Information AboutT-conorm |
| CATEGORIES ABOUT T-NORM | |
| fuzzy logic | |
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DEFINITION A t-norm is a Function T: × [0, 1 → [0, 1] which satisfies the following properties:
The defining conditions of the t-norm are exactly those of the partially ordered Abelian monoid on the real unit interval {Link without Title} . '' (Cf. Ordered Group .)'' The monoidal operation of any partially ordered Abelian monoid ''L'' is therefore by some authors called a ''triangular norm on L''. Motivations and applications T-norms are a generalization of the usual two-valued Logical Conjunction , studied by classical logic, for Fuzzy Logic s. Indeed, the classical Boolean conjunction is both commutative and associative. The monotonicity property ensures that the Truth Value of conjunction does not decrease if the truth values of conjuncts increase. The requirement that 1 be an identity element corresponds to the interpretation of 1 as ''true'' (and consequently 0 as ''false''). Continuity, which is often required from fuzzy conjunction as well, expresses the idea that, roughly speaking, very small changes in truth values of conjuncts should not macroscopically affect the truth value of their conjunction. T-norms are also used to construct the Intersection of Fuzzy Set s or as a basis for aggregation operators (see Fuzzy Set Operations ). In Probabilistic Metric Space s, t-norms are used to generalize Triangle Inequality of ordinary metric spaces. Individual t-norms may of course frequently occur in further disciplines of mathematics, since the class contains many familiar functions. Classification of t-norms A t-norm is called ''continuous'' if it is Continuous as a function, in the usual interval topology on {Link without Title} 2 (similarly for ''left-'' and ''right-continuity'').
The usual partial ordering of t-norms is pointwise, i.e., : T1 ≤ T2 if T1(''a'', ''b'') ≤ T2(''a'', ''b'') for all ''a'', ''b'' in {Link without Title} . As functions, pointwise larger t-norms are sometimes called ''stronger'' than those pointwise smaller. In the semantics of fuzzy logic, however, the larger a t-norm, the ''weaker'' the conjunction it represents. PROMINENT EXAMPLES
:: b & \mbox{if }a=1 \ a & \mbox{if }b=1 \ 0 & \mbox{otherwise.} \end{cases} :The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm (see the Properties Of T-norms below). It is a right-continuous Archimedean t-norm.
:: \min(a,b) & \mbox{if }a+b > 1 \ 0 & \mbox{otherwise} \end{cases} :is a standard example of a t-norm which is left-continuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent t-norm.
:: 0 & \mbox{if } a=b=0 \ rac{ab}{a+b-ab} & \mbox{otherwise} \end{cases} :is a strict Archimedean t-norm, and an important representative of the parametric classes of Hamacher T-norms and Schweizer–Sklar T-norms . PROPERTIES OF T-NORMS The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm: : for any t-norm and all ''a'', ''b'' in {Link without Title} . For every t-norm T, the number 0 acts as null element: T(''a'', 0) = 0 for all ''a'' in {Link without Title} . A t-norm T has Zero Divisor s if and only if it has Nilpotent elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent elements is an interval or [0, ''a''), for some ''a'' in [0, 1 . Properties of continuous t-norms Although real functions of two variables can be continuous in each variable without being continuous on this is not the case with t-norms: a t-norm T is continuous if and only if it is continuous in one variable, i.e., if and only if the functions ''fy''(''x'') = T(''x'', ''y'') are continuous for each ''y'' in [0, 1 . Analogous theorems hold for left- and right-continuity of a t-norm. A continuous t-norm is Archimedean if and only if 0 and 1 are its only Idempotents . A continuous Archimedean t-norm T is nilpotent if and only if each ''x'' < 1 is a nilpotent element of T. Thus with a continuous Archimedean t-norm T, either all or none of the elements of (0, 1) are nilpotent. If it is the case that all elements in (0, 1) are nilpotent, then the t-norm is isomorphic to the Łukasiewicz t-norm; i.e., there is a strictly increasing function ''f'' such that : If on the other hand it is the case that there are no nilpotent elements of T, the t-norm is isomorphic to the product t-norm. In other words, all nilpotent t-norms are isomorphic, the Łukasiewicz t-norm being their prototypical representative; and all strict t-norms are isomorphic, with the product t-norm as their prototypical example. The Łukasiewicz t-norm is itself isomorphic to the product t-norm undercut at 0.25, i.e., to the function ''p''(''x'', ''y'') = max(0.25, ''x'' · ''y'') on {Link without Title} 2. For each continuous t-norm, the set of its idempotents is a closed subset of {Link without Title} . Its complement — the set of all elements which are not idempotent — is therefore a union of countably many non-overlapping open intervals. The restriction of the t-norm to any of these intervals (including its endpoints) is Archimedean, and thus isomorphic either to the Łukasiewicz t-norm or the product t-norm. For such ''x'', ''y'' that do not fall into the same open interval of non-idempotents, the t-norm evaluates to the minimum of ''x'' and ''y''. These conditions actually give a characterization of continuous t-norms, called the Mostert–Shields theorem, since every continuous t-norm can in this way be decomposed, and the described construction always yields a continuous t-norm. The theorem can also be formulated as follows: :A t-norm is continuous if and only if it is isomorphic to an Ordinal Sum of the minimum, Łukasiewicz, and product t-norm. A similar characterization theorem for non-continuous t-norms is not known (not even for left-continuous ones), only some non-exhaustive methods for the Construction Of T-norms have been found. RESIDUUM For any left-continuous t-norm , there is a unique binary operation on {Link without Title} such that : if and only if for all ''x'', ''y'', ''z'' in {Link without Title} . This operation is called the ''residuum'' of the t-norm. In prefix notation, the residuum to a t-norm is often denoted by or by the letter R. The interval {Link without Title} equipped with a t-norm and its residuum forms a category. In the standard semantics of t-norm based fuzzy logics, where conjunction is interpreted by a t-norm, the residuum plays the role of implication (often called ''R-implication''). Basic properties of residua If is the residuum of a left-continuous t-norm , then : Consequently, for all ''x'', ''y'' in the unit interval, : if and only if and :
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\max(x, y) & = & \min((x \Rightarrow y)\Rightarrow y, (y \Rightarrow x)\Rightarrow x). \end{array}
Residua of prominent left-continuous t-norms If ''x'' ≤ ''y'', then R(''x'', ''y'') = 1 for any residuum R. The following table therefore gives the values of prominent residua only for ''x'' > ''y''. T-CONORMS T-conorms (also called '''S-norms''') are dual to t-norms under the order-reversing operation which assigns 1 – ''x'' to ''x'' on {Link without Title} . Given a t-norm, the complementary conorm is defined by : This generalizes De Morgan's Laws . It follows that a t-conorm satisfies the following conditions, which can be used for an equivalent axiomatic definition of t-conorms independently of t-norms:
T-conorms are used to represent Logical Disjunction in Fuzzy Logic and Union in Fuzzy Set Theory . Examples of t-conorms Important t-conorms are those dual to prominent t-norms:
:: b & \mbox{if }a=0 \ a & \mbox{if }b=0 \ 1 & \mbox{otherwise,} \end{cases} :dual to the drastic t-norm, is the largest t-conorm (see the Properties Of T-conorms below).
:: \max(a,b) & \mbox{if }a+b < 1 \ 1 & \mbox{otherwise.} \end{cases}
:: :is a dual to one of the Hamacher T-norms . Properties of t-conorms Many properties of t-conorms can be obtained by dualizing the properties of t-norms, for example:
::, for any t-conorm and all ''a'', ''b'' in {Link without Title} . Further properties result from the relationships between t-norms and t-conorms or their interplay with other operators, e.g.:
::T(''x'', S(''y'', ''z'')) = S(T(''x'', ''y''), T(''x'', ''z'')) for all ''x'', ''y'', ''z'' in {Link without Title} , :if and only if S is the maximum t-conorm. Dually, any t-conorm distributes over the minimum, but not over any other t-norm. SEE ALSO REFERENCES
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