A novel probabilistic quantifier fuzzification mechanism for information retrieval

A novel probabilistic quantifier fuzzification mechanism for information retrieval
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  A novel probabilistic quantifier fuzzification mechanismfor information retrieval Félix Díaz-Hermida David E. Losada Alberto Bugarín Senén Barro Intelligent Systems GroupDepartment of Electronics and Computer ScienceUniversity of Santiago de Compostela15782 Santiago de Compostela{felixdh,dlosada,alberto,senen} Abstract In this work, a novel quantifier fuzzi-fication mechanism is proposed. Thismethod is deeply rooted in the the-ory of probability and skips the nestedassumption for crisp representatives,which is often taken by other probabili-tistic approaches to quantification. Thenew proposal takes into account all pos-sible crisp representatives which yieldsto a natural and intuive strategy for in-formation retrieval tasks. Furthermore,preliminar analysis of the formal prop-erties of the fuzzification mechanismpermits us to advance that the applica-tion of this method in other domains isalso promising. 1 Introduction Fuzzy quantifiers have been extensively appliedin diverse fields such as expert systems, monitor-ing and control of processes, database systems,etc. [2, 12]. These fuzzy tools have played a keyrole in such domains because the linguistic state-ments a human expert uses are naturally modeledand, hence, the expressiveness of the system isenriched.Fuzzy quantifiers can be defined in a direct way,e.g. proposing a form of combination of themembership values of the elements belonging tothe involved fuzzy set(s). Nevertheless, given acertain linguistic expression, it is often difficultto achieve consensus about the most appropriatequantified definition. In order to avoid such in-conveniences, indirect definitions of fuzzy quan-tifiers have been introduced [9], based on the con-cept of semi-fuzzy quantifier, which is a half-way point between classic quantifiers and fuzzyquantifiers. Fuzzy quantifiers are intuitively de-fined from semi-fuzzy quantifiers through a so-called quantifier fuzzification mechanism. Be-cause semi-fuzzy quantifiers are closer to thewell-known classic quantifiers, the implementa-tion of a linguistic expression in terms of semi-fuzzy quantifiers results more natural and intu-itive.In the information retrieval (IR) literature, fuzzyquantification has been applied for designing flex-ible query languages [1]. Sincethe connection be-tween query term semantics and document con-tents is inherently vague, the retrieval processcan be naturally modelled in terms of fuzzy sets.Fuzzy quantification supplies appropriate formaltools for handling linguistic expressions whichenrich the query languages of the IR system. Thisaids users to establish additional constraints in theretrieval process (e.g. retrieved documents shouldmatch at least 3 of the query terms).The importance of fuzzy quantifiers for IR wasempirically demonstrated for large collections of documents in [11]. Nevertheless, this practicaldeployment of fuzzy quantifiers also revealed thata new class of quantifier fuzzification mecha-nisms may be beneficial. This motivated us todefine here a new fuzzification method which isevaluated for a retrieval task. Furthermore, themost relevant properties the model fulfills are pre-sented, advancing the adequacy of this new ap-proach for other domains.Therest ofthis paper isorganized asfollows. Sec-  tion 2 reports briefly some background conceptson fuzzy quantification. Section 3 explains thenew quantification proposal and section 4 appliesthe new quantification framework for handling in-formation retrieval. The paper ends with someconclusions. 2 Fuzzy quantifiers The formal notions of classic quantifier, fuzzyquantifier and semi-fuzzy quantifier have beenused to associate meaning to quantified sentences[9]. Formally, a classic s-ary quantifier on abase or referential set    is a mapping ¡¢£¥¤  §¦©¨"! , where £¥¤  §¦ is the powersetof     . Throughout this work we assume the refer-ential set    to be finite, which is sufficient from apractical perspective.An s-ary fuzzy quantifier #¡ on a base set   %$&(' is a mapping #¡)¢#£0¤  1¦2¨35476 which to eachchoice of  8@9BABABA 8¨0C#£¥¤  §¦ assigns a gradualresult #¡¤8D9BABABA 8¨¦C476 ( #£¥¤  §¦ is the fuzzypowerset of     ).In many cases, it is not easy to achieve consen-sus on an intuitive and generally applicable ex-pression for implementing a given quantified sen-tence. To overcome this problem, the conceptof semi-fuzzy quantifier was introduced [9]. Asemi-fuzzy quantifier is a half-way point betweenclassic quantifiers and fuzzy quantifiers, which isvery close to the idea of Zadeh’s linguistic quan-tifier [14]. Semi-fuzzy quantifiers are similar toclassic quantifiers, but they allow variation of theresults in 476 . Formally, an s-ary semi-fuzzyquantifier ¡ on a base set 5$&E' is a mapping ¡F¢£¥¤  §¦¨476 which assigns a gradualresult ¡¤8@9BABABA 8¨¦C4G76 to each choice of crisp 89BABABAH 8¨C£¥¤  §¦ .Semi-fuzzy quantifiers are much more intuitiveand easier to define than fuzzy quantifiers, butthey do not resolve the problem of evaluatingfuzzy quantified sentences. In order to do soquantifier fuzzification mechanisms are needed[9] that enable us to transform semi-fuzzy quan-tifiers into fuzzy quantifiers, i.e. mappings withdomain in the universe of semi-fuzzy quantifiersand range in the universe of fuzzy quantifiers: IQPSRUTVPXWYRa`cbedgfihprqts©uvbwfyxTQP WRa`bedgfihprqsu 2.1 Quantifier fuzzification mechanisms Several methods for evaluating quantified sen-tences have been proposed in the literature [9, 5].In [5] two models for the evaluation of fuzzyquantified sentences are proposed. These modelsshow avery consistent behaviour and arebased ona voting model interpretation of fuzzy sets [10,6].If the universe of discourse    is finite and expres-sions are unary (i.e. involve a single fuzzy set)then both models collapse into the same: "§3BXV 2UjHk mlnoXo" (1) where &9 & and z9|{ABABA{} denote the membership values in descending or-der of the elements in    to the fuzzy set 8 and ¤8¦~X stands for the  -cut of level  of  8 , i.e.the crisp set containing the elements of     whosedegree of membership in 8 is greater or equalthan  . For the unary and binary cases, expres-sion 1 is equivalent to the quantification modeldefined in [3] 1 . Moreover, for the case of non-decreasing unary quantifiers, itis equivalent to thequantification method based on ordered weightedoperators [13].In equation 1, the value w9 can be inter-preted as the probability that ¤8¦B~X is selectedas the crisp representative for the fuzzy set 8 .Therefore, the semi-fuzzy quantifier is applied forevery crisp representative of  8 and those valuesare weighted by the probability of each crisp rep-resentative. In this formulation, the use of   -cutsmakes that, given the fuzzy set 8 , the crisp rep-resentatives ¤8¦~X are nested.In [11], equation 1 was empirically evaluated forthe basic IR task. Although this experimentationmade evident the benefits that IR might obtainfrom fuzzy quantifiers, it also revealed that thenested assumption may not always be appropri-ate, as it will be seen in the following sections.This motivated us to propose a novel probabilistic 1 keeping aside differences related to representative nor-malization.  method that skips the nested assumption. In sec-tion 4 we will enter into details on the adequacyof the new quantification proposal for IR. Alongthis paper, the approach sketched in equation 1will be referred to as NVM (standing for NestedVoting Model) approach. 3 A probabilistic interpretation of fuzzysets Given a fuzzy set 8C#£0¤  1¦ , the process thatselects a number of elements in    to belong toa crisp representative of  8 can be viewed as arandom process in which    mutually independent  binary decisions are made (    & ¢¡    ¡ ). Every in-dividual decision involving an element £ C   may be viewed as a Bernoulli trial whose prob-ability of success (i.e. the probability of se-lecting e for representing X) is equal to ¤¦¥ ¤ £ ¦ .Hence, for every possible crisp representative of  8 , § C£g¤  1¦ , we can estimate its probabil-ity as follows. Given a discrete random variable ¨£©££  !"$#%£ ¥ which takes values on £z¤  §¦ ,the probability that ¨£©££  !"$#%£¥ results inY is equal to: & R ('0)132 4)456)87@9$AB9DCFEG)  IHQP b HRSTUV R W) b XRS`YTU Rms baV R W) bb For simplicity, we introduce the fol-lowing compact notation: c¥ ¤ § ¦& d ¤ ¨£e©fg£hB£4  3"$#%£¥ & § ¦ .In the next section this definition is used for de-signing a novel quantifier fuzzification mecha-nism based on this independence assumption. 3.1 A new fuzzification mechanism Following the previous definition, a new fuzzifi-cation method in which all possible crisp repre-sentatives of a given fuzzy set 8 are consideredarises in a natural way. This contrasts with theNVMapproach inwhich only thenested crisp setsobtained by successive  -cuts on 8 are taken intoaccount. Definition 1 ( ip ) Let  ¡¢£0¤  1¦¨476 bea semi-fuzzy quantifier. We define the quantifier  fuzzification mechanism iqp as:  fr z  es$t$t$tus  v mX wyx  88G t$t$twyxv88G`  U  t$t$tv  Uv  U  s$tut$t$s Uv  es$t$t$t$s  vT      Following this definition, all the crisp representa-tives are handled independently and no crisp rep-resentative is disregarded a priori. The superindex  was chosen to stress that all crisp representa-tives are considered.Unfortunately, in the general case iqp is not com-putable in polynomial time. Nevertheless, when quantitative semi-fuzzy quantifiers (i.e, thosewhich can be expressed as a function of the car-dinalities of the involved sets 2 ) are handled, it ispossible to develop polynomial time algorithms.This is very important because quantitative quan-tifiers are the most interesting from a practicalview [4,7]and, indeed, sufficient for ourpurposesin IR.Now we will sketch the procedure for solvingquantitative unary quantifiers. Algorithms forsolving higher arity quantitative quantifiers can bedesigned using similar ideas.We will denote by   & £ X9BABABAH £ B! a referentialcontaining " elements. By § C£     $ ( 8C#£      ) we will denote a crisp (fuzzy) set on thisreferential.Let us consider a unary semi-fuzzy quantitativequantifier: ¡§9 §   &  9 h§   8C£      where  9 is a function with the form  9¢  4G76 .For this case the independence quantification ex-pression becomes: 2 More specifi cally, those which can be expressed asa function of the cardinalities of the arguments and theirboolean combinations[8].   r Sel x  8      U l U l x  8    ¡ £¢ x  ¢ H   U l U l ¥¤ t$tut ¤  x  `    ¡£¢ x  ¢ H   U l U l x  8    ¡ £¢ x  ¢ H   U l §¦  ©¨  ¤ t$t$t ¤  x  `    ¡£¢ x  ¢ H   U l §¦    If we denote   "! w $#&%£%  '%(0) c¥  §   by 132  ¤ 4 3 5 ¥ & 76 ¦ 3 we can rewrite the previous ex-pression as follows:  r ml 98@BADC¡EGFIH   9¨IP¦  ©¨  ¤ t$t$t ¤¤8@AC¡EQFRH   9P¦   X S a 8@AC¡EQFRH   UTP¦  £T  It can be proved that the values 1V2  ¤ 4 ! 5 ¥ & W6 ¦ can be obtained with a complexity X ¤    `Y ¦ .The next example clarifies the use of the newquantification approach. First, probabilities of allpossible crisp representatives are computed and,next, the previous expression is applied. Example 1 Let us consider the evaluation of thequantified sentence "almost all students are tall".Suppose that we model the property tall for anumber of individuals   & £ 9 £ Y  £ ba ! throughthe fuzzy set   dcc &HA ©egf £ r9©A ©hgf £ Y  bf £ a ! and we support the quantified expression "almost all"by means of the following semi-fuzzy quantifier: ¡¤8¦&  9¤ ¡ 8 ¡ ¦  9¤    ¦& pi    qsrY First, we compute the probabilities 132  ¤ 4 3 5tDuwv©v & W6 ¦  for every value of  6 : 3 This value can be interpreted as the probability that thefuzzy set x  is represented by a crisp set whose size is y . 8@  C¡EQFRHR£  9¨ i x  8`G  ¢ x  ¢   "&£ " U l  "&£ 7 © X 9¨ t  n ¨ t  n ¨  9¨8@  C¡EQFRHR£   i x  8`G  ¢ x  ¢ "  "&£ " U l  "&£ 7  S  I  ¤  "&£ t " S ¡  ¤¤  "&£   S ¡ X 9¨ t  n ¨ t  n ¨¤¤¨ t  n ¨ t  n ¨¤¨ t  n ¨ t  n   9¨ t ¨8@  C¡EQFRHR£   i x  8`G  ¢ x  ¢    "&£ " U l  "&£ 7  S  esS ¡  ¤  "&£ t " S  esS   ¤¤  "&£   S  sS ¡ X 9¨ t  n ¨ t  n ¨¤¤¨ t  n ¨ t  n ¤¨ t  n ¨ t  n   ¨ t ¨¤¨ t &  9¨ t 8@  C¡EQFRHR£   i x  8`G  £¢ x  ¢    "&£  U l  "&£   S  sS  sS ¡ X ¨ t  n ¨ t  n   9¨ t   And then,  r  $ER %   S aH 8@  C¡EGFIHR"&£  UT  ¦  £T  8@  "C¡EQFIHR"&£  9¨  ¦  ©¨  ¤¤8@  "C¡EQFIH "&£    ¦    ¤¤8@  "C¡EQFIHR"&£    ¦    ¤¤8@  "C¡EQFIHR"&£    ¦   X ¨ n ¨¤¨ t ¨ n b¤¨ t I n b¤¤¨ t  n   9¨ t  It is interesting to see how the NVM approachproceeds against the same example. Given thefuzzy set  cjc , the  values (equation 1) are &g9& Y &A ©h  a &A ©e  lk & and the fuzzification process runs as follows: cB ©$EQ %  H§ ©$ER jHklnoo© ©$ER jHktlnoo ¤¤  ©$ER jHk©lnoo   ¤¤ © ©$ER jHk  lno  o   ¤¤ © ©$ER jHk  lno  o bm X3 n S  n     ¤ 3 " S  n   ¨ t   ¤¤ 3 n S  sS ' n ©¨ t   ¨ t   ¤¤ 3 n S  sS ' sS  n ©¨ t   ¨ X  n ¨ t ¤ n ¨ t ¤ n ¨ t   ¨ t IoIo This example clarifies the differences betweenboth methods. For instance, the NVM approach  estimates the odds that exactly two individuals aretall by means of the two elements of   cjc havingthe largest degrees ofmembership, £ Y  £ a . Itisim-plicitly assumed that, if only two individuals areconsidered tall, these should be £ Y and £ ba manda-torily. This assumption is completely reasonableif the fuzzy set  cjc was built from measures of height of the individuals involved. In that case, itis difficult to imagine a situation in which, for in-stance, £ ba and £ 9 are considered as tall individualswhereas £ Y is not considered a tall person. Never-theless, think that the fuzzy set  cjc may representpredictions about the height of future descendantsfor three couples (e.g. estimated from the heightof the members of each couple) and, hence, itmight be the case that £ a and £ X9 do finally producetall descendants whereas £ Y produces a short one.Summing up, it may be adequate to consider allpossible two-sized crisp representatives for com-puting the odds that exactly two elements do com-ply with the property formalized by the fuzzy set.This is precisely what the ip method does. Theodds that exactly two individuals are tall are com-puted taking into account the odds that only £ 9 and £ Y are tall, the odds that only £ Y and £ a aretall and the odds that only £ r9 and £ a are tall.As we will detail later, IR is foundated on a num-ber of useful heuristics that have played a fun-damental role to enhance retrieval performance,e.g. tf/idf to weight the importance of a term fora given document. Of course, this class of heuris-tics is not perfect and, hence, one can never besure that the two terms which are the most sig-nificative in the context of a given document arethose ones have the higher tf/idf values. As a con-sequence, the ip approach is a good support forour application of fuzzy quantifiers in IR. 3.2 Properties of the model A formal analysis of the properties fulfilled bythe new fuzzification approach is currently un-dergoing. In this study we follow the axiomaticframework presented in [9]. We can advance thatthe model is well-behaved because it fulfills theproperties of  correct generalization of crisp ex- pressions, induced operations, external negation,internal negation, duality, internal meets, mono-tonicity in arguments , monotonicity in quantifiersand coherence with logic. This assures that thequantification method proposed yields a naturaland intuitive modeling of quantified expressions,as depicted briefly in the following examples. Forinstance, if external negation is not fulfilled sen-tences such as "at most 10 tall individuals areblonde" and "not more than 10 tall individualsare blonde" are not considered equivalent. Thesentences "all tall individuals are blonde" and "notall individual is not blonde" are only equivalentwhen the quantification model complies with in-ternal negation. Duality assures that "some tall in-dividuals are blonde" and "not all tall individualsare not blonde" are equivalent and the equivalencebetween "some tall individuals are blonde" and"there is some individual who is tall and blonde"is guaranteed by the property of internal meets.Monotonicity in quantifiers assures that the resultof evaluating an expression such as "about 80% ormore of the tall individuals are blonde" is less orequal than the result obtained from "about 60% ormore of the tall individuals are blonde". These ex-amples show clearly that unacceptable and coun-terintuitive situations might arise when the quan-tification approach does not comply with someof these fundamental properties. Since the ip method complies with such properties, its appli-cation in a wide range of domains is promising. 4 Application in information retrieval The adequacy of fuzzy quantifiers for IR was al-ready anticipated in [1]. In a recent work [11],a query language enriched with quantified state-ments was empirically tested. This evaluationrevealed that fuzzy quantifiers are beneficial interms of retrieval performance. The proposal en-closed in [11] designs a general framework basedon theNVMmethod in whichquantifiers withdif-ferent degrees of expressiveness can be handled.In the experimental setting quantified expressionswere handled through unary quantifiers. This ap-proach subsumes the quantification model basedon ordered weighted operators [13] and, as arguedbefore, it falls into the nested assumption for crisprepresentatives.Next paragraphs sketch the use and convenienceof the ip model in the context of the basic IRtask.Consider a query formulation with the form
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