A Probabilistic Quantifier Fuzzification Mechanism: The Model and Its Evaluation for Information Retrieval Felix Díaz-Hemida, David E. Losada, Alberto Bugarín, and Senén Barro Present by Chia-Hao Lee

2 outline Introduction Fuzzy Quantifiers –Probabilistic Quantifier Fuzzification Mechanisms New View in Crisp Representatives –FA Quantifier Fuzzification mechanism –Properties of the Model Applying the FA Quantifier Fuzzificaiton Mechanism for Information Retrieval –Fuzzy Quantifiers and Information Retrieval –Example Information Retrieval Experiments Conclusion

3 Introduction The ability of fuzzy quantifiers to model linguistic statements in a natural way has proved useful in diverse areas such as expert systems, data mining, control systems, database systems, etc. In the information retrieval (IR) field, fuzzy quantification has been proposed for handling expressive queries giving rise to flexible query language.

4 Introduction Fuzzy quantification is a linguistic granulation technique capable of expressing the global characteristics of a collection of individuals, or a relation between individuals, through meaningful linguistic summaries. Granular computing attempts to manage complex, large- scale problems by organizing these into different levels of detail. It is understood that each sub-problem should be solved at its appropriate level of granularity, and there are effective transformations which mediate between these levels.

5 Introduction The need for such transformation process not only arises in the technical problem areas tackled by computers. It is hence not surprising that natural language ( NL ) provides a class of expressions specifically designed to express accumulative properties and to summarize information: natural language quantifiers. NL quantifiers, and in particular their approximate variety (“ almost all ”, “ a few ” etc.), provide flexible means for expressing accumulative properties of collections and can also describe global aspects of relationships between individuals.

6 Introduction Fuzzy set theory attempts to model NL quantifiers by operators called fuzzy quantifiers. –Interpretation : the development of methods for evaluating quantifying expressions which capture the meaning of natural language quantifiers. –Summarization : the development of processes for constructing quantifying statements, which succintly describe a collection of observations and/or relationships between a large number of observations (find domain concepts X and Y and a quantifier Q such that “ Q X’s are Y’s is true ”). –Reasoning : the development of methods which deduce further knowledge from a set of rules and/or facts involving fuzzy quantifiers.

7 Fuzzy Quantifiers Two-valued quantifier: input : crisp input output: crisp output Fuzzy quantifier: Input : fuzzy input Output : fuzzy output Semi-fuzzy quantifier: Input : crisp input Output : fuzzy output

8 Definition 1 (Classic Quantifier or Two valued Quantifier) : An n-ary generalized quantifier on a base set is a mapping Q : A two-valued quantifier hence assigns to each n -tuple of crisp subsets a two-valued quantification result. Fuzzy Quantifiers : the powerset of E : the fuzzy powerset of E

9 Fuzzy Quantifiers Well-known examples A typical example of a classic quantifier is the following definition of an all statement which can be used for sentences such as “ ” :

10 Fuzzy Quantifiers For example : Let us consider the evaluation of the sentence “80% or more of students are Spanish” in the reference where the properties “students” and “Spanish” are, respectively, defined as X 1 (students)={1,0,1,0,1,0,1,1} (true : 1, false : 0) X 2 (Spanish)={1,0,1,0,1,0,0,0} and “80% or more” is defined as in (1). Then Logic “and”

11 Definition 2 (Fuzzy Quantifier) : An n-ary fuzzy quantifier on a base set is a mapping which to each n-tuple of fuzzy subsets of E assigns a gradual result An example of a fuzzy quantifier is, which can defined as a fuzzy extension of 1 using a typical definition for the fuzzy inclusion operator: Fuzzy Quantifiers

12 For example : Let us consider the evaluation of sentence “all tall people are blond” in the referential set. Let us assume that properties “tall” and “blond” are, respectively, defined as Using expression (2) then: In many cases, it is not easy to achieve consensus on an intuitive and generally applicable expression for implementing a given quantified expression. Fuzzy Quantifiers

13 Fuzzy Quantifiers Definition 3 (Semi-fuzzy Quantifier) : An n-ary semi-fuzzy quantifier on a base set is a mapping which to each n-tuple of crisp subsets of E assigns a gradual result..

14 Fuzzy Quantifiers Examples of semi-fuzzy quantifier are :

15 Fuzzy Quantifiers For example : Let us consider the evaluation of the sentence “about 80% or more of the students are Spanish”. Let us assume that properties “students” and “Spanish” are, respectively, defined as X 1 (students)={1,0,1,0,1,0,1,1}, X 2 (Spanish)={1,0,1,0,1,0,0,0} then

16 Fuzzy Quantifiers Semi-fuzzy quantifiers are half-way between two-valued quantifiers and fuzzy quantifiers because they have crisp input and fuzzy output. In particular, every two-valued quantifier of TGQ (theory of generalized quantifiers) is a semi-fuzzy quantifier by definition. Being half-way between two-valued generalized quantifiers and fuzzy quantifiers, semi-fuzzy quantifiers do not accept fuzzy input, and we have to make use of a fuzzification mechanism which transports semi-fuzzy quantifiers to fuzzy quantifiers.

17 Fuzzy Quantifiers Probabilistic Quantifier Fuzzification Mechanisms : In the universe of discourse E is finite and expressions and unary then both expressions collapse into the same discrete expression The value can be interpreted as the probability that ( ) is selected as the crisp representative for the fuzzy set X.

18 Fuzzy Quantifiers Let be a set of individuals for which the set represents the fulfillment of the property “being all”. It is reasonable for X to arise on the basis of a consonant vote. The intuitive ordering of the elements of the referential on the basis of their height is. The focal elements and their associated probability masses are :

19 Fuzzy Quantifiers It should also be noted that where denoted the α-cut of X ;

20 Fuzzy Quantifiers For example : Let us consider the evaluation of the quantified sentence “almost all students are tall.” Suppose that we model the property tall for a referential set of students through the fuzzy set tall and we support the quantified expression “almost all” by means of the following semi-fuzzy quantifier : the feature “tall”

21 Fuzzy Quantifiers given the fuzzy set tall, the values are and the fuzzification process runs as follows:

22 New View on Crisp Representatives Given a fuzzy set, the process that selects a number of elements in E to be included in a crisp representative of X can be viewed as a random process in which n mutually independent binary decisions are made. Every individual decision involving an element may be viewed as a Bernoulli trial whose probability of success equals. A random variable X has a Bernoulli distribution with parameter p ( 0<p<1 ) if X take only the values 0 and 1. The p.f. f ( · | p ) of X can be written in the form

23 New View on Crisp Representatives Definition 4 ( ) ： We define the probability that a crisp set is a crisp representative of X as Definition 5 ( ) ： Let be a semi-fuzzy quantifier. For simplicity, fuzzification process :

24 New View on Crisp Representatives We will denote by a referential containing m elements. By we will denote a crisp (fuzzy) set on this referential. (so we have subsets) Let us consider a unary semi-fuzzy quantitative quantifier : a function with the form

25 New View on Crisp Representatives For this case, the expression becomes And we instead of

26 New View on Crisp Representatives Example of the approach

27 New View on Crisp Representatives It can be proved that all the value can be obtained with a complexity

28 New View on Crisp Representatives We can advance that the model is well-behaved because it fulfills the properties of correct generalization of crisp expressions, induced operations, external negation, internal negation, duality, internal meets, monotonicity in arguments monotonicity in quantifiers and coherece with logic.

29 Applying the FA Quantifier Fuzzificaiton Mechanism for Information Retrieval IR is the science concerned with the effective and efficient retrieval of information for the subsequent use by interested parties. IR models differ in the way in which documents and queries are represented and matched. The proposal designs a general framework based on the NVM method in which quantifiers with different degrees of expressiveness can be handled.

30 Applying the FA Quantifier Fuzzificaiton Mechanism for Information Retrieval Consider a query with the form. Given a document of the document base, every query term produces a score which represents the connection between the document’s semantics and the term. Formally, every document induces a fuzzy set on the set of query terms which is defined applying the popular weighting strategy : the raw frequency of term in the document : the maximum raw frequency computed over all terms mentioned by the document

31 Applying the FA Quantifier Fuzzificaiton Mechanism for Information Retrieval The fuzzy set models the connection between the document and every query component. Quantification can now be applied on for evaluating the quantified symbol all.

32 Applying the FA Quantifier Fuzzificaiton Mechanism for Information Retrieval Example : Let us suppose that we apply the following power function for supporting a given query quantification symbol Q : Imagine a query and consider a document whose fuzzy set induced on the query components is Applying now the fuzzification process explained along this paper, the query-document matching is assigned a score n : the number of query terms

33 Applying the FA Quantifier Fuzzificaiton Mechanism for Information Retrieval Let us now apply the NVM approach to handle the same example. The score assigned is equal to It follows that the final value yielded by the NVM method is:

34 Information Retrieval Experiment We ran experiments against the Wall Street Journal (WSJ) documents, which are about 173,000 news articles (from 1987 to 1992). Natural language documents are preprocessed as follow: –First, common words such as prepositions, articles, etc. are eliminated. –Second, terms are reduced to their syntactical root by applying the popular Porter’s stemmer.

35 Information Retrieval Experiment We tried out different semi-fuzzy quantifiers for relaxing the interpretation of the quantified statement all and, for each semi-fuzzy quantifier, both the fuzzification approach and the NVM approach were applied. We experimented with power functions and exponential functions, both of them normalized in the interval as follows :

36 Information Retrieval Experiment

37 Information Retrieval Experiment

38 Information Retrieval Experiment

39 Information Retrieval Experiment

40 Information Retrieval Experiment

41 Information Retrieval Experiment

42 Conclusion In the paper, we present a new probabilistic quantifier fuzzification mechanism, its efficient implementation and its application for the basic information retrieval task.