Fuzzy Quantifiers in Fuzzy Logic¶
This document provides a short overview of the fuzzy quantifiers implemented in frsutils.core.fuzzy_quantifiers.
1. Fuzzy Quantifier Overview¶
The implemented quantifiers map values from [0, 1] to [0, 1]. Both use parameters alpha and beta with:
0 <= alpha < beta <= 1
They are non-decreasing and satisfy Q(0) = 0 and Q(1) = 1.
2. Fuzzy Quantifier Table¶
| Name | Formula | Parameters | Aliases | Reference |
|---|---|---|---|---|
| Linear | Piecewise-linear quantifier shown below | alpha, beta |
linear | [1], Eq. (7) |
| Quadratic | Quadratic S-shaped quantifier shown below | alpha, beta |
quadratic, quad | [2], Eq. (12) |
The linear quantifier is:
\[
Q_{\alpha,\beta}(x) =
\begin{cases}
0, & x \leq \alpha \\
\dfrac{x-\alpha}{\beta-\alpha}, & \alpha < x < \beta \\
1, & x \geq \beta
\end{cases}
\]
The quadratic quantifier is:
\[
Q_{\alpha,\beta}(x) =
\begin{cases}
0, & x \leq \alpha \\
\dfrac{2(x-\alpha)^2}{(\beta-\alpha)^2}, & \alpha < x \leq \dfrac{\alpha+\beta}{2} \\
1 - \dfrac{2(x-\beta)^2}{(\beta-\alpha)^2}, & \dfrac{\alpha+\beta}{2} < x \leq \beta \\
1, & x > \beta
\end{cases}
\]
3. Notes¶
- Both implementations support NumPy arrays and backend-aware computation.
- The quantifiers are used by VQRS to transform interim approximation ratios.
- Input validation requires finite values in
[0, 1]unless disabled explicitly.
4. References¶
- F. Nasirzadeh, M. Khanzadi, and H. Mianabadi (2013) — A Fuzzy Group Decision Making Approach to Construction Project Risk Management. International Journal of Industrial Engineering & Production Research, 24(1), 71–80. The piecewise-linear fuzzy linguistic quantifier is given in Eq. (7). Free PDF
- R. Jensen and C. Cornelis (2011) — Fuzzy-Rough Nearest Neighbour Classification and Prediction. Theoretical Computer Science, 412(42), 5871–5884. The quadratic fuzzy quantifier used in VQRS is given in Eq. (12). Free PDF