Construction of t-norms explained

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 \} & for f non-decreasing \\ \sup \{ x\in[a,b] \mid f(x) > y \} & for f non-increasing. \end{cases}

Additive generators

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

Let f: [0,&nbsp;1] → [0,&nbsp;+&infin;] 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,&nbsp;1]. Then the function T: [0,&nbsp;1]² → [0,&nbsp;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 ⇒ 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,&nbsp;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,&nbsp;1] → [0,&nbsp;+&infin;] 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,&nbsp;+∞] and multiplication on [0,&nbsp;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,&nbsp;1] → [0,&nbsp;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,&nbsp;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,&nbsp;1] → [0,&nbsp;+∞] 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,&nbsp;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

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 ^{SS} _{p(x,y)} = \begin{cases} T _{min} (x,y) & if p = -inf ty \\ (x ^{p} + y ^{p} - 1 ) ^{1/p} & if -inf ty < p < 0 \\ T _{prod} (x,y) & if p = 0 \\ ( max ( 0 , x ^{p} + y ^{p} - 1 )) ^{1/p} & if 0 < p < +inf ty \\ T _{D} (x,y) & if p = +inf ty. \end{cases}

A Schweizer–Sklar t-norm

T ^{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 [−∞,&nbsp;+∞]. An additive generator for

T ^{SS} _{p} for −∞ < p < +∞ isf ^{SS} _{p} (x) = \begin{cases} -log x & if p = 0 \\ (1- x ^{p})/(p) & 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 ^{H} _{p} (x,y) = \begin{cases} T _{D} (x,y) & if p = +inf ty \\ 0 & if p = x = y = 0 \\ (xy)/(p + ( 1- p)(x + y - xy)) & otherwise. \end{cases} The t-norm T ^{H} _{0} is called the Hamacher product.

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

T ^{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 ^{H} _{p} for p < +∞ isf ^{H} _{p(x)} = \begin{cases} (1- x)/(x) & if p = 0 \\ log (p + ( 1- p)x)/(x) & 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 ^{F} _{p(x,y)} = \begin{cases} T _{min} (x,y) & if p = 0 \\ T _{prod} (x,y) & if p = 1 \\ T _{Luk} (x,y) & if p = +inf ty \\ log _{p\left(1} + ((p ^{x} -1 )(p ^{y} -1 ))/(p -1) \right) & otherwise. \end{cases}

The Frank t-norm

T ^{F} _{p} is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for T ^{F} _{p} isf ^{F} _{p(x)} = \begin{cases} -log x & if p = 1 \\ 1 - x & if p = +inf ty \\ log (p -1)/(p ^{x} -1) & otherwise. \end{cases}

Yager t-norms

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

T ^{Y} _{p} (x,y) = \begin{cases} T _{D} (x,y) & if p = 0 \\ max \left( 0 , 1 - (( 1 - x) ^{p} + ( 1 - y) ^{p)} ^{1/p} \right) & if 0 < p < +inf ty \\ T _{min} (x,y) & if p = +inf ty \end{cases}

The Yager t-norm

T ^{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 ^{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 ^{Y} _{p} for 0 < p < +∞ isf ^{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 ^{AA} _{p} (x,y) = \begin{cases} T _{D} (x,y) & if p = 0 \\ e ^{- \left(| -log x| ^{p} + | -log y| ^{p\right)} ^{1/p}} & if 0 < p < +inf ty \\ T _{min} (x,y) & if p = +inf ty \end{cases}

The Aczél–Alsina t-norm

T ^{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 ^{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 ^{AA} _{p} for 0 < p < +∞ isf ^{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 ^{D} _{p} (x,y) = \begin{cases} 0 & if x = 0 or y = 0 \\ T _{D} (x,y) & if p = 0 \\ T _{min} (x,y) & if p = +inf ty \\ (1)/(1 + \left( \left( (1- x)/(x) \right) ^{p} + \left( (1- y)/(y) \right) ^{p} \right) ^{1/p}) & otherwise. \\ \end{cases}

The Dombi t-norm

T ^{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 ^{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 ^{D} _{p} for 0 < p < +∞ isf ^{D} _{p(x)} = \left( (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 ^{SW} _{p} (x,y) = \begin{cases} T _{D} (x,y) & if p = -1 \\ max \left( 0 , (x + y -1+ pxy)/(1+ p) \right) & if -1 < p < +inf ty \\ T _{prod} (x,y) & if p = +inf ty \end{cases}

The Sugeno–Weber t-norm

T ^{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 ^{SW} _{p} for 0 < p < +∞ [sic] isf ^{SW} _{p(x)} = \begin{cases} 1 - x & if p = 0 \\ 1 - log _{1} ( 1 + px) & 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,&nbsp;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,&nbsp;1]. Then the function T: [0,&nbsp;1]² → [0,&nbsp;1] defined as

T(x, y) = \begin{cases} a _{i} + (b _{i} - a _{i)} ⋅ T _{i\left( (x - a _{i})/(b _{i} - a _{i})} , (y - a _{i})/(b _{i} - a _{i}) \right) & if x, y \in [a _{i,} b ^{2} _{i]} \\ min (x, y) & 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

T = o plus\nolimits _{i\in} (T _{i,} a _{i,} b _{i),} or (T _{1,} a _{1,} b _{1)} ⊕ ... ⊕ (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,&nbsp;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 = o plus\nolimits _{i\in} (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 & if x \le y \\ a _{i} + (b _{i} - a _{i)} ⋅ R _{i\left( (x - a _{i})/(b _{i} - a _{i})} , (y - a _{i})/(b _{i} - a _{i}) \right) & if a _{i} < y < x \le b _{i} \\ y & 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,&nbsp;''p''] for a parameter p in [0,&nbsp;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,&nbsp;''p''] for a parameter p in [0,&nbsp;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,&nbsp;1] → [0,&nbsp;1] the function that assigns 1 − x to x and t = 0.5. Let T₁ be the linear transformation of T into [''t'',&nbsp;1] and

R _{T _{1}} (x,y) = \sup\{z \mid T _{1(z,x)\le} y\}. Then the functionT _{rot} = \begin{cases} T _{1(x,} y) & if x, y \in (t, 1 ] \\ N(R _{T _{1}} (x, N(y))) & if x \in (t, 1 ] and y \in [ 0 , t] \\ N(R _{T _{1}} (y, N(x))) & if x \in [ 0 , t] and y \in (t, 1 ] \\ 0 & 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,&nbsp;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,&nbsp;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,&nbsp;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. .
• Fodor, János (2004), "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), "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), "Triangular norms and conorms", Scholarpedia http://www.scholarpedia.org/.