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.

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

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

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

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.)

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

${\displaystyle \lim _{p\to p_{0}}T_{p}=T_{p_{0}}}$

for all values p₀ of the parameter.

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

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

A Schweizer–Sklar t-norm ${\displaystyle T_{p}^{\mathrm {SS} }}$ 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 ${\displaystyle T_{p}^{\mathrm {SS} }}$ for −∞ < p < +∞ is

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

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

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

The t-norm ${\displaystyle T_{0}^{\mathrm {H} }}$ is called the Hamacher product.

Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm ${\displaystyle T_{p}^{\mathrm {H} }}$ 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 ${\displaystyle T_{p}^{\mathrm {H} }}$ for p < +∞ is

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

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:

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

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

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

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

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

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

${\displaystyle f_{p}^{\mathrm {Y} }(x)=(1-x)^{p}.}$

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

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

The Aczél–Alsina t-norm ${\displaystyle T_{p}^{\mathrm {AA} }}$ 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 ${\displaystyle T_{p}^{\mathrm {AA} }}$ for 0 < p < +∞ arises from the product t-norm by raising its additive generator to the power of p. An additive generator of ${\displaystyle T_{p}^{\mathrm {AA} }}$ for 0 < p < +∞ is

${\displaystyle f_{p}^{\mathrm {AA} }(x)=(-\log x)^{p}.}$

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

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

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

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

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

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

The Sugeno–Weber t-norm ${\displaystyle T_{p}^{\mathrm {SW} }}$ 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 ${\displaystyle T_{p}^{\mathrm {SW} }}$ for 0 < p < +∞ [sic] is

${\displaystyle f_{p}^{\mathrm {SW} }(x)={\begin{cases}1-x&{\text{if }}p=0\\1-\log _{1+p}(1+px)&{\text{otherwise.}}\end{cases}}}$

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

${\displaystyle T(x,y)={\begin{cases}a_{i}+(b_{i}-a_{i})\cdot T_{i}\left({\frac {x-a_{i}}{b_{i}-a_{i}}},{\frac {y-a_{i}}{b_{i}-a_{i}}}\right)&{\text{if }}x,y\in [a_{i},b_{i}]^{2}\\\min(x,y)&{\text{otherwise}}\end{cases}}}$

is a t-norm.

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

${\displaystyle T=\bigoplus \nolimits _{i\in I}(T_{i},a_{i},b_{i}),}$

or ${\displaystyle (T_{1},a_{1},b_{1})\oplus \dots \oplus (T_{n},a_{n},b_{n})}$ 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 ${\displaystyle T=\bigoplus \nolimits _{i\in I}(T_{i},a_{i},b_{i})}$ is a left-continuous t-norm, then its residuum R is given as follows:

${\displaystyle R(x,y)={\begin{cases}1&{\text{if }}x\leq y\\a_{i}+(b_{i}-a_{i})\cdot R_{i}\left({\frac {x-a_{i}}{b_{i}-a_{i}}},{\frac {y-a_{i}}{b_{i}-a_{i}}}\right)&{\text{if }}a_{i}<y<x\leq b_{i}\\y&{\text{otherwise.}}\end{cases}}}$

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 ${\displaystyle R_{T_{1}}(x,y)=\sup\{z\mid T_{1}(z,x)\leq y\}.}$ Then the function

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

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

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).

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

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