Construction of t-norms

In mathematics, t-norms are a special kind of binary operations on the real unit interval [0, 1]. Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms. This is important, e.g., for finding counter-examples or supplying t-norms with particular properties for use in engineering applications of fuzzy logic. The main ways of construction of t-norms include using generators, defining parametric classes of t-norms, rotations, or ordinal sums of t-norms.

Relevant background can be found in the article on t-norms.

Generators of t-norms

The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (most often, addition or multiplication) into a t-norm.

In order to allow using non-bijective generators, which do not have the inverse function, the following notion of pseudo-inverse function is employed:

Let f: [a, b] → [c, d] be a monotone function between two closed subintervals of extended real line. The pseudo-inverse function to f is the function f ⁽⁻¹⁾: [c, d] → [a, b] defined as

$f^{(-1)}(y) = \begin{cases} \sup \{ x\in[a,b] \mid f(x) < y \} & \text{for } f \text{ non-decreasing} \\ \sup \{ x\in[a,b] \mid f(x) > y \} & \text{for } f \text{ non-increasing.} \end{cases}$

Additive generators

The construction of t-norms by additive generators is based on the following theorem:

Let f: [0, 1] → [0, +∞] be a strictly decreasing function such that f(1) = 0 and f(x) + f(y) is in the range of f or equal to f(0⁺) or +∞ for all x, y in [0, 1]. Then the function T: [0, 1]² → [0, 1] defined as

T(x, y) = f ^(-1)(f(x) + f(y))

is a t-norm.

Alternatively, one may avoid using the notion of pseudo-inverse function by having $T(x,y)=f^{-1}\left(\min\left(f(0^+),f(x)+f(y)\right)\right)$. The corresponding residuum can then be expressed as $(x \Rightarrow y) = f^{-1}\left(\max\left(0,f(y)-f(x)\right)\right)$. And the biresiduum as $(x \Leftrightarrow y) = f^{-1}\left(\left|f(x)-f(y)\right|\right)$.

If a t-norm T results from the latter construction by a function f which is right-continuous in 0, then f is called an additive generator of T.

Examples:

• The function f(x) = 1 – x for x in [0, 1] is an additive generator of the Łukasiewicz t-norm.
• The function f defined as f(x) = –log(x) if 0 < x ≤ 1 and f(0) = +∞ is an additive generator of the product t-norm.
• The function f defined as f(x) = 2 – x if 0 ≤ x < 1 and f(1) = 0 is an additive generator of the drastic t-norm.

Basic properties of additive generators are summarized by the following theorem:

Let f: [0, 1] → [0, +∞] be an additive generator of a t-norm T. Then:

• T is an Archimedean t-norm.
• T is continuous if and only if f is continuous.
• T is strictly monotone if and only if f(0) = +∞.
• Each element of (0, 1) is a nilpotent element of T if and only if f(0) < +∞.
• The multiple of f by a positive constant is also an additive generator of T.
• T has no non-trivial idempotents. (Consequently, e.g., the minimum t-norm has no additive generator.)

Multiplicative generators

The isomorphism between addition on [0, +∞] and multiplication on [0, 1] by the logarithm and the exponential function allow two-way transformations between additive and multiplicative generators of a t-norm. If f is an additive generator of a t-norm T, then the function h: [0, 1] → [0, 1] defined as h(x) = e^−f (x) is a multiplicative generator of T, that is, a function h such that

• h is strictly increasing
• h(1) = 1
• h(x) · h(y) is in the range of h or equal to 0 or h(0+) for all x, y in [0, 1]
• h is right-continuous in 0
• T(x, y) = h ⁽⁻¹⁾(h(x) · h(y)).

Vice versa, if h is a multiplicative generator of T, then f: [0, 1] → [0, +∞] defined by f(x) = −log(h(x)) is an additive generator of T.

Parametric classes of t-norms

Many families of related t-norms can be defined by an explicit formula depending on a parameter p. This section lists the best known parameterized families of t-norms. The following definitions will be used in the list:

• A family of t-norms T_p parameterized by p is increasing if T_p(x, y) ≤ T_q(x, y) for all x, y in [0, 1] whenever p ≤ q (similarly for decreasing and strictly increasing or decreasing).
• A family of t-norms T_p is continuous with respect to the parameter p if

$\lim_{p\to p_0} T_p = T_{p_0}$

for all values p₀ of the parameter.

Schweizer–Sklar t-norms

Graph (3D and contours) of the Schweizer–Sklar t-norm with p = 2

The family of Schweizer–Sklar t-norms, introduced by Berthold Schweizer and Abe Sklar in the early 1960s, is given by the parametric definition

$T^{\mathrm{SS}}_p(x,y) = \begin{cases} T_{\min}(x,y) & \text{if } p = -\infty \\ (x^p + y^p - 1)^{1/p} & \text{if } -\infty < p < 0 \\ T_{\mathrm{prod}}(x,y) & \text{if } p = 0 \\ (\max(0, x^p + y^p - 1))^{1/p} & \text{if } 0 < p < +\infty \\ T_{\mathrm{D}}(x,y) & \text{if } p = +\infty. \end{cases}$

A Schweizer–Sklar t-norm $T^{\mathrm{SS}}_p$ is

• Archimedean if and only if p > −∞
• Continuous if and only if p < +∞
• Strict if and only if −∞ < p ≤ 0 (for p = −1 it is the Hamacher product)
• Nilpotent if and only if 0 < p < +∞ (for p = 1 it is the Łukasiewicz t-norm).

The family is strictly decreasing for p ≥ 0 and continuous with respect to p in [−∞, +∞]. An additive generator for $T^{\mathrm{SS}}_p$ for −∞ < p < +∞ is

$f^{\mathrm{SS}}_p (x) = \begin{cases} -\log x & \text{if } p = 0 \\ \frac{1 - x^p}{p} & \text{otherwise.} \end{cases}$

Hamacher t-norms

The family of Hamacher t-norms, introduced by Horst Hamacher in the late 1970s, is given by the following parametric definition for 0 ≤ p ≤ +∞:

$T^{\mathrm{H}}_p (x,y) = \begin{cases} T_{\mathrm{D}}(x,y) & \text{if } p = +\infty \\ 0 & \text{if } p = x = y = 0 \\ \frac{xy}{p + (1 - p)(x + y - xy)} & \text{otherwise.} \end{cases}$

The t-norm $T^{\mathrm{H}}_0$ is called the Hamacher product.

Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm $T^{\mathrm{H}}_p$ is strict if and only if p < +∞ (for p = 1 it is the product t-norm). The family is strictly decreasing and continuous with respect to p. An additive generator of $T^{\mathrm{H}}_p$ for p < +∞ is

$f^{\mathrm{H}}_p(x) = \begin{cases} \frac{1 - x}{x} & \text{if } p = 0 \\ \log\frac{p + (1 - p)x}{x} & \text{otherwise.} \end{cases}$

Frank t-norms

The family of Frank t-norms, introduced by M.J. Frank in the late 1970s, is given by the parametric definition for 0 ≤ p ≤ +∞ as follows:

$T^{\mathrm{F}}_p(x,y) = \begin{cases} T_{\mathrm{min}}(x,y) & \text{if } p = 0 \\ T_{\mathrm{prod}}(x,y) & \text{if } p = 1 \\ T_{\mathrm{Luk}}(x,y) & \text{if } p = +\infty \\ \log_p\left(1 + \frac{(p^x - 1)(p^y - 1)}{p - 1}\right) & \text{otherwise.} \end{cases}$

The Frank t-norm $T^{\mathrm{F}}_p$ is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for $T^{\mathrm{F}}_p$ is

$f^{\mathrm{F}}_p(x) = \begin{cases} -\log x & \text{if } p = 1 \\ 1 - x & \text{if } p = +\infty \\ \log\frac{p - 1}{p^x - 1} & \text{otherwise.} \end{cases}$

Yager t-norms

Graph of the Yager t-norm with p = 2

The family of Yager t-norms, introduced in the early 1980s by Ronald R. Yager, is given for 0 ≤ p ≤ +∞ by

$T^{\mathrm{Y}}_p (x,y) = \begin{cases} T_{\mathrm{D}}(x,y) & \text{if } p = 0 \\ \max\left(0, 1 - ((1 - x)^p + (1 - y)^p)^{1/p}\right) & \text{if } 0 < p < +\infty \\ T_{\mathrm{min}}(x,y) & \text{if } p = +\infty \end{cases}$

The Yager t-norm $T^{\mathrm{Y}}_p$ is nilpotent if and only if 0 < p < +∞ (for p = 1 it is the Łukasiewicz t-norm). The family is strictly increasing and continuous with respect to p. The Yager t-norm $T^{\mathrm{Y}}_p$ for 0 < p < +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of p. An additive generator of $T^{\mathrm{Y}}_p$ for 0 < p < +∞ is

$f^{\mathrm{Y}}_p(x) = (1 - x)^p.$

Aczél–Alsina t-norms

The family of Aczél–Alsina t-norms, introduced in the early 1980s by János Aczél and Claudi Alsina, is given for 0 ≤ p ≤ +∞ by

$T^{\mathrm{AA}}_p (x,y) = \begin{cases} T_{\mathrm{D}}(x,y) & \text{if } p = 0 \\ e^{-\left(|\log x|^p + |\log y|^p\right)^{1/p}} & \text{if } 0 < p < +\infty \\ T_{\mathrm{min}}(x,y) & \text{if } p = +\infty \end{cases}$

The Aczél–Alsina t-norm $T^{\mathrm{AA}}_p$ is strict if and only if 0 < p < +∞ (for p = 1 it is the product t-norm). The family is strictly increasing and continuous with respect to p. The Aczél–Alsina t-norm $T^{\mathrm{AA}}_p$ for 0 < p < +∞ arises from the product t-norm by raising its additive generator to the power of p. An additive generator of $T^{\mathrm{AA}}_p$ for 0 < p < +∞ is

$f^{\mathrm{AA}}_p(x) = (-\log x)^p.$

Dombi t-norms

The family of Dombi t-norms, introduced by József Dombi (1982), is given for 0 ≤ p ≤ +∞ by

$T^{\mathrm{D}}_p (x,y) = \begin{cases} 0 & \text{if } x = 0 \text{ or } y = 0 \\ T_{\mathrm{D}}(x,y) & \text{if } p = 0 \\ T_{\mathrm{min}}(x,y) & \text{if } p = +\infty \\ \frac{1}{1 + \left( \left(\frac{1 - x}{x}\right)^p + \left(\frac{1 - y}{y}\right)^p \right)^{1/p}} & \text{otherwise.} \\ \end{cases}$

The Dombi t-norm $T^{\mathrm{D}}_p$ is strict if and only if 0 < p < +∞ (for p = 1 it is the Hamacher product). The family is strictly increasing and continuous with respect to p. The Dombi t-norm $T^{\mathrm{D}}_p$ for 0 < p < +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of p. An additive generator of $T^{\mathrm{D}}_p$ for 0 < p < +∞ is

$f^{\mathrm{D}}_p(x) = \left(\frac{1-x}{x}\right)^p.$

Sugeno–Weber t-norms

The family of Sugeno–Weber t-norms was introduced in the early 1980s by Siegfried Weber; the dual t-conorms were defined already in the early 1970s by Michio Sugeno. It is given for −1 ≤ p ≤ +∞ by

$T^{\mathrm{SW}}_p (x,y) = \begin{cases} T_{\mathrm{D}}(x,y) & \text{if } p = -1 \\ \max\left(0, \frac{x + y - 1 + pxy}{1 + p}\right) & \text{if } -1 < p < +\infty \\ T_{\mathrm{prod}}(x,y) & \text{if } p = +\infty \end{cases}$

The Sugeno–Weber t-norm $T^{\mathrm{SW}}_p$ is nilpotent if and only if −1 < p < +∞ (for p = 0 it is the Łukasiewicz t-norm). The family is strictly increasing and continuous with respect to p. An additive generator of $T^{\mathrm{SW}}_p$ for 0 < p < +∞ [sic] is

$f^{\mathrm{SW}}_p(x) = \begin{cases} 1 - x & \text{if } p = 0 \\ 1 - \log_{1 + p}(1 + px) & \text{otherwise.} \end{cases}$

Ordinal sums

The ordinal sum constructs a t-norm from a family of t-norms, by shrinking them into disjoint subintervals of the interval [0, 1] and completing the t-norm by using the minimum on the rest of the unit square. It is based on the following theorem:

Let T_i for i in an index set I be a family of t-norms and (a_i, b_i) a family of pairwise disjoint (non-empty) open subintervals of [0, 1]. Then the function T: [0, 1]² → [0, 1] defined as

$T(x, y) = \begin{cases} a_i + (b_i - a_i) \cdot T_i\left(\frac{x - a_i}{b_i - a_i}, \frac{y - a_i}{b_i - a_i}\right) & \text{if } x, y \in [a_i, b_i]^2 \\ \min(x, y) & \text{otherwise} \end{cases}$

is a t-norm.

Ordinal sum of the Łukasiewicz t-norm on the interval [0.05, 0.45] and the product t-norm on the interval [0.55, 0.95]

The resulting t-norm is called the ordinal sum of the summands (T_i, a_i, b_i) for i in I, denoted by

$T = \bigoplus\nolimits_{i\in I} (T_i, a_i, b_i),$

or $(T_1, a_1, b_1) \oplus \dots \oplus (T_n, a_n, b_n)$ if I is finite.

Ordinal sums of t-norms enjoy the following properties:

• Each t-norm is a trivial ordinal sum of itself on the whole interval [0, 1].
• The empty ordinal sum (for the empty index set) yields the minimum t-norm T_min. Summands with the minimum t-norm can arbitrarily be added or omitted without changing the resulting t-norm.
• It can be assumed without loss of generality that the index set is countable, since the real line can only contain at most countably many disjoint subintervals.
• An ordinal sum of t-norm is continuous if and only if each summand is a continuous t-norm. (Analogously for left-continuity.)
• An ordinal sum is Archimedean if and only if it is a trivial sum of one Archimedean t-norm on the whole unit interval.
• An ordinal sum has zero divisors if and only if for some index i, a_i = 0 and T_i has zero divisors. (Analogously for nilpotent elements.)

If $T = \bigoplus\nolimits_{i\in I} (T_i, a_i, b_i)$ is a left-continuous t-norm, then its residuum R is given as follows:

$R(x, y) = \begin{cases} 1 & \text{if } x \le y \\ a_i + (b_i - a_i) \cdot R_i\left(\frac{x - a_i}{b_i - a_i}, \frac{y - a_i}{b_i - a_i}\right) & \text{if } a_i < y < x \le b_i \\ y & \text{otherwise.} \end{cases}$

where R_i is the residuum of T_i, for each i in I.

Ordinal sums of continuous t-norms

The ordinal sum of a family of continuous t-norms is a continuous t-norm. By the Mostert–Shields theorem, every continuous t-norm is expressible as the ordinal sum of Archimedean continuous t-norms. Since the latter are either nilpotent (and then isomorphic to the Łukasiewicz t-norm) or strict (then isomorphic to the product t-norm), each continuous t-norm is isomorphic to the ordinal sum of Łukasiewicz and product t-norms.

Important examples of ordinal sums of continuous t-norms are the following ones:

• Dubois–Prade t-norms, introduced by Didier Dubois and Henri Prade in the early 1980s, are the ordinal sums of the product t-norm on [0, p] for a parameter p in [0, 1] and the (default) minimum t-norm on the rest of the unit interval. The family of Dubois–Prade t-norms is decreasing and continuous with respect to p..
• Mayor–Torrens t-norms, introduced by Gaspar Mayor and Joan Torrens in the early 1990s, are the ordinal sums of the Łukasiewicz t-norm on [0, p] for a parameter p in [0, 1] and the (default) minimum t-norm on the rest of the unit interval. The family of Mayor–Torrens t-norms is decreasing and continuous with respect to p..

Rotations

The construction of t-norms by rotation was introduced by Sándor Jenei (2000). It is based on the following theorem:

Let T be a left-continuous t-norm without zero divisors, N: [0, 1] → [0, 1] the function that assigns 1 − x to x and t = 0.5. Let T₁ be the linear transformation of T into [t, 1] and $R_{T_1}(x,y) = \sup\{z \mid T_1(z,x)\le y\}.$ Then the function

$T_{\mathrm{rot}} = \begin{cases} T_1(x, y) & \text{if } x, y \in (t, 1] \\ N(R_{T_1}(x, N(y))) & \text{if } x \in (t, 1] \text{ and } y \in [0, t] \\ N(R_{T_1}(y, N(x))) & \text{if } x \in [0, t] \text{ and } y \in (t, 1] \\ 0 & \text{if } x, y \in [0, t] \end{cases}$

is a left-continuous t-norm, called the rotation of the t-norm T.

The nilpotent minimum as a rotation of the minimum t-norm

Geometrically, the construction can be described as first shrinking the t-norm T to the interval [0.5, 1] and then rotating it by the angle 2π/3 in both directions around the line connecting the points (0, 0, 1) and (1, 1, 0).

Rotations of the Łukasiewicz, product, nilpotent minimum, and drastic t-norm

The theorem can be generalized by taking for N any strong negation, that is, an involutive strictly decreasing continuous function on [0, 1], and for t taking the unique fixed point of N.

The resulting t-norm enjoys the following rotation invariance property with respect to N:

T(x, y) ≤ z if and only if T(y, N(z)) ≤ N(x) for all x, y, z in [0, 1].

The negation induced by T_rot is the function N, that is, N(x) = R_rot(x, 0) for all x, where R_rot is the residuum of T_rot.

See also

• T-norm
• T-norm fuzzy logics

References

• Klement, Erich Peter; Mesiar, Radko; and Pap, Endre (2000), Triangular Norms. Dordrecht: Kluwer. ISBN 0-7923-6416-3.
• Fodor, János (2004), "Left-continuous t-norms in fuzzy logic: An overview". Acta Polytechnica Hungarica 1(2), ISSN 1785-8860
• Dombi, József (1982), "A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators". Fuzzy Sets and Systems 8, 149–163.
• Jenei, Sándor (2000), "Structure of left-continuous t-norms with strong induced negations. (I) Rotation construction". Journal of Applied Non-Classical Logics 10, 83–92.
• Mirko Navara (2007), "Triangular norms and conorms", Scholarpedia .