Natural language heavily depends on quantifying constructions.
These often involve fuzzy concepts like "tall",
and they frequently refer to fuzzy quantities in agreement
"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
Barro et.al 1999,
Delgado et.al 2000,
hindered the spread of these `traditional' approaches into commercial
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.
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