modelo de representación 2-tupla. un enfoque computacional simbólico charlas sinbad 2 extensiones...

Modelo de Representación 2-tupla.Un enfoque computacional simbólico

Charlas Sinbad2

EXTENSIONES Y APLICACIONES EN TOMA DE

DECISION LINGÜÍSTICA

OUTLINE

• INTRODUCTION– DECISION MAKING AND PREFERENCE MODELLING– FUZZY LINGUISTIC APPROACH AND CWW

• LINGUISTIC 2-TUPLE MODEL• EXTENSIONS– MULTIGRANULAR LINGUISTIC INFORMATION– HETEROGENOUS INFORMATION– UNBALANCED LINGUISTIC INFORMATION

• HESITANT FUZZY LINGUISTIC TERM SETS• CONCLUSIONS

INTRODUCTION• DECISION MAKING

Decision making is a core area of different research pursuits such as engineering, both theory and practice, management, medicine and alike. It tries to make the best selection among a set of feasible solutions

– SELECTION PROCESS• Aggregation phase• Exploitation phase

– Solution set of alternative/s

AGGREGATION EXPLOITATIONPREFERENCES SOLUTION

INTRODUCTION

Basic Elements of a Classical Decision Problem

A set of alternatives or available decisions: A set of states of nature that defines the framework of the problem:

A set of utility values, , each one associated to a pair composed of an alternative and a state of nature:

A function that establishes the expert’s preferences regarding the plausible results.

},...,{ 1 maaA

},...,{ 1 nssS iju

s1 ... sN

Alternative 1 u11 u12 ... u1N

Alternative 2 u21 u22 ... u2N

... ... ... … ...

Alternative M uM1 uM2 ... uMN

jiij sau ,:

INTRODUCTION

• DECISION PROBLEMS

– EXPERTS PREFERENCES – ASPECTS OR CRITERIA• NATURE

– QUANTITATIVE» How tall is John ?

– QUALITATIVE» How comfortable is that chair ?

INTRODUCTION

• DECISION PROBLEMS

– QUANTITATIVE• NUMERICAL INFORMATION

– CRISP– INTERVALS

– QUALITATIVE ASPECTS• SUBJECTIVITY• VAGUENESS• IMPRECISION

NUMBERS ARE NOT ADEQUATEDHARD TO EXPRESS NUMERICALLY

INTRODUCTION

• REAL WORLD DECISION PROBLEMS

– UNCERTAINTY

• PROBABILISTIC– PROBABILITY BASED MODELS– DECISION THEORY

• NON PROBABILISTIC– CHALLENGE– EXPERTS: LINGUISTIC DESCRIPTORS

INTRODUCTION

• DECISION MAKING

Issues related to decision making have been traditionally handled either by deterministic or by probabilistic approaches. The first one completely ignores uncertainty, while the second one assumes that any uncertainty can be represented as a probability distribution. However in real-world problems (say, engineering, scheduling, and planning) decisions should be made under circumstances with vague, imprecise and uncertain information. Commonly, the uncertainty could be of non-probabilistic nature. Among the appropriate tools to overcome these difficulties are fuzzy logic and fuzzy linguistic approach. The use of linguistic information enhances the reliability and flexibility of classical decision models.

INTRODUCTION• DECISION PROBLEMS

– NON-PROBABLISTIC UNCERTAINTY– LINGUISTIC INFORMATION• FUZZY LOGIC• FUZZY LINGUISTIC APPROACH

INTRODUCTION

FUZZY LINGUISTIC APPROACH

Linguistic variables differ from numerical variables in that their values are not numbers but are words or phrases in a natural or artificial language (Zadeh, 1975).

Very low Low Medium High Very high

Linguistic terms

Semantic rule

VariableLinguistic variable

INTRODUCTION

• COMPUTING WITH WORDS

– LINGUISTIC COMPUTING MODELS• Based on Membership Functions• Based on Ordinal Scales• 2-Tuple based computational model

– INTERPRETABILITY– ACCURACY– EXTENSIONS

LACK OF ACCURACY IN RETRANSLATION

FOUNDATIONS:LINGUISTIC 2-TUPLE REPRESENTATION

LINGUISTIC 2-Tuple• BIBLIOGRAPHY

– F. Herrera and L. Martínez. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8(6):746-752, 2000

– F. Herrera, L. Martínez. An Approach for Combining Numerical and Linguistic Information based on the 2-tuple fuzzy linguistic representation model in Decision Making. International Journal of Uncertainty , Fuzziness and Knowledge -Based Systems. 8.5 (2000) 539-562

– F. Herrera, L. Martínez. The 2-tuple Linguistic Computational Model. Advantages of its linguistic description, accuracy and consistency. International Journal of Uncertainty , Fuzziness and Knowledge-Based Systems. 2001, Vol 9 pp. 33-48

– F. Herrera, L. Martínez. A model based on linguistic 2-tuples for dealing with multigranularity hierarchical linguistic contexts in Multiexpert Decision-Making. IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics, 2001, Vol 31 Num 2 pp. 227.234.

– F. Herrera, L. Martínez. P.J. Sánchez. Managing non-homogeneous information in group decision making. European Journal of Operational Research 166:1(2005) pp. 115-132

– F. Herrera, E. Herrera-Viedma, L. Martínez, A Fuzzy Linguistic Methodology To Deal With Unbalanced Linguistic Term Sets. IEEE Transactions on Fuzzy Systems 2008. Page(s): 354-370. Volume: 16, Issue: 2.

– M. Espinilla, J. Liu, L. Martínez. An extended hierarchical linguistic model for decision-making problems. Computational Intelligence. In press. 2011

LINGUISTIC 2-Tuple

• Linguistic representation:– Model based on the symbolic approach.

• Linguistic Domain: Continuous–Linguistic representation any symbolic computation

Arith_Mean(L,VL,VH,P)=(2+1+5+6)/4=3,25

LINGUISTIC 2-Tuple

• Linguistic Representation based on pair of values

• Symbolic Translation

Sss iii ),,( )5.0,5.0[i

LINGUISTIC 2-Tuple

• 2-tuple Functions– From a numerical value in the interval of granularity

into a 2-tuple

– Example

)5.0,.5.0(,0: Sg

)5,0,5.0[

)(),,()(

roundiswiths i

)25.0,()75.2( 3 s

)25.0,()25.2( 2s

LINGUISTIC 2-Tuple

• 2-tuple Functions– It inverse

– Example

gS ,05.0,5,0:1

isi ),(1

75.2)25.0,( 31 s

2-Tuple Computational Model

• Negation Operator

• Example)),((),( 1 ii sgsNeg

)25.0,()25.5())25.0,(6()25.0,( 1 VHVLVLNeg

LINGUISTIC 2-Tuple

2-tuple Computational Model

• Aggregation 2-tuple operators

– To use and compute as in the numerical models– To use and transform in a 2-tuple– Aggregation operators

• Arithmetic mean

• Weighting average

• OWA operator

LINGUISTIC 2-Tuple

2-tuple Computational Model

• Comparison: Lexico-graphic order Let and be two 2-tuples

If k < l then is less than

If k = l then: • If then and are equal

• If then is less than • If then is greater than

),( 1ks ),( 2ls

),( 1ks),( 2ls

),( 1ks

),( 1ks),( 2ls

),( 2ls

LINGUISTIC 2-Tuple

LINGUISTIC 2-Tuple• Applications

– Decision Making and Decision Analysis• Multi-Criteria Decision Making• Group Decision Making

– Consensus Reaching Processes• Evaluation

– Sensory Evaluation– Performance Appraisal

– Internet Based Services– Recommender Systems– Information Retrieval

– Genetic Fuzzy Systems

LINGUISTIC 2-Tuple• Problems–Complex frameworks–Different degrees of knowledge• Multiple linguistic scales

–Information of different nature• Quantitative aspects• Qualitative aspects

–Non-symmetrically distributed linguistic information• Unbalanced Linguistic Information

2-tuple EXTENSIONS

LINGUISTIC 2-TUPLE EXTENSIONS

2-Tuple EXTENSIONS

• MULTIGRANULAR LINGUISTIC INFORMATION

– FUSION APPROACH– LINGUISTIC HIERARCHIES– EXTENDED LINGUISTIC HIERARCHIES

• HETEROGENOUS INFORMATION

• UNBALANCED LINGUISTIC INFORMATION

MULTI-GRANULARLINGUISTIC

INFORMATION

MULTI-GRANULAR LINGUISTIC INFORMATION

• Real World Problems– Multiple Sources of information– Different degree of uncertainty– Different degree of knowledge

• Linguistic Information– Necessity of Multiple scales

• Different Approaches– Based on membership functions– Probabilistic– Symbolic

MULTI-GRANULAR LINGUISTIC INFORMATION

• BIBLIOGRAPHY1. Herrera, F., Herrera-Viedma, E., and Martínez, L. (2000). A fusion approach for managing multi-

granularity linguistic term sets in decision making. Fuzzy Sets and Systems, 114(1), 43-58.

2. Herrera, F. and Martínez, L. (2001). A model based on linguistic 2-tuples for dealing with multigranularity hierarchical linguistic contexts in multiexpert decision-making. IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics, 31(2), 227-234.

3. Huynh, V. and Nakamori, Y. (2005). A satisfactory-oriented approach to multiexpert decision-making with linguistic assessments. IEEE Transactions On Systems Man And Cybernetics Part B-Cybernetics, 35(2), 184-196.

4. Chen, Z. and Ben-Arieh, D. (2006). On the fusion of multi-granularity linguistic label sets in group decision making. Computers and Industrial Engineering, 51(3), 526-541.

5. Chang, S., Wang, R., and Wang, S. (2007). Applying a direct multi-granularity linguistic and strategy-oriented aggregation approach on the assessment of supply performance. European Journal of Operational Research, 117(2), 1013-1025.

6. M. Espinilla, J. Liu, L. Martínez. An extended hierarchical linguistic model for decision-making problems. Computational Intelligence. In press. 2011

MULTI-GRANULAR LINGUISTIC FUSION

APPROACH

LINGUISTIC HIERARCHIES•MULTI-GRANULAR LINGUISTIC CONTEXTS

• PROBLEMS

– Multiple Experts or criteria– Different degree of Knowledge– Linguistic modelling– Multiple Linguistic term sets

•INTERPRETABILITY

– LINGUISTIC RESULTS

FUSION APPROACH•FEATURES

– MEMBERSHIP BASED COMPUTATIONS– LACK OF ACCURACY

FUSION APPROACH

– MULTIPLE EXPERTS

– DIFFERENT LINGUISTIC TERM SETS

FUSION APPROACH

• COMPUTATIONAL MODEL

– SELECTING A BASIC LINGUISTIC TERM SET ST

– UNIFICATION PHASE

– COMPUTATIONAL PHASE– FUZZY ARITHMETIC

FUSION APPROACH

• COMPUTATIONAL MODEL

– LINGUISTIC RESULTS

)/())(((

],0[)(: gSF T

))((( TSF ( )47.,()47.0,(53.0)) 1 VLs

LINGUISTIC HIERARCHIES

LINGUISTIC HIERARCHIES•MULTI-GRANULAR LINGUISTIC CONTEXTS

• PROBLEMS

– Multiple Experts or criteria– Different degree of Knowledge– Linguistic modelling– Multiple Linguistic term sets

•INTERPRETABILITY•ACCURACY

– AVOID LOSS OF INFORMATION

• Linguistic Hierarchies– LH:• A set of levels

– Level:• A linguistic term set with different granularity to

the remaining ones l(t,n(t))

– The linguistic term set of a LH of the level t:

• Linguistic Hierarchy

– The label sets of a hierarchy

•Semantics: triangular membership functions

•Uniformly and symmetrically distributed in [0,1]

•Odd granularity

•Middle label stands for indifference

• Linguistic Hierarchy Basic Rules

Rule 1: To preserve all former modal points of the membership functions of each linguistic term from one level to the following one.

Rule 2: To make smooth transitions between successive levels. The aim is to build a new linguistic term set, Sn(t+1). A new linguistic term will be added between each pair of terms belonging to the term set of the previous level t. To carry out this insertion, we shall reduce the support of the linguistic labels in order to keep place for the new one located in the middle of them.

)1)(2,1())(,( tntltntl

l (1,3)

l (2,5)= l (2,(2*3)-1)

l (3,9)= l (3,(2*5)-1)

)1)(2,1())(,( tntltntl

l (1,3)

l (2,5)= l (2,(2*3)-1)

l (3,9)= l (3,(2*5)-1)

F. Herrera and L. Martínez. A Model Based on Linguistic 2-Tuples for Dealing with Multigranular Hierarchical Linguistic Context in Multi-Expert Decision Making. IEEE Transactions on SMC - Part B: Cybernetics 31 (2001) 227-234.

• Computational Model

COMPUTING WITH WORDS

MULTIPLE LINGUISTIC SCALES

LINGUISTIC HIERARCHIES• Transformation functions

– One to One mapping– Without loss of information– Computing based on:

• 2-tuple computational model• Transformation functions

• Computational Model– Example

• Computational Model

– Translation• Unification phase

– Computations– Retranslation

• Transformation• Different levels

Strong limitation!!To deal with

some linguistic term sets

LH l (t,n(t)) l (t,n(t))

t=1 l (1,3) l (1,7)

t=2 l (2,5) l (2,13)

t=3 l (3,9)

)1)(2,1())(,( tntltntl

• LIMITATIONS

– Definition framework

•It is not possible the use of any linguistic term set

–5 and 7 linguistic term sets are not possible with a LH

•CHALLENGE

– New structure able to deal with any linguistic term set

•EXTENDED LINGUISTIC HIERARCHIES

EXTENDED LINGUISTIC

HIERARCHIES

EXTENDED LINGUISTIC HIERARCHIES

• Extended Linguistic Hierarchies (ELH)

– Flexible evaluation framework– Accuracy Desirable Features!– Results in the framework

• Flexible evaluation framework – 3,5,7– 5,7,9– Etc.

l(1,3)

l(2,5)

l(3,7)

M. Espinilla, J. Liu, L. Martínez. An extended hierarchical linguistic model for decision-making problems. Computational Intelligence. Computational Intelligence, Vol. 27, Issue 3, pp. 489-512 50

• Extended Hierarchical Rules

Extended Rule 1• Include a finite number of the levels t={1,…,m}• Not necessary to keep the former modal points one to another.

Extended Rule 2• Add a new level t’ that keeps all the former modal points of all the previous levels• Granularity level t’

n(t’) = (LCM( n(t)-1, n(t)-1, …., n(t)-1)+1t={1,…,m}

l(1,3)

l(2,5)

l(3,7)

LCM(2,4,6)+1=13

l(4,13)

l(1,3)

l(2,5)

l(3,7)

l(4,13) LCM(2,4,6)+1=13

l(1,3)

l(2,5)

l(3,7)

l(4,13) LCM(2,4,6)+1=13

l(1,3)

l(2,5)

l(3,7)

l(4,13) LCM(2,4,6)+1=13

CW in ELH – Without loss of information– Use• 2-tuple computational model• Transformation functions

– The information cannot be unified in any level.– t={1,…,m} and t’=m+1

• Unification of the information

• Transformation Functions• Level t’

• Computations and Results

– Aggregation of the information

– Results• Initial linguistic term sets

HETEROGENEOUSINFORMATION

HETEROGENEOUS INFORMATION

•Non Homogeneous contexts

– Representation Structures point of view•Preference Relations•Utility Vectors•Ordered preferences

– Representation Models point of view•Numerical•Interval-Valued•Linguistic

• Heterogenous framework:– Numerical– Linguistic– Linguistic-MG– Interval-valued

– operate directly Different domains

•To Operate with Non Homogenous Information– To Make information Uniform– Basic Linguistic Term Set (BLTS)– Fuzzy Sets (FSs)

•Transformation Functions– Numerical Information into a FS in the BLTS– Linguistic Information into a FS in the BLTS– Interval-valued Information into a FS in the BLTS– FS in the BLTS to a 2-tuple in BLTS

SELECTING THE BASIC LINGUISTIC TERM SET

•It context dependent

– It should keep the level of discrimination used by the experts• Granularity: Maximum

– Transformation Functions without loss of information:• Fuzzy Partition• Semantics: Triangular fuzzy membership functions

– To make the information uniform in the BLTS that we note as ST

• Measures of comparison

•Numerical Information into a FS in the BLTS– Let a numerical value in [0,1]– Its transformation into a FS in ST is carried out as:

)(1,0: TNS SFT

1,0,,/)(0

iiiNS Sss

bsaifcb

sSupportsifi

TRANSFORMATION FUNCTIONS

Linguistic Information into a FS in the BLTS– Let a linguistic value in S– Its transformation into a FS in ST is carried out as :

)(: TSS SFST

LiSS ScSscs

,,/)(0

)}(),(min{max yyk

•Interval-Valued Information into a FS in the BLTS– Let be an interval valued in I([0,1])– Before transforming in a FS. The interval-value will be

represented as:

– Its transformation into a FS in ST is carried out as:

)(: TIS SFIT

kIS cI

)}(),(min{max yykcIy

•Unified Information– Fuzzy sets in the BLTS

•To operate over the FSs by means of the Extension Principle

–Membership functions– Limitations, difficulties

•To Transform FSs into:– 2-tuples

•To Transform a Fuzzy set into a 2-tuple

•Information unified by means of 2-tuples•2-tuple computational model

gSF T ,0)(:

)/())((

),()( is

•Example

INPUTS

UNIFICATION

AGGREGATION AND

2-TUPLE73

•Numerical and Linguistic

•Linguistic Multi-granular

•Numerical, Interval-Valued and Linguistic

•Numerical, Interval-Valued and Linguistic Multi-Granular

CONTEXTS

UNBALANCED LINGUISTIC

INFORMATION

•Linguistic Scales– Usually: Symmetrical

– Sometimes: non-symmetrical Unbalanced

• How to manage, Representation and computations ?

UNBALANCED LINGUISTIC INFORMATION

TotalAbsence

BarelyPerceptible Good

Average Great

TotalAbsence

BarelyPerceptible

Slight

Average Great

F. Herrera, E. Herrera-Viedma, L. Martínez, A Fuzzy Linguistic Methodology To Deal With Unbalanced Linguistic Term Sets. IEEE Transactions on Fuzzy Systems 2008. Page(s): 354-370. Volume: 16, Issue: 2

•Methodology

– Representation• Semantic Algorithm– Linguistic 2-tuple– Linguistic Hierarchies

– Computations: CW• Symbolic• Accurate• Interpretable

•OUTLINE

– Basic Ideas

– Algorithm• Representation

– Computational model• Computing with Words

UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm

•Basic ideas:

TotalAbsence

BarelyPerceptible

Slight

Average Great

LS RSCS

RCL SSSS

TotalAbsence

BarelyPerceptible

Slight

Average Great

•Basic ideas: Total

AbsenceBarely

PerceptibleSlight

Average Great

LS RSCS

TotalAbsence

BarelyPerceptibl

Slight

AverageGreat

One level

Two levels

•Basic ideas: One level in the hierarchy

TotalAbsence

BarelyPerceptible

Slight Average Great

What side?

In Level (1,3)

Represent andRS

)(tnRR SS

)(tnCC ss

• Representation using one level:– It is analogous to :

)(tnLL SS

)(tnCC ss

LS RSCS

F D C B A

• Representation using two levels:

–What levels?

– In • Level (2,5) • Level (3,9)

LS RSCS

1)1()(#

• Representation using two levels:– The right side of the levels contains the assignable labels to

LS RSCS

},...,{ )1(1)1((

)1(1)2/)1)1(((

)1()1(

tnR ssSAS

},,,{ 98

9 ssssASR

},...,{ )(1)((

)(1)2/)1)(((

)()( tntn

tnR ssSAS

},{ 54

5 ssASR

• Representation using two levels:

LS RSCS

MiddleDensityRS

labels close to the centre labels close to the extreme

},{}{ ABCSSS RERCR LS RS

RESRCS

LS RSCS

ExtremeDensityRS

•Assigning labels from two levels in the LH

)(#1 Rtt Slablab

)(#)2/)1)1((( Rt Stnlab

3)(# RS

21 tlab

CS RESRCS

THENis represented onis represented on ELSE is represented onis represented on

extremedensityRS

)1( tnRAS

)(tnRAS

•Assigning from two levels: Brigdes

–S must be a fuzzy partition:• Bridging Gaps

LS RSCS

CS RESRCS

Brigde

extremedensityRS

ssss tnkjump

tnijump

; )1()(

ssss tnkjump

tnijump

; )1()(

LS RSCS

•Assigning from two levels: Central label

–S must be a fuzzy partition:

extremedensityRS

)(tnCC ss

)1( tnCC ss

LS RSCS

•Algorithm

UNBALANCED LINGUISTIC INFORMATION Computing with Words

Outputs: Five subsets :

Sets of levels: Table:

{tLE, tLC, tRC ,tRE} = {1,1,2,3}

SLE = SLC= SL= {F}Sc = {D}SRC = {C,B}SRE = {A}

F D C B A

I(i) label index G(i) level in the LH n(t)

•Ouputs– Semantics and representation

TotalAbsence

BarelyPerceptible

Slight

AverageGreat

• Representation Model:– Semantics is represented using different levels of the

• Computational Model– Operate with unbalanced linguistic term sets• Without loss of information.

• Context:– Linguistic Hierarchy– 2-tuple Linguistic Accurate Computational Model

• Define transformation functions:– Unbalanced term Linguistic term into LH

)5.0,5.0[)5.0,5.0[: LHSLH

siGiIi

iGiIii

)( ),,(),(

))5.0,5.0[(),(

• Define transformation functions

F D C B A

)1(0)0,( ssF n LH

• Define transformation functions:– Unbalanced term Linguistic term into LH

)5.0,5.0[)5.0,5.0[: SLH-1LH

LHtntlSs

LHstntn

))(,(,|

))5.0,5.0[(),()()(

• Define transformation functions

F D C B A

)0,()( )1(0

30 Fss n -1-1 LHLH

• Computational Model Scheme1. Unbalanced Linguistic Assessments in S

2. Unbalanced Linguistic Assessments in LH

3. Unbalanced Linguistic Assessments in Sn(t’)

4. Result in Sn(t’)

5. Result in S

Sss iii ),,(

),( )()( iiGiIs

),( )'(j

),( )'(f

),( hhs

),( iis LH

),( )(' k

tt sTF

torFtupleOpera2

),( )'(f

tnfs -1LH

• Computational Model Scheme)0,(),0,( FA

),( )()( iiGiIs

),( )'(j

),( )'(f

),( hhs

),( iis LH

),( )(' k

tt sTF

),( )'(f

tnfs -1LHF D C B A

Arithmetic Mean 2T

HESITANT SITUATIONS INDECISION MAKING

Hesitant Fuzzy Sets (HFS)

• Hesitant fuzzy sets (Torra 2010)– Fulfil the management of decision situations– Quantitative contexts

• Decision makers• Among different values• Assess criteria or alternatives

– HFS• A function that returns a subset of values in [0,1]

• In terms of the union of their membership degree to set a fuzzy sets

Linguistic Hesitant Situations

• Qualitative Setting– Hesitant Fuzzy Linguistic Term Sets (HFLTS)

• Objectives– Improve the flexibility of the elicitation– Experts hesitate among different linguistic values

• New linguistic expressions– Closer to human beings expressions– Context-free grammar

Hesitant Fuzzy Linguistic Term Sets (HFLTS)

• Similarly to the HFS• Qualitative context– Decision makers– Among different linguistic values

• To manage such situations– HFLTS FLA and HFS

Let S={s0,…,sg} be a linguistic term set, a HFLTS, Hs, is an ordered finite subset of consecutive linguistic terms of S

Example:S={s0:nothing, s1:very_low, s2:low, s3:medium, s4: high, s5:very_high, s6:perfect}

HS={high, very_high, perfect}

Hesitant Fuzzy Linguistic Term Sets (HFLTS)

• Define two operators–Maximum and minimum bounds of a HFLTS

– Compare two HFLTS• Envelope of a HFLTS• It is a linguistic interval

Upper bound

Lower bound

Context-free Grammar

• Let GH be a context-free grammar and S={s0,…,sg} a linguistic term set. The elemets of

Context-free Grammar

• Transformation function, – Obtain HFLTS from the linguistic expressions

– Linguistic expressions are transformed:

•Linguistic Information– Computing with words– Symbolic Approaches

•Linguistic 2-tuple– Accuracy– Interpretability

•Extensions•Hesitant information

CONCLUSIONS

THANKS A LOTFOR YOUR ATTENTION

QUESTIONS

modelo de representación 2-tupla. un enfoque computacional simbólico charlas sinbad 2 extensiones...

decision lingstica slide

cww linguistic

linguistic descriptors

use of linguistic information

models decision theory

introduction computing

classical decision problem

uncertain information

Documents

codemotion madrid 2013 - [php] desarrollo de extensiones en...

capital simbólico, dominación y legitimidad. las raíces...

corrÊa, mariza - traficantes do simbólico [caps. 1 e 3]

extensiones de las funciones...

o ohar simbólico no consumo brasileiro #smwsp

interaccionismo simbólico eq6

"interaccionismo simbólico" hans joas

lo fantÁstico y lo simbÓlico como espacios para la

interaccionismo simbólico h mead

pierre bourdieu – capital simbólico y clases sociales

a natureza do interacionismo simbólico - h blumer ·...

formatos y extensiones

extensiones de cabello de miamixtensions pop

cómputo simbólico y gráfico: estudio del sistema...

extensiones de navegadores

revista extensiones

a natureza do interacionismo simbólico - h blumer ·...

extensiones remotas seguras en elastix

s3-us-east-2.amazonaws.com · extensiones con acceso a...

teoría de señales, interacción estratégica y capital...