Construction of t-norms

In mathematics, t-norms are a special kind of binary operations on the real unit interval , 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
::$f^(y) = \begin \sup \ & \text f \text \\ \sup \ & \text f \text \end$

Additive generators

The construction of t-norms by additive generators is based on the following theorem:
: Let ''f'': , 1→ , +∞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 , 1 Then the function ''T'': , 1sup>2 → , 1defined 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 $T(x,y)=f^\left(\min\left(f(0^+),f(x)+f(y)\right)\right)$. The corresponding residuum can then be expressed as $(x \Rightarrow y) = f^\left(\max\left(0,f(y)-f(x)\right)\right)$. And the biresiduum as $(x \Leftrightarrow y) = f^\left(\left, f(x)-f(y)\\right)$.

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 , 1is 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'': , 1→ , +∞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 , +∞and multiplication on , 1by 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'': , 1→ , 1defined 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 , 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'': , 1→ , +∞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 , 1whenever ''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
::$\lim_ T_p = T_$
:for all values ''p''₀ 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
:$T^_p(x,y) = \begin T_(x,y) & \text p = -\infty \\ (x^p + y^p - 1)^ & \text -\infty < p < 0 \\ T_(x,y) & \text p = 0 \\ (\max(0, x^p + y^p - 1))^ & \text 0 < p < +\infty \\ T_(x,y) & \text p = +\infty. \end$

A Schweizer–Sklar t-norm $T^_p$ 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 $T^_p$ for −∞ < ''p'' < +∞ is
:$f^_p (x) = \begin -\log x & \text p = 0 \\ \frac & \text \end$

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'' ≤ +∞:
:$T^_p (x,y) = \begin T_(x,y) & \text p = +\infty \\ 0 & \text p = x = y = 0 \\ \frac & \text \end$
The t-norm $T^_0$ is called the ''Hamacher product.''

Hamacher t-norms are the only t-norms which are rational functions.
The Hamacher t-norm $T^_p$ 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 $T^_p$ for ''p'' < +∞ is
:$f^_p(x) = \begin \frac & \text p = 0 \\ \log\frac & \text \end$

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:
:$T^_p(x,y) = \begin T_(x,y) & \text p = 0 \\ T_(x,y) & \text p = 1 \\ T_(x,y) & \text p = +\infty \\ \log_p\left(1 + \frac\right) & \text \end$

The Frank t-norm $T^_p$ is strict if ''p'' < +∞. The family is strictly decreasing and continuous with respect to ''p''. An additive generator for $T^_p$ is
:$f^_p(x) = \begin -\log x & \text p = 1 \\ 1 - x & \text p = +\infty \\ \log\frac & \text \end$

Yager t-norms

The family of ''Yager t-norms'', introduced in the early 1980s by Ronald R. Yager, is given for 0 ≤ ''p'' ≤ +∞ by
:$T^_p (x,y) = \begin T_(x,y) & \text p = 0 \\ \max\left(0, 1 - ((1 - x)^p + (1 - y)^p)^\right) & \text 0 < p < +\infty \\ T_(x,y) & \text p = +\infty \end$

The Yager t-norm $T^_p$ 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 $T^_p$ for 0 < ''p'' < +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of ''p''. An additive generator of $T^_p$ for 0 < ''p'' < +∞ is
:$f^_p(x) = (1 - x)^p.$

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
:$T^_p (x,y) = \begin T_(x,y) & \text p = 0 \\ e^ & \text 0 < p < +\infty \\ T_(x,y) & \text p = +\infty \end$

The Aczél–Alsina t-norm $T^_p$ 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 $T^_p$ for 0 < ''p'' < +∞ arises from the product t-norm by raising its additive generator to the power of ''p''. An additive generator of $T^_p$ for 0 < ''p'' < +∞ is
:$f^_p(x) = (-\log x)^p.$

Dombi t-norms

The family of ''Dombi t-norms'', introduced by József Dombi (1982), is given for 0 ≤ ''p'' ≤ +∞ by
:$T^_p (x,y) = \begin 0 & \text x = 0 \text y = 0 \\ T_(x,y) & \text p = 0 \\ T_(x,y) & \text p = +\infty \\ \frac & \text \\ \end$

The Dombi t-norm $T^_p$ 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 $T^_p$ for 0 < ''p'' < +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of ''p''. An additive generator of $T^_p$ for 0 < ''p'' < +∞ is
:$f^_p(x) = \left(\frac\right)^p.$

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
:$T^_p (x,y) = \begin T_(x,y) & \text p = -1 \\ \max\left(0, \frac\right) & \text -1 < p < +\infty \\ T_(x,y) & \text p = +\infty \end$

The Sugeno–Weber t-norm $T^_p$ 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 $T^_p$ for 0 < ''p'' < +∞ icis
:$f^_p(x) = \begin 1 - x & \text p = 0 \\ 1 - \log_(1 + px) & \text \end$

Ordinal sums

The ordinal sum constructs a t-norm from a family of t-norms, by shrinking them into disjoint subintervals of the interval , 1and 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 , 1 Then the function ''T'': , 1sup>2 → , 1defined as
::T(x, y) = \begin
a_i + (b_i - a_i) \cdot T_i\left(\frac, \frac\right)
& \text x, y \in _i, b_i2 \\
\min(x, y) & \text
\end
: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
:$T = \bigoplus\nolimits_ (T_i, a_i, b_i),$
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 , 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 $T = \bigoplus\nolimits_ (T_i, a_i, b_i)$ is a left-continuous t-norm, then its residuum ''R'' is given as follows:
:$R(x, y) = \begin 1 & \text x \le y \\ a_i + (b_i - a_i) \cdot R_i\left(\frac, \frac\right) & \text a_i < y < x \le b_i \\ y & \text \end$
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 , ''p''for a parameter ''p'' in , 1and 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 , ''p''for a parameter ''p'' in , 1and 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'': , 1→ , 1the function that assigns 1 − ''x'' to ''x'' and ''t'' = 0.5. Let ''T''₁ be the linear transformation of ''T'' into 't'', 1and $R_(x,y) = \sup\.$ Then the function
::$T_ = \begin T_1(x, y) & \text x, y \in (t, 1] \\ N(R_(x, N(y))) & \text x \in (t, 1] \text y \in$, t\\
N(R_(y, N(x))) & \text x \in , t\text y \in (t, 1] \\
0 & \text x, y \in , t\end
: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 .5, 1and 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 , 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 , 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 1785-886

* 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"
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Fuzzy logic