A T-norm , often also a small t-norm , is a mathematical function that has become important in the field of multi-valued logics , especially in fuzzy logic . The term is derived from the English triangular norm , in German Dreiecksnorm , and comes from the fact that a T-norm describes a triangular-like surface in .
A T-norm is defined on the unit interval [0,1]
and must have the following properties (for an exact definition of these properties, see the table on T-Norm and T-Conorm at the end of this article):
- Associativity : T ( a , T ( b , c )) = T (T ( a , b ), c )
- Commutativity : T ( a , b ) = T ( b , a )
- Monotony : T ( a , b ) ≤ T ( c , d ), if a ≤ c and b ≤ d
- 1 is a neutral element : T ( a , 1) = a
The T standard is used for valued logics a generalized Konjunktions - operator to provide. The properties mentioned above are, as it were, the most general properties of such an operator: Associativity and commutativity are self-evident. The monotony guarantees a certain regularity in the structure of the definition and target set. The "1" as a neutral element enables conjunctions, the result of which only depends on one operand.
Complementary to T-norms, T-Conormen (also called S-Norms ) are used, as an identifier is usually ⊥ or S:
Generalization: It may be a different than the standard negator
be used. This generalizes the above relationship to
The minimum requirements for an negator are generally: monotony (falling), n (0) = 1, n (1) = 0.
In this context, however, strict monotony and involutivity n (n ( x )) = x, ie n = n −1 , are required:
The triple is then called the De Morgan triplet.
Common T-Norms and T-Conormen
The specified T-norms are each related to the standard negation N (x) = 1-x to the corresponding T-norm dual, i.e. linked via De Morgan's laws. With other involutive negations, other T-conorms generally also result.
The former is the most widely used because of its simplicity and its properties mentioned below. The 3rd T-Norm and its T-Conorm come from the calculation of probability . The following relationships also apply:
This means that the drastic T-norm (T -1 ) is the smallest and the minimum T-norm the largest. The opposite applies to the T-Conorm. T (a, b) or ⊥ (a, b) stands for any T-Norm or T-Conorm.
Relationship between T-Norm and T-Conorm
Due to the De Morgan laws already mentioned, the following complementary relationships arise:
- 1-⊥ (a, b) = T (1-a, 1-b) and 1-T (a, b) = ⊥ (1-a, 1-b)
The above axioms for T-norms correspond to the following conditions for a T-conorm:
|Zero element:||T (0, a) = T (a, 0) = 0||⊥ (a, 1) = ⊥ (1, a) = 1|
|Neutral element:||T (a, 1) = T (1, a) = a||⊥ (0, a) = ⊥ (a, 0) = a|
|Associativity:||T (a, T (b, c)) = T (T (a, b), c)||⊥ (a, ⊥ (b, c)) = ⊥ (⊥ (a, b), c)|
|Commutativity:||T (a, b) = T (b, a)||⊥ (a, b) = ⊥ (b, a)|
|Monotony:||a ≤ b ⇒ T (a, c) ≤ T (b, c)||a ≤ b ⇒ ⊥ (a, c) ≤ ⊥ (b, c)|
These relationships apply not only to the standard negator, but to any De Morgan triplet.
Relationship between T-norm and copula
A T-norm has the positive rectangle property if :
Every T-norm with a positive rectangular property is a bivariate copula (see Grabisch et al. 2009). Copulae of the above examples are at the same time, but not.
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