Dominance in the family of Sugeno-Weber t-norms
DDOMINANCE IN THE FAMILY OF SUGENO-WEBER T-NORMS
MANUEL KAUERS, VERONIKA PILLWEIN, AND SUSANNE SAMINGER-PLATZ
Abstract.
The dominance relationship between two members of the family of Sugeno Weber t-norms is proven by using a quantifer elimination algorithm. Further it is shown that dominanceis a transitive, and therefore also an order relation, on this family of t-norms. Introduction
Dominance is a functional inequality which arises in different application fields. It most oftenappears when discussing the preservation of properties during (dis-)aggregation processes like,e.g., in flexible querying, preference modelling or computer-assisted assessment [5, 7, 20, 23]. Itis further crucial in the construction of Cartesian products of probabilistic metric and normedspaces [8, 31, 36] as well as when constructing many-valued equivalence and order relations [2, 5,6, 39].Introduced in 1976 in the framework of probabilistic metric spaces as an inequality involving twotriangle functions (see [36] and [31] for an early generalization to operations on a partially orderedset), it was soon clear that dominance constitutes a reflexive and antisymmetric relation on the setof all t-norms. That it is not a transitive relation has been proven much later in 2007 in [30]. Thisnegative answer to a long open question has, to some extent, been surprising. In particular sinceearlier results showed that for several important single-parametric families of t-norms, dominanceis also a transitive and therefore an order relation [10, 21, 24, 29, 33] (see also [25] for an overviewand additional results on families of t-norms and copulas).The family of Sugeno-Weber t-norms has been one of the more prominent families of t-norms forwhich the dominance has not been completely characterized so far. First partial results could beachieved in [22] by invoking results on different sufficient conditions derived from a generalizationof the Mulholland inequality and involving the additive generators of the t-norms, their pseudo-inverses and their derivatives (for more details on the differential conditions see [22], for thegeneralization of the Mulholland inequality look at [26]).The purpose of this paper is to close this gap. We present a proof for a complete characterizationof dominance in the family of Sugeno-Weber t-norms. This is interesting because from all thefamilies of t-norms discussed in Section 4 in the monograph [10], the family of Sugeno-Webert-norms was, until now, the only family for which no complete classification result was available.But there are further aspects which make our results interesting:First, the solution sets are of a completely different form than witnessed before for other families.So far dominance in single-parametric families has either been in complete accordance with theordering in the family (i.e., dominance constitutes a linear order on the family of t-norms) ordominance has rarely appeared between family members, i.e., holds only in the trivial cases ofself-dominance or when involving maximal or minimal elements of the family. For the family ofSugeno-Weber t-norms neither is the case.Second, although the solution sets look different, dominance is a transitive relation on the familyof Sugeno-Weber t-norms.Third, the results have been achieved by the use of symbolic computation algorithms. More explic-itly a quantifier elimination algorithm for real closed fields (Cylindrical Algebraic Decomposition)has, after several transformation steps, been applied to logical equivalent formulations of the orig-inal problems. The present contribution is therefore also an example of a successful application ofcomputer algebra and symbolic computation for solving polynomial inequalities. a r X i v : . [ m a t h . F A ] J u l MANUEL KAUERS, VERONIKA PILLWEIN, AND SUSANNE SAMINGER-PLATZ
The following preliminaries shall clarify the necessary notions and summarize basic facts about thefamily of Sugeno-Weber t-norms. Dominance as well as some basic aspects of quantifier eliminationalgorithms will be explained. We then provide and prove the main results — the characterizationof dominance between two t-norms of the family of Sugeno-Weber t-norms and transitivity ofdominance in the family. We finally discuss the results in more detail.2.
Preliminaries
Triangular norms.
We briefly summarize some basic properties of t-norms for a thoroughunderstanding of this paper. Excellent overviews on and discussions of triangular norms (includinghistorical accounts, further details, proofs, and references) can be found in the monographs [1, 10],the edited volume [9] and the articles [11, 12, 13].
Definition 1. A triangular norm (briefly t-norm ) T : [0 , → [0 , is a binary operation on theunit interval which is commutative, associative, increasing and has neutral element . Speaking in more algebraic terms, T turns the unit interval into an ordered Abelian semigroupwhose neutral element is 1.The most prominent examples of t-norms are the minimum T M , the product T P , the (cid:32)Lukasiewiczt-norm T L and the drastic product T D . They are defined by T M ( u, v ) = min( u, v ), T P ( u, v ) = u · v , T L ( u, v ) = max( u + v − , T D ( u, v ) = (cid:40) min( u, v ) , if max( u, v ) = 1 , , otherwise.Obviously, the basic t-norms T M , T P and T L are continuous, whereas the drastic product T D isnot. The comparison of two t-norms is done pointwisely, i.e., if, for all x, y ∈ [0 , T ( x, y ) ≥ T ( x, y ), then we say that T is stronger than T and denote it by T ≥ T . Theminimum T M is the strongest of all t-norms, the drastic product T D is the weakest of all t-norms.Moreover, the four basic t-norms are ordered in the following way: T D ≤ T L ≤ T P ≤ T M . Definition 2.
A t-norm T is called(i) Archimedean if for all u, v ∈ ]0 , there exists an n ∈ N such that T ( u, . . . , u (cid:124) (cid:123)(cid:122) (cid:125) n times ) < v . (ii) A t-norm T is called strict if it is continuous and strictly monotone, i.e., for all u, v, w ∈ [0 , it holds that T ( u, v ) < T ( u, w ) whenever u > and v < w . (iii) A t-norm T is called nilpotent if it is continuous and if each u ∈ ]0 , is a nilpotent elementof T , i.e., there exists some n ∈ N such that T ( u, . . . , u (cid:124) (cid:123)(cid:122) (cid:125) n times ) = 0 . Note that for a strict t-norm T it holds that T ( u, v ) > u, v ∈ ]0 , T it holds that for every u ∈ ]0 ,
1[ there exists some v ∈ ]0 ,
1[ such that T ( u, v ) = 0 (each u ∈ ]0 ,
1[ is a so-called zero divisor ). Therefore for a nilpotent t-norm T and a strict t-norm T it can never hold that T ≥ T . OMINANCE IN THE FAMILY OF SUGENO-WEBER T-NORMS 3
The family of Sugeno-Weber t-norms.
In 1983 S. Weber proposed the use of this particularfamily for modelling the intersection of fuzzy sets [40]. Since then, its dual operations, the Sugeno-Weber t-conorms, defined for all λ ∈ [0 , ∞ ] and all u, v ∈ [0 ,
1] by S SW λ ( u, v ) = 1 − T SW λ (1 − u, − v ),have played a prominent role for generalized decomposable measures [14, 15, 19, 41], in particular,since they already appeared as possible generalized additions in the context of λ -fuzzy measuresin [35].The family of Sugeno-Weber t-norms ( T SW λ ) λ ∈ [0 , ∞ ] is, for all u, v ∈ [0 , T SW λ ( u, v ) = T P ( u, v ) , if λ = 0 ,T D ( u, v ) , if λ = ∞ , max(0 , (1 − λ ) uv + λ ( u + v − , if λ ∈ ]0 , ∞ [ . The family is of particular interest since all but two of its members are nilpotent t-norms. Param-eters λ ∈ ]0 , ∞ [ lead to nilpotent t-norms (with T SW = T L as special case), while T SW = T P isthe only strict member. For λ ∈ [0 , ∞ [, the Sugeno-Weber t-norms are continuous Archimedeant-norms [16, 35, 40], for λ ∈ [0 ,
1] they can be interpreted as convex combinations of T L and T P ,and are therefore also copulas (for more details on copulas see also [18]).For λ ∈ ]0 , T is a Sugeno-Weber t-norm T SW λ with λ ∈ ]0 ,
1[ if and only if T is nilpotent, has an additivegenerator such that for each w ∈ [0 , T ( x, . ) of T is a straightline segment (for more details on the additive generators and construction methods for t-normswe again refer to the monographs [1, 10], the edited volume [9] and the articles [11, 12, 13]).2.2. Dominance.
The dominance relation has, as t-norms do, its roots in the field of probabilisticmetric spaces [31, 36]. It was originally introduced for associative operations (with common neutralelement) on a partially ordered set [31], and has been further investigated for t-norms [21, 29, 30, 37]and aggregation functions [20, 23, 17]. For more recent results on dominance between trianglefunctions resp. operations on distance distribution functions see also [27].We state the definition for t-norms only.
Definition 3.
Consider two t-norms T and T . We say that T dominates T (or T is dominatedby T ), denoted by T (cid:29) T , if, for all x, y, u, v ∈ [0 , , it holds that (1) T ( T ( x, y ) , T ( u, v )) ≥ T ( T ( x, u ) , T ( y, v )) . As mentioned already earlier, the dominance relation, in particular between t-norms, plays an im-portant role in various topics, such as the construction of Cartesian products of probabilistic metricand normed spaces [8, 31, 36], the construction of many-valued equivalence relations [5, 6, 39] andmany-valued order relations [2], the preservation of various properties during (dis-)aggregationprocesses in flexible querying, preference modelling and computer-assisted assessment [5, 7, 20, 23].Every t-norm, in fact every function non-decreasing in each of its arguments, is dominated by T M . Moreover, every t-norm dominates itself and T D . Since all t-norms have neutral element1, dominance between two t-norms implies their comparability: T (cid:29) T implies T ≥ T . Theconverse does not hold.Due to the induced comparability it also follows that dominance is an antisymmetric relation onthe class of t-norms. Associativity and symmetry ensure that dominance is also reflexive on theclass of t-norms.Although dominance is not a transitive relation on the set of continuous, and therefore also not onthe set of all, t-norms (see the results by Sarkoci [30, 28] and also [24]), it is transitive on severalsingle-parameteric families of t-norms (see also Table 1).It is interesting to see that in all the cases displayed in Table 1, dominance is either in completeaccordance with the ordering in the family (i.e., dominance constitutes a linear order on the familyof t-norms) or dominance rarely appears among family members, i.e., holds only in the trivialcases of self-dominance and dominance involving maximal or minimal elements of the family (foran overview on known results and referential details see [25]). MANUEL KAUERS, VERONIKA PILLWEIN, AND SUSANNE SAMINGER-PLATZ
Family of t-norms T λ (cid:29) T µ Hasse-DiagramsSchweizer-Sklar ( T SS λ ) λ ∈ [ −∞ , ∞ ] λ ≤ µ (Sherwood, 1984) Acz´el-Alsina ( T AA λ ) λ ∈ [0 , ∞ ] λ ≥ µ Dombi ( T D λ ) λ ∈ [0 , ∞ ] Yager ( T Y λ ) λ ∈ [0 , ∞ ] (Klement, Mesiar, Pap, 2000) ( T λ ) λ ∈ [0 , ∞ ] λ ≥ µ ( T λ ) λ ∈ [0 , ∞ ] ( T λ ) λ ∈ [0 , ∞ ] ( T λ ) λ ∈ [0 , ∞ ] (Saminger-Platz, 2009) Frank ( T F λ ) λ ∈ [0 , ∞ ] λ = 0, λ = µ , µ = ∞ ... Hamacher ( T H λ ) λ ∈ [0 , ∞ ] (Sarkoci, 2005) ( T λ ) λ ∈ [0 , ∞ ] λ = ∞ , λ = µ , µ = 0 (Saminger-Platz, 2009) Mayor-Torrens ( T MT λ ) λ ∈ [0 , λ = 0 , λ = µ ... Dubois-Prade ( T DP λ ) λ ∈ [0 , (Saminger, De Baets, De Meyer, 2005) Table 1.
Dominance relation in several families of t-normsWe will show below that dominance between two members of the family of Sugeno-Weber t-normsis of a complete different type, although finally dominance also turns out to be a transitive relationon this family.2.2.1.
The family of Sugeno-Weber t-norms.
The members of the family form a decreasing se-quence of t-norms with respect to their parameter, i.e., T SW λ ≥ T SW µ if and only if λ ≤ µ . Sincedominance induces order on the t-norms involved, it is therefore clear that a necessary conditionfor T SW λ (cid:29) T SW µ is that λ ≤ µ .Dominance among the family members has further been studied in [22] by invoking results ondifferent sufficient conditions derived from a generalization of the Mulholland inequality and in-volving the additive generators of the t-norms, their pseudo-inverses and their derivatives (formore details on the differential conditions see [22], for the generalization of the Mulholland in-equality see [26]). The results obtained did not lead to a full characterization of dominance inthe whole family, but already indicated that the dominance structure might be of a completelydifferent structure than the dominance relationship laid bare in any other family before. We quotethe result from [22]. Proposition 1.
Consider the family of Sugeno-Weber t-norms ( T SW λ ) λ ∈ [0 , ∞ ] . For all λ, µ ∈ [0 , ∞ ] such that one of the following condition holds (i) λ ≤ min(1 , µ ) , (ii) 1 < λ ≤ µ ≤ r ∗ , with r ∗ = 6 . denoting the second root of log ( t ) + log( t ) − t + 1 ,it follows that T SW λ (cid:29) T SW µ . OMINANCE IN THE FAMILY OF SUGENO-WEBER T-NORMS 5
Quantifier Elimination and Cylindrical Algebraic Decomposition.
In contrast tomany other families of t-norms, the dominance relation for Sugeno-Weber t-norms does not involveany logarithms or exponentials but can be formulated by addition, multiplication and the max-operation only. This is a striking structural advantage, because there are algorithms availablefor proving this kind of formulas automatically. The first algorithm for proving formulas aboutpolynomial inequalities was already given by Tarski in the early 1950s [38] but his algorithm wasonly of theoretical interest. Nowadays, modern implementations [3, 32, 34] of Collins’ algorithmfor Cylindrical Algebraic Decomposition (CAD) [4] make it possible to actually perform nontrivialcomputations within a reasonable amount of time. They are meanwhile established as valuabletools for solving problems about polynomial inequalities.In general, the input to CAD is a formula of the formQ x ∈ R · · · Q n x n ∈ R : A ( x , . . . , x n , y , . . . , y m )where the Q i are quantifiers (either ∀ or ∃ ) and A is a boolean combination of polynomial equationsand inequalities in the variables x i and y i . The variables x i are bound by quantifiers, the variables y i are free. Given such a formula, the algorithm computes a quantifier free formula B ( y , . . . , y m )which is equivalent to the input formula.A simple example is given by ∀ x ∈ R ∃ y ∈ R : ( x − y − > ⇔ x + y − z > , where A ( x, y, z ) is the formula ( x − y − > ⇔ x + y − z >
0, the bound variables are x, y, and z is a free variable. Applied to this formula, the CAD algorithm may return the quantifierfree formula B ( z ) = z ≤ − ∨ z ≥
1. This formula is equivalent to the quantified formula in thesense that for every real number z ∈ R Main results
Theorem 2.
Consider the family of Sugeno-Weber t-norms ( T SW λ ) λ ∈ [0 , ∞ ] . Then, for all λ, µ ∈ [0 , ∞ ] , T SW λ dominates T SW µ , T SW λ (cid:29) T SW µ , if and only if one of the following conditions holds:(i) λ = 0 ,(ii) µ = ∞ ,(iii) λ = µ ,(iv) < λ < µ ≤
17 + 12 √ ,(v)
17 + 12 √ < µ and < λ ≤ (cid:16) − √ µ −√ µ (cid:17) . MANUEL KAUERS, VERONIKA PILLWEIN, AND SUSANNE SAMINGER-PLATZ √
217 + 12 √ µλ ∞ ∞ impossible λ > µ (iv) (v) Figure 1.
Relationship between parameters λ and µ for T SW λ dominating T SW µ In Fig. 2, we have illustrated for which parameters λ for a given parameter µ it holds that T SW λ dominates T SW µ . In Section 4 the solution sets will be discussed in more detail.Based on the characterization of dominance between two members of the family of Sugeno-Webert-norms, another CAD computation directly applied to the problem of transitivity in the familyasserts the transitivity of the relation on the family within less than two seconds. Note also thatan alternative proof of the transitivity of dominance in the family of Sugeno-Weber t-norms isgiven in Section 4.3. In any case, we can state: Proposition 3.
Dominance is a transitive, and therefore an order, relation on the set of allSugeno-Weber t-norms.
The remainder of this section is devoted to the proof of Theorem 2, to the necessary steps ofreformulating intermediate proof steps, exploiting properties of subparts of the problem, andfinding an (equivalent) formulation of the original problem computable and solvable by CAD inreasonable time.Consider some λ, µ ∈ [0 , ∞ ]. In case that λ = 0, µ = ∞ , or λ = µ the result is trivially true since T M = T SW dominates all, T D = T SW ∞ is dominated by all t-norms, and every t-norm dominatesitself. Since dominance induces order, we may assume w.l.o.g. that 0 < λ < µ < ∞ . Moreover,the dominance inequality T SW λ ( T SW µ ( x, y ) , T SW µ ( u, v )) ≥ T SW µ ( T SW λ ( x, u ) , T SW λ ( y, v ))is trivially fulfilled whenever 1 ∈ { x, y, u, v } or 0 ∈ { x, y, u, v } such that we can reformulate theremaining proof goal in the following way:Determine all λ, µ ∈ ]0 , ∞ [ with λ < µ such that for all x, y, u, v ∈ ]0 ,
1[ : T SW λ ( T SW µ ( x, y ) , T SW µ ( u, v )) ≥ T SW µ ( T SW λ ( x, u ) , T SW λ ( y, v ))or explicitly such that ∀ x, y, u, v ∈ ]0 ,
1[ :max(0 , (1 − λ ) max(0 , (1 − µ ) uv + µ ( u + v − , (1 − µ ) xy + µ ( x + y − λ (max(0 , (1 − µ ) uv + µ ( u + v − , (1 − µ ) xy + µ ( x + y − − ≥ max(0 , (1 − µ ) max(0 , (1 − λ ) ux + λ ( u + x − , (1 − λ ) vy + λ ( v + y − µ (max(0 , (1 − λ ) ux + λ ( u + x − , (1 − λ ) vy + λ ( v + y − − . As mentioned earlier this problem is so that a final result can in principle be obtained directly byapplication of a quantifier elimination algorithm (like, e.g., CAD) for real closed fields. However,in practice this computation would take very long. By a series of appropriate simplifications we
OMINANCE IN THE FAMILY OF SUGENO-WEBER T-NORMS 7 can reduce the computation time tremendously. The following simplification steps result in anequivalent quantified formula for which quantifier elimination takes a few minutes only.(1)
Eliminate the outer maxima.
The body of the formula in question has the form max(0 , A ) ≥ max(0 , B ). It is readily confirmed by hand, or by CAD, thatmax(0 , A ) ≥ max(0 , B ) ⇐⇒ B ≤ ∨ A ≥ B ⇐⇒ B ≤ ∨ A ≥ B >
A, B . Applying the last equivalence and making the range restrictions on x, y, u, v and λ, µ explicit, we arrive at the equivalent formulation ∀ x, y, u, v ∈ R : 0 < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ < v < ⇒ (cid:16) (1 − µ ) max(0 , (1 − λ ) ux + λ ( u + x − , (1 − λ ) vy + λ ( v + y − µ (max(0 , (1 − λ ) ux + λ ( u + x − , (1 − λ ) vy + λ ( v + y − − ≤ ∨ (1 − λ ) max(0 , (1 − µ ) uv + µ ( u + v − , (1 − µ ) xy + µ ( x + y − λ (max(0 , (1 − µ ) uv + µ ( u + v − , (1 − µ ) xy + µ ( x + y − − ≥ (1 − µ ) max(0 , (1 − λ ) ux + λ ( u + x − , (1 − λ ) vy + λ ( v + y − µ (max(0 , (1 − λ ) ux + λ ( u + x − , (1 − λ ) vy + λ ( v + y − − > (cid:17) (2) Eliminate the inner maxima.
The new formula still contains four different maximumexpressions:max(0 , (1 − µ ) uv + µ ( u + v − , (1 − µ ) xy + µ ( x + y − A ;max(0 , (1 − λ ) ux + λ ( u + x − , (1 − λ ) vy + λ ( v + y − B .To get rid of these, observe that if Φ( X ) is a formula depending on a real variable X , thenthe following equivalences are validΦ(max(0 , X )) ⇐⇒ ( X ≤ ∨ X > ∧ Φ(max(0 , X )) ⇐⇒ ( X ≤ ∧ Φ(0)) ∨ ( X > ∧ Φ( X )) . For a formula depending on several real variables, this rewriting yieldsΦ(max(0 , X ) , max(0 , X ) , max(0 , X ) , max(0 , X )) ⇐⇒ (cid:0) X ≤ ∧ X ≤ ∧ X ≤ ∧ X ≤ ∧ Φ(0 , , , ∨ X > ∧ X ≤ ∧ X ≤ ∧ X ≤ ∧ Φ( X , , , ∨ X ≤ ∧ X > ∧ X ≤ ∧ X ≤ ∧ Φ(0 , X , , ∨ X > ∧ X > ∧ X ≤ ∧ X ≤ ∧ Φ( X , X , , ∨ X > ∧ X > ∧ X > ∧ X > ∧ Φ( X , X , X , X ) (cid:1) . Applying these considerations to our problem, we first put X = (1 − λ ) ux + λ ( u + x − ,X = (1 − λ ) vy + λ ( v + y − ,X = (1 − µ ) uv + µ ( u + v − ,X = (1 − µ ) xy + µ ( x + y − , so that A = (1 − λ ) max(0 , X ) max(0 , X ) + λ (max(0 , X ) + max(0 , X ) − ,B = (1 − µ ) max(0 , X ) max(0 , X ) + µ (max(0 , X ) + max(0 , X ) − , MANUEL KAUERS, VERONIKA PILLWEIN, AND SUSANNE SAMINGER-PLATZ and then arrive at the following equivalent formulation of our problem ∀ x, y, u, v ∈ R : 0 < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ < v < ⇒ (cid:16)(cid:0) X ≤ ∧ X ≤ ∧ (1 − µ )0 0 + µ (0 + 0 − ≤ ∨ X > ∧ X ≤ ∧ (1 − µ ) X µ ( X + 0 − ≤ ∨ X ≤ ∧ X > ∧ (1 − µ )0 X + µ (0 + X − ≤ ∨ X > ∧ X > ∧ (1 − µ ) X X + µ ( X + X − ≤ (cid:1) ∨ (cid:0) X ≤ ∧ X ≤ ∧ X ≤ ∧ X ≤ ∧ (1 − λ )0 0 + λ (0 + 0 − ≥ (1 − µ )0 0 + µ (0 + 0 − > ∨ X > ∧ X ≤ ∧ X ≤ ∧ X ≤ ∧ (1 − λ )0 0 + λ (0 + 0 − ≥ (1 − µ ) X µ ( X + 0 − > ∨ · · ·∨ X > ∧ X > ∧ X > ∧ X > ∧ (1 − λ ) X X + λ ( X + X − ≥ (1 − µ ) X X + µ ( X + X − > (cid:1)(cid:17) B ≤ A ≥ B> . (3) Discard redundant clauses.
The number of clauses in this last formula can be reducedconsiderably. For example, from0 < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ < v < ∧ X ≤ ∧ X ≤ − µ )0 0 + µ (0 + 0 −
1) = − µ ≤ X ≤ ∧ X ≤ X > ∧ X ≤ X ≤ ∧ X >
0, respectively. (CAD computations confirm these assertions quickly.) The last literalsof the first three clauses dropped, we can simplify the first four clauses, corresponding to B ≤
0, to( X ≤ ∧ X ≤ ∨ ( X > ∧ X ≤ ∨ ( X ≤ ∧ X > ∨ ( X > ∧ X > ∧ (1 − µ ) X X + µ ( X + X − ≤ ⇐⇒ ¬ ( X > ∧ X > ∨ ( X > ∧ X > ∧ (1 − µ ) X X + µ ( X + X − ≤ ⇐⇒ X ≤ ∨ X ≤ ∨ (1 − µ ) X X + µ ( X + X − ≤ . The simplification of the remaining 16 clauses is complementary: here, all but the lastsimplify to false, and thus these clauses can be dropped altogether. For verifying that aclause 0 < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ < v < ∧ X (cid:5) ∧ X (cid:5) ∧ X (cid:5) ∧ X (cid:5) ∧ A ≥ B > , is unsatisfiable, with (cid:5) denoting either ≤ or > , it is sufficient to show unsatisfiability of theclause with A ≥ B > A ≥ B ≥
0. For 15 of the16 clauses, a CAD computation quickly yields false for at least one of these two choices.The only surviving clause is the one corresponding to X > ∧ X > ∧ X > ∧ X > ∀ x, y, u, v ∈ R : 0 < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ < v < ⇒ (cid:0) X ≤ ∨ X ≤ ∨ (1 − µ ) X X + µ ( X + X − ≤ ∨ X > ∧ X > ∧ X > ∧ X > ∧ (1 − λ ) X X + λ ( X + X − ≥ (1 − µ ) X X + µ ( X + X − > (cid:1) . OMINANCE IN THE FAMILY OF SUGENO-WEBER T-NORMS 9 (4)
Apply some logical simplification.
First of all, we may drop the conditions X > X > X ≤ X ≤ < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ < v < ∧ (1 − µ ) X X + µ ( X + X − > ⇒ X i > i = 1 , , , X > X > − µ ) X X + µ ( X + X − > ⇒ X i > i = 1 ,
2) is equivalent to X i ≤ ⇒ (1 − µ ) X X + µ ( X + X − ≤ i = 1 ,
2) which allows us to discard X ≤ X ≤ > − µ ) X X + µ ( X + X − ≤ ∀ x, y, u, v ∈ R : 0 < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ < v < ⇒ (cid:0) (1 − µ ) X X + µ ( X + X − ≤ ∨ (1 − λ ) X X + λ ( X + X − ≥ (1 − µ ) X X + µ ( X + X − (cid:1) . (5) Apply some algebraic simplification.
In terms of x, y, u, v we have for the second inequality (cid:0) (1 − λ ) X X + λ ( X + X − (cid:1) − (cid:0) (1 − µ ) X X + µ ( X + X − (cid:1) = ( µ − λ ) (cid:0) ( µ + λ (1 − µ ))( u − v − x − y − − (( u − y − u )(( v − x − v ) + 1 (cid:1) ≥ , from which the factor ( µ − λ ) can be discarded because 0 < λ < µ < ∞ is part of theassumptions. Doing in addition the substitutions x (cid:55)→ − x , y (cid:55)→ − y , u (cid:55)→ − u , v (cid:55)→ − v , the last inequality becomes uy + vx (1 − (1 − µ )(1 − λ ) uy ) ≥ . The substitutions leave the conditions 0 < x <
1, 0 < y <
1, 0 < u <
1, 0 < v < − µ ) X X + µ ( X + X − ≤ u (( λ − x + 1)(( µ − λ − vy + v + y ) + 1)+ ( µ − x (( λ − vy + v + y ) + (( λ − vy + v + y ) + x − ≥ . This can be simplified further by replacing the subexpression ( λ − vy + v + y by a newvariable ˜ v , viz. by making the additional substitution v (cid:55)→ (˜ v − y ) / (1 + ( λ − y ). Thisturns the first inequality into u (( λ − x + 1)(( µ − v + 1) + ( µ − vx + ˜ v + x − ≥ vx (1 − ( λ − µ − uy ) + y (( λ − uy (( µ − x + 1) + u − x )( λ − y + 1 ≥ λ − y + 1 > < v < y < ˜ v < λy . Puttingthings together, we arrive at the equivalent formulation ∀ x, y, u, ˜ v : 0 < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ y < ˜ v < λy ⇒ (cid:0) u (( λ − x + 1)(( µ − v + 1) + ( µ − vx + ˜ v + x − ≥ ∨ ˜ vx (1 − ( λ − µ − uy ) + y (( λ − uy (( µ − x + 1) + u − x ) ≥ (cid:1) . With this last formulation, the quantifier elimination problem can be completed automaticallywithin a reasonable amount of time, at least if it is properly input. The order of the quantifiers,while logically irrelevant, has a dramatic influence on the runtime. We found that a feasibleorder is µ, λ, u, y, ˜ v, x . Mathematica’s command Resolve unfortunately reorders the quantifiersinternally, in this case not to the advantage of the performance. So we have to do the elimination by resorting to the low-level CAD command. It is also advantageous to consider the negation ofthe whole formula, and then taking the complement of the result to obtain the desired region for µ, λ . We thus consider the quantified formula ∃ x, y, u, ˜ v : 0 < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ y < ˜ v < λy ∧ u (( λ − x + 1)(( µ − v + 1) + ( µ − vx + ˜ v + x − < ∧ ˜ vx (1 − ( λ − µ − uy ) + y (( λ − uy (( µ − x + 1) + u − x ) < . To further improve the performance, we consider the cases 0 < λ ≤ λ > < λ ≤
1, the body of the existentially quantified formula is unsatisfiable, even when the first biginequality is dropped: A CAD computation quickly asserts that0 < λ < ∧ λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ y < ˜ v < λy ∧ ˜ vx (1 − ( λ − µ − uy ) + y (( λ − uy (( µ − x + 1) + u − x ) < λ >
1, we proceed in two steps. First we compute a CAD only for1 < λ < µ < ∞ ∧ < x < ∧ < y < ∧ < u < ∧ y < ˜ v < λy ∧ u (( λ − x + 1)(( µ − v + 1) + ( µ − vx + ˜ v + x − < . This takes about a minute and then gives something which is trivially equivalent to1 < λ < µ < ∞ ∧ < u < ∧ < y < ˜ v < − u µ − u ∧ < x < − u − ˜ v − ( µ − u ˜ v (1 + ( λ − u )(1 + ( µ − v ) . Denoting this latter formula by Φ, we then compute the CAD ofΦ ∧ ˜ vx (1 − ( λ − µ − uy ) + y (( λ − uy (( µ − x + 1) + u − x ) < . This takes about three minutes and then returns µ >
17 + 12 √ ∧ (cid:16) − √ µ − √ µ (cid:17) < λ < µ < ∞ ∧ ( . . . )where ( . . . ) is some messy formula involving u, y, ˜ v, x . The specification of the CAD algorithmimplies now that the existentially quantified formula above is valid if and only if µ and λ satisfythis formula with the ( . . . ) part removed. Intersecting the complement of this region with theregion where 0 < λ < µ < ∞ (another quick CAD computation), we finally obtain (cid:16) < µ ≤
17 + 12 √ ∧ < λ < µ < ∞ (cid:17) ∨ (cid:16) µ >
17 + 12 √ ∧ < λ ≤ (cid:16) − √ µ − √ µ (cid:17) (cid:17) as claimed in the beginning. 4. Discussion of results
Equivalent results.
It is interesting to see that Theorem 2 can be expressed in the follow-ing equivalent way. This equivalent result shows that the partial results obtained by the sufficientconditions related to the generalized Mulholland inequality as displayed in Proposition 1 alreadycovered the first four conditions. Condition (v) now closes the missing gap for a full characteriza-tion of dominance between two Sugeno-Weber t-norms.
Theorem 4.
Consider the family of Sugeno-Weber t-norms ( T SW λ ) λ ∈ [0 , ∞ ] . Then, for all λ, µ ∈ [0 , ∞ ] , T SW λ dominates T SW µ if and only if one of the following conditions holds:(i) λ = 0 ,(ii) µ = ∞ ,(iii) λ = µ ,(iv) < λ < min( µ, ,(v) < λ < µ and √ λµ ≤ √ λ + √ µ ) . OMINANCE IN THE FAMILY OF SUGENO-WEBER T-NORMS 11 µλ ∞−∞−∞ ∞ impossible λ > µ λ ≤ µ µλ ∞ ∞ µλ
100 1
Schweizer-Sklar t-norms Frank, Hamacher t-norms Mayor-Torrens, Dubois-Prade t-norms µλ ∞ ∞ λ ≥ µ impossible λ < µ µλ ∞ ∞ √ √ µλ ∞ ∞ impossible λ >µ (iv) (v) Acz´el-Alsina t-norms and others T Sugeno-Weber t-norms
Figure 2.
Schematics for the solution sets for the parametric families of t-normsdisplayed in Tab. 1 and for the family of Sugeno-Weber t-norms.4.2.
Solution sets and their properties.
It is further worth to look a bit in more detail at therelationship between the parameters of t-norms T β being dominated by some T α for some given α ofa parametric family of t-norms ( T λ ) λ ∈ I . Figure 1 visualizes the set of all pairs of parameters ( λ, µ )such that T SW λ dominates T SW µ . We call such a set solution set S , i.e., S = { ( λ, µ ) | T λ (cid:29) T µ } .In a completely analogous way we have illustrated the solution sets for the parametric families oft-norms as summarized in Table 1. The results are displayed in Figure 2 and it is immediatelyobvious that the solution set of the family of Sugeno-Weber t-norms is much more complex thanthe ones for the other families. Note that for the other families we even do have nice Hassediagrams whereas for the family of Sugeno-Weber t-norms a nice graphic is, at least so far, stillmissing.Therefore we inspect the solution set a bit in more detail. The following function is important forthe description of the solution set, so that we briefly discuss its properties: Corollary 5.
Consider the function f : ]9 , ∞ [ → R defined, for all x ∈ ]9 , ∞ [ , by (2) f ( x ) := (cid:18) − √ x − √ x (cid:19) . Then f is strictly decreasing. Moreover, f is involutive, i.e., for all x ∈ ]9 , ∞ [ we have f ( f ( x )) = x .Proof. The function f is continuous and differentiable. Its first derivative can be computed, forall x ∈ ]9 , ∞ [, as f (cid:48) ( x ) = − − √ x ) √ x (3 − √ x ) . For x > √ x >
3, 1 − √ x <
0, and (3 − √ x ) < f (cid:48) ( x ) < x ∈ ]9 , ∞ ], i.e., f is strictly decreasing. By simple computations it can be easily verified that indeed f is an involution, i.e., if f ( x ) = y then also x = f ( y ) and therefore x = f ( y ) = f ( f ( x )) for arbitrary x ∈ ]9 , ∞ [. (cid:3) Note that since f is continuous and strictly decreasing, its range is a convex set. The boundarylimits can be computed as lim x → f ( x ) = ∞ and lim x →∞ f ( x ) = 9 such that Ran f = ]9 , ∞ [.Moreover, 17 + 12 √ f .Let us now turn to a series of results which emphasizes different aspects in the dominance rela-tionship between the members of the family of Sugeno-Weber t-norms. Corollary 6.
For all α ∈ [0 , , it holds that T SW α dominates T SW β for all β ≥ α .Proof. Let α be an arbitrary real number from [0 ,
9] and choose an arbitrary β ≥ α . If α = β or β = ∞ , the dominance relationship trivially holds. We therefore assume that α < β < ∞ . If β ≤
17 + 12 √ T SW α dominates T SW β because of Theorem 2 (iv). If β >
17 + 12 √
2, then f ( β ) > f ( β ) > α such that T SW α dominates T SW β because of Theorem 2 (v). (cid:3) Therefore for all α ∈ [0 ,
9] the set D α = { β | T SW α (cid:29) T SW β } is of the form [ α, ∞ ]. In case that α ∈ (cid:3) ,
17 + 12 √ (cid:2) , D α equals [ α, f ( α )] ∪ {∞} as the following Corollary shows. Corollary 7.
For all α ∈ (cid:3) ,
17 + 12 √ (cid:2) there exists some β α ≥
17 + 12 √ such that(i) ∀ γ ∈ [ α, β α ] : T SW α (cid:29) T SW γ ,(ii) ∀ δ > β α : T SW α (cid:29) T SW δ ⇐⇒ δ = ∞ .Proof. Consider some α ∈ (cid:3) ,
17 + 12 √ (cid:2) . Define β α := f ( α ).Since f is continuous and strictly decreasing it obtains its minimal value at its upper boundary.Since α ≤
17 + 12 √ β α = f ( α ) ≥
17 + 12 √ α ∈ (cid:3) ,
17 + 12 √ (cid:2) .(i) Consider some γ ∈ [ α, β α ]. If α ≤ γ ≤
17 + 12 √ T SW α dominates T SW γ because ofTheorem 2 (iii) and (iv). For 17 + 12 √ < γ ≤ β α the decreasingness and involutivness of f imply that α = f ( f ( α )) = f ( β α ) ≤ f ( γ ) such that T SW α (cid:29) T SW γ due to Theorem 2 (v).(ii) Consider some δ > β α then if δ = ∞ , T SW α (cid:29) T SW δ trivially holds. Vice versa if T SW α dominates T SW δ , then necessarily δ = ∞ , since f ( δ ) < α . (cid:3) Note that for all α ∈ (cid:3) ,
17 + 12 √ (cid:2) it holds that T SW α dominates T SW β α , since β α = f ( α ) >f (17 + 12 √
2) = 17 + 12 √ > α . For α = 17 + 12 √ β α = 17 + 12 √
2, such that T SW α dominates T SW β α since α = β α .Finally for all α ≥
17 + 12 √ D α just consists of α and ∞ . Corollary 8.
For all α , α ≥
17 + 12 √ : T SW α (cid:29) T SW α ⇒ α = α ∨ max( α , α ) = ∞ . From the decreasingness of f we immediately conclude: Corollary 9.
For all α , α ∈ (cid:3) ,
17 + 12 √ (cid:2) : α ≥ α ⇒ β α ≤ β α . These results allow for an alternative proof of the transitivity of dominance in the family ofSugeno-Weber t-norms.
OMINANCE IN THE FAMILY OF SUGENO-WEBER T-NORMS 13
Alternative proof for transitivity.
Consider three members T SW a , T SW b , T SW c of thefamily of Sugeno-Weber t-norms with arbitrary a, b, c ∈ [0 , ∞ ]. We assume w.l.o.g. that a (cid:54) = b (cid:54) = c (cid:54) = a . Assume that T SW a (cid:29) T SW b and T SW b (cid:29) T SW c then a ≤ b ≤ c due to the ordering. Weadditionally assume that c < ∞ for which T SW a (cid:29) T SW c trivially holds. For showing that indeedalso T SW a dominates T SW c we distinguish the following cases: Case 1.: If b ≥
17 + 12 √
2, then c ≥
17 + 12 √ b = c or c = ∞ , the latter being a contradiction. Case 2.:
If 9 < b <
17 + 12 √
2, then there exists, because of Corollary 7, some β b such that c ∈ [ b, β b [. Since a ≤ b , it follows that β a ≥ β b (Corollary 9), and therefore c ∈ [ b, β b [ ⊆ [ a, β a [ such that T SW a (cid:29) T SW c due to Corollary 7. Case 3.: If b ≤
9, then also a ≤ T SW a dominates T SW b .In all cases T SW a dominates T SW c such that the transitivity of dominance in this family is proven. Acknowledgements
The authors would like to thank Peter Paule for indicating that CAD might be helpful for solvingthe problem of dominance in the family of Sugeno-Weber t-norms and as such initiating thecollaboration between the authors and Peter Sarkoci for the idea of representing dominating t-norms by solution sets and for fruitful discussions during an early stage of these investigations.
References [1] C. Alsina, M.J. Frank, and B. Schweizer.
Associative Functions: Triangular Norms and Copulas . WorldScientific Publishing Company, Singapore, 2006.[2] U. Bodenhofer.
A Similarity-Based Generalization of Fuzzy Orderings , volume C 26 of
Schriftenreihe derJohannes-Kepler-Universit¨at Linz . Universit¨atsverlag Rudolf Trauner, 1999.[3] Chris W. Brown. QEPCAD B – a program for computing with semi-algebraic sets.
Sigsam Bulletin , 37(4):97–108, 2003.[4] G.E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In
Automatatheory and formal languages (Second GI Conf., Kaiserslautern, 1975) , pages 134–183. Lecture Notes in Com-put. Sci., Vol. 33. Springer, Berlin, 1975.[5] B. De Baets and R. Mesiar. T -partitions. Fuzzy Sets and Systems , 97:211–223, 1998.[6] B. De Baets and R. Mesiar. Metrics and T -equalities. J. Math. Anal. Appl. , 267:331–347, 2002.[7] S. D´ıaz, S. Montes, and B. De Baets. Transitivity bounds in additive fuzzy preference structures.
IEEE Trans.Fuzzy Systems , 15:275–286, 2007.[8] B. Lafuerza Guill´en. Finite products of probabilistic normed spaces.
Rad. Mat. , 13(1):111–117, 2004.[9] E. P. Klement and R. Mesiar, editors.
Logical, Algebraic, Analytic, and Probabilistic Aspects of TriangularNorms . Elsevier, Amsterdam, 2005.[10] E. P. Klement, R. Mesiar, and E. Pap.
Triangular Norms , volume 8 of
Trends in Logic. Studia Logica Library .Kluwer Academic Publishers, Dordrecht, 2000.[11] E. P. Klement, R. Mesiar, and E. Pap. Triangular norms. Position paper I: basic analytical and algebraicproperties.
Fuzzy Sets and Systems , 143:5–26, 2004.[12] E. P. Klement, R. Mesiar, and E. Pap. Triangular norms. Position paper II: general constructions and param-eterized families.
Fuzzy Sets and Systems , 145:411–438, 2004.[13] E. P. Klement, R. Mesiar, and E. Pap. Triangular norms. Position paper III: continuous t-norms.
Fuzzy Setsand Systems , 145:439–454, 2004.[14] E. P. Klement and S. Weber. Generalized measures.
Fuzzy Sets and Systems , 40:375–394, 1991.[15] E. P. Klement and S. Weber. Fundamentals of a generalized measure theory. The Handbook of Fuzzy SetsSeries, chapter 11, pages 633–651. Kluwer Academic Publishers, Boston, 1999.[16] G. Mayor. Sugeno’s negations and t-norms.
Mathware Soft Comput. , 1:93–98, 1994.[17] R. Mesiar and S. Saminger. Domination of ordered weighted averaging operators over t-norms.
Soft Computing ,8:562–570, 2004.[18] R.B. Nelsen.
An introduction to copulas . Springer Series in Statistics. Springer, New York, second edition,2006.[19] E. Pap, editor.
Handbook of Measure Theory . Elsevier Science, Amsterdam, 2002.[20] S. Saminger.
Aggregation in Evaluation of Computer-Assisted Assessment , volume C 44 of
Schriftenreihe derJohannes-Kepler-Universit¨at Linz . Universit¨atsverlag Rudolf Trauner, 2005.[21] S. Saminger, B. De Baets, and H. De Meyer. On the dominance relation between ordinal sums of conjunctors.
Kybernetika , 42(3):337–350, 2006. [22] S. Saminger, B. De Baets, and H. De Meyer. Differential inequality conditions for dominance between contin-uous archimedean t-norms.
Mathematical Inequalities & Applications , 12(1):191–208, 2009.[23] S. Saminger, R. Mesiar, and U. Bodenhofer. Domination of aggregation operators and preservation of transi-tivity.
Internat. J. Uncertain. Fuzziness Knowledge-Based Systems , 10/s:11–35, 2002.[24] S. Saminger, P. Sarkoci, and B. De Baets. The dominance relation on the class of continuous t-norms from anordinal sum point of view. In H. de Swart, E. Orlowska, M. Roubens, and G. Schmidt, editors,
Theory andApplications of Relational Structures as Knowledge Instruments II . Springer, 2006.[25] S. Saminger-Platz. The dominance relation in some families of continuous archimedean t-norms and copulas.
Fuzzy Sets and Systems , 160:2017–2031, 2009. (doi:10.1016/j.fss.2008.12.009).[26] S. Saminger-Platz, B. De Baets, and H. De Meyer. A generalization of the mulholland inequality for continuousarchimedean t-norms.
J. Math. Anal. Appl. , 345:607–614, 2008.[27] S. Saminger-Platz and C. Sempi. A primer on triangle functions II. (submitted).[28] P. Sarkoci. Dominance of ordinal sums of (cid:32)Lukasiewicz and product t-norm. (submitted).[29] P. Sarkoci. Domination in the families of Frank and Hamacher t-norms.
Kybernetika , 41:345–356, 2005.[30] P. Sarkoci. Dominance is not transitive on continuous triangular norms.
Aequationes Mathematicae , 75:201–207, 2008.[31] B. Schweizer and A. Sklar.
Probabilistic Metric Spaces . North-Holland, New York, 1983.[32] A. Seidl and T. Sturm. A generic projection operator for partial cylindrical algebraic decomposition. In J.R.Sendra, editor,
ISAAC 2003. Proceedings of the 2003 international symposium on symbolic and algebraiccomputation, Philadelphia, PA, USA , pages 240–247, New York, NY, 2003. ACM Press.[33] H. Sherwood. Characterizing dominates on a family of triangular norms.
Aequationes Math. , 27:255–273, 1984.[34] Adam Strzebo´nski. Solving systems of strict polynomial inequalities.
Journal of Symbolic Computation , 29:471–480, 2000.[35] M. Sugeno.
Theory of fuzzy integrals and its applications . PhD thesis, Tokyo Institute of Technology, 1974.[36] R. M. Tardiff. Topologies for probabilistic metric spaces.
Pacific J. Math. , 65:233–251, 1976.[37] R. M. Tardiff. On a generalized Minkowski inequality and its relation to dominates for t-norms.
AequationesMath. , 27:308–316, 1984.[38] A. Tarski.
A decision method for elementary algebra and geometry . University of California Press, Berkeley.,2nd edition, 1951.[39] L. Valverde. On the structure of F -indistinguishability operators. Fuzzy Sets and Systems , 17:313–328, 1985.[40] S. Weber. A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms.
Fuzzy Sets and Systems , 11:115–134, 1983.[41] S. Weber. ⊥ -decomposable measures and integrals for Archimedean t-conorms ⊥ . J. Math. Anal. Appl. ,101:114–138, 1984.
Research Institute for Symbolic Computation, Johannes Kepler University Linz, Altenbergerstrasse69, A-4040 Linz, Austria
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