WATCC

IV WORKSHOP ASPECTOS TEÓRICOS DE CIENCIA DE LA

COMPUTACIÓN- WATCC -

XIX Congreso Argentino de Ciencias de la Computación - CACIC 2013 : Octubre 2013,

Mar del

Plata, Argentina : organizadores : Red de Universidades con Carreras en Informática

RedUNCI, Universidad CAECE / Armando De Giusti ... [et.al.] ; compilado por Jorge

Finochietto ; ilustrado por María Florencia Scolari. - 1a ed. - Mar del Plata : Fundación

de Altos Estudios en Ciencias Exactas, 2013.

E-Book.

ISBN 978-987-23963-1-2

1. Ciencias de la Computación. I. De Giusti, Armando II. Finochietto, Jorge, comp. III.

Scolari, María Florencia, ilus.

CDD 005.3

Fecha de catalogación: 03/10/2013

AUTORIDADES DE LA REDUNCI

Coordinador TitularDe Giusti Armando (UNLP) 2012-2014

Coordinador AlternoSimari Guillermo (UNS) 2012-2014

Junta DirectivaFeierherd Guillermo (UNTF) 2012-2014Padovani Hugo (UM) 2012-2014Estayno Marcelo (UNLZ) 2012-2014Esquivel Susana (UNSL) 2012-2014Alfonso Hugo (UNLaPampa) 2012-2013Acosta Nelson (UNCPBA) 2012-2013Finochietto, Jorge (UCAECE) 2012-2013Kuna Horacio (UNMisiones) 2012-2013

SecretariasSecretaría Administrativa: Ardenghi Jorge (UNS)Secretaría Académica: Spositto Osvaldo (UNLaMatanza)Secretaría de Congresos, Publicaciones y Difusión: Pesado Patricia (UNLP)Secretaría de Asuntos Reglamentarios: Bursztyn Andrés (UTN)

AUTORIDADES DE LA UNIVERSIDAD CAECE

RectorDr. Edgardo Bosch

Vicerrector AcadémicoDr. Carlos A. Lac Prugent

Vicerrector de Gestión y Desarrollo EducativoDr. Leonardo Gargiulo

Vicerrector de Gestión AdministrativaMg. Fernando del Campo

Vicerrectora de la Subsede Mar del Plata:Mg. Lic. María Alejandra Cormons

Secretaria Académica: Lic. Mariana A. Ortega

Secretario Académico de la Subsede Mar del PlataEsp. Lic. Jorge Finochietto

Director de Gestión Institucional de la Subsede Mar del PlataEsp. Lic. Gustavo Bacigalupo

Coordinador de Carreras de Lic. e Ing. en SistemasEsp. Lic. Jorge Finochietto

COMITÉ ORGANIZADOR LOCAL

PresidenteEsp. Lic. Jorge Finochietto

MiembrosEsp. Lic. Gustavo BacigalupoMg. Lic. Lucia MalbernatLic. Analía VarelaLic. Florencia ScolariC.C. María Isabel MeijomeCP Mayra FullanaLic. Cecilia PelleriniLic. Juan Pablo VivesLic. Luciano Wehrli

Escuela Internacional de Informática (EII)DirectoraDra. Alicia Mon

CoordinaciónCC. María Isabel Meijome

COMITÉ ACADÉMICO

Universidad Representante

Universidad de Buenos Aires

Universidad Nacional de La Plata

Universidad Nacional del Sur

Universidad Nacional de San Luis

Universidad Nacional del Centro de la

Provincia de Buenos Aires

Universidad Nacional del Comahue

Universidad Nacional de La Matanza

Universidad Nacional de La Pampa

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Universidad Nacional Patagonia San Juan

Bosco

Universidad Nacional de Entre Ríos

Universidad Nacional del Nordeste

Universidad Nacional de Rosario

Universidad Nacional de Misiones

Universidad Nacional del Noroeste de la

Provincia de Buenos Aires

Universidad Nacional de Chilecito

Universidad Nacional de Lanús

Echeverria, Adriana (Ingeniería) – Fernández

Slezak, Diego (Cs. Exactas)

De Giusti, Armando

Simari, Guillermo

Esquivel, Susana

Acosta, Nelson

Vaucheret, Claudio

Spositto, Osvaldo

Alfonso, Hugo

Estayno, Marcelo

Feierherd, Guillermo

Gil, Gustavo

Márquez, María Eugenia

Leone, Horacio

Otazú, Alejandra

Aranguren, Silvia

Buckle, Carlos

Tugnarelli, Mónica

Dapozo, Gladys

Kantor, Raúl

Kuna, Horacio

Russo, Claudia

Carmona, Fernanda

García Martínez, Ramón

COMITÉ ACADÉMICO

Universidad Representante

Universidad Nacional de Santiago del Estero

Escuela Superior del Ejército

Universidad Nacional del Litoral

Universidad Nacional de Rio Cuarto

Universidad Nacional de Córdoba

Universidad Nacional de Jujuy

Universidad Nacional de Río Negro

Universidad Nacional de Villa María

Universidad Nacional de Luján

Universidad Nacional de Catamarca

Universidad Nacional de La Rioja

Universidad Nacional de Tres de Febrero

Universidad Nacional de Tucumán

Universidad Nacional Arturo Jauretche

Universidad Nacional del Chaco Austral

Universidad de Morón

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Universidad Católica Argentina - Rosario

Universidad del Salvador

Universidad del Aconcagua

Universidad Gastón Dachary

Universidad del CEMA

Universidad Austral

Durán, Elena

Castro Lechstaler Antonio

Loyarte, Horacio

Arroyo, Marcelo

Brandán Briones, Laura

Paganini, José

Vivas, Luis

Prato, Laura

Scucimarri, Jorge

Barrera, María Alejandra

Nadal, Claudio

Cataldi, Zulma

Luccioni, Griselda

Morales, Martín

Zachman, Patricia

Padovani, Hugo René

De Vincenzi, Marcelo

Guerci, Alberto

Foti, Antonio

Bournissen, Juan

Finochietto, Jorge

Ditada, Esteban

Grieco, Sebastián

Zanitti, Marcelo

Gimenez, Rosana

Belloni, Edgardo

Guglianone, Ariadna

Robiolo, Gabriela

COMITÉ CIENTífICOCoordinación

Armando De Giusti (UNLP) - Guillermo Simari (UNS)

Abásolo, María José (Argentina)Acosta, Nelson (Argentina)Aguirre Jorge Ramió (España)Alfonso, Hugo (Argentina)Ardenghi, Jorge (Argentina)Baldasarri Sandra (España)Balladini, Javier (Argentina)Bertone, Rodolfo (Argentina)Bría, Oscar (Argentina)Brisaboa, Nieves (España)Bursztyn, Andrés (Argentina)Cañas, Alberto (EE.UU)Casali, Ana (Argentina)Castro Lechtaller, Antonio (Argentina)Castro, Silvia (Argentina)Cechich, Alejandra (Argentina)Coello Coello, Carlos (México)Constantini, Roberto (Argentina)Dapozo, Gladys (Argentina)De Vicenzi, Marcelo (Argentina)Deco, Claudia (Argentina)Depetris, Beatriz (Argentina)Diaz, Javier (Argentina)Dix, Juerguen (Alemania)Doallo, Ramón (España)Docampo, DomingoEchaiz, Javier (Argentina)Esquivel, Susana (Argentina)Estayno, Marcelo (Argentina)Estevez, Elsa (Naciones Unidas)Falappa, Marcelo (Argentina)Feierherd, Guillermo (Argentina)Ferreti, Edgardo (Argentina)Fillottrani, Pablo (Argentina)Fleischman, William (EEUU)García Garino, Carlos (Argentina)García Villalba, Javier (España)Género, Marcela (España)Giacomantone, Javier (Argentina)Gómez, Sergio (Argentina)Guerrero, Roberto (Argentina)Henning Gabriela (Argentina)

Janowski, Tomasz (Naciones Unidas)Kantor, Raul (Argentina)Kuna, Horacio (Argentina)Lanzarini, Laura (Argentina)Leguizamón, Guillermo (Argentina)Loui, Ronald Prescott (EEUU)Luque, Emilio (España)Madoz, Cristina (Argentina)Malbran, Maria (Argentina)Malverti, Alejandra (Argentina)Manresa-Yee, Cristina (España)Marín, Mauricio (Chile)Motz, Regina (Uruguay)Naiouf, Marcelo (Argentina)Navarro Martín, Antonio (España)Olivas Varela, José Ángel (España)Orozco Javier (Argentina)Padovani, Hugo (Argentina)Pardo, Álvaro (Uruguay)Pesado, Patricia (Argentina)Piattini, Mario (España)Piccoli, María Fabiana (Argentina)Printista, Marcela (Argentina)Ramón, Hugo (Argentina)Reyes, Nora (Argentina)Riesco, Daniel (Argentina)Rodríguez, Ricardo (Argentina)Roig Vila, Rosabel (España)Rossi, Gustavo (Argentina)Rosso, Paolo (España)Rueda, Sonia (Argentina)Sanz, Cecilia (Argentina)Spositto, Osvaldo (Argentina)Steinmetz, Ralf (Alemania)Suppi, Remo (España)Tarouco, Liane (Brasil)Tirado, Francisco (España)Vendrell, Eduardo (España)Vénere, Marcelo (Argentina)Villagarcia Wanza, Horacio (Arg.)Zamarro, José Miguel (España)

IV Workshop Aspectos teórIcosde cIencIA de lA computAcIón

- WATCC -

Trabajo AutoresID

A Complexity Lower Bound Based On Software Engineering Concepts

Andrés Rojas Paredes (UBA)5756

On Aggregation Process in Linguistic Decision Making Framework

M. Giménez, S. Gramajo (UTN)5822

A Complexity Lower Bound Based On

Software Engineering Concepts

Andres Rojas Paredes

Universidad de Buenos Aires,Facultad de Ciencias Exactas y Naturales, Departamento de Computacion.

Pabellon 1, Ciudad Universitaria, Buenos Aires, [email protected]

Abstract. We consider the problem of polynomial equation solving alsoknown as quantifier elimination in Effective Algebraic Geometry. Thecomplexity of the first elimination algorithms were double exponential,but a considerable progress was carried out when the polynomials wererepresented by arithmetic circuits evaluating them. This representationimproves the complexity to pseudo–polynomial time.

The question is whether the actual asymptotic complexity of circuit–based elimination algorithms may be improved. The answer is no whenelimination algorithms are constructed according to well known soft-ware engineering rules, namely applying information hiding and takinginto account non–functional requirements. These assumptions allows toprove a complexity lower bound which constitutes a mathematically cer-tified non–functional requirement trade–off and a surprising connectionbetween Software Engineering and the theoretical fields of Algebraic Ge-ometry and Computational Complexity Theory.

Keywords: Non-functional requirement trade–off, information hiding,arithmetic circuit, complexity lower bound, polynomial equation solving,quantifier elimination in algebraic geometry

1 Introduction

The main issue of this paper is to describe the Software Engineering aspects ofthe mathematical computation model introduced in [9]. This model captures thenotion of a circuit–based elimination algorithm in order to solve a thirty years oldproblem in algebraic complexity theory (see e.g. [8], [10]): in arithmetic circuit–based effective elimination theory the elimination of a single existential quantifierblock in the first order theory of algebraically closed fields of characteristic zerois intrinsically hard (i.e. it has an exponential complexity lower bound). Thisconclusion may also be expressed in terms of a trade–off between two non–functional requirements: on one hand we have a complexity requirement and onthe other a property of mathematical functions called geometrical robustness.This complexity lower bound in terms of software engineering concepts appears

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2 A. Rojas Paredes

for first time in [5] in the context of polynomial interpolation. In this work westudy a more general case in the context of quantifier elimination.

Complexity lower bounds are undoubtedly theoretical research. But there isalso a practical aim behind that. Consider the process in software design where asoftware architecture is developed in order to solve a certain computational prob-lem. Assume also that one of the non–functional requirements of the softwaredesign project consists of a restriction on the run time computational complex-ity of the program which is going to be developed (this was the case during theimplementation of the polynomial equation solver Kronecker by G. Lecerf, see[11]). Our practical aim is to provide the software engineer with an efficient toolwhich allows him to answer the question whether his software design process isentering at some moment in conflict with the given complexity requirement. Ifthis is the case, the software engineer will be able to change at this early stagehis design and may look for an alternative software architecture. The followingexample illustrates this description.

Example 1 (Finite Set). Suppose that our task is to implement a finite set S

of cardinality n, e.g. a subset of the natural numbers N, and that we have tosatisfy the requirement that membership to the finite set S is decided using onlyO(log n) comparisons. If the set S is implemented by an unordered array, wewill be unable to satisfy our complexity requirement. So we are forced to thinkin alternative implementations of the abstract concept of a finite set, e.g. byordered arrays, special trees or any other data type which is well suited for ourtask.

Example 1 represents a case where it may be impossible to satisfy a givencomplexity requirement by means of a previously fixed software architecture.Our aim is to formalise such impossibility by means of a complexity lower boundwhich is usually difficult to infer when the number of components of the systemunder consideration is large or when the predicate to decide or the functionto compute becomes more sophisticated like in polynomial equation solving.This leads to the idea to fix in advance only a small selection of architecturalfeatures, e.g. the abstraction levels or part of the language of our system (notthe algorithms themselves). The computation model we are going to explain infollowing sections takes into account these considerations.

This work is organised as follows: in Section 2 we introduce quantifier elim-ination as the subject of our complexity studies and the algorithmic approachwhich is based on the transformation of arithmetic circuits. In Section 3 we de-scribe the tool used to obtain the announced complexity lower bound. Our toolis a computation model which captures the notion of non–functional requirementin circuit–based elimination algorithms. Finally we present the new result in thiswork: we make the following question: What does it happen if our algorithmsare not circuit–based and we found a representation which is more efficient thancircuits? The answer is that our complexity results are valid for arbitrary contin-uous representations if the algorithms follow the principle of information hiding.We illustrate this conclusion with a relevant example from the theory of AbstractData Types (see, e.g. [13] and [12]).

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A Complexity Lower Bound Based On Software Engineering Concepts 3

In the rest of the paper we shall use notions and notations from algebraic ge-ometry and algebraic complexity theory which are all standard (see for example[14] and [3]).

2 Quantifier elimination and its implementation

2.1 Quantifier Elimination

We start with the subject of our complexity studies. The subject is quantifierelimination in the particular case of elementary algebraic geometry over C. LetΦ be an existentially quantified formula. In general terms, the quantifier elimina-tion problem consists in obtaining a quantifier free formula Ψ which is logicallyequivalent to Φ (this means that Ψ and Φ define the same set). In the partic-ular case of elementary algebraic geometry over C, the formulas Φ and Ψ arecomposed by polynomial equations. In this context we are going to considerexclusively the polynomials of these equations.

Let n and r be natural numbers. Let T , U := (U1, . . . , Ur) be parametersand X := (X1, . . . , Xn) be variables subject to quantification. We focus ourattention to polynomials G1(X), . . . , Gn(X) and H(T, U,X) which belong toC[X ] and to C[T, U,X ] respectively. These polynomials constitute a so called FlatFamily of Elimination Problems given by the polynomial equation system G1 =0, . . . , Gn = 0 and the polynomial H (see, e.g. [4] and [9] for details). In generalterms this system represents the quantified formula Φ : (∃X1)(∃Xn)(G1 = 0 ∧. . . ∧Gr = 0 ∧ Y −H = 0).

On the other hand, there exists a polynomial F ∈ C[T, U, Y ] of minimaldegree, called the associated Elimination Polynomial, such that the equationF = 0 represents a quantifier–free formula Ψ which is equivalent to Φ.

Thus, we arrive to a functional requirement where the flat family of elimina-tion problems given by G1 = 0, . . . , Gn = 0 and H becomes transformed into theelimination polynomial F . This transformation is carried out by a mathematicalfunction f as Fig. 1 illustrates.

Φ : (∃X1) . . . (∃Xn)(G1 = 0 ∧ . . . ∧Gn = 0 ∧H − Y = 0)︸︷︷︸

QuantifiedFormula

Ψ : F = 0︸︷︷︸

Quantifier–freeFormula

Fig. 1: Quantifier elimination problem.

At this abstract level we do not know, for example, how the polynomials areimplemented in the computer. We define now these implementation details.

An implementation option is to represent polynomials by their coefficients.Unfortunately the coefficient representation in some elimination polynomials

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4 A. Rojas Paredes

may conduce to complexity blow ups, e.g. the Pochhammer polynomial∏

0≤j<2n

(Y − j)

which has 2n terms (see [7] for open questions in complexity theory related to thispolynomial). This circumstance suggests to represent in elimination algorithmspolynomials not by their coefficients but by arithmetic circuits. This idea becamefully realised by the “Kronecker” algorithm for the resolution of polynomialequation systems over algebraically closed fields. The algorithm was anticipatedin [6] and implemented in a software package of identical name (see [11]). Thefollowing example illustrates the notion of arithmetic circuit.

Fig. 2: Arithmetic circuit and Hornerscheme.

q(X) := a1 + (a2 + a3X)X

+

∗

+

∗

a1 a2 a3 X

Abs

Example 2 (Horner scheme). Leta1, a2, a3 be constants and X be avariable. Consider the polynomialp(X) = a1 + a2X + a3X

2 and theHorner scheme of this polynomialwhich is q(X) = a1 + (a2 + a3X)X .From this scheme we have a directedacyclic graph where each node is anarithmetic operation +, ∗, a constanta1, a2, a3 or a variable X . This arith-metic circuit is a concrete object im-plementing the abstract object q(X).Fig. 2 illustrates the relation betweenq(X) and its implementing circuit bymeans of an abstraction function Abs.

2.2 Implementation of quantifier elimination

To understand the role of arithmetic circuits in elimination algorithms we fix thenotion of polynomials in terms of abstract data types and classes implementingthem. Here we follow the terminology in [13].

Suppose that we have an abstract data type specification for polynomials interms of query and creator functions (observers and constructors in the termi-nology of [12]). Thus the elimination problem of Fig. 1 may be expressed as aspecification in terms of abstract data types.

Consider now the classes implementing the abstract data type of polynomials.We have a class for polynomials and a class for circuits. The connection betweenthese two classes is that the class of circuits is a private part of the class ofpolynomials. This private part is used to implement the interface of the classof polynomials in terms circuits. In this context polynomials are encapsulatedcircuits which are mapped into instances of the abstract data type of polynomialsby an abstraction function Abs.

Now recall our functional requirement: transform en elimination problemgiven by polynomials G1, . . . , Gn and H into an elimination polynomial F . Since

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polynomials become implemented by circuits, an elimination algorithm worksdirectly with circuits taking care of satisfy class invariants and the abstractionfunction Abs. In this sense, an elimination algorithm A transforms an inputcircuit β representing G1, . . . , Gn, H into an output circuit γ representing theelimination polynomial F as Fig. 3 illustrates.

G1, . . . , Gn,Hf

−−−−−−−−−−−−−→ F

Abs

x

xAbs

βA

−−−−−−−−−−−−−→ γ

Fig. 3: Elimination problem and its implementation.

The transformation of β into γ is carried out by means of circuit opera-tions, e.g. join of circuits which mimics the composition of functions (see alsounion of circuits and recursive routine in [9]). If we require the algorithm A tobe branching parsimonious (see Section 3.1 below), then A captures all knowncircuit–based elimination algorithms including the polynomial equation solverKronecker.

At this point the question is how we measure the complexity of algorithmA. We shall mainly be concerned with the size of the output circuit γ. Herewe refer with “size” to the number of internal nodes which count for the givencomplexity measure. Our basic complexity measure is the non–scalar one (alsocalled Ostrowski measure) over the ground field C. This means that we count,at unit costs, only essential multiplications and divisions (see [3] for details).

3 Software engineering–based approaches to complexity

lower bounds

3.1 A circuit–based computation model

The polynomials G1, . . . , Gn, H and F described before belong to mathematicalstructures C[X ], C[T, U,X ] and C[T, U, Y ] respectively. In these mathematicalstructures polynomials have a natural property called geometrical robustnesswhich interpreted as a non–functional requirement constitutes a key ingredientin our complexity result (see Theorem 1 below). This property is invisible ifwe only consider abstract data type specifications in the sense of [13]. Thus,in order to include geometrical robustness in the specification of eliminationproblems we model the notion of abstract data type of polynomials with thecorresponding mathematical structure and we call this structure an abstractdata type. For example, the polynomials G1, . . . , Gn, H will be instances of theabstract data type O ⊂ C[T, U,X ] and F will be an instance of the abstract

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6 A. Rojas Paredes

data type O∗ ⊂ C[T, U, Y ] and the elimination problem will be specified by ageometrically robust map f : O → O∗.

Geometrical robustness The map f is a function (mathematical application)which we require to be constructible, i.e. definable by a boolean combinationof polynomial equations. The mapping is called geometrically robust if it iscontinuous (see [9] and [5] for an algebraic characterisation of robustness). Sincegeometrical robustness is a property belonging to the specification level of ourelimination task, we have to describe how this non–functional property is realisedby the circuit–based algorithms implementing the elimination.

Branching parsimoniousness The intuitive meaning of geometrical robust-ness is reflected by the algorithmic notion of branching parsimoniousness. Wecall an algorithm branching parsimonious if it avoids unnecessary branchings.We may restrict branchings by means of only considering division–free circuits,or circuits where divisions by zero were replaced by suitable limits and divisionsmay only involve parameter nodes (nodes without variables). In this sense ourcircuits are essentially division–free and will be called robust if all intermediateresults (functions represented by each node) are geometrically robust.

The notion of branching parsimoniousness as a tactic In the context ofsoftware architecture, the satisfaction of quality attributes requires techniqueswhich are called tactics. For example, a system is easily modified when it is struc-tured, modularised and well documented. A tactic is, according to [2], a designdecision that influences the control of a quality attribute response. Consideringthis definition we may describe branching parsimoniousness as a tactic for elim-ination algorithms. We require an algorithm to be branching parsimonious inorder to achieve the non–functional requirement of geometrical robustness. Inthis sense we say that branching parsimoniousness is a tactic to achieve geometri-cal robustness. For example, the reader may identify branching parsimoniousnesswith modularity which is a tactic to achieve the modifiability quality attribute.

Now recall our elimination algorithm A in Fig. 3 which transform the circuitβ (representing G1, . . . , Gn, H) into circuit γ (representing F ). The eliminationalgorithm A implements the additional property of geometrical robustness if werequire A to be banching parsimonious.

Thus in the input we have an essentially division–free, robust parameterizedarithmetic circuit β of size O(n) with basic parameters T , U := U1, . . . , Un

and input X := X1, . . . , Xn which computes polynomials G1, . . . , Gn ∈ C[X ]and H ∈ C[T, U,X ] constituting a flat family of zero–dimensional eliminationproblems with associated elimination polynomial F ∈ C[T, U, Y ].

The branching parsimoniousness allows to affirm that each circuit operationgives as result a robust circuit. Thus we conclude that the property of geometricalrobustness is transmitted from the input β to the output γ. Then γ := A(β) isan essentially division–free, robust parameterized arithmetic circuit with basicparameters T, U1, . . . , Un and input Y representing the elimination polynomialF .

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These notations and assumptions, in particular the property of robustness inthe output γ, allows to conclude the following theorem.

Theorem 1 ([9], Theorem 10). The circuit γ has, as ordinary arithmeticcircuit over C, non–scalar size at least Ω(2n).

Theorem 1 corresponds to circuit–based algorithms, now we ask what doesit happen if we found a representation which is more efficient than arithmeticcircuits? We argue that Information Hiding–based algorithms have the samecomplexity status. This implies that our complexity results are valid for arbitrarycontinuous representations. This is part of future work but we give preliminaryresults in the following section.

3.2 Towards an Information Hiding–based computation model

Since polynomials G1, . . . , Gn, H and F are objects belonging to suitable ab-stract data types, we may define the function f of Fig. 3 in terms of query andcreator functions (observers and constructors) of the given abstract data typespecification obtaining a transformation which does not involve circuits directlybecause they become encapsulated. To illustrate this kind of transformation con-sider the following example.

Example 3. Suppose a case where f is the identity function of binary trees. Inthis context let us consider the following abstract functions of the correspondingabstract data type specification: root(), left(), right() and isNil?() as queryfunctions (observers) and bin() and nil() as creator functions (constructors).Then, we propose the following definition for f :

f(X) =

nil() if isNil?(X)bin(root(X), id(left(X)), id(right(X))) otherwise

(1)

This specification of function f may be implemented in such a way that, at anabstract level the implementation is the identity function of binary trees, whereasat a concrete level the implementation is a transformation of the representationof binary trees (compare this with the transformation of circuits in eliminationalgorithm A). This hidden transformation is carried out by the classes imple-menting the abstract data type of binary trees where the implementation of fmay be called f. Notice that we write the implementation in verbatim font inorder to distinguish the difference with abstract data type expressions which wewrite in cursive font.

Let Tree<E> be a class implementing the abstract data type of binary trees.Let Tree1<E> and Tree2<E> be subclasses of class Tree<E> with the followingproperty: Tree1<E> implements trees as arrays (the internal representation oftrees is given by arrays) and Tree2<E> implements trees as nested nodes. Letroot, left, right and isNil be routines in the class Tree<E> implementing thecorresponding query functions (observers) in the abstract data type specification.

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8 A. Rojas Paredes

Let p1 be a variable of type E and p2 y p3 be variables of type Tree2<E>, thenTree2<E>() and Tree2<E>(p1,p2,p3) are constructors of the class Tree2<E>

implementing the creator functions nil() and bin() respectively. Then the imple-mentation in java code is as follows:

Tree<E> f(Tree<E> t)if(t.isNil()) return new Tree2<E>();

else return new Tree2<E>(t.root(),

(Tree2<E>) f(t.left()),

(Tree2<E>) f(t.right()) );

(2)

Notice that the effective transformation of the representation is carried outwhen an instance of Tree1<E> is passed as parameter and the constructor of theother class is applied, say the constructor of Tree2<E>.

Equation 2 illustrates the definition of an algorithm in terms of observersand constructors. In the case of elimination problems such an algorithm has asimilar structure but we do not exhibit an example here. This is left for a futurework (see [1]) where the notion of information hiding is modelled in full detail.Such a model allows to conclude the following:

– if the complexity measure is given by the number of parameters instead ofthe size of circuits, we obtain an exponential complexity lower bound for thisquantity which implies the result in Theorem 1,

– this allows to conclude that elimination algorithms programmed with infor-mation hiding, i.e. hiding the circuits or any other representation of polyno-mials, have the same complexity status.

Final comments The circuit–based computation model described in Section3.1 corresponds to the tool for the software engineer we described at the intro-duction. Of course this model cannot be applied to any software project sinceit is restricted to the particular case of elimination. However, it gives the keyingredients for the definition of a computation model suitable for complexityquestions where another non–functional requirement must be considered.

On the other hand, our description of an Information Hiding–based compu-tation model in Section 3.2 constitutes an stronger result which together withTheorem 1, allows to conclude that the Kronecker algorithm is asymptoticallyoptimal. This suggest that the Kronecker is a good option to use in applicationsof scientific computing where polynomial equation solving is needed.

Finally, a computation model which captures algorithms constructed in aprofessional way, namely applying software engineering concepts, in combinationwith the complexity lower bound obtained in Section 3.1 allows to conclude thefollowing idea which we repeat from [9]: neither mathematicians nor softwareengineers, nor a combination of them will ever produce a practically satisfactory,generalistic software for elimination tasks in Algebraic Geometry. This is a jobfor hackers which may find for particular elimination problems specific efficientsolutions.

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Acknowledgements The author thanks Joos Heintz for his insistent encour-agement to finish this work and Pablo Barenbaum, Gaston Bengolea Monzon,Mariano Cerrutti, Carlos Lopez Pombo, Hvara Ocar and Alejandro Scherz, Uni-versidad de Buenos Aires, for discussions about the topic of this paper and/orcomments and ideas on earlier drafts.

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Berlin Heidelberg, New York (1994)

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On Aggregation Process in Linguistic Decision Making

Framework

M. Gimenez, S. Gramajo

Artificial Intelligence Research Group. National Technological University, French 414, Resistencia, 3500, Argentina

[email protected], [email protected]

Abstract. When solving a problem, human beings must face situations in which they should choose among different alternatives by means of reasoning and mental processes. Many of these decision problems are under uncertain envi-ronments including vague, imprecise and subjective information that is usually modeled by fuzzy linguistic approach. This approach uses linguistic infor-mation or natural language words and its relation to mental reasoning processes of the experts when expressing their assessments. In a decision process multiple criteria can be evaluated which involving multiple experts with different de-grees of knowledge. Such process can be modeled by using Multi-granular Lin-guistic Information (MGLI) and Computing with Words (CW) processes to solve the related decision problems. Once decision makers (experts) provided their opinions, it is necessary to combine all these opinions to obtain a single overall result that can be interpreted. An aggregation operator allows accom-plishing this objective calculating a global value in different ways. In this paper we study the use of aggregation operators in multi-criteria decision-making processes comparing them and obtaining conclusions about their use in our framework. Furthermore, we propose a new aggregation operator taking into account the criteria importance to evaluate the alternatives, and then an illustra-tive example shows its outcomes.

Keywords: Multi-granular Linguistic Information, Computing with Words, Aggregation operator, Decision Making.

1 Introduction

The decision making is a day-to-day activity for human beings. The multiple facets of real world decision problems are well addressed by Multi-Criteria Decision Making (MCDM) [1]. The crucial point of interest within the MCDM is the analysis and the modeling of the multiple decision makers’ preferences giving rise to Multi-Expert Decision Making (MEDM) [2].

In many situations, context involves vague and probably incomplete information. In these cases, information cannot be assessed precisely in a quantitative form; ex-perts may feel more comfortable employing other approaches. To overcome this prob-lem, information is normally modeled by using a fuzzy linguistic approach [3][4][5]

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allowing the experts to express their opinions with words rather than numbers (e.g. when evaluating the comfort or design of a car, terms like good, medium, bad can be used). Therefore, the fuzzy linguistic approach is a technique that represents qualita-tive information as linguistic values by means of linguistic variables [3], that is, vari-ables whose values are no numbers but words or sentences in a natural language. Each linguistic value is characterized by its syntax (label) and semantic (meaning). The label is a word or a sentence belonging to a linguistic term set and the meaning is a fuzzy subset in a universe of discourse. The concept of linguistic variables provides an estimated measure since words are less precise than numbers. This is more effec-tive because the experts may feel more comfortable using words they really know and understand in accordance with the context of use of these words. Also, when offering different expression domains or different linguistic term sets (multi-granular infor-mation) to the experts, this solution would be suitable to adjust the degree of experi-ence of each one [6][7].

An important aspect of the MCDM is the aggregation process. In order to obtain a unique final result, the assessments of each expert involved must be taken in account. An aggregation operator allows accomplishing this objective calculating a global value. The aggregation is the operation that transforms a set of elements, such as indi-vidual opinions on a set of alternatives, into a single element that is representative of the whole. Different ways of carrying out the combination of preferences have led many authors to study and design different aggregation operators. Depending on the problem different types of aggregation operators can be used.

In this paper, we focus on the aggregation process when dealing with complex de-cisions under uncertainty using decision analysis process. We will study the results of applying different aggregation operators on the same decision problem in order to obtain relevant conclusions about their use in complex decision systems.

This paper is organized as follows. Section 2 reviews basic concepts about linguis-tic background that will be used to model uncertain information and multi-granular information in our framework. Section 3 presents the phases in order to analyze deci-sions, with special emphasis on aggregation process. Then, section 4 proposes an example of use on investment decisions in a company. Finally, section 5 shows some conclusions.

2 Preliminaries

Normally the decision analysis depends highly on subjective, vague and ill-structured information must have a model to manage this kind of information. Therefore, we consider the use of the fuzzy linguistic approach [3] to model and manage the inher-ent uncertainty in this kind of problems and the 2-tuple linguistic model to represent linguistic information [8]. Additionally, it is useful to manage multiple linguistic scales (multi-granular information) giving more flexibility to the different experts involved in the problem and, to manage this, we use Extended Linguistic Hierarchies (ELH) method. For this reason, in this section we review in short the concepts and

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methods such as the fuzzy 2-tuple linguistic model, ELH and its computational meth-od (aggregation process).

2.1 The 2-tuples linguistic model

When using linguistic information to solve a problem it is necessary the use of com-puting with words CW. The main limitation with this approach is the “loss of infor-mation” suffered in the most used computational techniques that implies the lack of precision in the final results. These computational models are: The semantic model [9] and the symbolic model [10]. In these two models an approximation process must be developed to express the result in the initial expression domain, here is when the information gets lost.

The 2-tuples linguistic model [11] is a representation model that overcomes the loss of information. It represents the linguistic information with a pair of values, that we call 2-tuple, composed by a linguistic term and a number. Definition 1. The Symbolic Translation of a linguistic term 0 ,..., i gs S s s is a

numerical value assessed in 0.5,0.5 that supports the “difference of information”

between an amount of information 0, g and the closest value in 0,..., g that

indicates the index of the closest linguistic term in iS s , being 0, g the interval of granularity of S .

From this concept a new linguistic representation model was developed, which rep-resents the linguistic information by means of a linguistic 2-tuple. It consists of a pair of values namely, ( , ) [ 0.5,0.5)is S S , being

is S a linguistic term and [ 0.5,0.5) a numerical value representing the symbolic translation. This repre-

sentation model defined a set of transformation functions between numeric values and linguistic 2-tuples to facilitate linguistic computational processes. Definition 2. Let 0 ,..., gS s s be a linguistic terms set and [0, ]g a value sup-porting the result of a symbolic aggregation operation. The 2-tuple set associated with S is defined as [ 0.5,0.5)S S . A 2-tuple that expresses the equivalent infor-mation to is then obtained as follow:

: 0,

, ( )( ) ( , ),

, [ 0.5,0,5)

i

i

g S

s i round

s with

i

(1)

being round (·) the usual round operation, i the index of the closest label, is , to “

”, and “ ”the value of the symbolic translation. It is noteworthy to point out that is a one to one mapping and 1 : [0, ]S g is

defined by 1( , )is i . In this way the 2-tuple of S is identified by a numerical

value in the interval 0, g .

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Remark 1. The transformation of a linguistic term into a linguistic 2-tuples consists of adding value 0 as symbolic translation: ( ,0)i is S s S . On other hand,

( ) ( ,0)ii s and 1( ,0) , 0,1,..,is i i g . If 3.25 is the value representing the result of a symbolic aggregation operation

on the set of labels, 0 1 2 3 4 5 6 , , , , , , S s Nothing s VeryLow s Low s Mediums s High s VeryHigh s Perfect

, then the 2-tuple that expresses the equivalent information to is ( ,.25)medium . This model has a linguistic computational technique based on the functions and1 , for a further detailed see Ref. [12].

2.2 Extended Linguistic Hierarchies:

A flexible expression domain with several linguistic scales is necessary to express the assessments for experts according to their degree of knowledge about the problem. Different approaches dealing with multi-granular linguistic information have been proposed. In this paper shall use the ELH [13] approach to model and manage multi-granular linguistic information because of its features of flexibility and accuracy in the processes of computing with words (CW) in multi-granular linguistic contexts. An ELH is a set of levels, where each level represents a linguistic term set with different granularity from the remaining levels of the ELH. Each level belongs to an ELH is denoted as ( , ( ))l t n t being t a number that indicates the level of the ELH and ( )n t the granularity of the terms set of the level t . To build an ELH have been proposed a set of extended hierarchical rules:

Rule 1: A finite set of levels, ( , ( ))l t n t with 1,..,t m , that defines the multi-granular linguistic context required by experts to express their assessments are includ-ed.

Rule 2: to obtain an ELH a new level, * *( , ( ))l t n t with * 1t m , should be added. This new level must have the following granularity:

( *) ( . . .( (1) 1,..., ( ) 1)) 1n t LC M n n m (2)

being L.C.M. the Least Common Multiple. ELH building process then consists of two processes: i) It adds m linguistic scales

used by the experts to express their information. And ii) then it adds the term set ( *, ( *))l t n t , with 1t m , according to Eq. (2). Therefore, the ELH is the union of

all levels required by the experts plus the new level * *( , ( ))l t n t . 1

1

( ( , ( )))t m

t

ELH l t n t

The use of multi-granular linguistic information makes the processes of CW more complex. ELH computational model needs to make a three-step process.

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1. Unification phase. The multi-granular linguistic information is conducted into only one linguistic term set, that in ELH is always ( *)n tS , by means of a transfor-mation function (·)a

bTF :

Definition 3. Let ( ) ( ) ( )0 ( ) 1 ,..., n a n a n a

n aS s s and ( ) ( ) ( )0 ( ) 1 ,..., n b n b n b

n bS s s be two linguistic term sets, with a b . The linguistic transformation function is defined as:

( ) ( )

1 ( ) ( )( ) ( )

( ) ( )

:

( , )·( ( ) 1)( , )

( ) 1

( , )

a n a n b

b

n a n a

ja n a n a

b j j S

n b n b

k k

TF S S

s n bT F s

n a

s

(2)

2. Computational process. Once the information is expressed in only one expres-sion domain ( *)n tS , the computations are carried out by using the linguistic 2-tuple model.

3. Expressing results. In this step the results might be transformed into any level, t

, of ELH in a precise way by using Eq. (3) to improve the understanding of the results if necessary.

Remark 2. In the processes of CW with information assessed in an ELH, the lin-guistic transformation function, a

bTF , performed in the unification phase, a , might be any level in the set 1,.., t m and the computational processes are carried out in the level b that it is always the level *t (See Eq. (3)).

It was proved in [13] that the transformation functions between linguistic terms in different levels of the Extended Linguistic Hierarchy are carried out without loss in-formation.

2.3 Aggregation process:

Aggregation operators allow accomplishing a global value from a set of values in order to obtain a unique final value. Here we have analyzed four kinds of aggregation operators, Geometric Mean Aggregation Operator (GMAO), Arithmetic Mean Ag-gregation Operator (AMAO) Weighted Aggregation Operator (WAO). WAO is based on the weight of the experts (WAO) or criteria (WAOC). Definition 6. GMAO. Let 1 1(( , ),.., ( , )) m

m ml l S be a 2-tuples linguistic vector, geometric mean operator is defined as follows: : mG S S

1 1

11 1

1 1

: (( , ),.., ( , )) ( , )m mm m

m m i i i

i i

G l l l

(3)

Definition 5. AMAO: Let 1 1(( , ),.., ( , )) n

n nl l S be a 2-tuples linguistic vector,

arithmetic mean operator is defined as follows: : nG S S

11 1

1 1

1 1[( , ),.., ( , )] ( ( , )) ( )n n

n

j j

n j j jGn n

l l r

(4)

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A rational assumption about the resolution process could be associating more im-portance to the experts who have more “knowledge” or “experience”. These values can be interpreted as importance degree, competence, knowledge or ability of the experts. In addition some experts could have some difficulties in giving all their as-sessments due to lack of knowledge about part of the problem. Besides the use of different scales, the expert should be carried out in different way with weighted ag-gregation operator. Definition 6. WAO: Let 1 1(( , ),.., ( , ) m

m ml l S be a vector of linguistic 2-tuples, and

1,..., [0,1]m

mw w w be a weighting vector such that1

1m

iiw

. The 2-tuple WAO

associated with w is the function :m

wG S S defined by

1 11

[( , ),.., ( , )]m

w

m m s i i

i

G l l w

(5)

In the same way that experts have importance, criteria also may have it. In this sense we use the process of obtaining the importance of criteria based on the poten-cies method. This method takes in account the importance for each criterion in the problem solution using a vector of importance with defined values for every criterion involved. When working with linguistic information we just don’t have a method for comparing criteria in order to obtain this vector of importance. According to this, it is necessary to obtain the comparison matrix between criteria and then calculate the weighted vector based on criteria importance. The matrix

nxnA that represents the

matrix comparison between criteria is obtained from the experts judgments about criteria. Then the weighted vector that represents the weight held by each criterion in the decision process and is obtained using

nxnA as explained in [14].

Definition 7. WAOC: Let 1 1(( , ),.., ( , ) n

n nl l S be a vector of linguistic 2-tuples,

and 1,..., [0,1]n

n be a weighting vector based on the criteria importance

such that1

1j

n

j

. The 2-tuple aggregation operator associated with is the func-

tion :n

G S S defined by:

1 11

[( , ),.., ( , )]n

sn n j j

j

G l l w

(6)

3 Decision analysis process

Linguistic decision analysis process consists of several phases described below: Phase 1. Data definition: It defines the evaluation context in which experts will ex-press their preferences. Linguistic descriptors and their semantics are chosen as well as each problem potential solution (alternative) is identified. It also determines the criteria to evaluate every alternative and the experts who are involved in decision

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process. In order to allow different expression domain for multiple experts, linguistic terms sets used are organized into an ELH. Therefore, let consider:

A finite set of alternatives , 1,..., kX x k q . A finite set of criteria , 1,...,

jC c j n .

A finite set of experts , 1,..., iE e i m that express their assessments by using different linguistic scales of information in ELH. Phase 2. Information gathering: Experts provide their linguistic assessments in utility vectors for each criterion of the evaluated alternatives. The experts express their as-sessments on every criterion considering every alternative using their linguistic term set in ELH. Due to the fact that our Framework will use linguistic 2-tuple computing model the linguistic preferences provided by the experts will be transformed into linguistic 2-tuples according to the Remark 1. Phase 3. Computational process: This phase consists of three steps to obtain a global value for each alternative:

-Unification of MGLI. Due to experts provide their assessments in different lin-guistic scales; it is necessary to transform each assessment in a unique expression domain so called *t whose granularity is given by Eq. (2). Thus, transformation must be the last level of the ELH according to Eq. (3). Once the information has been uni-

fied, it will be expressed by means of linguistic 2-tuples in *n t

S . In order to obtain the global value for each alternative the information must be ag-

gregated. In our framework we use four different aggregation operators and the pro-cess is performed in two levels:

- Expert Aggregation Level: The first aggregation step it obtains a collective value for all experts’ assessments. Here is possible to choose between GMAO and WAO.

- Criteria Aggregation Level: The second one computes a global value for each al-ternative from results obtained in previous step. Here is possible to choose between AMAO and WAOC. Figure 1 shows the possible combinations of operators.

Fig. 1. Framework aggregation operators

Phase 4. Results presentation: Final values are presented in an ordered scale as a ranking of preferences from the most suitable to the less convenient alternative.

Expert aggregation level

GMAO

GMAO

WAO BASED ON EXPERTS

WAO BASED ON EXPERTS

Criteria aggregation level

AMAO

WAO BASED ON CRITERIA

AMAO

WAO BASED ON CRITERIA

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4 Illustrative example

We consider the decision process to acquire software in one organization. The deci-sion from where get software implies decide the supply channel option. There are advantages and drawbacks for particular acquire channels and experts many times do not reach to an agreement. In order to satisfy this need, the CEO has arranged a meet-ing with the three main experts in software solution in the organization: CIO, Head of development department and Head of data management department. The objective of this meeting is to determine which one of the three channels available for software supplying is the most suitable for the company. There are three main channels to ob-tain software: internal development, external development and buy a standard packet.

In the Internal development, the organization IT department builds the needed software solution.

The External development means acquire by external software development con-sulting.

Buy a standard package. One of the fastest way for satisfying software needs is by acquiring a standard software package of general purpose. To obtain software 4 crite-ria should be evaluated, how well it meets the necessary requirements, ease of chang-es and growths and development time.

Therefore, in phase 1 we have the following: 1 2

3

CIO, Head of development department,A set of experts

Head of data management department

E E

E

1 2

3

Internal development, External development,A set of alternatives

Buy a standard package

A A

A

1 2

3

Satisfied requirements, Facility implementing changes,A set of criteria

Development time

C C

C

An ELH with two linguistic term sets: 1 Very Bad, Bad, Medium, Good, Very GoodS VB B M G VG

2

Worst, Very Bad, Bad, Medium, Good,

Very Good, Excellent

W VB B M GS

VG E

Besides a new level, *t , in accordance with Eq. (2).

Table 1. Phase 2. Information gathering

Experts Assessments

1A 2A 3A

1C 2C 3C 1C 2C 3C 1C 2C 3C

1E (E,0) (VG,0) (B,0) (VG,0) (M,0) (VG,0) (M,0) (W,0) (E,0)

2E (VG,0) (VG,0) (VB,0) (E,0) (G,0) (M,0) (M,0) (B,0) (VG,0)

3E (VG,0) (G,0) (VB,0) (G,0) (M,0) (M,0) (M,0) (VB,0) (VG,0)

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Table 2. Criteria comparison

1E 2E 3E

1C 2C 3C 1C 2C 3C 1C 2C 3C

1C 1 5 1 / 3 1 7 5 1 7 4

2C 1 / 5 1 1 / 9 1 / 7 1 1 / 2 1 / 7 1 1 / 2

3C 3 9 1 1 / 5 2 1 1 / 4 2 1

Table 3. Criteria weight vector

vector

Criterion Weight

1C 0.2654

2C 0.0629

3C 0.6716

Table 4. Experts weight vector

w vector

Expert Weight

1E 0.5

2E 0.3

3E 0.2

Table 5. GMAO and AMAO results

Alternative Percentage

2A

38,54%

1A

33,67%

3A

27,79%

Table 6. WAO with experts weighting and AMAO results


2A

37,12%

1A

33,73%

3A

29,15%

Table 7. GMAO and WAO based on crite-ria importance


3A

44,34%

2A

38,25%

1A

17,42%

Table 8. WAO with experts weighting and WAO based on criteria importance


3A

42,62%

2A

35,98%

1A

21,41%

Bearing in mind the first step of aggregation, our framework allows use GMAO and WAO in accordance with the weighting vector showed in Table 4. Then, the se-cond aggregation steps we use AMAO and WAO based on criteria importance (see Table 3).

From Table 5 to Table 8 results are expressed in percentage way to better under-standing. When it compute GMAO and AMAO (see Table 5) the results are similar to the Table 6 that uses WAO with experts weighting and AMAO (see Table 6). Howev-er, a slight difference it can be seen between both but the order of importance is the same. Weighted operator (WAO) introduces a new parameter, the weight of im-portance of experts, allowing greater differentiation between the final results to elimi-

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nate equal importance between opinions. Thus, decision makers have more accurate values with better differentiation between them.

On the contrary, in Tables 7 and 8, the operator for the second level of aggregation used was the weighted vector based on criteria importance. Here, the priority ranking changes significantly. It is because importance vector modifies last criteria aggrega-tion step, allocating highest values for the most important criterion and reducing val-ues for the others. Furthermore, obtained values in Table 8 take into account the weight of the experts.

5 Conclusion

Aggregation refers to the process of combining several values into a single one, so that the final result of aggregation takes into account in a given manner all the indi-vidual values. Such an operation is used in many well-known disciplines such as Mul-ti-Criteria Multi-Expert Decision Making. In order to reach good results for decision process, classical synthesizing functions have been proposed: arithmetic mean, geo-metric mean, median and many others. In this papers we present a linguistic frame-work developed that allows analyze different decision results by using several aggre-gation operators. In this regard we also propose compute criteria importance based on the potencies method with Saaty scale.

Currently, the framework computation capability is expanded by using different aggregation operators such as Ordered Weighted Averaging (OWA) aggregation op-erators’ family. In addition, we are comparing different methodologies and decision making approaches such as Analytic Hierarchy Process (AHP).

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(1995) 3. Zadeh, L.: The concept of a linguistic variable and its applications to approximate reason-

ing. Information Sciences, Part I, II, III (8,9) (1975) 199–249,301–357,43–80 4. Dong, Y., Xu, Y., Yu, S.: Linguistic multiperson decision making based on the use of mul-

tiple preference relations. Fuzzy Sets and Systems 160(5) (2009) 603–623 5. Delgado, M., Verdegay, J., Vila, M.: Linguistic decision-making models. International

Journal of Intelligent Systems 7(5) (1992) 479–492 6. Herrera, F., Herrera-Viedma, E., Martínez, L.: A fusion approach for managing multi-

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9. Degani, R., Bortolan, G.: The problem of linguistic approximation in clinical decision making. International Journal of Approximate Reasoning 2 (1988) 143–162

10. Delgado, M., Verdegay, J., Vila, M.: On aggregation operations of linguistic labels. Inter-national Journal of Intelligent Systems 8(3) (1993) 351–370

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Declarado de Interés Municipal por el Honorable Concejo Deliberante del Partido de General Pueyrredon

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