@output beha vior is designing a simpler
n@dimensional system (n<<N) that supports almost the same input@output behavior
@ a technique known as order reduction. However, in such case, the practical
n−dimensional implementation S Pn and the theoretical N@dimensional model S TN
are still highly cointensive, whereas the practical n@dimensional implementation
SPn and the theoretical n@dimensional reduced model S Tn are still totally
cointensive, and therefore, the classical possibili ty@probability relations (3), (4),
(7), (8), (14) and (15) still hold true in the cons istent experimental framework
(X,R,Y,S Tn ,ξ) when ξ is S Pn based.
3.1 Imperfect experimental frameworks in iris recogniti on

One of the worst but still manageable (hence accept able) degree of imperfection is
when the required (target) input@output (i@o) behav ior is known as being possible,
but all practical implementations S P attempted up to some point for mimicking the
target i@o behavior are almost failed resulting in practical i@o behaviors that are
very different from the target i@o behavior. This i s currently the case of all iris
recognition results and biometric systems belonging to the statistical paradigm
pioneered by Daugman [3], [4]. The target i@o behav ior assigned to an a priori
unknown target system S T that the user intends to design (proved possible b y
interrogating a qualified human agent during a Turi ng test while using good
quality eye images) can be statistically illustrate d as a 0@1 histogram of correct
biometric decisions (0 @ for identifying a pair of iris images taken for different
eyes, 1 @ for identifying a pair of iris images tak en for the same eye), as in
Figure 1, whereas the i@o behavior of implemented s ystem is statistically
illustrated in Figure 2. As exemplified in Figure 1 and Figure 2, plotting the
statistics of the results recorded in a Turing test is a way of visually quantifying
the loss in precision occurred for the implemented system in comparison with the
target i@o behavior. The lack of cointension betwee n target i@o behavior and
implemented i@o behavior is obvious. Figure 1 illus trates a logically consistent,
crisp, artificial and binary understanding (logical , crisp, binary and lossless
human@precisiation of meaning) for two concepts who se extension are not just
disjoint but complementary in the input space of th e recognition system: ‘imposter
pairs’ (IP) and ‘genuine pairs’ (GP). Figure 2 illu strates a lossy compression of the
original meaning of the two concepts, a fuzzy artif icial perception and a fuzzy

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 89 – machine@precisiation of meaning for the two origina l human@precisiated concepts.
The artificially perceived fuzzy concepts (further denoted as f@IP and f@GP or as f@
imposter and f@genuine) are no longer disjoint. The re are pairs of irides qualifying
simultaneously and equally as f@genuine and f@impos ter pairs, while others qualify
ambiguously but with different probabilities. A bet ter situation in terms of
cointension can be observed by comparing Figure 1 f rom here to Figure 2 from
[19], Figure 4 from [20] and Figure 5 from [22].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91
imposter pairs (imposter comparisons)
genuine pairs (genuine comparisons)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.010.020.030.040.050.060.07
IMPOSTER SCORES PDF
GENUINE SCORES PDF

Figure 1
Human@precisiation of iris recognition: PDFs
corresponding to the correct biometric
decisions given by a qualified human agent
during a Turing test of iris recognition
(diamond for imposter pairs, square for
genuine pairs)
Figure 2
Machine@precisiation of iris recognition:
PDFs of the imposter scores (diamond) and
genuine scores (square) collected during a
Turing test of iris recognition while
interrogating an artificial agent implementig
statistical iris recognition.

The imperfection of such an experimental framework illustrated in Figure 1 and
Figure 2 is further commented in terms of possibili ty@probability relation: in
theoretical target system corresponding to Figure 1 the confusion between an
imposter and a genuine pair is impossible, whereas in the implemented system it is
possible, indeed. Forcing the correct recognition o f all imposter pairs within the
implemented system comes at the price of not recogn izing correctly all genuine
pairs. This trade@off is inherent in classical stat istical iris recognition. Therefore,
despite the original concepts consistently support negation, the fuzzy concepts f@IP
and f@GP cannot be consistently negated.

The consistent experimental frameworks are contexts in which the cointension
between a target formal theory and the actual imple mented system transports
probability@possibility relation between theory and practice in the same manner in
which continuity transports convergence from the ar gument space to the image
space in the framework of topology. On the contrary , in the imperfect
experimental frameworks the cointension between the ory and practice (or between
target and implemented system) can be so weak that even theoretically but still
logically impossible events could appear as practic ally probable (hence practically
possible). However, the fact that cointension is we ak enough and consequently
unable to establish a consistent bridge between wha t is practically probable and
theoretically possible does not mean that the possi bility@probability relation is

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 90 – broken inside the set of events concerning the impl emented system alone:
confusion between the two artificially perceived co ncepts (imposter and genuine
pairs) is practically p@probable (1>p>0) and theref ore practically possible. On the
other hand, when logical inconsistency gets its pla ce inside an imperfect
experimental framework, the right contra@measure to take is attempting to recover
logical consistency, not attempting to redefine pos sibility@probability relation,
because otherwise, contradiction being explosive, s ooner or later, anything can be
proved and some logically impossible events could a ppear as proved/supported by
“experimental evidence” to be probable (hence possi ble) events. In short, when it
comes to test or to design systems in imperfect exp erimental frameworks, there is
no sound argument to extrapolate experimental evide nce to expectations for real
life without precautions, regardless if the channel on which the extrapolation is
made is possibility@to@probability or possibility@t o@fuzzy membership relation.
3.2 A possibility – probability – fuzzy membership rela tion
for Gaussian events
The most important aspect revealed by the compariso n between Figure 1 and
Figure 2 is the lack of cointension between what is maximally achievable
(Figure 1) and what is currently achieved in implem entation (Figure 2). The
relation between what is theoretically possible and practically probable is broken,
whereas the relation between what is practically po ssible and practically probable
still stays consistent. Translating probability to fuzzy membership is possible in
many ways. However, this operation is closed in the semantics of implemented
system, is an operation whose result is at most an equivalent of some facts already
known, i.e. it is at most a rule for rewriting know n facts, not an attempt to improve
the actual implemented system. Regardless the way c hosen to express fuzzy
membership based on statistics of experimental data , in the absence of
cointension, there is no instrument to carry this i nformation back and forth
between theory and practice. Besides, as the follow ing example illustrates, forcing
the meaning of the data to conform to a pattern tha t it is not really exhibited as an
observable behavior, inevitably brings more inconsi stency to an already imperfect
experimental framework. In short, interpreting the experimental data does not
solve the problems, but only point out to them.

Since in the case investigated here, the errors are inherent to the implemented
system (Figure 2), computing right@to@left CDF unde r the imposter distribution
(FAR @ False Accept Rate, diamond markers in Figure 3 and Figure 4) and left@to@
right CDF under the genuine distribution (FRR @ Fal se Reject Rate, square
markers in Figure 3 and Figure 4) define a way of i nterpreting fuzzy membership
of the recognition scores obtained by the input iri s pairs to two fuzzy sets
corresponding to the two artificially perceived con cepts f@imposter and f@genuine
scores (see Figure 3 and Figure 4). However, accord ing to this interpretation, 0 is
an imposter score (a convenable interpretation, giv en the target behavior in
Figure 1), fact that is neither confirmed experimen tally (Figure 2) nor confirmed

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 91 – theoretically in statistical iris recognition. More over, the means of the two
Gaussian variables in Figure 2 have the degrees of membership to the fuzzy sets f@
imposter and f@genuine scores expressed as 0.5, whi ch is clearly counter@intuitive,
the mean being the most representative sample of a Gaussian signal.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−1010−810−610−410−2100

Figure 3
Machine@precisiation of statistical iris
recognition: False Accept Rate (FAR,
diamond markers) and False Reject Rate
(square markers) represented with liniear y
coordinate; zoom to membership degrees that
the means of the two Gaussians (Figure 2)
have in f@imposter and f@genuine fuzzy sets.
Figure 4
Machine@precisiation of statistical iris
recognition: False Accept Rate (FAR,
diamond markers) and False Reject Rate
(square markers) represented with logarithmic
y coordinate @ zoom to the Equal Error Point
(indicated above by the continuous horizontal
line situated slightly under 1E@2).

Experimental data can be interpreted using fuzzy if @then rules. For example, the
ensemble of fuzzy if@then rules producing the fuzzy @membership assignments
represented in Figure 3 as FAR and FRR curves is th e following:
@ fewer successors a score has within the imposter di stribution, weaker its
degree of membership to f@imposter fuzzy set is;
@ fewer predecessors a score has within the genuine d istribution, weaker its
degree of membership to f@genuine fuzzy set is;
However, the same interpretation can easily fail in contradiction or in counter@
intuitive facts, as illustrated above.
A different interpretation of experimental data can be made accordingly to the
following ensemble of fuzzy if@then rules, which is better suited for describing
Figure 2 (probability) in terms of fuzzy membership as in Figure 5:
@ for scores situated on the right/left side of the m ean, fewer successors /
predecessors a score has within the (imposter or ge nuine) distribution, weaker
its degree of membership to the corresponding fuzzy set (f@imposter or f@
genuine) is;
@ recognition score equals to the mean of (imposter o r genuine) distribution has
unitary membership degree with respect to the corre sponding fuzzy set (f@
imposter or f@genuine);

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 92 –
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−810−710−610−510−410−310−210−1100

Figure 5
Machine@precisiation of statistical iris recognitio n: fuzzy membership assignment of the
recognition scores obtained experimentally and repr esented in Figure 2 as the two fuzzy sets f@
imposter (diamond markers) and to f@genuine (square markers).
Given the symmetry of Gaussian variables, the fuzzy membership function should
also be symmetric. For any given distribution and a ny given score from Figure 2
there is a pair of left@ and right@side cumulatives under that distribution and the
sub@unitary ratio of these two cumulatives is a pla usible fuzzy membership
assignment of the given score to the corresponding fuzzy set (f@imposter or f@
genuine) that satisfies the two fuzzy if@then rules from above. Fuzzy membership
functions constructed this way are illustrated in F igure 5, which represents a fuzzy
decisional model associated to the statistics of ex perimental data within Figure 2.
This second interpretation of the experimental data within Figure 2 is better that
the previous one (illustrated in Figure 3 and Figur e 4) at least for three reasons:
the means are scored with a unitary degree of membe rship, membership is not
arbitrarily extended to scores that are not obtaine d experimentally, and at last, but
not the least, the abscise of the Equal Error Point is preserved from Figure 4 to
Figure 5 (i.e. the point of equal error expressed i n Figure 5 in terms of fuzzy
memberships corresponds exactly to the point of equ al error expressed in Figure 4
in statistical terms of FAR and FRR).

Summarizing, even in imperfect experimental framewo rks, iris recognition
experimental results can be expressed in terms of p ractical possibility @ probability
@ fuzzy membership on the experimental side of the framework, because a
recognition event ‘e s’ with a given score ‘s’ recorded experimentally (i n Figure 2)
is practically possible, practically probable and m aps the input pair into the two
fuzzy sets f@imposter through the fuzzy membership assignment illustrated in
Figure 5. However, in an imperfect experimental fra mework there are at least two
unsolved problems:

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 93 – @ the formal logic involving experimentally recorde d recognition events is
contextual and temporal because all true sentences that can be derived on the
experimental side of the framework are true at some specific time , in some specific
case , accordingly to a specific experiment , accordingly to specific experimental
data . This situation emphasizes the importance of condi tion (9) within the
Definition 1.
@ the huge difference between what is achievable (i n human@precisiated iris
recognition Figure 1) and what is achieved (in mach ine@precisiated statistical iris
recognition, Figure 2, Figure 5), a gap that can be filled through creativity only,
not through interpretation.

On the other hand, Gaussian events that vanish at s ome left@ and right@side points
(for example, the recognition of imposter pairs in statistical iris recognition and
the corresponding Gaussian distributed numbers) def ine specific fuzzy sets as
follows: if s g / e g is a Gaussian distributed number / event vanishing at some left
and right points, its membership to any continuous or (discretized) interval
considered within the vanishing left and right limi ts can be expressed as a fuzzy
degree of membership (Figure 2, Figure 5):
FM(s g) = min(L C(s g)/R C(s g), R C(s g)/L C(s g)), (17)
where L C(s g) and R C(s g) are the left@ and right@side cumulatives under th e given
Gaussian distribution to the left and right vanishi ng points, respectively.
3.3 Natural fuzzification of a continuous distribution
Theorem 1 : For any 1@dimensional, real@valued random variabl e s that is
continuously distributed on a real interval and van ishes at some left and right
points, the fuzzy membership FM defined in formula (17) is bounded in [0,1] and
has a single global maximum point s M for which FM(s M)=1;

Proof : When thinking at the upper bound of FM, the most favorable situation is
for that point s M where the right@side cumulative equals the left@si de cumulative.
Indeed, if such point s M exists, FM(s M)=1. Since the left@side cumulative on the
given continuous distribution computed in the curre nt point s with respect to the
left vanishing point increases with s and since the value of the left@side cumulative
evolves continuously between 0 and 1, the point s M exists and it is defined by the
abscise where the left@side and right@side cumulati ves are equal to 0.5. Hence, FM
has upper bound that is also its maximum value 1. G iven the definition (17), FM
strictly increases with s @ when s is between left@ side vanishing point and s M, and
strictly decreases with s @ when s is between s M and the right@side vanishing point
of the given distribution, hence, (s M, 1) is the only local and global maximum
point of FM, whereas its lower bound is given by th e values of FM in the left and
right vanishing points of the distribution, namely zero.

Formula (17) establishes the degree of fuzzy member ship (FM) as being the most
pessimistic degree of interiority that s g has with respect to the left and right

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 94 – vanishing points of the distribution. FM is bounded between 0 and 1, and has
always a single maximum point. In the case of Gauss ian events, this maximum
point corresponds to the mean and is the first poin t around which long enough
experiments throws enough neighboring values such t hat to emulate topological
density in a given precision (a property that does not hold true for bimodal
Gaussian distributed numbers – for example).
Searching for the right context in which the relati on between possibility,
probability and fuzzy membership can be consistentl y expressed for the particular
case of experimentally obtained iris recognition re sults brought us inevitably to a
natural (canonic) and universal fuzzification proce dure available for an entire
class of continuous distributions, for which the fo rmula (17) is formally correct (is
actually making sense) due to Riemann integrability of all continuous
distributions. Formula (17) is a meeting point wher e classical logic, modal logic,
fuzzy logic, probability theory, measure theory, sy stem theory and topology shake
their hands explaining consistently the complex but natural provability –
observability / reachability @ possibility @ probab ility @ fuzzy membership @
integrability relation without pointing out to a need for weakening the
σ−additivity condition within the definition of probability up to the for mula (2).
The σ−additivity condition is actually the true principle of possibility-probability
consistency . Conversely, not any fuzzy membership assignment t hat we would
wish to operate with can be consistently mapped ont o a continuous distribution,
especially when it is not compatible with the σ−additivity condition and
consequently, the interpretation given by such fuzz y membership assignments can
be neither confirmed nor infirmed by measurable exp eriments (organized in
consistent experimental frameworks). Therefore, the reader should figure out if
using such fuzzy membership assignments in connecti on with measurable things
and spaces is a matter of excessive oratorical tale nt or a matter of logic and sound
science.
3.4 The future of statistics and fuzzy logic
In the context of measurable spaces and consistent experimental frameworks, the
σ−additivity condition ensures that the answer to the question “ is there a need of
fuzzy logic? ” (Zadeh, [31]) is “ yes, in the same degree in which there is a need fo r
statistics, classical logic, modal logic, system th eory, measure theory and
topology ”. Otherwise, operating fuzzy logic while weakening σ−additivity
condition results in a contradiction, namely the fi nding of a hypothetically
consistent way (science/theory) of quantifying thin gs that by their nature are not
theoretically (mathematically) measurable and pract ically (physically /
experimentally) measurable, a theory that unfortuna tely, cannot be confirmed with
instruments of statistics, classical logic, modal l ogic, system theory, measure
theory and topology. On the contrary, formula (17) shows there is at least one way
of consistently expressing an agreement point for a ll these sciences/theories

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 95 – (including fuzzy logic) while maintaining σ−additivity condition. The lack of
interconnection between probability theory and fuzz y logic evidenced here when
fuzzy logic accepts the weakening of the σ−additivity condition is just the small
empty part of a bottle of very old wine: given the huge diversity of fuzzy
membership assignments and the natural parity that exists between at least a part
of them and the continuous distributions, a real ex plosion of new probability
distribution models is expected to happen in the ye ars to come, marking a point
from where fuzzy logic and statistics will further develop side@by@side exactly
because of the bridge established in between them b y the σ−additivity condition.
3.5 Negation of fuzzy membership
As far as we know, expressing probable events (like those represented in Figure 2)
as fuzzy membership (Figure 5) is a matter of interpretation , a matter of rewriting
some facts from a dialect of statistics to a dialec t of fuzzy logic using a rewriting
rule, for example formula (17) – in our specific ca se considered here. Therefore,
in this context, the negation operates at a semanti c level: for example, if 0.5 is
interpreted as a certain imposter score @ and this interpretat ion is allowed by
formula (17) because FM(0.5)=1 (despite the fact th at the statistics of
experimental data says nothing more than 0.5 score is the most probable imposter
score), then logically, should be impossible for th e same 0.5 score to be
interpreted as a genuine score, fact that is indeed true, despite some false
appearances in Figure 2 and Figure 5. Indeed, the i ndex of genuine pairs can be
intoxicated accidentally by wrong segmentation resu lts or by the impossibility that
the implemented recognition system to deal successf ully with the variability of
acquired iris instances. This is also another facet of the imperfect experimental
frameworks where the job of solving apparently conf licting information obtained
through interpretation of experimental data must be done carefully by a qualified
human operator / system administrator.
3.6 Blind negation of fuzzy membership
If no attention is given to the semantic of actuall y implemented system and to the
actual input@output relation, negation can be made by applying fuzzy complement
[29]. For example, if FM is defined by (17) as the fuzzy membership of all scores
within [0,1] interval with respect to the imposter distribution, 1@FM is a
complementary fuzzy membership assignment defining a sort of negation in
which a single element within [0,1] interval is cer tainly a non@imposter score,
namely 0.5 – the mean of the imposter distribution. This fact is, of course,
counter@intuitive with respect to the practical pro blem. Besides, we are talking
again about many values situated outside the set of experimentally observed
imposter scores. This operation is meaningless with respect to the actual
implemented system. This is why we called this sort of negation, blind negation.

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 96 – On the contrary, if we remember that negation by co mplement should have a
meaning with respect to the practical problem (impl emented system) and should
make a real sense inside the set of reachable score s, the situation changes
completely: since f@imposter and f@genuine are two fuzzy granules covering the set
of reachable scores, the assertion ‘ s is an f-imposter score ’ means accordingly to
the actual experimental data that s is the Gaussian distributed number as illustrated
in Figure 2 (i.e. is 0.5 with maximal probability, 0.51 with other probability and so
on) whereas its negation should be ‘ s is an f-genuine score ’ whose interpretation
should be derived by analogy from the previous one.
4 Negation in biometrics
The important remark to make here is that since the fuzzy membership assignment
is an interpretation of data, there is not a unique way of negating it, as already
exemplified above. Even at this stage it is not cap itally important to find a
negation operator meaningful with respect to the pr actical problem, we will try
this in a future paper, eventually. Bell that saved us this time is the fact that the
final stage in obtaining a practicable crisp decisi onal model from the fuzzy model
within Figure 5 is defuzzification of f@imposter an d f@genuine fuzzy sets to
classical disjoint intervals labeled ‘imposter’ and ‘genuine’ and partitioning [0,1]
interval.
4.1 Negation as a Boolean algebraic operator
After defuzzification, the Boolean logic of recogni tion can be expressed through
isomorphism with the Boolean algebra generated by t he empty set, [0,1] interval
and the two intervals ‘imposter’ and ‘genuine’, whe reas negation is simply a
transcript of complement operation within this Bool ean algebra. This is a way of
implementing a binary recognition function and a bi nary decisional model for
biometrics, a model in which the recognition error rates are hopefully stationary
(there is no proof for that, whereas increased reco gnition errors over time is
already documented under the wrong name of “templat e ageing” (critical analyses
of this concept can be found also in [6] and [9]). A ternary recognition function
and a ternary decisional model for biometrics is ob tained if the two intervals
‘imposter’ and ‘genuine’ are separated by a third o ne labeled ‘uncertain’ and
covering ambiguous score values.

By investigating the consistency of the concepts of “template ageing” and
“biometric menagerie” [12], [23] we found an improv ed quinary recognition
function and a quinary decisional model for iris bi ometrics, which is obtained
while practicing iris recognition on intelligent ir is verifier systems with stored
digital identities [22]. In such systems, it is pos sible that the lowest scores are

Acta Polytechnica Hungarica Vol. 11, No. 4, 2014
– 97 – imposter scores, followed by a class of degraded imposter s cores obtained by
dishonest users when claiming (actively hunting) di fferent identities than they
actually have ( hyena ), followed by a class of uncertain scores centered in 0.5, and
further to the right by a class of degraded genuine scores ( goats ). The rightmost
class is that of genuine scores. All of these models are illustrated in Fig ure 6,
whereas the binary and the ternary models are appli cable also for classical
statistical iris recognition (Figure 2, Figure 5).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105101520253035
IMPOSTER HYENA UNCERTAIN GOAT GENUINEGENUINE UNCERTAIN IMPOSTER GENUINE IMPOSTER

Figure 6
A better machine@precisiation of statistical iris r ecognition with distant imposter and genuine
score classes; From top to bottom: binary, ternary, respectively quinary decisional models.

For all decisional models within Figure 6, negation is expressed within a Boolean
logic induced by a partial power set Boolean algebr a (in a similar manner as that
described in [21]): the negation of the assertion ‘ s is an impostor score ’ is
intuitively and naturally expressed as ‘ s is either hyena, or uncertain, or goat, or
genuine score ’, for example. This is why we did not insist on fi nding the negation
operator in the previous step (prior to defuzzifica tion). Besides, between an
interpretation and a Boolean logic, the latter is c learly the appropriate context of
negation.
4.2 Strong negation in biometrics
Blind negation of fuzzy membership by fuzzy complem ent (see section 3.6) is a
negation practiced in the codomain of the fuzzy mem bership function. On the
contrary, the negation by complement applied to ind ividual recognition scores s,
N(s) = 1@s, (18)
acts in the domain of that fuzzy membership functio n. The negation N in (18)
satisfies boundary condition N(0)=1, is continuous, involutive and decreasing with

N. Popescu-Bodorin & V.E. Balas . Fuzzy Membership, Possibility, Probability and Neg ation in Biometrics
– 98 – respect to the score s, hence it is a strong negati on [7], as all instances of Sugeno
λ@ complement [26], [32]. The effect of this negatio n is a rewriting of all
recognition results from terms of similarity (proxi mity) in terms of non@similarity
(distance). For example, applying the strong negati on (18) for all available
Hamming distance scores, expresses the same experim ental results in terms of
Hamming similarity (the complement of Hamming dista nce) and vice versa.
Graphically, the effect of this negation is that th e score distributions are all
symmetrized against the vertical pointing in 0.5, r egardless how many they are (2,
3 or 5). Hence, strong negation on all recognition scores has a meaningful effect
with respect to the practical problem of recognitio n. However, the appropriate
context of applying it is prior to defuzzification. After defuzzification, depending
on the number of values chosen for the recognition function, the decisional model
is still binary, ternary or quinary, but the order of the intervals reverses: genuine,
imposter – for the binary case, genuine, uncertain, imposter – for the ternary case,
and finally, genuine, goat, uncertain, hyena, impos ter – for the quinary case.
Conclusions
This paper proposed a new formalization of the clas sical probability@possibility
relation, which was further confirmed as a much com plex, but natural relation
between provability, observability, reachability, p ossibility, probability, fuzzy
membership and Riemann integrability. Searching for the right context in which
this relation can be consistently expressed for the particular case of experimentally
obtained iris recognition results brought us inevit ably to a natural (canonic) and
universal fuzzification procedure @ formula (17) @ available for an entire class of
continuously distributed random numbers, as a confl uence point of statistics,
classical logic, modal logic, system theory, measur e theory and topology. The
applications were initially intended for iris recog nition scenarios and can be easily
extrapolated anywhere else where there is a need of expressing the relation
possibility @ probability @ fuzzy membership withou t weakening the σ@additivity
condition within the definition of probability, whi ch as this paper suggested is
actually the true principle of consistent possibili ty@probability translation. .
Acknowledgement
This work was supported by the University of South@ East Europe Lumina
(Bucharest, Romania) and the Lumina Foundation (Buc harest, Romania).

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