Fuzzy Quantifiers: A Computational Theory

466 Pages·2006·4.542 MB·English

by  Dr. Ingo Glöckner (auth.)

IngoGlöckner
FuzzyQuantifiers
StudiesinFuzzinessand SoftComputing, Volume193
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Ingo Glöckner
Fuzzy Quantifiers
A Computational Theory
ABC
Dr.IngoGlöckner
FernUniversitätinHagen
LGPrakt.Inf.VII
Informatikzentrum
Universitätsstr.1
58084Hagen
Germany
E-mail:[e-mail]
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Preface
From a linguistic perspective, it is quantification which makes all the differ-
ence between “having no dollars” and “having a lot of dollars”. And it is the
meaning of the quantifier “most” which eventually decides if “Most Ameri-
cans voted Kerry” or “Most Americans voted Bush” (as it stands). Natural
language(NL)quantifierslike“all”,“almostall”,“many”etc.serveanimpor-
tant purpose because they permit us to speak about properties of collections,
as opposed to describing specific individuals only; in technical terms, quan-
tifiers are a ‘second-order’ construct. Thus the quantifying statement “Most
AmericansvotedBush”assertsthatthesetofvotersofGeorgeW.Bushcom-
prisesthemajorityofAmericans,while“Bushsneezes”onlytellsussomething
about a specific individual. By describing collections rather than individuals,
quantifiers extend the expressive power of natural languages far beyond that
of propositional logic and make them a universal communication medium.
Hencelanguageheavilydependsonquantifyingconstructions.Theseoften
involvefuzzyconceptslike“tall”,andtheyfrequentlyrefertofuzzyquantities
in agreement like “about ten”, “almost all”, “many” etc. In order to exploit
this expressive power and make fuzzy quantification available to technical
applications, a number of proposals have been made how to model fuzzy
quantifiers in the framework of fuzzy set theory. These approaches usually
reduce fuzzy quantification to a comparison of scalar or fuzzy cardinalities
[197, 132]. Thus for understanding a quantifying proposition like “About ten
persons are tall”, we must know the degree to which we have j tall people in
thegivensituation,andthedegreetowhichthecardinalj qualifiesas“about
ten”. However, the results of these methods can be implausible in certain
cases[3,35,54],i.e.theycanviolatesomebasiclinguisticexpectations.These
problems certainly hindered the spread of the ‘traditional’ approaches into
commercial applications.
This monograph proposes a new solution which shows itself immune
against the deficiencies of existing approaches to fuzzy quantification. The
theory covers all aspects of fuzzy quantification which are necessary for a
comprehensive treatment ofthe subject,ranging fromabstracttopics like the
VI Preface
proper semantical description of fuzzy quantifiers to practical concerns like
robustness of interpretations and efficient implementation. The novel theory
of fuzzy quantification is
• a compatible extension of the Theory of Generalized Quantifiers [7, 9];
• not limited to absolute and proportional quantifiers;
• a genuine theory of fuzzy multi-place quantification;
• not limited to finite universes of quantification;
• not limited to quantitative (automorphism-invariant) examples of quanti-
fiers;
• based on a rigid axiomatic foundation which enforces meaningful and co-
herent interpretations;
• fully compatible with the formation of negation, antonyms, duals, and
other constructions of linguistic relevance.
To accomplish this, the theory introduces
• a formal framework for analysing fuzzy quantifiers;
• asystemofformalpostulatesforthosemodelsoffuzzyquantificationthat
are semantically conclusive;
• a number of theorems showing that these models are plausible from a
linguistic point of view and also internally coherent;
• adiscussionofseveralconstructiveprinciplesalongwithananalysisofthe
generated models;
• acollectionofalgorithmsforimplementingfuzzyquantifiersinthemodels
and examples for their application.
Specifically,thisworkapproachesthesemanticsoffuzzyNLquantificationby
formalizing the essential linguistic expectations on plausible interpretations,
and it also presents examples of models which conform to these semantical
requirements.
Thebookfurtherdisclosesageneralstrategyforimplementing quantifiers
in a model of interest, and it develops a number of supplementary methods
which will optimize processing times. In order to illustrate these techniques,
the basic procedure will then be detailed for three prototypical models, and
the complete algorithms for implementing the relevant quantifiers in these
models will be presented. Hence the proposed solution is not only theoret-
ically appealing, but it also avails us with computational models of fuzzy
quantification which permit its use in practical applications. The quantifiers
covered by the new algorithms include the familiar absolute and proportional
types known from existing work on fuzzy quantification, but they also com-
prise some other cases of linguistic relevance. In particular, the treatment of
quantifiers of exception (“all except k”) and cardinal comparatives (“many
more than”) is innovative in the fuzzy sets framework.
The material presented in the book will be of interest to those working at
thecrossroadsofnaturallanguageprocessingandfuzzysettheory.Therange
of applications comprises fuzzy querying interfaces, where fuzzy quantifiers
Preface VII
support more flexible queries to information systems; systems for linguistic
data summarization, where fuzzy quantifiers are used to build succinct lin-
guistic summaries of given databases; decision-support systems, where fuzzy
quantifiers implement multi-criteria decisionmaking, and many others. In all
of these applications, the ‘old’ models of fuzzy quantification can easily be
replaced with the improved models developed here. I hope that by removing
the ‘plausibility barrier’ faced by earlier techniques and rendering possible a
more reliable use of fuzzy quantifiers, the new theory of fuzzy NL quantifica-
tion will contribute to a break-through of these techniques into commercial
applications.∗
Hagen
September 2005 Ingo Glo¨ckner
∗ The monograph is based on the author’s PhD thesis titled “Fuzzy Quantifiers
inNaturalLanguage:SemanticsandComputationalModels” [52]whichwaspub-
lished by Der Andere Verlag, Osnabru¨ck (now T¨onning), Germany under ISBN
3-89959-205-0. Kind permission of Der Andere Verlag is acknowledged to repub-
lish this improved version of the manuscript.
Overview of the Book
Thebookisorganizedasfollows.Thefirstchapter presentsageneralintroduc-
tionintothehistoryandmainissuesoffuzzynaturallanguagequantification.
Tothisend,thebasiccharacteristicsoflinguisticquantifiersarefirstreviewed
inordertodistillsomerequirementsonaprincipledtheoryoflinguisticquan-
tification. Following this, we consider the problem of linguistic vagueness and
its modelling in terms of continuous membership grades, which is fundamen-
tal to fuzzy settheory. Zadeh’s traditional framework for fuzzy quantification
[197, 199] is then explained. It is argued that the framework is too narrow
for a comprehensive description of fuzzy NL quantification, and the partic-
ular approaches that evolved from it are shown to be inconsistent with the
linguistic data.
The second chapter introduces a novel framework for fuzzy quantifica-
tion in which the linguistic phenomenon can be studied with the desired
comprehensiveness and formal rigor. The framework comprises: fuzzy quan-
tifiers, which serve as the operational model or target representation of the
quantifiers of interest; semi-fuzzy quantifiers, which avail us with a uniform
and practical specification format; and finally QFMs (quantifier fuzzification
mechanisms), which establish the link between specifications and operational
quantifiers.Thebasicrepresentationsunderlyingtheproposedframeworkare
directlymodelledafterthegeneralizedquantifiersfamiliarinlinguistics[7,9].
Comparedtothetwo-valuedlinguisticconcept,semi-fuzzyquantifiersaddap-
proximate quantifiers like “almost all”, while fuzzy quantifiers further admit
fuzzyarguments(“tall”,“young”etc.).Thus,semi-fuzzyquantifiersandfuzzy
quantifiers are the apparent extension of the original generalized quantifiers
to Type III and Type IV quantifications in the sense of Liu and Kerre [108].
The organization of my approach into specification and operational layers
greatly simplifies its application in practice. The commitments intrinsic to
this analysis of fuzzy quantification are discussed at the end of the chapter,
which also judges the actual coverage of the proposed approach compared to
the phenomenon of linguistic quantification in its full breadth.
X Overview of the Book
The third chapter is concerned with the investigation of formal criteria
whichcharacterizeaclassofplausiblemodelsoffuzzyquantification.Theba-
sicstrategycanbelikenedtotheaxiomaticdescriptionoft-norms[148],which
constitutetheplausibleconjunctionsinfuzzysettheory.Thusmyapproachis
essentially algebraic. The basic idea is that of formalizing the intuitive expec-
tations on plausible interpretations in order to avoid the notorious problems
ofexistingapproaches.Tothisend,thechapterintroducesasystemofsixba-
sic requirements which distill a larger catalogue of linguistic desiderata. The
criteria are chosen such that they capture independent aspects of plausibil-
ity and that, taken together, they identify a class of plausible models which
answers the relevant linguistic expectations.
The fourth chapter gives evidence that the proposed axioms indeed cap-
ture a class of plausible models. To this end, an extensive list of criteria will
be considered which are significant to the linguistic plausibility of the com-
puted interpretations and to their expected coherence. All of these criteria
are validated by the proposed models. This supports my choice of axioms
even if some of these might appear rather ‘abstract’ at first sight. Apart from
this purpose of justifying the proposed class of models, the formalization of
semantical criteria is also a topic of independent interest. By investigating
such criteria, we can further our knowledge about quantifier interpretations
in natural languages.
The space of possible models will be further explored in the fifth chapter.
Specificallywewillpayattentiontocertainsubclassesofmodels,i.e.classesof
models with some common structure or joint properties. The relative homo-
geneity of the models within these classes permits the definition of important
concepts e.g. regarding the specificity of results. We shall further identify the
classofstandardmodelswhichcomplywiththestandardchoiceofconnectives
in fuzzy set theory. The role of these standard models to fuzzy quantification
can be likened to that of Abelian groups vs. general groups in mathematical
group theory.
The sixth chapter, then, is devoted to the study of some additional prop-
erties, like continuity (smoothness), which are ‘nice to have’ from a practical
perspective,butnotalwaysusefulfortheoreticalinvestigations,orsometimes
even awkward in this context. This is why these properties should not be in-
cluded into the core requirements on plausible models, but they can of course
be used to further restrict the considered models if so desired. The chapter
further discusses some ‘critical’ properties which cannot be satisfied by the
models because they contradict some of the core requirements. (usually these
properties even fail in much weaker systems). The existence of such cases is
not surprising, of course, because fuzzy logic, as a rule, can never satisfy all
axiomsofBooleanalgebra.Thesedifficultieswillusuallyberesolvedbyshow-
ing that the critical property conflicts with very basic requirements, and by
pruning the original postulate to a compatible ‘core’ requirement.
Having laid these theoretical foundations, we shall proceed to the issue of
identifying prototypical models, which are potentially useful to applications.
Overview of the Book XI
The investigation of various constructive principles will give us a grip on
such concrete examples. Existing research has typically tried to explain fuzzy
quantification in terms of cardinality comparisons, based on some notion of
cardinalityforfuzzysets.However,suchreductionisnotpossibleforarbitrary
quantifiers.Henceacomprehensiveinterpretationoflinguisticquantifiersmust
rest on a more general conception.
Chaptersseventotenwilldescribesomesuitablechoicesforsuchconstruc-
tions which result in the increasingly broader classes of MB-models (chap-
ter seven), F -models (chapter eight), and finally F or F -models (chapter
ξ Ω ω
nine). All of these models rest on a generalized supervaluationist approach
based on a three-valued cutting mechanism. The tenth chapter, by contrast,
will present a different mechanism based on the extension principle, which
results in the constructive class of F -models. This class of models which are
ψ
definableintermsofthestandardextensionprinciple,isthenshowntoprovide
a different perspective on the class of F -models, to which it is coextensive.
Ω
Apart from introducing these classes of prototypical models, it will also be
shown how how important properties of the models, like continuity, can be
expressed in terms of conditions imposed on the underlying constructions.
This facilitates the test whether a model of interest is sufficiently robust, its
comparison of models with respect to their specificity etc. In particular, this
analysis reveals that all practical F - or F -models belong to the F -type.
ψ Ω ξ
The eleventh chapter develops the algorithmic part of the theory, and is
thus concerned with the issue of efficient implementation. Obviously, it only
makes sense to consider practical (i.e. sufficiently robust) models. Thus, we
can confine ourselves to analyzing models of the F -type. The general strate-
ξ
gies for efficient implementation described in this chapter will be instantiated
for three prototypical models. The considered quantifiers include the familiar
absolute and proportional types, as well as quantifiers of exception and car-
dinal comparatives. Some application examples are also discussed at the end
of the chapter.
The twelvth chapter proposes an extension of the basic framework for
fuzzy quantification. The generalization will cover the most powerful notion
of quantifiers developed by mathematicians, so-called Lindstr¨om quantifiers
[106].Thesequantifiersarealsoofpotentiallinguisticrelevance,andIexplain
how they can be used to model certain reciprocal constructions in natural
language which give rise to branching quantification.
The thirteenth chapter will resume the main contributions of this work
and propose some directions for future research.
There are two appendices.
AppendixApresentsthestudyofexistingapproachescitedintheintroduc-
tion. It proposes an evaluation framework for approaches to fuzzy quantifica-
tion based on Zadeh’s traditional framework. This analysis makes it possible
toapplythesemanticalcriteriadevelopedinthemainpartofthebooktothe
existing approaches to fuzzy quantification described in the literature.