T-norm

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In mathematics, a T-norm (or t-norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and AND in logic. The name derives from triangular norm, which refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize triangle inequality of ordinary metric spaces.

Contents

1 Definition
2 Applications
3 Notions about t-norms
4 Residuum
5 T-conorms
6 Examples
7 References

Definition

A T-norm is a function T : [0,1] × [0,1] → [0,1] with the following properties:

Commutativity: T(a, b) = T(b, a);
Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d;
Associativity: T(a, T(b, c)) = T(T(a, b), c);
Null element: T(a, 0) = 0;
1 acts as Identity element: T(a, 1) = a.

Applications

T-norms are a generalization of the usual logical conjunction operator for fuzzy logics. Indeed, the usual conjunction is obviously associative and commutative. The monotonicity property guarantees a certain regularity in the structure of the set [0,1] of truth values. Finally, the last two properties state that truth values 0 and 1 correspond to false and true, respectively.

T-norms are also used to construct the union of fuzzy sets, see fuzzy set operations.

Notions about t-norms

A t-norm is called Archimedean if 0 and 1 are its only idempotents. An Archimedean t-norm is called strict if 0 is its only nilpotent element.

Residuum

For any continuous t-norm, there is a unique operation [x \Rightarrow y] such that for all [x, y, z \in [0,1]], we have [ T(x,z) \le y \iff z \le (x \Rightarrow y)]. This operation is called the residuum, and [(x \Rightarrow y) = \max\ ].

see Residuated lattice

T-conorms

T-conorms (also called S-norms) are in a certain sense dual to T-norms. Given a T-norm, the complementary conorm is defined by

[ \bot(a,b) = 1-T(1-a, 1-b). ]

This generalizes De Morgan's laws.

It follows that a T-conorm satisfies the following relations:

Commutativity: ⊥(a, b) = ⊥(b, a);
Monotonicity: ⊥(a, b) ≤ ⊥(c, d) if a ≤ c and b ≤ d;
Associativity: ⊥(a, ⊥(b, c)) = ⊥(⊥(a, b), c);
Null element: ⊥(a, 1) = 1;
Identity element: ⊥(a, 0) = a.

The T-conorm is used to represent logical disjunction in fuzzy logic and intersection in fuzzy set theory.

Examples

The following T-norms and T-conorms are often used:

[\begin\mathrm}(a, b) &=& \min \ &\mathrm}(a, b) &=& \max \ \\ \\\mathrm}(a, b) &=& \max \ &\mathrm}(a, b) &=& \min \ \\ \\\mathrm}(a, b) &=& a \cdot b &\mathrm}(a, b) &=& a+b- a \cdot b \\ \\\mathrm}(a, b) &=& \left\a, & \mboxb=1 \\ b, & \mboxa=1 \\ 0, & \mbox\end \right. &\mathrm}(a, b) &=& \left\a, & \mboxb=0 \\ b, & \mboxa=0 \\ 1, & \mbox\end\right.\end]

The first T-norm and T-conorm are used most often, as they are simple and have some special properties (see below). The third T-norm and the corresponding T-conorm derive from probability theory.

Furthermore, the following relationships hold for any T-norm:

[\begin\mathrm}(a, b) & \le & \top(a, b) & \le & \mathrm}(a, b) \\\mathrm}(a, b) & \le & \bot(a, b) & \le & \mathrm}(a, b).\end]

In other words, every T-norm lies between the drastic T-norm (T_-1) and the minimum T-norm (T_min). Conversely, every T-conorm lies between maximum T-conorm and the drastic T-conorm.

References

Much of the content of this article comes from the [equivalent German-language wikipedia article] (retrieved 24 June, 2005).
Erich Peter Klement, Radko Mesiar and Endre Pap, Triangular Norms. Kluwer, Dordrecht, 2000. ISBN 0-7923-6416-3.
Petr Hájek, Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998.
Roberto L.O. Cignoli, Itala M.L. D'Ottaviano, Daniele Mundici, Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht, 2000.

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