Fuzzy implicators¶
A fuzzy implicator generalizes Boolean implication to truth degrees in \([0,1]\). In fuzzy-rough lower approximations, the first argument is normally a relation value and the second argument is a class-membership value. Standard implicators are non-increasing in the first argument, non-decreasing in the second, and satisfy the classical corner conditions \(I(0,0)=I(0,1)=I(1,1)=1\) and \(I(1,0)=0\).
Implemented implicators¶
| Name | Formula | Registered aliases |
|---|---|---|
| Łukasiewicz | \(\min(1,1-a+b)\) | lukasiewicz, luk |
| Gödel | \(1\) if \(a\le b\); otherwise \(b\) | goedel |
| Kleene–Dienes | \(\max(1-a,b)\) | kleenedienes, kleene, kd |
| Reichenbach | \(1-a+ab\) | reichenbach |
| Goguen | \(1\) if \(a\le b\); otherwise \(b/a\) | goguen, product |
| Rescher | \(1\) if \(a\le b\); otherwise \(0\) | rescher |
| Yager | \(b^a\), with \(I(0,0)=1\) | yager |
| Weber | \(b\) if \(a=1\); otherwise \(1\) | weber |
| Fodor | \(1\) if \(a\le b\); otherwise \(\max(1-a,b)\) | fodor |
The public implementations are vectorized and backend-aware; users do not need
numpy.vectorize.
Binary class-membership note¶
For a crisp decision class, the consequent \(b\) is either zero or one. Under that restriction:
- Łukasiewicz, Kleene–Dienes, and Reichenbach return the same values;
- Gödel and Goguen return the same values.
These equivalences do not hold for arbitrary fuzzy consequents. Other implicators can still produce different lower approximations even with crisp class labels.
References¶
- Baczyński, M., & Jayaram, B. (2008). Fuzzy Implications. Springer. https://doi.org/10.1007/978-3-540-69082-5
- Radzikowska, A. M., & Kerre, E. E. (2002). A comparative study of fuzzy rough sets. Fuzzy Sets and Systems, 126(2), 137–155. https://doi.org/10.1016/S0165-0114(01)00032-X