Natural language heavily depends on quantifying constructions.
These often involve fuzzy concepts like "tall",
and they frequently refer to fuzzy quantities in agreement
like
"about ten", "almost all", "many" etc.
In order to exploit this expressive power and make fuzzy
quantification available to technical applications,
fuzzy set theory has been
enriched with various techniques which reduce fuzzy quantification
to a comparison of scalar or fuzzy cardinalities
(Zadeh 1983, Ralescu 1986, Yager 1988/1993).
However, it soon became clear that the results of these methods
fail to be plausible in some cases
(Arnould/Ralescu 1993,
Barro et.al 1999,
Delgado et.al 2000,
Dubois/Prade 1985,
Glöckner 1999,
Ralescu 1986/1995,
Yager 1993).
This certainly
hindered the spread of these `traditional' approaches into commercial
applications.
Only recently a new solution
has been pursued
which showed itself immune against
the pitfalls of existing approaches to fuzzy quantification.
The goal of this report is to compile the material
on this new theory known as the DFS theory of fuzzy
quantification,
which until now was
scattered
across
several publications
(Glöckner 1997-2001).
It will
stipulate
a canonical terminology and notation,
thus introducing
the theory as a consistent whole, including
latest
developments.
The present report covers the
fundamental elements of fuzzy quantification
which comprise
the formal framework for describing fuzzy quantifiers;
the axioms imposed on plausible models;
my results on derived properties of these models;
a catalogue of additional adequacy requirements;
a discussion of
several constructive principles
along with an analysis of the generated models;
an extensive list of
algorithms for implementing fuzzy quantifiers
in the models; and finally
some
illustrative examples.
In particular,
the
missing
elements have now been added,
which were still
needed
to complete
the proposed solution and
make it
really
useful in practice.
Until recently, there was a clear focus on theoretical
topics and beyond some example algorithms
sketched in
(Glöckner/Knoll 2000, Glöckner et.al 1998),
there was
no
systematical discussion of implementation issues.
In order to close this gap,
the report now
presents an
in-depth treatise of the computational aspects of DFS theory.
The report discloses
a general strategy for implementing
quantifiers
in a model of interest,
and it further develops a number of supplementary
techniques
which
will
optimize
processing
times.
In particular, I describe an analysis of fuzzy
quantifiers in terms of cardinality coefficients, which can be computed
from histogram information.
In order to illustrate the implementation strategy,
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.
The quantifiers covered by this method not only include the familiar
absolute and proportional quantifiers
known from
existing
work on fuzzy quantification.
They also include some further important types
like quantifiers of exception ("all except k") and
cardinal comparatives ("many more than")
which are
well-known to linguists,
but innovative in the fuzzy sets framework.
I. Glöckner
Fundamentals of Fuzzy Quantification: Plausible Models,
Constructive Principles, and Efficient Implementation
Technical Report TR2002-07,
Technische Fakultät, Universität Bielefeld,
PO-Box 100131, 33501 Bielefeld, Germany,
March 2003, 571 pages.
Portable Document Format PDF, 571 pages, 2.6MB