The FA Quantifier Fuzzification Mechanism: analysis of convergence and efficient implementations

TL;DR

This paper proves the convergence of the FA fuzzy quantification model to Zadeh's model for large input sets, and provides efficient algorithms for practical implementation of the model.

Contribution

It establishes a convergence theorem linking FA to Zadeh's model and offers exact and approximate algorithms for practical evaluation of FA.

Findings

Proves convergence of FA model to Zadeh's model as input size grows.

Provides exact algorithms for common linguistic quantifiers.

Introduces Monte Carlo approximation for FA evaluation.

Abstract

The fuzzy quantification model FA has been identified as one of the best behaved quantification models in several revisions of the field of fuzzy quantification. This model is, to our knowledge, the unique one fulfilling the strict Determiner Fuzzification Scheme axiomatic framework that does not induce the standard min and max operators. The main contribution of this paper is the proof of a convergence result that links this quantification model with the Zadeh's model when the size of the input sets tends to infinite. The convergence proof is, in any case, more general than the convergence to the Zadeh's model, being applicable to any quantitative quantifier. In addition, recent revisions papers have presented some doubts about the existence of suitable computational implementations to evaluate the FA model in practical applications. In order to prove that this model is not only a theoretical approach, we show exact algorithmic solutions for the most common linguistic quantifiers as well as an approximate implementation by means of Monte Carlo. Additionally, we will also give a general overview of the main properties fulfilled by the FA model, as a single compendium integrating the whole set of properties fulfilled by it has not been previously published.