The structure of n-contractive t-norms

Archimedean components of t-norms are shown to determine the degree of contractivity of the investigated t-norms. It is shown that Archimedean components play an important role also in the case of torsion t-norms. A special class of generated n-contractive left-continuous t-subnorms is introduced, thus allowing to construct left-continuous n-contractive t-norms by means of ordinal sums. Several examples of 3- and 4-contraction t-norms are given.     

Cancellativity properties for t-norms and t-subnorms

On the one hand, cancellativity properties are mainly used to express to which extend the partial functions of a t-subnorm T are injective. On the other hand, the zooms of T corresponding to its non-trivial Archimedean components are t-subnorms that largely determine T. Fixing one out of four basic types of cancellativity (cancellativity, conditional cancellativity, weak cancellativity and weak conditional cancellativity) we figure out which less restrictive type of cancellativity expresses that all maximal Archimedean zooms of T satisfy the given cancellativity property. We lay bare the mutual relationships between all these types of cancellativity and solve an open problem posed by Klement et al.     

Solution of an open problem on pseudo-Archimedean t-norms

An open problem on pseudo-Archimedean t-norms, recently posed by B. De Baets and R. Mesiar [B. De Baets, R. Mesiar, Triangular norms on product lattices, Fuzzy Sets and Systems 104 (1) (1999) 61-75], is discussed.     

t-norms induced by metrics on Boolean algebras

Let d _ν be the metric associated with a strictly positive submeasure ν on some Boolean algebra . If d _ν is bounded from above by 1, E _ν=1−d _ν is a (fuzzy) similarity relation on at least w.r.t. the Lstrok ukasiewicz t-norm, but possibly also w.r.t. numerous further t-norms.In this paper, we show that under certain assumptions on and ν, we may associate with ν in a natural way a continuous t-norm w.r.t. which E _ν is a similarity relation and which, in a certain sense, is the weakest such t-norm. Up to isomorphism, every continuous t-norm arises in this way     

Domination of ordered weighted averaging operators over t-norms

The fusion of transitive fuzzy relations preserving the transitivity is linked to the domination of the involved aggregation operator. The aim of this contribution is to investigate the domination of OWA operators over t-norms whereas the main emphasis is on the domination over the ukasiewicz t-norm. The domination of OWA operators and related operators over continuous Archimedean t-norms will also be discussed.This work was partly supported by network CEEPUS SK-42, COST Action 274 TARSKI and project APVT 20-023402.     

Migrative t-norms with respect to continuous ordinal sums

In this paper we study and characterize continuous α-migrative t-norms T with respect to a continuous t-norm T₀. Depending on whether α is an idempotent element of T₀ or not, the (α, T₀)-migrative property restricts the ordinal sum structure of T especially “locally”, i.e., at α or around it. Outside this well-defined neighbourhood of α, the t-norm T can be arbitrary, under the only condition of keeping it continuous. The investigation exploits the ordinal sum structure of continuous t-norms and our former results related to the migrative property.     

Interval additive generators of interval t-norms and interval t-conorms

The aim of this paper is to introduce the concepts of interval additive generators of interval t-norms and interval t-conorms, as interval representations of additive generators of t-norms and t-conorms, respectively, considering both the correctness and the optimality criteria. The formalization of interval fuzzy connectives in terms of their interval additive generators provides a more systematic methodology for the selection of interval t-norms and interval t-conorms in the various applications of fuzzy systems. We also prove that interval additive generators satisfy the main properties of additive generators discussed in the literature.     

On the representation of intuitionistic fuzzy t-norms and t-conorms

Intuitionistic fuzzy sets form an extension of fuzzy sets: while fuzzy sets give a degree to which an element belongs to a set, intuitionistic fuzzy sets give both a membership degree and a nonmembership degree. The only constraint on those two degrees is that their sum must be smaller than or equal to 1. In fuzzy set theory, an important class of triangular norms and conorms is the class of continuous Archimedean nilpotent triangular norms and conorms. It has been shown that for such t-norms T there exists a permutation /spl phi/ of [0,1] such that T is the /spl phi/-transform of the Lukasiewicz t-norm. In this paper we introduce the notion of intuitionistic fuzzy t-norm and t-conorm, and investigate under which conditions a similar representation theorem can be obtained.     

Modifiers based on some t-norms in fuzzy logic

Modifiers generated by n-placed functions are considered. The subject matter of modifiers are fuzzy sets, i.e. membership functions defined on the unit interval I = [0, 1]. Two sets of modifiers are considered as example cases. One of them is a set of modifiers generated by t-norms and t-conorms. Here different dual pairs of norms create modifiers of different grade of strength. These norms are examples of two-placed functions. Another case is to generate a series of modifiers using only one DeMorgan class of norms. Norms are generalized to be n-placed functions. The place number n takes effect to the strength of a modifier. Two different DeMorgan classes are taken into the consideration. The first steps to the direction of many-valued modifier logics are taken.     

Solutions of fuzzy relation equations based on continuous t-norms

This study is concerned with fuzzy relation equations with continuous t-norms in the form ATR = B, where A and B are the fuzzy subsets of X and Y, respectively; R ⊂ X × Y is a fuzzy relation, and T is a continuous t-norm. The problem is how to determine A from R and B. First, an “if and only if” condition of being solvable is presented. Novel algorithms are then presented for determining minimal solutions when X and Y are finite. The proposed algorithms generate all minimal solutions for the equations, making them efficient solving procedures.     

广义三角模和广义剩余蕴涵之间的对应关系

进一步讨论了广义三角模和广义剩余蕴涵的对应性质,完善和修正了过去已有的一些定理结论,证明广义三角模和广义剩余蕴涵两者之间存在着两大类的对应关系,一类是可以构成伴随对的,而另一类则不能构成伴随对。     

Compatibility of t-norms with the concept of ?-partition

In this paper we study the behaviour of a kind of partitions formed by fuzzy sets, the ?-partitions, with respect to three important operations: refinement, union and product of partitions. In the crisp set theory, the previous operations lead to new partitions: every refinement of a partition is also a partition; the union of partitions of disjoint sets is a partition of the union set; the product of two partitions of two sets is a partition of the intersection of the partitioned sets. It has been proven that ?-partitions extend the three previous properties when the intersection of fuzzy sets is defined by the minimum t-norm and the union by the maximum t-conorm. In this paper we consider any t-norm defining the intersection of fuzzy sets and we characterize those t-norms for which refinements, unions and products of ?-partitions are ?-partitions. We pay special attention to these characterizations in the case of continuous t-norms.     

Basic Fuzzy Logic is the logic of continuous t-norms and their residua

In this paper we prove that Basic Logic (BL) is complete w.r.t. the continuous t-norms on [0, 1], solving the open problem posed by Hájek in [4]. In fact, Hájek proved that such completeness theorem can be obtained provided two new axioms, B1 and B2, were added to the original axioms of BL. The main result of the paper is to show that B1 and B2 axioms are indeed redundant. We also obtain an improvement of the decomposition theorem for saturated BL-chains as ordinal sums whose components are either MV, product or Gödel chains, in an analogous way as for continuous t-norms. Finally we provide equational characterizations of the variety of BL-algebras generated by the three basic BL subvarieties, as well as of the varieties generated by each pair of them, together with completeness results of the calculi corresponding to all these subvarieties.     

Mining closed strict episodes

Discovering patterns in a sequence is an important aspect of data mining. One popular choice of such patterns are episodes, patterns in sequential data describing events that often occur in the vicinity of each other. Episodes also enforce in which order the events are allowed to occur. In this work we introduce a technique for discovering closed episodes. Adopting existing approaches for discovering traditional patterns, such as closed itemsets, to episodes is not straightforward. First of all, we cannot define a unique closure based on frequency because an episode may have several closed superepisodes. Moreover, to define a closedness concept for episodes we need a subset relationship between episodes, which is not trivial to define. We approach these problems by introducing strict episodes. We argue that this class is general enough, and at the same time we are able to define a natural subset relationship within it and use it efficiently. In order to mine closed episodes we define an auxiliary closure operator. We show that this closure satisfies the needed properties so that we can use the existing framework for mining closed patterns. Discovering the true closed episodes can be done as a post-processing step. We combine these observations into an efficient mining algorithm and demonstrate empirically its performance in practice.     

A theory of strict P-completeness

A serious limitation of the theory of P-completeness is that it fails to distinguish between those P-complete problems that do have polynomial speedup on parallel machines from those that don't. We introduce the notion of strict P-completeness and develop tools to prove precise limits on the possible speedups obtainable for a number of P-complete problems.     

基于凸多边形的凸壳算法

确定平面点集的凸壳问题在计算机图形学、图像处理、CAD／CAM、模式识别等众多领域中有广泛的应用。本文根据凸多边形的性质构建了一种新的基于凸多边形的凸壳算法，该算法利用X、y坐标的极值将凸多边形分为几个段，应用凸壳顶点有序性，分段计算凸壳的顶点而得到凸壳。理论分析和实验结果表明，该算法运行速度快效率高，具有较强的实用性。     

The complexity of strict serializability revisited

A well-known result by Sethi (1982) states that strict view- and final-state serializability are NP-complete in general, but decidable in polynomial time if useless values are absent. This paper falsifies one of these claims by proving that the absence of useless values is not sufficient to make strict final-state serializability decidable in polynomial time. Moreover, we show that a slightly stronger condition is sufficient, namely the absence of dead values. All claims refer to the two-step transaction model which is used by Bernstein et al. (1979), Kelter (1985, 1986) and Papadimitriou (1979). These results give new insights into the differences between strict view- and final-state serializability and between uselessness and deadness of values.