Aggregation operators based on indistinguishability operators

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  • Aggregation Operators Based onIndistinguishability OperatorsJ. Jacas,1, J. Recasens2,*1Sec Matemtiques i Informtica, ETS Arquitecura de Barcelona,Universitat Politcnica de Catalunya, Diagonal 649, 08028 Barcelona, Spain2Secci Matemtiques i Informtica, ETS Arquitectura del Valls,Universitat Politcnica de Catalunya, Pere Serra 1-15, 08190 Sant Cugatdel Valls, Spain

    This article gives a new approach to aggregating assuming that there is an indistinguishabilityoperator or similarity defined on the universe of discourse. The very simple idea is that when wewant to aggregate two values a and b we are looking for a value l that is as similar to a as to b or,in a more logical language, the degrees of equivalence of l with a and b must coincide. Inter-esting aggregation operators on the unit interval are obtained from natural indistinguishabilityoperators associated to t-norms that are ordinal sums. 2006 Wiley Periodicals, Inc.

    1. INTRODUCTION

    When we aggregate two values a and b we may want to get a number l that isas similar to a as to b or, in other words, l should be equivalent to both values. Soif we have defined some kind of similarity E on our universe, the aggregation l ofa and b should satisfy E~a,l! E~b,l!. We will develop this idea when the uni-verse is the unit interval @0,1# and the similarity is the natural T-indistinguishabilityoperator ET associated to a ~continuous! t-norm T. In particular we will show thatfor an Archimedean t-norm T with additive generator t the aggregation operatorassociated with ET is the quasi-arithmetic mean m generated by t ~m~x, y! t1 ~@t~x! t~ y!#/2!!. This can give a justification for using a concrete quasi-arithmetic mean in a real problem, because it will be related to a logical systemhaving T as conjunction ~and ET as bi-implication!. If T is an ordinal sum, inter-esting aggregation operators are obtained because the way they aggregate two val-ues varies locally: For points in a piece @ai , bi # 2 where we have a copy of anArchimedean t-norm with additive generator ti , their aggregation is related to the

    *Author to whom all correspondence should be addressed: e-mail: j.recasens@upc.edu.e-mail: joan.jacas@upc.edu.

    INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 21, 857873 ~2006! 2006 Wiley Periodicals, Inc. Published online in Wiley InterScience~www.interscience.wiley.com!. DOI 10.1002/int.20165

  • quasi-arithmetic mean generated by ti . Points outside these pieces with the smallestcoordinate c in some @ai , bi # have

    ai ~bi ai !ti1 tc ai

    bi ai

    2

    as aggregation whereas the aggregation of the rest of the points is their smallestcoordinate.

    This idea can be easily generalized to weighted aggregations and aggrega-tions of more than two objects. Roughly speaking, if the weights are p and q 1 p, then we can replace E~a,l! and E~b,l! by T p~E~a,l!! and T q~E~b,l!!,respectively. If we want to aggregate n numbers a1 a2 {{{ an , we can findl satisfying T ~E~a1,l!, . . . , E~ai ,l!! T ~E~ai1,l!, . . . , E~an ,l!! for some i . Inthis case we can have some problems if the t-norm has nilpotent elements. Then itwould be possible that both sides of the previous equality were 0. This problem issolved in this article by softening the indistinguishability operator E, replacing itby E 1/n .

    2. PRELIMINARIES

    This section contains some results on t-norms and indistinguishability opera-tors that will be needed later on in the article. Besides well-known definitions andtheorems, the power T n of a t-norm is generalized to irrational exponents in Def-inition 2 and given explicitly for continuous Archimedean t-norms in Propo-sition 1. Theorem 4 and Corollary 1 give new representations of the naturalT-indistinguishability operator ET associated to a t-norm T.

    Though many results remain valid for arbitrary t-norms and especially forleft continuous ones, for the sake of simplicity we will assume continuity for thet-norms throughout the article.

    Definition 1. A continuous t-norm is a continuous map T : @0,1# @0,1#r @0,1#satisfying for all x, y, z, x ', y ' @0,1#

    (1) T ~x, T ~ y, z!! T ~T ~x, y!, z! (Associativity)(2) T ~x, y! T ~ y, x! (Commutativity)(3) If x x ' and y y ', then T ~x, y! T ~x ', y ' ! (Monotonicity)(4) T ~1, x! x

    Because a t-norm T is associative, we can extend it to an n-ary operation in thestandard way:

    T ~x! x

    T ~x1, x2, . . . , xn ! T ~x1, T ~x2, . . . , xn !!

    858 JACAS AND RECASENS

    International Journal of Intelligent Systems DOI 10.1002/int

  • In particular,

    T ~

    n timesAssDssG

    x, x, . . . , x !

    will be denoted by xT~n!

    .

    If T is continuous, the nth root xT~1/n!

    of x wrt T is defined by

    xT~1/n! sup$z @0,1# 6 zT

    ~n! x%

    and for m, n N, xT~m/n! ~xT

    ~1/n! !T~m!

    .

    Lemma 1.1 If k, m, n N, k, n 0, then xT~km/kn! xT~m/n! .

    Lemma 2. Let x1, . . . , xn ~0,1# and n N. T ~x1T~1/n!

    , . . . , xnT~1/n! ! 0.

    Proof. Let xiMin~x1, . . . , xn !. Then

    T ~x1T~1/n!

    , . . . , xnT~1/n! ! T ~

    n timesAsssssDsssssGxiT~1/n!

    , . . . , xiT~1/n!! ~xiT

    ~1/n! !T~n! xiT

    ~n/n! xi 0

    Assuming continuity for the t-norm T, the powers xT~m/n!

    can be extended toirrational exponents in a straightforward way.

    Definition 2. If r R is a positive real number, let $an %nN be a sequence ofrational numbers with limnr` an r. For any x @0,1# , the power xT

    ~r! is

    xT~r! limnr` xT

    ~an !

    Continuity assures the existence of last limit and independence of the sequence$an %nN .

    Proposition 1. Let T be an Archimedean t-norm with additive generator t, x @0,1# , and r R. Then

    xT~r! t @1# ~rt~x!!

    Proof. Due to continuity of t we need to prove it only for rational r.If r is a natural number m, then trivially xT

    ~m! t @1#~mt~x!!.If r 1/n with n N, then xT

    ~1/n! z with zT~n! x or t @1#~nt~z!! x and

    xT~1/n! t @1#~t~x!/n!.

    AGGREGATION OPERATORS 859

    International Journal of Intelligent Systems DOI 10.1002/int

  • For a rational number m/n,

    xT~m/n! ~xT

    ~1/n! !T~m! t @1# ~mt~xT

    ~1/n! !!

    t @1#mtt @1# t~x!n

    t @1#mn

    t~x! Let E~T ! $x @0,1# 6 xT

    ~2! x% be the set of idempotent elements of T andNIL~T ! $x @0,1# 6 xT

    ~n! 0 for some n N % the set of nilpotent elements of T.

    Definition 3. A continuous t-norm T is Archimedean if and only if E~T ! $0,1%.T is called strict when NIL~T ! @0,1!. Otherwise it is called nonstrict andNIL~T ! $0%.

    Theorem 1.2 A continuous t-norm T is Archimedean if and only if there exists acontinuous decreasing map t : @0,1#r @0,`# with t~1! 0 such that

    T ~x, y! t @1# ~t~x! t~ y!!

    where t @1# stands for the pseudo-inverse of t defined by

    t @1# ~x! 1 if x 0t1~x! if x @0, t~0!#0 otherwise

    T is strict if t~0!` and nonstrict otherwise.

    t is called an additive generator of T, and two additive generators of the samet-norm differ only by a multiplicative constant.

    The next theorem states that all continuous t-norms can be built from Archi-medean ones.

    Theorem 2.1,3 T is a continuous t-norm if and only of there exists a familyof continuous Archimedean t-norms ~Ti !iI ~I finite or countably infinite) and~#ai , bi @!iI a family of nonempty pairwise disjoint open subintervals of @0,1#such that

    T ~x, y! ai ~bi ai !Tix ai

    bi ai,

    y aibi ai

    if ~x, y! @ai , bi # 2Min~x, y! otherwise

    T is called the ordinal sum of the summands ^@ai , bi # , Ti &, i I and will bedenoted T ~^@ai , bi # , Ti &!iI .

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    International Journal of Intelligent Systems DOI 10.1002/int

  • Proposition 2.4 Let T be a continuous t-norm. There exists a family of strictlydecreasing and continuous maps ti : @ai , bi # r @0,`# with ti ~bi ! 0, i I suchthat

    T ~x, y! ti@@1## ~ti ~x! ti ~ y!! if ~x, y! @ai , bi # 2Min~x, y! otherwisewhere here ti

    @@1##are defined by

    ti@@1## ~x!

    bi if x 0ti1 ~x! if x @0, t~ai !#

    ai otherwise

    Proposition 3.1 Let T ~^ @ai , bi # , Ti &!iI be the ordinal sum of the summands^@ai , bi # , Ti &, i I and let ti be an additive generator of Ti for all i I. Then

    T ~x, y! hi1 ~hi ~x! hi ~ y!! if ~x, y! @ai , bi # 2Min~x, y! otherwisefor each i I hi : @ai , bi #r @0,`# is given by

    hi ~x! ti x aibi aiDefinition 4. The residuation

  • ET is indeed a special kind of ~one-dimensional! T-indistinguishability oper-ator ~Definition 6!,5 and in a logical context where T plays the role of the conjunc-tion, ET is interpreted as the bi-implication associated to T.6

    For readers interested in a deeper study of indistinguishability operators, werecall its definition below and refer them to Ref. 5.

    Definition 6. Given a t-norm T, a T-indistinguishability operator E on a set Xis a fuzzy relation E : X Xr @0,1# satisfying for all x, y, z X

    (1) E~x, x!1 (Reflexivity)(2) E~x, y! E~ y, x! (Symmetry)(3) T ~E~x, y!, E~ y, z!! E~x, z! ~T-transitivity)

    Example 1.

    ~1! If T is the Lukasiewicz t-norm, then ET ~x, y!1 6x y 6 for all x, y @0,1# .~2! If T is the Product t-norm, then ET ~x, y! Min~x/y, y/x! for all x, y @0,1# where

    z/01.~3! If T is the Minimum t-norm, then

    ET ~x, y! Min~x, y! if x y1 otherwiseIt is important to note that even if the t-norm T is continuous, ET need not be

    so. It is easy to see, for instance, that for the Minimum t-norm, ET is not continu-ous in the diagonal D of @0,1# ~D $~x, x! 6 x @0,1#%! and for a strict Archime-dean t-norm T ET is not continuous in ~0,0!.

    More generally, if T is an ordinal sum of ~Ti !iI , then the points of disconti-nuity of ET lie on the diagonal of @0,1# as follows directly from the next theorem