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In mathematics, [[t-norm]]s 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-example]]s 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-norm]]s. | |||
== 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 [[affinely extended real number system|extended real line]]. The ''pseudo-inverse function'' to ''f'' is the function ''f'' <sup>(−1)</sup>: [''c'', ''d''] → [''a'', ''b''] defined as | |||
::<math>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}</math> | |||
=== 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<sup>+</sup>) or +∞ for all ''x'', ''y'' in [0, 1]. Then the function ''T'': [0, 1]<sup>2</sup> → [0, 1] defined as | |||
::''T''(''x'', ''y'') = ''f'' <sup>(-1)</sup>(''f''(''x'') + ''f''(''y'')) | |||
: is a t-norm. | |||
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<sup>−''f'' (''x'')</sup> 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'' <sup>(−1)</sup>(''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''<sub>''p''</sub> parameterized by ''p'' is ''increasing'' if ''T''<sub>''p''</sub>(''x'', ''y'') ≤ ''T''<sub>''q''</sub>(''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''<sub>''p''</sub> is ''continuous'' with respect to the parameter ''p'' if | |||
::<math>\lim_{p\to p_0} T_p = T_{p_0}</math> | |||
:for all values ''p''<sub>0</sub> of the parameter. | |||
=== Schweizer–Sklar t-norms === | |||
[[Image:Schweizer-Sklar-2-Tnorm-graph-contour.png|thumb|270px|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 | |||
:<math>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}</math> | |||
A Schweizer–Sklar t-norm <math>T^{\mathrm{SS}}_p</math> 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 <math>T^{\mathrm{SS}}_p</math> for −∞ < ''p'' < +∞ is | |||
:<math>f^{\mathrm{SS}}_p (x,y) = \begin{cases} | |||
-\log x & \text{if } p = 0 \\ | |||
\frac{1 - x^p}{p} & \text{otherwise.} | |||
\end{cases}</math> | |||
=== 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'' ≤ +∞: | |||
:<math>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}</math> | |||
The t-norm <math>T^{\mathrm{H}}_0</math> is called the ''Hamacher product.'' | |||
Hamacher t-norms are the only t-norms which are rational functions. | |||
The Hamacher t-norm <math>T^{\mathrm{H}}_p</math> 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 <math>T^{\mathrm{H}}_p</math> for ''p'' < +∞ is | |||
:<math>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}</math> | |||
=== 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: | |||
:<math>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}</math> | |||
The Frank t-norm <math>T^{\mathrm{F}}_p</math> is strict if ''p'' < +∞. The family is strictly decreasing and continuous with respect to ''p''. An additive generator for <math>T^{\mathrm{F}}_p</math> is | |||
:<math>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} | |||
</math> | |||
=== Yager t-norms === | |||
[[Image:Yager-2-Tnorm-graph-contours.png|thumb|270px|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 | |||
:<math>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} | |||
</math> | |||
The Yager t-norm <math>T^{\mathrm{Y}}_p</math> 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 <math>T^{\mathrm{Y}}_p</math> for 0 < ''p'' < +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of ''p''. An additive generator of <math>T^{\mathrm{Y}}_p</math> for 0 < ''p'' < +∞ is | |||
:<math>f^{\mathrm{Y}}_p(x) = (1 - x)^p.</math> | |||
=== 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 | |||
:<math>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}</math> | |||
The Aczél–Alsina t-norm <math>T^{\mathrm{AA}}_p</math> 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 <math>T^{\mathrm{AA}}_p</math> for 0 < ''p'' < +∞ arises from the product t-norm by raising its additive generator to the power of ''p''. An additive generator of <math>T^{\mathrm{AA}}_p</math> for 0 < ''p'' < +∞ is | |||
:<math>f^{\mathrm{AA}}_p(x) = (-\log x)^p.</math> | |||
=== Dombi t-norms === | |||
The family of ''Dombi t-norms'', introduced by József Dombi (1982), is given for 0 ≤ ''p'' ≤ +∞ by | |||
:<math>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} | |||
</math> | |||
The Dombi t-norm <math>T^{\mathrm{D}}_p</math> 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 <math>T^{\mathrm{D}}_p</math> for 0 < ''p'' < +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of ''p''. An additive generator of <math>T^{\mathrm{D}}_p</math> for 0 < ''p'' < +∞ is | |||
:<math>f^{\mathrm{D}}_p(x) = \left(\frac{1-x}{x}\right)^p.</math> | |||
=== Sugeno–Weber t-norms === | |||
The family of ''Sugeno–Weber t-norms'' was introduced in the early 1980s by Siegfried Weber; the dual [[t-conorm]]s were defined already in the early 1970s by Michio Sugeno. It is given for −1 ≤ ''p'' ≤ +∞ by | |||
:<math>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} | |||
</math> | |||
The Sugeno–Weber t-norm <math>T^{\mathrm{SW}}_p</math> 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 <math>T^{\mathrm{SW}}_p</math> for 0 < ''p'' < +∞ [sic] is | |||
:<math>f^{\mathrm{SW}}_p(x) = \begin{cases} | |||
1 - x & \text{if } p = 0 \\ | |||
1 - \log_{1 + p}(1 + px) & \text{otherwise.} | |||
\end{cases}</math> | |||
== 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''<sub>''i''</sub> for ''i'' in an index set ''I'' be a family of t-norms and (''a''<sub>''i''</sub>, ''b''<sub>''i''</sub>) a family of pairwise disjoint (non-empty) open subintervals of [0, 1]. Then the function ''T'': [0, 1]<sup>2</sup> → [0, 1] defined as | |||
::<math>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}</math> | |||
:is a t-norm. | |||
[[Image:OrdSum-Luk-prod-graph-contours.png|thumb|270px|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''<sub>i</sub>, ''a''<sub>i</sub>, ''b''<sub>i</sub>) for ''i'' in ''I'', denoted by | |||
:<math>T = \bigoplus\nolimits_{i\in I} (T_i, a_i, b_i),</math> | |||
or <math>(T_1, a_1, b_1) \oplus \dots \oplus (T_n, a_n, b_n)</math> 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''<sub>min</sub>. 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''<sub>''i''</sub> = 0 and ''T''<sub>''i''</sub> has zero divisors. (Analogously for nilpotent elements.) | |||
If <math>T = \bigoplus\nolimits_{i\in I} (T_i, a_i, b_i)</math> is a left-continuous t-norm, then its residuum ''R'' is given as follows: | |||
:<math>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}</math> | |||
where ''R''<sub>i</sub> is the residuum of ''T''<sub>i</sub>, 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 (mathematician)|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 divisor]]s, ''N'': [0, 1] → [0, 1] the function that assigns 1 − ''x'' to ''x'' and ''t'' = 0.5. Let ''T''<sub>1</sub> be the linear transformation of ''T'' into [''t'', 1] and <math>R_{T_1}(x,y) = \sup\{z \mid T_1(z,x)\le y\}.</math> Then the function | |||
::<math>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}</math> | |||
:is a left-continuous t-norm, called the ''rotation'' of the t-norm ''T''. | |||
[[Image:NilpotentMinimum-as-rotation.png|thumb|120px|right|The [[T-norm#Nilpotent-minimum t-norm|nilpotent minimum]] as a rotation of the [[T-norm#Minimum t-norm|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|thumb|200px|left|Rotations of the [[T-norm#Lukasiewicz t-norm|Łukasiewicz]], [[T-norm#Product t-norm|product]], [[T-norm#Nilpotent-minimum t-norm|nilpotent minimum]], and [[T-norm#Drastic t-norm|drastic]] t-norm]] | |||
The theorem can be generalized by taking for ''N'' any ''strong negation'', that is, an [[Involution (mathematics)|involutive]] strictly decreasing continuous function on [0, 1], and for ''t'' taking the unique [[Fixed point (mathematics)|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''<sub>rot</sub> is the function ''N'', that is, ''N''(''x'') = ''R''<sub>rot</sub>(''x'', 0) for all ''x'', where ''R''<sub>rot</sub> is the residuum of ''T''<sub>rot</sub>. | |||
== 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), [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), [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. | |||
* Mirko Navara (2007), [http://www.scholarpedia.org/article/Triangular_Norms_and_Conorms "Triangular norms and conorms"], Scholarpedia [http://www.scholarpedia.org/]. | |||
[[Category:Fuzzy logic]] |
Revision as of 06:40, 28 April 2013
Template:No footnotes 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
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.
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 Tp parameterized by p is increasing if Tp(x, y) ≤ Tq(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 Tp is continuous with respect to the parameter p if
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
- 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 for −∞ < p < +∞ is
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 ≤ +∞:
The t-norm is called the Hamacher product.
Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm 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 for p < +∞ is
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:
The Frank t-norm is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for is
Yager t-norms
The family of Yager t-norms, introduced in the early 1980s by Ronald R. Yager, is given for 0 ≤ p ≤ +∞ by
The Yager t-norm 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 for 0 < p < +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of p. An additive generator of for 0 < p < +∞ is
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
The Aczél–Alsina t-norm 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 for 0 < p < +∞ arises from the product t-norm by raising its additive generator to the power of p. An additive generator of for 0 < p < +∞ is
Dombi t-norms
The family of Dombi t-norms, introduced by József Dombi (1982), is given for 0 ≤ p ≤ +∞ by
The Dombi t-norm 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 for 0 < p < +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of p. An additive generator of for 0 < p < +∞ is
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
The Sugeno–Weber t-norm 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 for 0 < p < +∞ [sic] is
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 Ti for i in an index set I be a family of t-norms and (ai, bi) a family of pairwise disjoint (non-empty) open subintervals of [0, 1]. Then the function T: [0, 1]2 → [0, 1] defined as
- is a t-norm.
The resulting t-norm is called the ordinal sum of the summands (Ti, ai, bi) for i in I, denoted by
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 Tmin. 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, ai = 0 and Ti has zero divisors. (Analogously for nilpotent elements.)
If is a left-continuous t-norm, then its residuum R is given as follows:
where Ri is the residuum of Ti, 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 Then the function
- 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 Trot is the function N, that is, N(x) = Rrot(x, 0) for all x, where Rrot is the residuum of Trot.
See also
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 [1]
- 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 [2].