CH 3 Quantum Fuzzy

8/11/2019 CH 3 Quantum Fuzzy

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CHAPTER 3

QUANTUM FUZZY CIRCUITS

Arushi Raghuvanshi and Marek Perkoski

This chapter explores various new mechanisms for modeling and creating humanoid

robot behaviors. We extend the binary quantum Braitenberg vehicles to fuzzy and a new

category of Quantum-uzzy-Braitenberg vehicles. !pplying and extending new logics toBraitenberg vehicles allowed the creation of methods to develop more complex and

realistic behaviors. "ext# Quantum circuits to model uzzy logic# a new development#

were used with genetic algorithms to evolve lip and text synchronization. The Quantum

uzzy $odel was then applied to demonstrate human emotions. inally# the concept ofQuantum circuits to model uzzy logic was defined and generalized.

3!"! INTR#$UCTI#N

%n this pro&ect# various research efforts and applications led to the development of

Quantum circuits to model uzzy logic. ig. '.( shows the evolution of the research wor)leading to the creation of a generalized Quantum uzzy model for humanoid robots

behavior. The concepts show the use of binary circuits# fuzzy circuits# and quantum

circuits for programming the behaviors of robots. *enetic algorithms are used to evolve

behaviors# represented by square boxes in the diagram. The experiments were conductedon a wide variety of robots such as Braitenberg vehicles# $r. +otato,ead and

chrdinger/s cat. +rogramming these robots with the logics described was an interesting

application as it created new behaviors.

Figure 3.1: A diagram for the generalized Quantum Fuzzy model for humanoid robots


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3!%! QUANTUM FUZZY &RAITEN&ER' (EHIC)ES

Braitenberg 0ehicles can now be even further extended to Quantum uzzy model# a newconcept created in 1xx2 and later generalized in 1ref2. This Quantum uzzy $odel can be

very useful because not only does it )eep some properties of quantum circuits such as

entanglement# but it also allows the input to be any real number between zero and one.

ig. '.(( presents the abstracted schematic of mobile Quantum Braitenberg 0ehicles and

stationary puppet robot 3$ister +otato ,ead4 15aghuvanshi672 which were described in

chapter 8. We call it the 9reactive architecture: as the robot reacts to the state ofenvironment. !ll input signals from sensors are transformed to binary quantum

initializations and all output signals come from 3simulated4 quantum measurement units

and are also binary. The output signals may be probabilistic or entangled. %nteger controls

are created as vectors of binary signals. ome examples use quantum ternary logic. Theinternal 9reasoning: of the robot controller is done in ,ilbert space and is deterministic.

%t is the measurement process 3with binary or ternary observables4 that creates discreteoutput signals used to control servos and other robot effectors. ;bserve however that

there is a possibility to have continuous signals from sensors and send continuous signals

to effectors. uzzy logic would be thus more appropriate not only for reasoning of therobot but also for its sensing and control of effectors. Quantum and uzzy


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When signals ( and 8 are not the same and there is no sound the robot turns left and

when there is sound the robot turns right. =hange of direction is done by red)in gate

controlled by sound sensor ". When there is no light from both sensors ( and 8 therobot is not moving. This circuit can be easily modified to a more interesting behavior of

the robot in fuzzy logic# shown in ig. ('b. "ow the robot smoothly combines rotation

with forward motion depending on the difference of lightning in left and right sensors andon the level of noise from sensor ". The quantum fuzzy equivalent of this circuit is

shown in ig. '.('a which is the same circuit for both binary and fuzzy quantum logic

3the difference is in initialization and may be also in the measurement4. %n ig. '.('arepresenting uzzy Quantum


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3!3! QUANTUM FUZZY M#$E) *ITH 'ENETIC A)'#RITHMS T#SYNCHR#NIZE TE+T AN$ SPEECH

We now use quantum logic and fuzzy logic to create realistic animations of behaviors and

human-robot interactions. %n previous sections we have presented the concept of aquantum controller designed as a behavior specification of a mobile robot. %n this section

we will combine these two approaches and introduce the concept of fuzzy quantum logic

15aghuvanshi6>2. This approach was to create efficient *enetic !lgorithm 3*!4approach for robotic learning of theatric and interactive behaviors# particularly for the

synchronization of speech and gesturesof puppet robots. The class of problems to be

solved was? how to use the *enetic !lgorithm together with Quantum uzzy $odel to

synchronize the gestures and speech sounds of a humanoid robot representing particularemotions.

We created an efficient *enetic !lgorithm for robotic learning to synchronize speech and

visual expressions of a robot. @sing *enetic !lgorithms is a significant improvement#over today/s static mapping methods for lip-speech synchronization. ;ur experiment used

an A5! robot which includes a set of motors to control expressions and lip motions. Byencoding lip shapes# using these encodings to create chromosomes# and applying *enetic

!lgorithms to evolve these chromosomes to match given text# my robot can develop lip

motions that correspond with the spo)en text. We experimented with various parametersof my algorithm to test how long it ta)es to synchronize text and lip motion for varying

lengths# crossover rates# mutation rates# number of generations# number of offspring# and

population size. We were surprised to find that robotic learning ta)es a relatively small

number of evolutionary stages. With feedbac) from multiple sources# some of thesestages can be further parallelized. Besides robots# this method can be applied to

animation in movies. The presented method is beneficial because it is fast# language

independent# and adapts to different spea)ing styles. This approach can be furtherextended by developing facial expressions# developing body movements# and

synchronizing the two.

! genetic algorithm is a search technique used in computer science and engineering to

find solutions to various search problems decision or optimization. *enetic algorithms

are a particular class of evolutionary algorithms that use techniques such as inheritance#

mutation# selection# and crossover. $utation involves changing the offspring in someway# selection is the selection of two parents# and crossover defines how the two parents

are combined to give an offspring genome. ! possible solution to a given problem can be

encoded into a sequence of codes called a chromosome 3or a genome4. %n our case# achromosome is a pattern that corresponds to a behavior of fuzzy quantum circuit. %n

*enetic !lgorithms# a set of random chromosomes are created to represent possible

solutions to a given problem. The goal of the algorithm is to evolve the best solution 3achromosome4 that matches the criteria. This can be done by formulating certain 9fitness

function: and 9stopping criteria:. The *!s are often used in con&unction with fuzzy logic

10ada))epat672 and recently also with binary and multi-valued quantum logic 1


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=urrent methods create a sequence of these codes to represent how the lips should move

when those words are spo)en. This static mapping is dependent on the selected language.or a different language or dialect# the mapping will have to be different. %n the simplest

variant there are many phonemes for one servo motion# thus the mapping is many to one.

,owever# a more sophisticated robot may have a one-to-one mapping. %n addition# arobot that pronounces phonemes based on its emotional state the mapping becomes

dependent on the robot/s emotion. inally# as humans have always slight gesture and lip

movement differences even remaining in the same emotional state# we add probabilistic

selection of motions becoming one-to-many probabilistic mapping# depending onemotion# or even for the same emotional state. This is a good application of uzzy

Quantum


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8. %t could also be used in reverse and convert lip motion into

text. This could be useful in documenting presentations# speeches# and even

court cases.'. %t could also be used to create subtitles in movies# or trying

to decode what is being said in an evidence tape without sound.

C. Body language# such as hand# arm and eye movements# canbe also developed in a similar way.D. !rtistic designs 3paintings# sculptures# installations# music#

dialog4# wal)ing and gymnastic robots# and other will use these )inds oftechniques.

3!,! QUANTUM FUZZY M#$E) T# MAP EM#TI#NS

%n existing literature# emotions have been mapped onto a two dimensional wheel of

emotions 1+lutchi)6>2. ! new idea presented in this chapter is that instead of mapping

emotions onto a two dimensional wheel# we can map them onto the Bloch sphere. Thenwe can simulate the application of quantum fuzzy literals. %n the Quantum uzzy model#

an output state before 3quantum4 measurement can represent the 9internal emotions: of a

robot while an output state after measurement can represent an emotion as perceived byan observer. urthermore# entanglement can represent how interactions between two

individuals 3programmed agents4 affect their emotions and the fuzziness of quantum

fuzzy logic allows emotions to have different intensities.

;ur new system is defined by a particular choice of con&unction# dis&unction and

complementation operations. We show that ";T and Toffoli gates )nown from binaryquantum circuits can be adapted as a set of universal gates to the limited fuzzy quantum

system in which one meridian of the Bloch phere is mapped to the 16#(2 interval of

fuzzy sets 15aghuvanshi6E2. %n this new approach# the logic is expanded to arguments

being arbitrary points on the Bloch phere# which is somewhat similar to complex fuzzylogic in which arguments are complex numbers 1Fic)6D2. ,owever# in contrast to

standard fuzzy logic in which the arguments and results are in interval 16#(2 or to

complex fuzzy logic in which arguments and results are complex numbers# the newextended model has the arguments as quantum states and the results as quantum states#

but in a ,ilbert space of a higher dimension. Thus the closure property of algebras of

logic is not satisfied in our new model. +lease note# that when externally observed3measured4# the new model reduces to the standard fuzzy set. But the internal

observation# as seen from the point of view of the observer located in ,ilbert space#

creates states that are not directly observable. Based on the ideas presented above# the

model presented by us is not a logic nor an algebra# but because it is related to fuzzy sets#logics and circuits# we call it a system with fuzzy properties realized in a quantum circuit

or a quantum fuzzy circuit. The created quantum states can represent the state of

reasoning or emotional state of a robot. This is illustrated in igure '.(D.The well-)nownconcept of 9wheel of emotions: 1+lutchi)6>2 can be mapped to both complex field space

and to Bloch phere. The second mapping proposed here gives additional possibilities to

describe relations between emotions and emotion-modifying dynamics of quantum agents3qubits and groups of qubits4. ;bserve that location of observables is basically arbitrary

on the Bloch phere which is reflected in which states the measurement observables are


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placed and in which they are not. By adding more measurement observables to repeated

identical states 3as in ensemble computers4 we get more information about the 9hidden

state: which however always remains not completely )nown to the external observer.

We will start with some example of different ways of modeling emotions using Bloch/s

sphere. ig. '.(D.! shows a simplified wheel of emotion that represents emotions alongGH axes of a complex iI& plane. ! combination of !ctiveI+assive behavior with

+ositiveInegative emotions provides a number of effective emotions characterized by

psychology research 1+lutchi)6>2. G axis represents active or passive mood. H axisrepresents negative or positive emotions. %f we traverse around the circumference of the

circle# any point will have some combination of these# represented by the pro&ection of

the point along G and H axis. ,owever# as we are restricting the representation along the

circumference of the circle only# and it is not an unconstrained plane# this limits the casesli)e Jfully active and fully positive/# since this would be outside the circle.

Fig. 3.15: Representing Simplified Wheel of Emotions using a complex plane and a Blochs Sphere. This

domain can have also many other interpretations of points and (fuzzy quantum) single-qubit operations on

them.

%n ig. '.(DB through ig. '.(D=# we add the third dimension to this set of logic values as

we represent this on the surface of the Bloch/s phere. %n ig. '.(DB# the Wheel of

Amotions from ig. '.(D! becomes the horizontal GH plane of the sphere# and a new Kaxis represents intensity. %n ig. '.(D= and ig. '.(DF# the Wheel of Amotions stays in

GK plane# while H represents the intensity.


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Fig. 3.16. Wheel of Emotions, from Plutchik [Plutchik08]

ig. '.(7 provides a more comprehensive picture of the quantum sphere of emotions.

This is an elaboration of the igure '.(D!# applied to complete the 9wheel of emotions:

1+lutchi)6>2 from ig. '.(L. ,ere the K axis represents the intensity# and the emotionsaround a concentric circle now become phases of the quantum states.

Fig. 3.17. Quantum Sphere of Emotions

The Quantum phere of Amotions significantly adds a lot to the Wheel of Amotions.

While in the Wheel of Amotions# only one emotion can have a high intensity# the

Quantum phere allows for existence of multiple emotions in the ,ilbert spacerepresented by the phases. !fter measurements# only one emotion will be observed. This


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is analogous to the human behavior where in the inside of human mind there are numbers

of emotions# but the actions typically represent one emotion.

The internal reasoning or dynamics of emotional state can be described now by a

quantum circuit built from counterparts of standard fuzzy operators 3";T# !"F

corresponding to IM# ;5 corresponding to MI4# eynman *ate as fuzzy logic AG;53additional operator defined in standard way based on MI# IM# and -# fuzzy membership

functions and other quantum operators serving as extensions. or instance# all )ind of

phase operators 3,adamard# 0# 0N# +auli and other rotations4. !s an illustrative examplelet us discuss igure '.(>a that presents the behavior of the prisoner. The prisoner is

willing to tal) when one cop is good and one cop is bad# otherwise his willingness level

does not change. ;bserve that if the behavior of cops is entangled# the prisoner will never

tal). This behavior can be made much more complicated when we put the behavior ofcops and the prisoner on three Bloch pheres and ta)e the phase information as in igure

'.(D into account and represent all emotions as in ig. '.(7. ig. '.(>b shows use of

general-purpose controlled operators such as =ontrolled-0 in ig. '.(>b. The behavior of

this circuit can be analyzed using methods from 15aghuvanshi67# 5aghuvanshi6>#5aghuvanshi6E# # "ielsen662. $ood of on is dependent on the activity level of

the parents. %t needs both ather and $other to be active to swap completely the mood ofon 3as 0O0 P ";T4. %f only ather or only $other is active then on is in ,ilbert pace

in state 9half happy and half-not-happy: 3(I843 6 N (4 3called the 9=at state: in1"ielsen6624. o when the lowest qubit in ig. '.(>b is measured then there is halfprobability that we observe on as happy and half probability that we observe on as

unhappy 3probability distribution is interpreted as a fuzzy set4. ingle measurement does

not tell us the emotional states of $other# ather or on. %f we repeat measurement many

times in separate copies of the circuit# as in ensemble computing# we would )now withhigh probability if the son is happy# unhappy or in the 9=at state:. Rnowing that the son

is in the cat state would mean that one of parents is active# but we would be not able tofind the state of $other. +lease note that these correlations of emotions specified byquantum states and operators are based on superposition and entanglements# and are

described by different mathematics than the classical correlations.

Fig. 3.18. Quantum circuits corresponding to emotional behaviors in two different situations. (a) behavior of

Prisoner based on behavior of Cops, and (b) mood of Son based on his parents activity level.


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Fig 3.19. Changes of quantum states as complex rotation operators on the

Bloch Sphere. (a) Negation, (b) rotations in the restricted case of only one

meridian representing all logic values.

@sing the )nown methods of quantum mechanics and quantum circuits 1"ielsen662 to

analyze our Quantum uzzy circuits# we can determine the effect of communication onthe emotions of all agents representing parts of robot/s brain or people. %n this way# the

communication between the two agents is analogous to the entanglement in quantumcircuits 1"ielsen662.


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this analysis becomes complicated with more than C qubits and a quantum simulator is

used.

or the last example# we can represent the type of emotion as the phase# and the intensity

of the emotion as a point from 6 to (. %n other words# the wheel of emotions is

represented as the central horizontal circle of the Bloch sphere.

,owever# in order to use this representation# we must be able to measure the phase of a

state on the Bloch sphere. We can do this by using Quantum tomography 1"ielsen662.;ne trivial example of quantum tomography is using the 0 3square-root of ";T4 gate for

approximate phase measurement. !pplying the 0 gate transformation two times# after

measurement we can distinguish between four points on the Bloch sphere. hown in the

calculations below# the 0 gate can be simplified to

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Fig. 3. 21. Using V gate for Quantum Tomography

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3!.! 'ENERA)IZIN' QUANTUM FUZZY M#$E)

!n important question to as) while generalizing a Quantum uzzy $odel is# 9as we can

create a quantum realization of standard fuzzy logic [Raghuvanshi08! can we use theseconcepts to e"tend the notions and applications of fuzzy quantum circuits#$! and %what

additional possibilities the fuzzy quantum circuit has over the standard fuzzy circuits#$

This new quantum fuzzy model uses con&unction# dis&unction and complement 3";T4

operations 15aghuvanshi6E2. =omplement is a basic one-argument operator. The truth

function of negationhas to be non-increasing 3and assign 6 to ( and vice versa4 thefunction " / x3Uu)asiewicz negation4 is the best )nown candidate used in most logics.

,owever# we apply the standard ";T operator used in quantum circuits# which is a

pro&ection 3+auli G rotation4 on a Bloch phere. %t operates then similar to fuzzy negationwhen the arguments are on one meridian of the phere with phase 6 15aghuvanshi6E2#

but it operates in a more general way when an arbitrary point on the Bloch/s sphere is its

argument. =on&unction 3t-norm# IM4 is a two-argument operator. %n Basic uzzy


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+ropositional


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AN$0A2 &1 or 40A2 &1 or A5&! =on&unction operator is defined in the same way as

quantum !"F# i.e. Toffoli gate with one qubit initialized to 6. The definition of uzzyQuantum =on&unction ;perator is thus the following?

*uzzy +uantum 56 2/0 7/2! 12/0 17/23 4 avio 2/0 7/2! 12/0 17/2!

03 4 2/0 7/23 9 12/0 17/23 /0.

This is shown in ig. '.8C and derived below. %nput is Rronec)er +roduct of three parallelinput qubits?

*iven this input# the output is calculated using the matrix of the Toffoli *ate.

*ig. :. 7:. ;alculations for quantum fuzzy conjunction operator.

The output therefore is ?

212/000 217/020 N 712/200N 717/222P 3212/00 217/02 N 712/203

6N 3717/223 /2 0e6ua7ion 3!"1

This implies that the probability of measurement of J(/ is 7177

*ig. :.7


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*ig. :.7>. ata for calculations for quantum fuzzy disjunction operator and its realization as a =offoli gate

where the avio output R is measured.

"ow we can do similar calculations are before?

*ig. :.7?. ;alculations for quantum fuzzy disjunction operator.

The output therefore is ?

717/002 712/022 N 217/202N 212/220P 3717/00 712/02 N 217/203

(N 3212/223 /0 0e6ua7ion 3!%1

This implies that the probability of measurement of J6/ is 2127 . Because 2127 N

2177 N 712/7N 7177 P ( thus the probability of measuring (is 2177 N 712/7N7177 as it should be to satisfy the fuzzy logic axioms. The internal quantum states do

not satisfy the rules of fuzzy logic. or instance# even commutativity does not hold. Theinternal states include much more information than their measured states but this

information is not easily accessible and is usually entangled. These facts# however# give

quite new interpretation possibilities of circuits/ behaviors as reasoning# perceptions#emotions or motions.

Based on the well-)nown quantum realization of Toffoli gate 1"ielsen662# only fivecontrolled gates 3two =ontrolled-0# one =ontrolled-0N and two eynman gates4 are

necessary to calculate not only the fuzzy quantum operator IM 3!"F4 but also other 8-

variable operators using only technology-realizable two-qubit primitives 1


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B# etc can be created using definitions of the basic operators defined in our new model.

=lassical binary gates of quantum logic such as red)in# $iller and eynman can be also

used in the proposed extended fuzzy quantum systems.


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values but remain in 3non-maximum4 entanglement state 3212/00 217/02 N 712/

203. This entangled state may be an argument input given to other gates 3and

measurements4 in other quantum agents processing the data qubits and .


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and + and probability/- "he state isin multi0dimensional1ilbert Space

2umber ofprocessors

One One Ensemble

Measurement +0+3eterministic

+0+3eterministic

#robabilistic

2umber andtype ofmeasurementapparatuses

+ + !ny numberand type ofmeasurements

Entanglement 2o 2o 4es

Internal state 2o 2o 4es

*ig. :.7C. escribes how the +uantum *uzzy ;ircuit model is different than other logic systems.

The fuzzy quantum circuits introduced in 15aghuvanshi6E2 and extended and explainedhere further are based on extended fuzzy concepts and do not satisfy internally some

axioms of any algebra of fuzzy logic. The logic of their operation is neither fuzzy logic1Trillas6(2 nor quantum logic 1Bir)choff'L2. This model is useful to simulate new typesof behaviors of fuzzy quantum circuits that are not possible with either standard fuzzy

logic or with complex fuzzy logic. Aspecially the concepts of entanglement and partial

measurement are new to modeling fuzzy systems. ig. '.8E discusses these aspects inmore detail. The new model was used by me to evolve behaviors of humanoid robots

with a genetic algorithm 15aghuvanshi6>2.

We plan to further investigate the mathematical properties of the fuzzy quantum systems.We plan also to compare them to various types of complex fuzzy logic 1Fic)6D2 and

quantum logic 1+y)aczEE# Titani6'2. The entangled versus non-entangled robot behaviors

with fuzzy and quantum fuzzy evolved controllers can be also compared and analyzed inour future wor). !nother research opportunity is that the controller is a quantum 3in

particular reversible4 circuit# so it can be not only evolved but also synthesized from

examples as an incompletely specified function or as a state machine 1Rumar6>#2 using inductive learning. These methods are similar to group theory

decompositions that we applied already to 5ubi)/s cube and other permutation games

15aghuvanshi6E!2# and we hope that our software can be adapted for these tas)s.

3!B! Con@=usion

This research created several realistic humanoid robots. The methodology of research

was the following? 3(4 learning from papers# boo)s# and the internet about various newconcepts related to quantum-ness# emotions and intelligence# 384 programming some of

these concepts# 3'4 modeling them on my robots and analyzing resulting robot behaviors#3C4 trying to find some theoretical unifying principles for the methods presented by other

authors# through experiments with robots and through software written and tested.

This chapter will be extended to find several practical applications in the entertainment

and animation industry. The models developed here are also valuable tools for learning


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and teaching in high schools. @sing robots to teach quantum logic to teens is a new idea

which will be further expanded 1ref papers from %srael2. !n entirely new concept of

modeling uzzy logic with Quantum =ircuits was created#modified# improved and testedon robots. =urrently# this model is applicable to mapping emotions and generating

realistic behaviors. Aspecially the concepts of entanglement and partial measurement are

new to fuzzy logic.This new logic to evolve behaviors of humanoid robots with a geneticalgorithm 15aghuvanshi6>2# will be applied to concrete quantum fuzzy circuits for more

more applications in future.

%t is well )nown that fuzzy logic has important applications in robotics# image processing#

decision ma)ing and industrial automation# and other 10ada))epat672. uzzy quantum

circuits can be even more useful. irst# we can observe that# realizing fuzzy logic in

quantum would be valuable as one would be able to integrate large fuzzy quantumcircuits into quantum computers# because of the predicted low power consumption and

high processing power of such computers. "ext# they could be integrated into generalized

quantum algorithms such as *rover. or instance# an example continuous version of

*rover was presented in 1Arma)ov682. The concepts of continuous variables in quantuminformation are presented in 1Braunstein6D2. !pplications in Quantum "eural "ets are

discussed in 1+urushothamanE72. Therefore generalizations will be created for fuzzyquantum circuits realizing standard fuzzy logic# as such circuits realize &ust a variant of

continuous quantum logic.

;ther applications of this new approach include creating a more efficient way of mapping

speech to lip motions and creating or displaying human-li)e emotions on a humanoid

robot. urthermore# robotics can be used to teach young )ids about complex ideas such as

Quantum circuits# computing and logic.

3!! *orks Ci7ed

1Brawn6L2 =h. Brawn# ". $etzger# S. Biamonte# $.


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acta @niversitatis# er. Alec. Ang. 0ol. 86# "o. '# Fecember 8667# pp. DL(-D>L.

1"ielsen662 $. !. "ielsen# and %.L41Broo)s>L2 1>2 5. Broo)s? ! 5obust


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1Arma)ov682 0. # "o. '# $ay (EE7# pp. L7E X LE'.

1+y)aczEE2 S. +y)acz# 92 5. 5ovatti and *. Baccarani# 9uzzy 5eversible 6'-C>L'-GIE>%AAA# from %nternet.

1Titani6'2 . Titani# and ,. Rozawa# 9Quantum et Theory#: %nternational S. of

Theoretical +hysics# 0ol. C8# "o. ((# "ov. 866'# pp. 8D7D X 8L68.1Trillas6(2 A.Trillas# A. 5enedo# . *uadarrama# ];n a new theory of fuzzy sets with &ust

one self-contradiction#: +roc. 866( %AAA %nternational uzzy ystems =onfernce# pp.

LD> X LL(.

10ada))epat672 +. 0ada))epat# G. +eng# B.R. Que)and# T. ,.

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