Level-based fuzzy generalized quantification

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Level-based fuzzy generalized quantification M. Dolores Ruiz ∗ , Daniel Sánchez, Miguel Delgado CITIC-UGR and Dept. Computer Science and A.I., University of Granada, Spain Received 15 September 2015; received in revised form 22 June 2017; accepted 26 June 2017

Abstract We propose a new type of generalized quantification using a recently developed representation of fuzziness by levels. This proposal integrates the existent models allowing the extension of crisp quantification to the fuzzy case keeping all the algebraic Boolean properties. Besides, the model covers new types of quantification not covered by the existent approaches. The advantages of our proposal are clear from the computational and practical perspective, since it allows a parallelization of the algorithm by taking into account a fixed number of levels. Some authors have proposed several properties that a good evaluation method should satisfy. We also prove that our proposal fulfills all the desired properties to be considered a good evaluation method. © 2017 Elsevier B.V. All rights reserved. Keywords: Fuzzy quantification; Generalized quantifiers; Quantified sentences; Representation by levels

1. Introduction Quantified sentences are a very powerful notion for modeling statements in Natural Language (NL). Particularly, they have been used in the framework of Computing with Words and Perceptions [24]. A quantified sentence is a linguistic expression modeling a quantification statement by employing the so-called quantifiers. Examples of quantifiers are all, a few, many, etc. Depending on the type of quantifier we can build different expressions, arising different kinds of quantification. The process of calculating the accomplishment degree of a quantified sentence is usually performed by using an evaluation method. Sentence evaluation methods have been widely applied in many fields. A good overview of the areas of application of evaluations methods can be found in [4]. To name some of them, information retrieval and linguistic summarization are some of the more active ones. Fuzzy quantified sentences arise when fuzzy quantifiers are used in the sentence. They offer a more realistic description of the sentence, allowing a relaxation when describing the quantities, having for instance “around a half” instead of “a half”. Some approaches for evaluating the accomplishment degree of fuzzy quantified sentences have been developed in the past years. We can distinguish between two different approaches: those following Zadeh’s framework [22,23] and those using the Theory of Generalized Quantifiers (TGQ) [3,15]. In fact, in [4] it is discussed * Corresponding author.

E-mail addresses: [email protected] (M.D. Ruiz), [email protected] (D. Sánchez), [email protected] (M. Delgado). http://dx.doi.org/10.1016/j.fss.2017.06.012 0165-0114/© 2017 Elsevier B.V. All rights reserved.

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how the quantified sentences described following the Zadeh’s framework can be considered as particular cases of sentences expressed using the modeling of TGQs. In both frameworks several properties have been proposed to be fulfilled by the quantification methods in order to obtain intuitive results from the human point of view. Depending on the followed framework we can find a different set of properties, but the majority of them correspond to the same idea. We refer the reader to [4] for a complete view of the different properties and the set of quantifiers satisfying them. Some of these properties are those of Boolean algebras or can be proved to be dependent on them and their fulfillment depends not only on the evaluation method, but on the operations employed on fuzzy sets. Fuzzy Set Theory (FST) has been extensively developed and some problems have arisen when some desirable properties were not fulfilled by an FST. For instance, the expected accomplishment degree of the following sentences should be 1: • all A are A ∧ A • all X are A ∨ ¬A • no X is A ∧ ¬A where ∧, ∨ and ¬ are calculated by means of an FST, the quantifier all is defined on [0,1] as all(α) = 1 iff α = 1, and 0 otherwise, and the quantifier no can be defined as the negation of the quantifier exists as no(α) = 1 iff α = 0 and 0 otherwise. But the accomplishment degree of 1 for an FST is satisfied if and only if the following three conditions hold simultaneously: • card(A ∧ A)/card(A) = 1, which implies A ∧ A = A (idempotency), • card(A ∨ ¬A)/card(X) = 1, which implies X = A ∨ ¬A (excluded middle), • card(A ∧ ¬A)/card(X) = 0, which implies A ∧ ¬A = ∅ (non-contradiction). However, this is not possible since Dubois and Prade showed in [6,7] that no standard FST can satisfy idempotency (as well as mutual distributivity) together with the laws of excluded middle and non-contradiction, although some of them can be fulfilled with particular choices of t-norms (like in the case of Lukasiewicz conjunction and the property A ⊗ ¬A = ∅). As a consequence, when using fuzzy sets, there are intuitive properties that cannot be satisfied by the evaluation of fuzzy quantified sentences. To solve all these problems we develop a proposal that 1. guarantees the fulfillment of Boolean algebra properties and therefore we obtain intuitive results when evaluating sentences like the ones introduced in the example above, 2. integrates previous proposals following the Zadeh’s framework and the Theory of Generalized Quantifiers, and 3. satisfies the desired properties for being a good evaluation method. Our proposal uses a recently developed model for representing fuzziness called Representation by Levels (RL). The RL theory model allows elements to satisfy a concept and its negation at the same time to some degree but keeping all Boolean properties [18,21]. In fact, the RL theory has the structure of a Boolean Algebra (see section 2.2). Fuzzy sets can also be represented in this theory but the results obtained when performing operations are different to those obtained when employing a FST. In particular, the result cannot be a fuzzy set. Moreover, our model is easily parallelizable by taking into account a fixed number of levels. This offers a clear advantage from the computational and practical perspective, feature that is very important in fields like fuzzy expert systems, natural language processing, fuzzy temporal knowledge representation and reasoning, etc. The structure of the paper is the following: first we present the preliminary concepts necessary for the comprehension of the paper, including a brief overview of Zadeh’s and TGQs frameworks and a description of the Representation by Levels (RL) theory. The core of this work is in Section 3 where our proposal for evaluating fuzzy quantified sentences using the RL theory is presented. Later in Section 4 we study the properties satisfied by our proposal. We finish with the conclusions and future research on this topic.

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2. Preliminaries 2.1. Zadeh’s and TGQ frameworks for fuzzy quantification and their relation Zadeh recognized the necessity of using fuzzy sets for representing more types of quantifiers than those used in First Order Logic (∃ and ∀). Quantifiers can represent notions such as “a few” or “most”, which are intrinsically imprecise, and fuzzy sets seem to be adequate for modeling them [22,23]. Zadeh gave a first classification into absolute and relative quantifiers modeling imprecise quantities and proportions respectively. “At least one”, “exactly 5” and “approximately 8” are examples of absolute quantifiers meanwhile “almost all”, “at least half” and “most” are relative ones. According to this, two kinds of sentences were considered: type I sentences (Q of X are A) and type II sentences (Q of D are A) where Q is a linguistic quantifier, X is a finite set and D, A are two fuzzy sets of X representing imprecise properties. Type I sentences are a particular case of type II sentences where D = X. Type I: Around 30 students are young Type II: Most of efficient students are young In general it is possible any choice of quantifier in both types of sentences, but as it was pointed in [16] the natural option is to use absolute quantifiers in type I sentences and relative ones in type II. In fact, a type II sentence with an absolute quantifier can be one of type I: Q of X are A ∩ D, that is, the sentence “around 30 efficient students are young” is semantically equivalent to “around 30 students are young and efficient”. Some types of quantified statements in natural language cannot be modeled using absolute or relative quantifiers. Glöckner proposed instead an approach that generalizes fuzzy quantification to any type of quantifier in the Theory of Generalized Quantifiers (TGQ) [9,10,12]. In the TGQ a quantifier (or determiner) like “all” is described by a mapping all : P(X)2 −→ {0, 1} where P(X) is the powerset of X, i.e. it contains all the possible subsets of X. Then the validity of a sentence of the type “all students are blond” is translated to the function all(X1 , X2 ) = 1 if and only if X1 ⊆ X2 and 0 otherwise, where X1 and X2 represent the set of students and the set of blond people respectively, which are subsets of X = people. Formally, a crisp n-ary quantifier is defined as a mapping QC : P(X)n −→ {0, 1} while fuzzy ones as mappings  n −→ [0, 1] where P(X)  QF : P(X) stands for the powerset of fuzzy subsets defined over X. One of the important features of the TGQ is the proposed procedure to generalize crisp quantifiers to their fuzzy analogous [9]. The benefit of using such procedure is that there is no necessity to decide case-by-case how to define the fuzzy quantifier. This procedure uses an intermediate step for extending crisp to fuzzy quantifiers [11]. For that aim, it is defined the concept of n-ary semi-fuzzy quantifier as a mapping Q : P(X)n −→ [0, 1]. The advantage of using semi-fuzzy quantifiers is that the usual crisp cardinality is applicable to their arguments. This procedure, called Quantifier Fuzzification Mechanism (QFM) and noted by F , assigns to each semi-fuzzy quantifier a fuzzy quantifier with the same arity and base set, i.e.  n → [0, 1]) F : (QC : P(X)n → [0, 1]) −→ (QF : P(X) QC

QF

(1)

where and are respectively an n-ary crisp quantifier and an n-ary fuzzy quantifier in the TGQ. In the literature we can find many evaluation proposals for fuzzy quantification. Some of them were defined following the Zadeh’s framework in terms of type I and type II sentences. We refer to [5,4] for some overview of the existing evaluation methods. Following the TGQ framework, some QFMs have been proposed by Glöckner and DíazHermida et al. in [9,11,2] for the evaluation of quantified sentences. The QFM proposed by Glöckner is a general max constructive method based on the cut range Tα (A) = {Y ⊆ X | Amin α ⊆ Y ⊆ Aα } ⊆ P(X) of a fuzzy subset A at min max level α, where Aα and Aα are expressed in terms of three different α-cuts. Some particular methods are obtained depending on the aggregation method used for Q(Y1 , . . . , Yn ) such that Yi ∈ Tα (Ai ). Díaz-Hermida et al. proposed several models based on voting models and probability theory. One of the models is defined using a function similar to a probability density function associated to the semi-fuzzy quantifier. Another model appears by taking a uniform probability density function. The third model, called independence model, assumes that the levels are independent, and the last model assumes that levels are “approximately equal” for the different properties [2]. A complete overview of the existing QFMs in the literature can be found in [4]. Although both theories deal with the evaluation of quantified sentences they have some differences in their formalization and nomenclature. In a recent paper [17] both frameworks have been analyzed and compared, estab-

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lishing a formal connection between the concepts defined in both theories. Therefore given a semi-fuzzy quantifier Q : P(X)n −→ [0, 1] we can obtain a fuzzy quantifier in the sense defined by Zadeh. And moreover, a Quantifier Fuzzification Mechanism is directly related to the evaluation methods developed for assessing fuzzy quantified sentences (in the sense of Zadeh) via the previous connection. For this reason our approach will be defined in terms of the Theory of Generalized Quantifiers, meaning that it can also be used to evaluate fuzzy quantified sentences following the Zadeh’s framework. 2.2. Representation by levels Level-based Representation or the Representation by Levels theory (RL for short) was introduced by Sánchez et al. in [21] and is akin to the notion of Gradual set by Dubois and Prade [8]. This theory provides an effective model for the representation of sets that are affected by some kind of fuzziness and for having desirable properties that can be relevant in diverse application domains, e.g. quantified sentence evaluation, syllogistic reasoning, computing with words, etc. The fuzziness in sets is modeled by means of an assignment function of crisp sets to levels of fulfillment of a property. RLs are not only conceived as representations of fuzzy concepts, but as more general structures having fuzzy sets as particular cases. The main difference with the representation of fuzzy sets by α-cuts is that the crisp sets corresponding to the different levels may not be nested with respect to inclusion in the usual way, as we will see later in some examples, and its main characteristic is that it allows a unique and straightforward extension of operations and definitions from the crisp case keeping all the Boolean properties. The levels are values in the unit interval meaning possible levels of relaxation of the property, where level 1 corresponds to the most restrictive strict view of the concept, 0 means being no strict at all and level 0.5 is halfway between being totally strict and no strict at all. We then consider the interval  = (0, 1] as the scale where different levels of relaxation will be considered. In practice, for computing purposes a finite set of levels is fixed (human beings are not able to distinguish an infinite amount of levels)  = {α1 , . . . , αm } satisfying that 1 = α1 > α2 > · · · > αm > αm+1 = 0, m ≥ 1. Definition 1. [21] An RL is a pair (, ρ) where  ⊆ (0, 1] such that 1 ∈  and ρ :  → P(X) is a function which applies each level in  into a crisp realization that represents the fuzzy concept for each level. In order to enable comparisons between two RLs the concept of extension of an RL is defined. Definition 2. [21] Let (, ρ) be an RL and  ⊆ ∗ . The RL (∗ , ρ ∗ ) is said to be an extension of (, ρ) provided that, for any α ∈ ∗ it holds ρ ∗ (α) = ρ(β), where β = inf{λ ∈ |λ > α}. For computational purposes, in practice a finite set of levels will be considered when performing calculations and comparisons between two or more RLs. In these cases, the information represented by two RLs can be compared by means of extending their definition to every α ∈  as follows: Lemma 1. Let (, ρ) be an RL with a finite set of levels  = {α1 , . . . , αm } satisfying that 1 = α1 > α2 > · · · > αm > 0. Then, it exists an RL (I, ρ ∗ ) which is an extension of (, ρ) where I = (0, 1]. Proof. The proof is immediate by defining ρ ∗ for all β ∈ I as follows: ρ ∗ (β) = ρ(αi ) where αi ≥ β > αi+1 for αi , αi+1 ∈ .

2

Now, we establish the equivalence of two RLs by employing their extensions.

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Table 1 RLs associated to A, ¬A, B and ¬B. αi

ρA (α)

ρ¬A (α)

ρB (α)

ρ¬B (α)

1 0.9 0.8 0.6 0.5 0.4

{x1 } {x1 } {x1 , x2 } {x1 , x2 } {x1 , x2 , x3 } {x1 , x2 , x3 , x5 }

{x2 , x3 , x4 , x5 } {x2 , x3 , x4 , x5 } {x3 , x4 , x5 } {x3 , x4 , x5 } {x4 , x5 } {x4 }

∅ {x1 } {x1 } {x1 , x3 } {x1 , x3 , x4 } {x1 , x3 , x4 }

X {x2 , x3 , x4 , x5 } {x2 , x3 , x4 , x5 } {x2 , x4 , x5 } {x2 , x5 } {x2 , x5 }

Definition 3. [21] Let (1 , ρ1 ) and (2 , ρ2 ) be two RLs on X. We say that both representations are equivalent, denoted by (1 , ρ1 ) ≡ (2 , ρ2 ), if and only if, (I, ρ1∗ ) = (I, ρ2∗ ), i.e. ρ1∗ (α) = ρ2∗ (α) for all α ∈ I = (0, 1]. Summarizing, only a finite set of levels are necessary for representing a fuzzy concept, but the representation extends to any other level in (0, 1]. Operations with RLs rely on the following ideas: (1) the meaning of a level is the same when applied to the representation of any fuzzy concept and, (2) crisp operations are extended to RLs by operating in each RL independently. Definition 4. [21] Let P , Q be two fuzzy concepts represented by (P , ρP ), (Q , ρQ ). Then, P ∧ Q, P ∨ Q and ¬P are fuzzy concepts represented by (P ∧Q , ρP ∧Q ), (P ∨Q , ρP ∨Q ) and (¬P , ρ¬P ) respectively, where P ∧Q = P ∨Q = P ∪ Q , ¬P = P and, for all α ∈ (0, 1], ρP ∧Q (α) = ρP (α) ∩ ρQ (α), ρP ∨Q (α) = ρP (α) ∪ ρQ (α),

(2)

ρ¬P (α) = ρP (α), where X is the usual complement of a crisp set X. Basic Boolean properties that cannot be fulfilled simultaneously by any standard fuzzy set theory (FST) hold simultaneously for RLs. It is well known that given the set X, P(X), ¬, ∨, ∧ is a Boolean Algebra since the ¬, ∨ and ∧ operations defined here correspond to the usual complement, union and intersection operations for crisp sets respectively. Since operations in each level define a Boolean algebra, and equality is defined by levels, operations on RLs define a Boolean algebra. Every fuzzy set is an RL by considering the RL (A , ρA ) given by A = {A(x) | x ∈ support (A)} ∪ {1} and ρA (α) = Aα = {x ∈ X | A(x) ≥ α} for all α ∈ A . Hence, for a concept defined by a fuzzy set A, the set of crisp representatives is the set of significant α-cuts of A. The contrary is not true, i.e. not every RL is a fuzzy set, since when negation is employed the usual nested α-cut representation is lost. In fact, given B a non-crisp fuzzy set, the RL that represents ¬B is not a fuzzy set like next example shows (see [21] for a more complete explanation and examples). Example 1. Let A, B be the following two fuzzy sets: A = 1/x1 + 0.8/x2 + 0.5/x3 + 0.4/x5 , B = 0.9/x1 + 0.6/x3 + 0.5/x4 defined over the referential X = {x1 , . . . , x5 }. We take their associated RLs (A , ρA ), (B , ρB ) (see Table 1) and we perform some operations involving negation. The RLs associated to A and B coincide with their α-cut representation, and therefore the representation by levels is nested. However when operating with complement the result has not to be nested anymore, and in consequence, the resulting RLs do not correspond to any fuzzy set. This is shown in Table 2 where (A∧¬B , ρA∧¬B ) is not the RL of any fuzzy set because the crisp sets are not nested when decreasing the level αi . It is remarkable that for subsequent operations involving ¬A or ¬B the obtained RLs will no longer correspond to any fuzzy set, but they will keep the main properties of Boolean algebras. RLs are suitable to represent and manage fuzzy information, but they are not easily expressed by means of a linguistic term. However, it is possible to obtain a fuzzy set that summarizes the information represented by an RL for being understandable for humans.

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Table 2 RLs of several concepts derived from A and B. αi

ρA∧B (α)

ρA∧¬B (α)

ρ¬A∧B (α)

ρ¬A∧¬B (α)

ρA∨B (α)

1 0.9 0.8 0.6 0.5 0.4

∅ {x1 } {x1 } {x1 } {x1 , x3 } {x1 , x3 }

{x1 } ∅ {x2 } {x2 } {x2 } {x2 , x5 }

∅ ∅ ∅ {x3 } {x4 } {x4 }

{x2 , x3 , x4 , x5 } {x2 , x3 , x4 , x5 } {x3 , x4 , x5 } {x4 , x5 } {x5 } ∅

{x1 } {x1 } {x1 , x2 } {x1 , x2 , x3 } {x1 , x2 , x3 , x4 } X

Definition 5. [21] Let (, ρ) be an RL and  = {ρ(α) | α ∈ } the set of crisp representatives of the RL. Then, the associated probability distribution ω :  → [0, 1] is ω(αi ) = αi − αi+1 . A probability assignment m :  → [0, 1] can also be associated to the crisp representatives of the RL as follows  m(Y ) = ω(αi ) (3) αi | Y =ρ(αi )

where Y must be the preimage of at least one α ∈ . The probability assignment mA gives information about the evidence associated to the crisp representatives of the RL A, by aggregating the information about the indexing structure. The representation by levels of a fuzzy real quantity (RL-real number for short [21]) is a pair (, ρ) in which  is a set of levels and ρ :  → R. This extension, which can also be employed for other kind of numbers like integers, complex numbers, etc., was first proposed in [8] and called gradual number. RL-numbers and gradual numbers are isomorphic, the only difference is the way they are summarized as fuzzy subsets of numbers (in the case of RL-numbers, by using the notion of numerical summary of Section 3.2). 3. Level-based fuzzy generalized quantification Existent quantification methods lie on the Fuzzy Set Theory and therefore the obtained evaluation methods may not fulfill all the desired properties. This is mainly due to the problems of FSTs to satisfy the algebraic properties of boolean algebras when the negation is involved. Our proposal uses instead the RL theory for defining what we call a level-based fuzzy generalized quantifier, which actually extends the notion of semi-fuzzy quantifier used in TGQ. Before introducing the concept of level-based fuzzy generalized quantifier we formalize some concepts to provide a theoretical framework for our proposal. Given the set of RL-truth degrees defined as the set of RL-numbers defined with values in [0, 1]: RLT () = {(, ρ) | ρ :  −→ [0, 1]} i.e., for each α ∈ , ρ(α) is a real number in the unit interval representing the degree of truthiness, where 1 represents the truth and 0 the lack of truth. Then we consider the set containing all the RL sets defined over the referential X and the scale  as follows: RLS(X, ) = {(, ρ) | ρ :  → P(X)}. The notion of extension of an RL allows the comparison and the definition of RLs in different scales, then to be consistent we need to extend RLT and RLS to be independent on the scale :  RLT = RLT () ⊆(0,1],1∈

RLS(X) =



RLS(X, )

⊆(0,1],1∈

An n-ary level-based fuzzy generalized quantifier (LbFGQ for short) can be defined as follows:

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 is a mapping Definition 6. An n-ary level-based fuzzy generalized quantifier on X, denoted by Q, : Q

n 

RLS(X) −→ RLT

i=1

such that  Q(( 1 , ρ1 ), . . . , (n , ρn )) = (, ρ) n with  = i=1 i . In particular, an n-ary level-based fuzzy generalized quantifier can be determined by a family D = {Qα | α ∈ (0, 1]} of semi-fuzzy quantifiers. Next definition distinguishes this special sub-class of level-based fuzzy generalized quantifiers: Definition 7. An n-ary level-based fuzzy generalized quantifier on X determined by a family D = {Qα | α ∈ (0, 1]} of D , is a mapping semi-fuzzy quantifiers, denoted by Q D : Q

n 

RLS(X) −→ RLT

i=1

defined as D ((1 , ρ1 ), . . . , (n , ρn )) = (, ρ) Q with =

n 

i

i=1

ρ(α) = Qα (ρ1∗ (α), . . . , ρn∗ (α)),

α ∈ ,

where (, ρi∗ ) denotes the extension of (i , ρi ) for i = 1, . . . , n. It is straightforward to see that semi-fuzzy quantifiers in the TGQ are special cases of level-based fuzzy quantifiers when the same semi-fuzzy quantifier appears in every level. In the following we present an evaluation method for these new types of quantifiers, and after that a method to summarize the evaluation, yielding, as in the classical sentence evaluation approaches, a value in the unit interval. We finish this section with some illustrative examples to compare the type of quantification handled in TGQ with the new types of quantifications that may arise when the new LbFGQ quantifiers are taken into account. 3.1. Evaluation method for LbFGQ, F RL The main idea of our proposal is to evaluate the sentence in every level of the representation, in which the sets are crisp. Besides, it is usual to provide a single value that represents the evaluation of the sentence, but following the ideas behind the representation by levels the final aggregation is not strictly necessary, and can be performed simply to be able to provide a numerical accomplishment degree as in classical methods. Instead, we propose to perform calculations in each level independently when the result of the evaluation has to be employed in a further calculation. And when no more calculations are necessary, we can yield a single value called the summary of the evaluation (see Section 3.2). Let us assume that we have fuzzy concepts A1 , . . . , An and a quantified sentence defined over these fuzzy concepts  In order to correctly operate Q  over these fuzzy concepts we by the n-ary level-based fuzzy generalized quantifier Q. have to consider the following injection  i : P(X) −→ RLS(X) that transforms a fuzzy set A into the corresponding RL (A , ρA ) as follows, i.e. i(A) = (A , ρA ) such that

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A = {α ∈ (0, 1] | ∃ x ∈ X : α = A(x)} ∪ {1} ρA (α) = {x ∈ X | A(x) ≥ α}. We can also define a map j : [0, 1] −→ RLT by j (α) = ({1}, ρα ) where ρα (1) = α for α ∈ [0, 1] which provides an embedding of [0, 1] into RLT . The definition of the injection i and the embedding j are useful to prove the following result:  n −→ [0, 1] be an n-ary fuzzy quantifier. Then there exists n-ary level-based fuzzy generTheorem 1. Let QF : P(X)   ◦ i ≡ j ◦ QF , i.e. for any fuzzy concepts A1 , . . . , An ∈ P(X),  alized quantifiers Q : RLS(X) −→ RLT such that Q it holds that F  Q(i(A 1 ), . . . , i(An )) ≡ j (Q (A1 , . . . , An ))

(4)

where ≡ means that both RL-truth values are equivalent in the sense of Definition 3.  can be defined as follows to fulfill such equivalence: Proof. Obviously, the level-based fuzzy generalized quantifier Q  Q(( 1 , ρ1 ), . . . , (n , ρn )) = (, ρ) n where  = i=1 i and for any α ∈   QF (A1 , . . . , An ), i(Ak ) = (k , ρk ), k = 1, . . . , n; ρ(α) = 0, otherwise

2

If we restrict ourselves to semi-fuzzy quantifiers we can also proved the following: Theorem 2. Let Q : P(X)n −→ [0, 1] be an n-ary semi-fuzzy quantifier. Then there exists an n-ary level-based fuzzy D : RLS(X) −→ RLT determined by a family D of semi-fuzzy quantifiers. generalized quantifier Q D , determined by the family D = {Qα | α ∈ (0, 1]} could be Proof. The level-based fuzzy generalized quantifier, Q defined as  D ((1 , ρ1 ), . . . , (n , ρn )) = Q(Y1 , . . . , Yn ), ∀j : Yj ∈ P(X), j = {1} and ρj (1) = Yj ; Q 0, otherwise where Q1 = Q and for any α ∈ (0, 1), Qα is an n-ary semi-fuzzy quantifier defined for any Y1 , . . . , Yn ∈ P(X) as follows: Qα (Y1 , . . . , Yn ) = 0. 2  1 , . . . , An ). In our case, the evaluation of the Our goal is thus to evaluate a quantified sentence of the form Q(A sentence will be an RL-truth value, i.e. an RL in RLT .  be an n-ary level-based fuzzy generalized quantifier on X determined by D = {Qα |α ∈ (0, 1]} Definition 8. Let Q  1 , . . . , An ) is defined and A1 , . . . , An be a set of fuzzy concepts defined over X. The evaluation of the sentence Q(A as the RL-truth value (E , ρE ), where E =

n 

Ai

(5)

i=1

and, for all α ∈ E ,  A1 , . . . , ρAn )(α) = Qα (ρA1 (α), . . . , ρAn (α)). ρE (α) = Q(ρ

(6)

From now on, we will refer to the associated QFM derived from the evaluation using level-based fuzzy generalized quantifiers by F RL , i.e. F RL assigns to each semi-fuzzy quantifier a level-based fuzzy quantifier with the same arity and base set

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Table 3 Some examples of different types of semi-fuzzy quantifiers. Quantified sentence

Quantifier

Type

At least one A1 is A2 All A1 are A2

Q∃ (A1 , A2 ) = 1 iff A1 ∩ A2 = ∅ Q∀ (A1 , A2 ) = 1 iff A1 ⊆ A2 |A ∩ A2 | Qp (A1 , A2 ) = 1 iff |A1 | = 0 |A1 | Qmore (A1 , A2 , A3 ) = 1 iff |A1 ∩ A3 | ≥ |A2 ∩ A3 | Qall except 1 (A1 , A2 ) = 1 iff |A1 \A2 | = 1

absolute absolute

Qp of A1 are A2 More A1 than A2 are A3 All except 1 of A1 are A2

: F RL : (Q : P(X)n → [0, 1]) −→ (Q

n 

proportional cardinal comparative exception

RLS(X) → RLT ),

i=1

D is the n-ary level-based fuzzy generalized quantifier where if Q is an n-ary semi-fuzzy quantifier, then F RL (Q) = Q determined by the family D = {Qα | α ∈ (0, 1]} via Theorem 2. The evaluation of a quantified sentence is given in this case by the evaluation of the level-based fuzzy generalized quantifier in each level, yielding, as a final result, an RL-truth value, i.e. in each level a value in [0, 1], which is the value of the quantifier in that level. As mentioned before, when a final numerical evaluation is required, we can compute a summary of the evaluation as detailed in next section. 3.2. Numerical evaluation for LbFGQ The classical evaluation methods of quantified sentences usually yield a number in the unit interval. We can obtain such summary of the evaluation when that is the final, expected result of our system; otherwise, following the ideas of RLs, we would proceed operating in each level independently. In [19,20] we propose to summarize the information given by the RL truth value of Definition 8 by means of the probability assignment of equation (3) as follows: Definition 9. The numerical summary S : RLT −→ [0, 1] of an RL-truth value (E , ρE ) is given by  S(E , ρE ) = mE (β) · β

(7)

β∈E

where E is the set of crisp representatives of (E , ρE ), i.e. E = {ρE (α) | α ∈ E }, mE : E → [0, 1] and therefore β = ρE (α) for a fixed α ∈ E . Let us remark that the presented summarization method is equivalent to that of using the Choquet integral for additive measures as it was proven in [5,21]. 3.3. Examples of LbFGQ Level-based fuzzy generalized quantifiers generalize semi-fuzzy quantifiers by setting the set of levels  = {1}, obtaining thus the semi-fuzzy quantifier in that level. Beside this, new types of fuzzy quantifiers may arise when in each level different semi-fuzzy quantifiers appear. The following cases present several scenarios for using LbFGQ together with their evaluations, and we compare them to semi-fuzzy quantifiers following the TGQ. Case 1: fuzzy quantification following TGQ’s framework using LbFGQ The TGQ deals with more types of quantification than those covered with absolute or relative quantifiers. In fact there is a tight relation between unary and 2-ary semi-fuzzy quantifiers and absolute/relative quantifiers defined in the sense of Zadeh [17]. For this reason, we present an example using semi-fuzzy quantifiers. Quantified sentences such as “more A than B are C” or “all except 1 of A are B” are represented by means of semi-fuzzy quantifiers like those in Table 3. Example 2. The semi-fuzzy quantifier Qmore representing “more A than B are C” in Table 3 can be extended to a suitable level-based fuzzy generalized quantifier, which is defined by a family of semi-fuzzy quantifiers

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Table 4 RLs associated to C and the derived formulas A ∧ C and B ∧ C. αi

ρC (α)

ρA∧C (α)

ρB∧C (α)

1 0.9 0.8 0.6 0.5 0.4

{x3 , x4 } {x3 , x4 } {x3 , x4 } {x3 , x4 } {x3 , x4 } {x1 , x3 , x4 }

∅ ∅ ∅ ∅ {x3 } {x1 , x3 }

∅ ∅ ∅ {x3 } {x3 , x4 } {x1 , x3 , x4 }

Table 5 RL-truth value (E , ρE ) representing the evaluation of the sentence “more A than B are C”. αi

|ρA∧C (α)|

|ρB∧C (α)|

Evaluation by levels

1 0.9 0.8 0.6 0.5 0.4

0 0 0 0 1 2

0 0 0 1 2 3

Qmore,1 (A, B, C) = 1 Qmore,0.9 (A, B, C) = 1 Qmore,0.8 (A, B, C) = 1 Qmore,0.6 (A, B, C) = 0 Qmore,0.5 (A, B, C) = 0 Qmore,0.4 (A, B, C) = 0

D = {Qmore,α | α ∈ (0, 1]}. The corresponding level-based quantifier by levels is then defined by setting the same semifuzzy quantifier in each level. For instance, let A, B, C be atomic concepts where A, B are those defined in Example 1 and C = 0.4/x1 + 1/x3 + 1/x4 . Then we have C = ¬C = {1, 0.4} and A∧B∧C = {1, 0.9, 0.8, 0.6, 0.5, 0.4}. If we want to evaluate the sentence “more A than B are C” we have to compute first the representation by levels of the sets A ∧ C and B ∧ C (see Table 4) and then to apply in each level the quantifier Qmore,α , obtaining in each level the values shown in the right column of Table 5. In order to compare the evaluation yielded using our approach, we first compute the evaluation of the sentence “more A than B are C”, which is represented in Table 5 and then we compute the numerical summary using the formula (7) as follows S(E , ρE ) = (1 − 0.9) · 1 + (0.9 − 0.8) · 1 + (0.8 − 0.6) · 1+ + (0.6 − 0.5) · 0 + (0.5 − 0.4) · 0 + 0.4 · 0 = 0.4

(8)

Example 3. To evaluate the quantified sentence “there are approximately a half of the elements of A in B” we first need to define a quantifier Q≈half A representing “Approximately a half of the elements of A” and then apply it to the ≈half A (B)”. Since A may be different each time that we evaluate the previous sentence, we are not able sentence “Q to define such quantifier in a general form, i.e. it will be defined once we know how A is defined. Let A and B be those sets defined in Example 1 and the LbFGQ representing approximately a half of the elements of A determined by the family D = {Q≈half A,α } where Q≈half A,α (B) = Taα , bα , cα (|Aα ∩ Bα |),

B ∈ P({x1 , . . . , x5 })

which can be defined by using the triangular function in Fig. 1 and Aα . The resulting level-based fuzzy generalized quantifier will depend on the number of elements in A in each level (see Table 6, where in each level the maximum degree of satisfiability 1 is reached when in that level the number of elements is exactly the half of elements of A) and it must be applied to |A ∩ B| in order to see the number of common elements between A and B. ≈half A (B) can then be computed using formula (7) as follows: The numerical summary of the sentence Q S(E , ρE ) = (1 − 0.9) · 0 + (0.9 − 0.8) · 0 + (0.8 − 0.6) · 1+ 2 + (0.6 − 0.5) · 1 + (0.5 − 0.4) · + 0.4 · 1 = 0.767. 3

(9)

In general, when the negation is not employed in the quantified sentence, the result obtained by the numeric summary coincides with the independence model F I presented by Díaz-Hermida et al. in [2] when the set of α-cuts

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⎧ ⎪ ⎪ ⎪0 ⎪ y−a ⎪ ⎪ ⎨ Ta,b,c (y) = b − ya − b ⎪ ⎪ 1− ⎪ ⎪ c−b ⎪ ⎪ ⎩0

11

y≤a a
Fig. 1. Triangular fuzzy set Ta,b,c in terms of a, b and c. Table 6 Example of level-based fuzzy generalized quantifier determined by the family D = {Q≈half A,α }: approximately a half of the elements of A. α

D Q

1 0.8 0.5 0.4

Q≈half A,1 = T0,0.5,1 Q≈half A,0.8 = T0,1,2 Q≈half A,0.5 = T0,1.5,3 Q≈half A,0.4 = T0,2,4

Table 7 RL-truth value (E , ρE ) representing the evaluation of the quantified sentence “There are approximately a half of elements of A in B” using LbFGQ. α

ρA∧B

Evaluation

1 0.9 0.8 0.6 0.5 0.4

∅ {x1 } {x1 } {x1 } {x1 , x3 } {x1 , x3 }

T0,0.5,1 (0) = 0 T0,0.5,1 (1) = 0 T0,1,2 (1) = 1 T0,1,2 (1) = 1 T0,1.5,3 (2) = 2/3 T0,2,4 (2) = 1

Table 8 RL-truth value (E , ρE ) representing the evaluation of “∃ C that are not C” using the LbFGQ. αi

ρC (α)

ρ¬C (α)

Evaluation

1 0.4

{x3 , x4 } {x1 , x3 , x4 }

{x1 , x2 , x5 } {x2 , x5 }

∃1 (C, ¬C) = 0 ∃0.4 (C, ¬C) = 0

of the involved sets is finite (see [4] for a complete explanation of this relation). This equivalence is not true when the negation is involved in the sentence as we explain in the following case. Case 2: fuzzy quantification involving negation using LbFGQ The properties of boolean Algebras are kept by the Representation Level theory and as a consequence the evaluation of sentences like “at least one C is not C” which is represented by the expression “∃ C that are not C” is the expected one. Let C be the fuzzy atomic concept defined in the previous case (see second column in Table 8) and ∃(A, B) the semi-fuzzy quantifier defined as 1 if and only if A ∩ B = ∅ and 0 otherwise. Then the evaluation of the sentence by levels is the right column of Table 8 and when computing the numerical summary the final result is 0, as the intuition suggests. Nevertheless, when employing the standard negation of fuzzy sets μ(¬x) = 1 − μ(x) the fuzzy set ¬C obtained is 0.6/x1 + 1/x2 + 1/x5 whose α-cuts are nested. In this case, if we use the minimum for the intersection, the obtained set C ∩ ¬C = 0.4/x1 and therefore, the evaluation of the sentence “∃ C that are not C” is not any longer zero.

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Table 9 ≈half A (B) which represents “Q≈half of Evaluation of the sentence Q A are B” is defined on equation (10). α

|ρA∧B (α)|

|ρA (α)|

Evaluation

1 0.9 0.8 0.6 0.5 0.4

0 1 1 1 2 2

1 1 2 2 3 4

T0, 0.5, 1 (0) = 0 T0, 0.5, 1 (1) = 0 T0, 0.5, 1 (1/2) = 1 T0, 0.5, 1 (1/2) = 1 T0, 0.5, 1 (2/3) = 2/3 T0, 0.5, 1 (1/2) = 1

Table 10 Example of level-based fuzzy generD determined by alized quantifier Q a family D of different semi-fuzzy quantifiers for each α. α

α Q

1 0.75 0.5 0.25

Q∃ Qf ew Q≈half Qmost

For some scenarios the constraints imposed by the nested assumption for crisp representatives can be inappropriate as this example highlights. In addition, there are several types of quantified sentences that have not been taken into account so far and that our approach using LbFGQ covers. The following case describes some examples. 3.4. Discussion A level-based fuzzy generalized quantifier allows to express different relaxation of the quantification according to the level α. The presented cases exemplify how it can be done. One option is to define a LbFGQ by using the same semi-fuzzy quantifier with different parameters setting, as shown in previous examples. In some cases, we can obtain an equivalent semi-fuzzy quantifier as it happens for instance with the LbFGQ representing Approximately a half of the elements of A defined in Table 6, that can be formulated using the family of semi-fuzzy quantifiers D = {Q≈half A,α } as follows:

 |Aα ∩ Bα | Q≈half A,α (B) = Taα , bα , cα . (10) |Aα | and it is straightforward to check that

 |Aα ∩ Bα | Taα , bα , cα = T0, |A| ,|A| (|Aα ∩ Bα |) , 2 |Aα |

B ∈ P({x1 , . . . , x5 })

following thus to the same results of Table 7 as it is depicted in Table 9. But in some other cases, we may want to quantify in each level separately by employing different semi-fuzzy quantifiers as depicted in Table 10, instead of defining it using the same semi-fuzzy quantifier with different parameters as shown in previous examples. In these cases the resulting LbFGQ does not coincide with any existent semi-fuzzy quantifier. The main problem of this type of LbFGQ is to adjust an appropriate semantics to it. Additionally, our proposal also allows to define quantifiers that may depend of previous calculations involving fuzzy or RL sets. This is extremely useful when we are “chaining” results from several previously computed evaluations in order to obtain a final assessment value for our sentence. This happens for instance when reasoning or doing inference involving prior knowledge in form of quantified sentences.

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4. Properties A collection of properties that any suitable method should satisfy has been proposed in several papers [1,2,5,10,14]. A joint view of all these properties is collected in [4]. As we mentioned before, absolute and relative quantifiers can be seen as special cases of binary semi-fuzzy quantifiers. For this reason we focus on properties defined for QFMs (see Table 11) but conveniently written for the case of level representations instead of fuzzy sets. Therefore, a QFM F here is understood as a mapping : F : (Q : P(X)n → [0, 1]) −→ (Q

n 

RLS(X) → RLT ),

i=1

which transforms semi-fuzzy quantifiers in level-based fuzzy generalized quantifiers of the same arity and based set. A QFM satisfying properties P.1 to P.6 is said to be a Determiner Fuzzification Scheme (DFS) [10].  the correFrom now on, we named by Q the semi-fuzzy quantifiers of the form Q : P(X)n → [0, 1] and by Q sponding level-based fuzzy generalized quantifier resulting from applying the QFM F . In particular, for F RL the resulting LbFGQ is defined via the injection i defined in formula (4) as described in Theorem 2. Additionally, the  injection i : P(X) −→ RLS(X) is also employed to transform crisp sets Yj into RLs since crisp sets are special cases of fuzzy sets via their characteristic function defined by 1 if x ∈ Yj and 0 otherwise. P.1. [10,13] Correct generalization. For any crisp sets Y1 , . . . , Ys ∈ P(X), it holds that F (Q)(i(Y1 ), . . . , i(Ys )) = (E , ρE ), where E = {1} and ρE (1) = Q(Y1 , . . . , Ys ). This property guarantees that the evaluation results are consistent with the crisp case. P. 2. [10,13] Membership assessment. Let x ∈ X and ∃x : P(X) −→ [0, 1] be the semi-fuzzy quantifier defined by ∃x (Y ) = 1 if x ∈ Y and ∃x (Y ) = 0 otherwise. Then, for any RL (, ρ) ∈ RLS(X), F (∃x )(, ρ) = (E , ρE ) such that E =  and ρE (α) = χρ(α) (x) where χρ(α) is the characteristic function of the subset ρ(α) in X, i.e.  1 if x ∈ ρ(α) χρ(α) (x) = 0 otherwise. This property assures that the fuzzification using RLs of classical existential quantifier produces the expected one from the perspective of fuzzy logic. P. 3. [10,13] Dualization. F must preserve dualization of quantifiers, i.e. for any crisp sets Y1 , . . . , Ys ∈ P(X), it holds that whenever Q (Y1 , . . . , Ys ) = ¬Q(Y1 , . . . , Ys−1 , ¬Ys ) then F (Q )(i(Y1 ), . . . , i(Ys )) = ¬F (Q)(i(Y1 ), . . . , i(Ys−1 ), i(¬Ys )). This property guarantees the satisfaction of duality in the fuzzy case, for instance the dual of the semi-fuzzy quantifier “all” is “at least one” since ¬Q∀ (X1 , ¬X2 ) = Q∃ (X1 , X2 ) (see in Table 3 their definitions). P. 4. [10,13] Union. F must preserve the unions of arguments, i.e. for any crisp sets Y1 , . . . , Ys ∈ P(X), it holds that F (Q )(i(Y1 ), . . . , i(Ys+1 )) = F (Q)(i(Y1 ), . . . , i(Ys−1 ), i(Ys ∪ Ys+1 )) whenever Q (Y1 , . . . , Ys+1 ) = Q(Y1 , . . . , Ys−1 , Ys ∪ Ys+1 ). This property assures the preservation of the union of arguments in the fuzzy case. Note that in the original definition the fuzzy operators ∪ and ∩ are related to the induced t-conorm and t-norm respectively. Nevertheless in our framework they are directly connected to the logic operators ∨ and ∧ of formula (2).

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P. 5. [10,13] Monotonicity in arguments, i.e. if Q is non-decreasing or non-increasing in the i-th argument, then F (Q) has the same property. It is said that Q is non-decreasing in its i-th argument 1 ≤ i ≤ s if and only if for each crisp Y1 , . . . , Ys such that Yi ⊆ Yi the following holds: Q(Y1 , . . . , Yi−1 , Yi , Yi+1 , . . . , Ys ) ≤ Q(Y1 , . . . , Yi−1 , Yi , Yi+1 , . . . , Ys ). P.6. [10,13] Functional application, i.e. for any crisp sets Y1 , . . . , Ys ∈ P(X), it holds that F (Q )(i(Y1 ), . . . , i(Ys )) = F (Q)(i(f1 (Y1 )), . . . , i(fs (Ys ))) whenever Q is defined by Q (Y1 , . . . , Ys ) = Q(f1 (Y1 ), . . . , fs (Ys )) where f1 , . . . , fs are obtained from the induced extension principle of F (see Definitions 20∼23 in [10]. If F induces the standard negation ¬x = 1 − x and the extension principle, then it is a standard DFS. When F is a DFS, i.e. fulfills the principles P.1∼P.6, a large number of properties are satisfied by F : A. F induces a reasonable set of fuzzy propositional connectives. B. F is compatible with the negation of quantifiers, i.e. for any crisp sets Y1 , . . . , Ys ∈ P(X), if Q (Y1 , . . . , Ys ) = ¬Q(Y1 , . . . , Ys ) then F (Q )(i(Y1 ), . . . , i(Ys )) = ¬F (Q)(i(Y1 ), . . . , i(Ys )). C. F is compatible with the formation of antonyms, i.e. for any crisp sets Y1 , . . . , Ys ∈ P(X), if Q (Y1 , . . . , Ys ) = Q(Y1 , . . . , Ys−1 , ¬Ys ) then F (Q )(i(Y1 ), . . . , (Ys )) = F (Q)(i(Y1 ), . . . , i(Ys−1 ), i(¬Ys )). D. F is compatible with intersections (analogous definition as in Property 4). E. F is compatible with argument permutations, i.e. if β is a permutation of {1, . . . , s}, for any crisp sets Y1 , . . . , Ys ∈ P(X), whenever Q (Y1 , . . . , Ys ) = Q(Yβ(1) , . . . , Yβ(s) ) then F (Q )(i(Y1 ), . . . , i(Ys )) = F (Q)(i(Yβ(1) ), . . . , i(Yβ(s) )). F. F is compatible with argument insertion, i.e. for any crisp sets Y1 , . . . , Ys , A ∈ P(X), if Q (Y1 , . . . , Ys ) = Q(Y1 , . . . , Ys , A) then F (Q )(i(Y1 ), . . . , i(Ys )) = F (Q)(i(Y1 ), . . . , i(Ys ), i(A)). Other properties have been proposed to be fulfilled by a QFM as for instance:  argument if and only if Q(Y1 , . . . , Ys ) ≥ min Q(Y1 , . . . , Yi−1 , P.7. [13] A semi-fuzzy quantifier Q is convex in its ith  Yi , Yi+1 , . . . , Ys ), Q(Y1 , . . . , Yi−1 , Yi , Yi+1 , . . . , Ys ) whenever Y1 , . . . , Ys , Yi , Yi ∈ P(X) and Yi ⊆ Yi ⊆ Yi . A QFM F is said to preserve convexity if for every convex Q, F (Q) is also convex. P.8. [13] A QF M F is Q-continuous if and only if for each semi-fuzzy quantifier Q and all > 0 there exists δ > 0 such that d(F (Q), F (Q )) < whenever Q semi-fuzzy quantifier satisfies d(Q, Q ) < δ where   d(Q, Q ) = sup{Q(Y1 , . . . , Ys ) − Q (Y1 , . . . , Ys ) : Y1 , . . . , Ys ∈ P(X)}   d(F (Q), F (Q )) = sup{F (Q)(i(Y1 ), . . . , i(Ys )) − F (Q )(i(Y1 ), . . . , i(Ys )) : i(Y1 ), . . . , i(Ys ) ∈ RLS(X)} We are going to study the fulfillment of these properties by our evaluation method, named F RL . At a first glance, we can see that those properties satisfied in the crisp case will be kept in general by our proposal since it make the computations separately in each level. This is an important feature of our proposal, since in general not every model satisfies them. Next proposition shows the properties satisfied by our evaluation method. The property about D convexity P.7 is not proved and it is easy to see that P.8 is fulfilled only in the finite case and when the LbFGQ Q is determined by a family of quantifiers where the same quantifier is used in every α ∈ . In this special case the continuity could be guaranteed when the levels are closer enough, since we work with a finite set of levels. Theorem 3. Properties P.1∼ P.6 are satisfied by the method F RL . Proof. P.1 Let Y1 , . . . , Ys be crisp sets. For i = 1, . . . , s the associated RLs are obtained via the injection i, i.e. Yi = {α ∈ (0, 1] | ∃x ∈ X : α = Yi (x)} ∪ {1} = {1} and ρYi by its α-cut. D (ρY1 , . . . , Then the evaluation F RL (Q)(i(Y1 ), . . . , (Ys )) is the RL (E , ρE ) where E = {1} and ρE (1) = Q ρYs )(1) = Q1 (Y1 , . . . , Ys ), where the family D = {Qα | α ∈ (0, 1]} only contains the semi-fuzzy quantifier Q1 = Q.

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Table 11 Compilation of the properties for a quantification model. Properties for a QFM P.1 Correct generalization P.2 Membership assessment P.3 Duality P.4 Internal meets P.5 Monotonicity in arguments P.6 Functional application P.7 Convexity P.8 Continuity or smoothness in behavior A Induced operators B External negation C Antonymy D Internal meets E Independence of order in the elements of the referential F Argument insertion

P.2 Let Y be a crisp set. The fulfillment of the sentence “exists x in Y ” is given by ∃x (Y ) = 1 if x ∈ Y and 0 otherwise. In our framework, for (, ρ) an RL, the F RL method transforms ∃x to the n-ary LbFGQ  ∃x such that  ∃x (, ρ) is via Definition 8 given by E =  and ρE (α) = ∃x (ρ(α)) = 1 if x ∈ ρ(α) and 0 otherwise, which coincides with the characteristic function of the subset ρ(α) in X. P.3 We have to prove that if Q (Y1 , . . . , Ys ) = ¬Q(Y1 , . . . , Ys−1 , ¬Ys ) for Yj crisp sets ∀j , then F (Q )(i(Y1 ), . . . , i(Ys )) = ¬F (Q)(i(Y1 ), . . . , i(Ys−1 ), i(¬Ys )). This property is fulfilled, since Y = ¬Y and our method applies the negation in an RL by applying it to the crisp representatives in every level. P.4 It is preserved since in our framework the union is calculated as the union of the crisp representatives in each level and Ys ∪Ys+1 = Ys ∪ Ys+1 . P.5 If Q is non-decreasing (non-increasing) in the i-th argument, then ρE (α) = Q(ρY1 (α), . . . , ρYi (α), . . . , ρYs (α)) and therefore ρE (α) has the same property as Q. P.6 From the way of defining ρE it is clearly fulfilled. 2 Proposition 3 also says that F RL is a standard DFS and therefore it fulfills properties A∼F. Note that property P.8 cannot be proved for F RL since it would be necessary to have an analogous definition taking into account the levels, but it is fulfilled in the finite case and when the family D is defined by the same semi-fuzzy quantifier Qα for every level. Another problem of some existing QFMs which is solved with our model is that of complexity. In general the computational complexity is quadratic or even more. In our case it is efficient in time since the evaluation is crisp (O(1)) in each level and the number of levels is finite (this will depend on either the amount of data or the precision considered). In the worst case, for type I and type II sentences, the complexity is O(n log n) as it is proved in [5] where n is the number of levels. For the methods proposed by Glöckner the complexity is studied depending on the quantifier used [11]. In this way, the algorithm goes from O(m) to O(mn2 ) where m is the number of relevant cutting levels and n = |X|. Moreover, our proposal has some intuitive properties that, to the best of our knowledge very few methods satisfy them1 : the evaluation of sentences of the kind “∃ A are ¬A” is definitely 0 (see Case 3 in Section 3.3). This goes beyond the development of QFMs, as it is related to the very nature of the fuzziness representation. The satisfiability of that property comes from the fact that A ∧ ¬A = ∅ in the RL model.

1 See [4] for a complete explanation about the different methods and the properties they satisfy.

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5. Conclusions Throughout this work we propose a generalization for the concept of quantifier called level-based fuzzy general quantifier (LbFGQ) and an evaluation method for computing the assessment of the sentence in each level. We also provide a summarization technique in order to give a numerical summary as it has been done in classical evaluation methods. This proposal covers the existent approaches of fuzzy quantification and keeps, at the same time, the properties of Boolean algebras, properties that are necessary for obtaining the expected results when evaluating sentences such as “at least one element of C is not C” that should be 0. Furthermore, the LbFGQs are suitable for defining new types of quantification not covered by the existent quantifiers by specifying different semi-fuzzy quantifiers in each level. Beside this, we study the properties of our model and give some examples for illustrating its applicability. The advantages of our model are clear from the computational and practical perspective, since it allows a parallelization of the algorithm by taking into account a fix number of levels. This is very important in fields like fuzzy expert systems, natural language processing, fuzzy temporal knowledge representation and reasoning, etc. Future studies will be aimed to the application of these techniques to fields such as computing with words and syllogistic reasoning, where fuzzy quantification plays an essential role. Acknowledgements The research reported in this paper was partially supported by the Andalusian Government (Junta de Andalucía) under projects P11-TIC-7460 and P10-TIC-6109 and from the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (FEDER) under project grants TIN2015-64776-C3-1-R and TIN2014-58227-P. References [1] S. Barro, A. Bugarín, P. Cariñena, F. Díaz-Hermida, A framework for fuzzy quantification models analysis, IEEE Trans. Fuzzy Syst. 11 (1) (2003) 89–99. [2] S. Barro, A. Bugarín, P. Cariñena, F. Díaz-Hermida, Voting-model based evaluation of fuzzy quantified sentences: a general framework, Fuzzy Sets Syst. 146 (1) (2004) 97–120. [3] J. Barwise, R. Cooper, Generalized quantifiers and natural language, Linguist. Philos. 4 (1981) 159–219. [4] M. Delgado, M.D. Ruiz, D. Sánchez, M.A. Vila, Fuzzy quantification: a state of the art, Fuzzy Sets Syst. 242 (2014) 1–30. [5] M. Delgado, D. Sánchez, M.A. Vila, Fuzzy cardinality based evaluation of quantified sentences, Int. J. Approx. Reason. 23 (2000) 23–66. [6] D. Dubois, H. Prade, New results about properties and semantics of fuzzy set-theoretic operators, in: Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems, 1980, pp. 59–75. [7] D. Dubois, H. Prade, An introduction to possibilistic and fuzzy logics, in: Non-Standard Logics for Automated Reasoning, 1988, pp. 287–315. [8] D. Dubois, H. Prade, Gradual elements in a fuzzy set, Soft Comput. 12 (2008) 165–175. [9] I. Glöckner, DFS – An Axiomatic Approach to Fuzzy Quantification, Technical Report TR97-06, Technical Faculty, University Bielefeld, Bielefeld, Germany, 1997. [10] I. Glöckner, Fundamentals of Fuzzy Quantification, Technical Report TR2002-07, Technical Faculty, University Bielefeld, Bielefeld, Germany, 2002. [11] I. Glöckner, Evaluation of quantified propositions in generalized models of fuzzy quantification, Int. J. Approx. Reason. 37 (2004) 93–126. [12] I. Glöckner, Fuzzy Quantifiers: A Computational Theory, Springer, 2006. [13] I. Glöckner, A. Knoll, A formal theory of fuzzy natural language quantification and its role in granular computing, in: W. Pedrycz (Ed.), Granular Computing: An Emerging Paradigm, Physica-Verlag, 2001. [14] I. Glöckner, A. Knoll, Fuzzy quantifiers: a natural language technique for data fusion, in: Proc. of the Fourth Int. Conference on Information Fusion, 2001. [15] E.L. Keenan, D. Westerståhl, Generalized quantifiers in linguistics and logic, in: J.V. Benthem, A.T. Meulen (Eds.), Handbook of Logic and Language, Elsevier, 1997, pp. 837–893, chapter 15. [16] Y. Liu, E. Kerre, An overview of fuzzy quantifiers part I (Interpretations) and II (Reasoning and applications), Fuzzy Sets Syst. 95 (1998) 1–21, 135–146. [17] M.D. Ruiz, D. Sánchez, M. Delgado, On the relation between fuzzy and generalized quantifiers, Fuzzy Sets Syst. 294 (2016) 125–135. [18] D. Sánchez, M. Delgado, M.A. Vila, A restriction level approach to the representation of imprecise properties, in: IPMU 2008, 2008, pp. 153–159. [19] D. Sánchez, M. Delgado, M.A. Vila, Fuzzy quantification using restriction levels, in: Proc. WILF 2009, in: LNCS, vol. 5571, Springer, 2009, pp. 28–35. [20] D. Sánchez, M. Delgado, M.A. Vila, An approach to general quantification using representation by levels, in: Proc. WILF 2011, in: LNAI, vol. 6857, Springer, 2011, pp. 50–57.

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