construction 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

::$f^\{(-1)\}(y)\; =\; \backslash begin\{cases\}$

\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}

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 in [f(0⁺), +∞] 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 $T(x,y)=f^\{-1\}\backslash left(\backslash min\backslash left(f(0^+),f(x)+f(y)\backslash right)\backslash right)$. The corresponding residuum can then be expressed as $(x\; \backslash Rightarrow\; y)\; =\; f^\{-1\}\backslash left(\backslash max\backslash left(0,f(y)-f(x)\backslash right)\backslash right)$. And the biresiduum as $(x\; \backslash Leftrightarrow\; y)\; =\; f^\{-1\}\backslash left(\backslash left|f(x)-f(y)\backslash right|\backslash right)$.

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.)

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.

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

::$\backslash lim\_\{p\backslash to\; p\_0\}\; T\_p\; =\; T\_\{p\_0\}$

:for all values p₀ of the parameter.

Image:Schweizer-Sklar-2-Tnorm-graph-contour.png

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

:$T^\{\backslash mathrm\{SS\}\}\_p(x,y)\; =\; \backslash begin\{cases\}$

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 $T^\{\backslash mathrm\{SS\}\}\_p$ 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 $T^\{\backslash mathrm\{SS\}\}\_p$ for −∞ < p < +∞ is

:$f^\{\backslash mathrm\{SS\}\}\_p\; (x)\; =\; \backslash begin\{cases\}$

-\log x & \text{if } p = 0 \\

\frac{1 - x^p}{p} & \text{otherwise.}

\end{cases}

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

:$T^\{\backslash mathrm\{H\}\}\_p\; (x,y)\; =\; \backslash begin\{cases\}$

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 $T^\{\backslash mathrm\{H\}\}\_0$ is called the Hamacher product.

Hamacher t-norms are the only t-norms which are rational functions.

The Hamacher t-norm $T^\{\backslash mathrm\{H\}\}\_p$ 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 $T^\{\backslash mathrm\{H\}\}\_p$ for p < +∞ is

:$f^\{\backslash mathrm\{H\}\}\_p(x)\; =\; \backslash begin\{cases\}$

\frac{1 - x}{x} & \text{if } p = 0 \\

\log\frac{p + (1 - p)x}{x} & \text{otherwise.}

\end{cases}

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:

:$T^\{\backslash mathrm\{F\}\}\_p(x,y)\; =\; \backslash begin\{cases\}$

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

:$f^\{\backslash mathrm\{F\}\}\_p(x)\; =\; \backslash begin\{cases\}$

-\log x & \text{if } p = 1 \\

1 - x & \text{if } p = +\infty \\

\log\frac{p - 1}{p^x - 1} & \text{otherwise.}

\end{cases}

Image:Yager-2-Tnorm-graph-contours.png

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

:$T^\{\backslash mathrm\{Y\}\}\_p\; (x,y)\; =\; \backslash begin\{cases\}$

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 $T^\{\backslash mathrm\{Y\}\}\_p$ 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 $T^\{\backslash mathrm\{Y\}\}\_p$ for 0 < p < +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of p. An additive generator of $T^\{\backslash mathrm\{Y\}\}\_p$ for 0 < p < +∞ is

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

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

:$T^\{\backslash mathrm\{AA\}\}\_p\; (x,y)\; =\; \backslash begin\{cases\}$

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 $T^\{\backslash mathrm\{AA\}\}\_p$ 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 $T^\{\backslash mathrm\{AA\}\}\_p$ for 0 < p < +∞ arises from the product t-norm by raising its additive generator to the power of p. An additive generator of $T^\{\backslash mathrm\{AA\}\}\_p$ for 0 < p < +∞ is

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

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

:$T^\{\backslash mathrm\{D\}\}\_p\; (x,y)\; =\; \backslash begin\{cases\}$

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 $T^\{\backslash mathrm\{D\}\}\_p$ 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 $T^\{\backslash mathrm\{D\}\}\_p$ for 0 < p < +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of p. An additive generator of $T^\{\backslash mathrm\{D\}\}\_p$ for 0 < p < +∞ is

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

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

:$T^\{\backslash mathrm\{SW\}\}\_p\; (x,y)\; =\; \backslash begin\{cases\}$

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 $T^\{\backslash mathrm\{SW\}\}\_p$ 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 $T^\{\backslash mathrm\{SW\}\}\_p$ for 0 < p < +∞ [sic] is

:$f^\{\backslash mathrm\{SW\}\}\_p(x)\; =\; \backslash begin\{cases\}$

1 - x & \text{if } p = 0 \\

1 - \log_{1 + p}(1 + px) & \text{otherwise.}

\end{cases}

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

::$T(x,\; y)\; =\; \backslash begin\{cases\}$

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.

Image:OrdSum-Luk-prod-graph-contours.png]

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

:$T\; =\; \backslash bigoplus\backslash nolimits\_\{i\backslash in\; I\}\; (T\_i,\; a\_i,\; b\_i),$

or $(T\_1,\; a\_1,\; b\_1)\; \backslash oplus\; \backslash dots\; \backslash oplus\; (T\_n,\; a\_n,\; b\_n)$ 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 $T\; =\; \backslash bigoplus\backslash nolimits\_\{i\backslash in\; I\}\; (T\_i,\; a\_i,\; b\_i)$ is a left-continuous t-norm, then its residuum R is given as follows:

:$R(x,\; y)\; =\; \backslash begin\{cases\}$

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.

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..

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 $R\_\{T\_1\}(x,y)\; =\; \backslash sup\backslash \{z\; \backslash mid\; T\_1(z,x)\backslash le\; y\backslash \}.$ Then the function

::$T\_\{\backslash mathrm\{rot\}\}\; =\; \backslash begin\{cases\}$

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.

Image:NilpotentMinimum-as-rotation.png 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).

Image:Rotation-Luk-prod-nM-drast-Tnorm-graphs.png, 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.

• T-norm
• T-norm fuzzy logics

• Klement, Erich Peter; Mesiar, Radko; and Pap, Endre (2000), Triangular Norms. Dordrecht: Kluwer. {{ISBN|0-7923-6416-3}}.
• Fodor, János (2004), [http://www.uni-obuda.hu/journal/Fodor_2.pdf "Left-continuous t-norms in fuzzy logic: An overview"]. Acta Polytechnica Hungarica 1(2), ISSN 1785-8860 [http://www.uni-obuda.hu/journal/]
• Dombi, József (1982), [https://web.archive.org/web/20120402164838/http://www.dopti.com/~dombi/publications/1982-J.Dombi---A_general_class_of_fuzzy_operators.pdf "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.
• Navara, Mirko (2007), [http://www.scholarpedia.org/article/Triangular_Norms_and_Conorms "Triangular norms and conorms"], Scholarpedia [http://www.scholarpedia.org/].

Category:Fuzzy logic