The FA Quantifier Fuzzification Mechanism: analysis of convergence and efficient implementations
This work addresses the need for practical implementations of the FA model in fuzzy quantification, making it more accessible for applications, though it is incremental in building on existing theoretical foundations.
The paper tackles the theoretical and practical challenges of the FA fuzzy quantification model by proving its convergence to Zadeh's model as input set sizes approach infinity and providing efficient computational implementations, including exact algorithms for common linguistic quantifiers and a Monte Carlo approximation.
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.