Crisp Relations Research Articles

Comparison of fuzzy implication operators by means of fuzzy relational products used for intelligent local path-planning of AUVs

This paper describes the best choice to fuzzy implication operator and α-cut that are proper to the heuristic search technique for real-time collision avoidance of autonomous underwater vehicles (AUVs). A fuzzy implication operator is applied to the computation of fuzzy triangle product that constructs a new fuzzy relation between two fuzzy relations. An α-cut transforms a fuzzy relation into a crisp relation which is represented as a matrix. Those are the theoretical basis of heuristic search technique. In this paper, we review briefly our previous work—a heuristic search technique using fuzzy relational products for the collision avoidance system of AUVs, and propose the selection of a fuzzy implication operator and α-cut which are the most suitable for the search technique. In order to verify the optimality and the efficiency of the selected fuzzy implication operator and α-cut, we simulate every case of α-cut for each fuzzy implication operator in view of the cost of path and the number of α -cut generating acceptable path to the goal.

A soft computing-based approach to spatio-temporal prediction

This paper aims to incorporate intelligent mechanisms based on Soft Computing in Geographical Information Systems (GIS). The proposal here is to present a spatio-temporal prediction method of forestry evolution for a sequence of binary images by means of fuzzy inference systems (FIS), genetic algorithm (GA) and genetic programming (GP). The main inference is based on a fuzzy system which processes a set of crisp/fuzzy relations and infers a crisp relation representing the predicted image at a predefined date. The fuzzy system is formed by a fixed fuzzy rule base and a partition set that may be defined by an expert or optimized by means of a GA. Genetic programming may also be adopted to generate the size of predicted area used in the final stage of the inference process. The developed methodology is applied in regions of Venezuela, France and Guatemala to identify their forestry evolution trends. The proposed approaches are compared with other techniques to validate the system.

Rough sets on fuzzy approximation spaces and applications to distributed knowledge systems

A fuzzy relation is an extension of crisp relation on any set U. Fuzzy proximity relations on U are much more general and abundant than equivalence relations. The fuzzy approximation space which depends upon a fuzzy proximity relation defined on a set U is a generalisation of the concept of knowledge base. So, rough sets defined on fuzzy approximation spaces extend the concept of rough sets on knowledge bases. The results of the present paper extend the basic properties of rough sets and results involving set theoretic operations on types of rough sets that have been established in Tripathy and Mitra (2008).

Descubrimiento de Conocimiento en Bases de Datos Utilizando Técnicas de Morfología Matemática Borrosa

In this paper the effect and usefulness of some filters associated to Fuzzy Mathematical Morphology operators, erosion, dilation, opening, closing, Top-Hat and Hit-or-Miss are analyzed. We propose the interpretation of those operators in processes of Database Knowledge Discovery. Using database information and binary crisp relations as structuring elements, some of these filters are implemented. Also, and using the morphological filters, it is verified that it is possible to analyze information constructed in table of records, interpreted as fuzzy subsets, obtaining not evident useful information. Finally, some properties of the defined operators are analyzed, illustrating the results with an example.

Compatible extensions of fuzzy relations

In 1930 Szpilrajn proved that any strict partial order can be embedded in a strict linear order. This theorem was later refined by Dushnik and Miller (1941), Hansson (1968), Suzumura (1976), Donaldson and Weymark (1998), Bossert (1999). Particularly Suzumura introduced the important concept of compatible extension of a (crisp) relation. These extension theorems have an important role in welfare economics. In particular Szpilrajn theorem is the main tool for proving a known theorem of Richter that establishes the equivalence between rational and congruous consumers. In 1999 Duggan proved a general extension theorem that contains all these results. In this paper we introduce the notion of compatible extension of a fuzzy relation and we prove an extension theorem for fuzzy relations. Our result generalizes to fuzzy set theory the main part of Duggan’s theorem. As applications we obtain fuzzy versions of the theorems of Szpilrajn, Hansson and Suzumura. We also prove that an asymmetric and transitive fuzzy relation has a compatible extension that is total, asymmetric and transitive. Our results can be useful in the theory of fuzzy consumers. We can prove that any rational fuzzy consumer is congruous, extending to a fuzzy context a part of Richter’s theorem. To prove that a congruous fuzzy consumer is rational remains an open problem. A proof of this result can somehow use a fuzzy version of Szpilrajn theorem.

Constructive and axiomatic approaches of fuzzy approximation operators

This paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used. In constructive approach, a pair of lower and upper generalized fuzzy rough (and rough fuzzy, respectively) approximation operators is first defined. The representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented. The connections between fuzzy (and crisp, respectively) relations and fuzzy rough (and rough fuzzy, respectively) approximation operators are further established. In axiomatic approach, various classes of fuzzy approximation operators are characterized by different sets of axioms. The minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators.

Topological relations between fuzzy regions: derivation of verbal terms

A method is presented to generate verbal terms about topological relations between fuzzy regions. The methodology relies on the fuzzy 4-intersection, which is a generalization of the crisp 4-intersection of Egenhofer and co-workers. The computation of the similarity between the fuzzy- and the crisp 4-intersection enables the verbal term, i.e., the linguistic variable, to be derived. The linguistic variable contains a semantic part which gives an immediate association to a crisp relation and a quantifier which indicates the strength of the relationship. Since the derivation of the linguistic variable depends on the definition of the boundary of the fuzzy regions, a method is presented to compute fuzzy boundaries. The approach here defines fuzzy boundaries so that each point in the fuzzy region is associated a partial membership in both the interior and the boundary of the region. This view is different from the boundary definition in crisp topology, but it agrees with the fuzzy set idea that elements can have partial membership in different sets. A simulation experiment demonstrates the properties of the proposed methodology, and it shows how the linguistic variable relates to an inclusion index. An example illustrates how some level of action can be associated to the linguistic variable, which is applicable in the course control of moving crafts, in military applications or in other kinds of operations where the level of warning or action depends on the topological relation between the fuzzy regions. The findings in this article are applicable to geographical information systems (GIS), the modelling of objects with indeterminate boundaries, in the reasoning about relations between geographical objects, or the evaluation of database queries. If the ideas in the present article are implemented in GIS, this will provide an enhanced user interface compared to most GIS today.

On the representation of fuzzy rules in terms of crisp rules

There are two classical ways of viewing a fuzzy set in terms of a combination of crisp sets; either as a weighted max-aggregation of its level-cuts, or as a convex combination of the latter. These views are applied to the particular case of fuzzy relations which are induced by fuzzy rules, where the rules are modeled by implication connectives (defined on a finite subset of the unit interval). The paper studies more closely the representation of such fuzzy rules in terms of a convex combination of gradual rules, a special type of implication-based fuzzy rule inducing a crisp relation. This representation, which might be interpreted in a probabilistic way, is shown to be unique on the assumption that the implication operator used for modeling the fuzzy rule does not satisfy a special kind of strict monotonicity condition. In this case, the crisp relations induced by the involved gradual rules correspond to level-cuts of the fuzzy relation associated with the fuzzy rule. However, other representations might exist if the aforementioned property is not satisfied. Under a slightly stronger (strict) monotonicity condition, the existence of further (non-consonant) representations is even guaranteed. Then, the crisp relations induced by gradual rules do not necessarily correspond to level-cuts of the underlying fuzzy relation. In this sense, the decomposition of a fuzzy rule into a class of crisp implication-based rules is more general than the classical decomposition of fuzzy sets.

Belief functions generated by fuzzy and randomized compatibility relations

The standard combinatoric model of belief functions can be generalized in two ways: (1) the crisp compatibility relation is replaced by a fuzzy relation and (2) the values of crisp compatibility relation are observed through a binary information channel with noise and, perhaps, subjected to a deformation, in other terms, the original compatibility relation is replaced by its randomized modification. Both the generalizations are investigated, separately, in more detail.

Fuzzy time-rank relations and order statistics

The rank ordering of samples is widely used in robust statistics and robust signal processing. Advances in these areas have focused on utilizing joint time-rank (TR) information. The TR information utilized to date is that resulting from a binary, or crisp, relation between the marginal time and rank ordering of samples. This crisp relation, while powerful, contains no information on sample values or spread. This paper generalizes the TR relation through fuzzy set theory. This generalization includes information on sample spread and leads to the concepts of fuzzy TR relations, fuzzy time and rank ordered samples, and fuzzy time and rank indices. These concepts are developed and analyzed through the derivation of fundamental properties. It is shown that the fuzzy TR relations, samples, and indices contain their crisp (standard) counterparts as special cases. These fuzzy generalizations constitute powerful tools that can be exploited in the design of signal processing algorithms.

Characterizing non-monotonic fuzzy relations

This paper is on &-T-monotonic Fuzzy Relations and on the particular case of &-Monotonic Fuzzy Relations (in the crisp case &-Weakly-monotonic and &-Strongly-monotonic Relations, respectively). It goes a little further of the previous work the authors did on this kind of Fuzzy and Crisp Relations by proving a Characterization Theorem of &-T-monotonicity for Reflexive Fuzzy Relations that contains all the former results. This theorem is not only particularized to the crisp case but it is shown that then only crisp Logical States are needed. Of course, also Non-&-T-monotonic Fuzzy Relations and Non-Weak-&-monotonic crisp Relations became characterized.

A characterization of ‘min’ as a procedure for exploiting valued preference relations and related results

AbstractThis paper is concerned with procedures which transform valued preference relations on a set of alternatives into crisp relations. We present a simple characterization of a procedure that ranks alternatives in decreasing order of their minimal performance. This is done by means of three axioms that are shown to be independent. Among other results, we characterize in a very similar manner a procedure called ‘leximin’ and investigate two families of procedures whose intersection is the ‘min’ procedure.

On the approximation of fuzzy preferences by exact relations

Abstract The generalized Hamming distance is commonly used in determining a closest crisp relation to a given fuzzy relation (which, in turn, is crucial in studying exact choices originating from fuzzy preferences). This note illustrates a difficulty attached to this practice, and proposes an alternative metric which is intuitively more appropriate in the case of fuzzy preferences. Some implications of this new metric on finding the nearest exact relation to a fuzzy preference ordering and on the understanding of the max-min transitivity property are also discussed.

Exploitation of a crisp relation in a ranking problem

The outranking methods usually contain two main steps: determination of a (sometimes valued) relation which aggregates the preferences and exploitation of this relation in order to choose, sort or rank the actions. In a ranking problem, the second step is the construction of a (complete or partial) preorder from a (valued) relation. We have considered it was interesting to list the ‘good properties’ such a construction should verify and to analyze some procedures in this context. We present here the first results of this study, which concern the construction of a complete preorder from a non valued relation.

Fuzzy congruences and compatible fuzzy partitions

The main purpose of this paper is to prove that fuzzy congruence relations and compatible fuzzy partitions on a fuzzy algebra f can be described with the help of crisp congruence relations and compatible partitions on the α-cuts of f.

Pool2: a generic system for cognitive map development and decision analysis

The authors present Pool2, a generic system for cognitive map development and decision analysis that is based on negative-positive-neutral (NPN) logics and NPN relations. NPN logics and relations are extensions of two-valued crisp logic, crisp (binary) relations, and fuzzy relations, NPN logics and relations assume logic values in the NPN interval (-1, 1) instead of values in (0, 1). A theorem is presented that provides conditions for the existence and uniqueness of heuristic transitive closures of an NPN relation. It is shown that NPN logic and NPN relations can be used directly to model a target world with a combination of NPN relationships of attributes and/or concepts for the purposes of cognitive map understanding, and decision analysis. Two algorithms are presented for heuristic transitive closure computation and for heuristic path searching, respectively. Basic ideas are illustrated by example. A comparison is made between this approach and others.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Family That Does Not Reproduce Itself

Mean family size in the industrial nations is less than the 2.1 children per couple needed for the population to remain constant over the long run. The countries of Western Europe have a mean family size of about 1.61 children per couple with West Germany as low as 1.42 Japan at 1.71 Europe as a whole at 1.9 and the US at 1.85. The decline of births is related to 1) contraception for the 1st time controlled by women; 2) womens employment outside the home; and 3) the democratization of decision making within couples. Work opportunities for women lower the birth rate but they do so by freeing women from the dictatorship of men. The activity of child rearing is compared with other uncompensated activities that occupy peoples leisure on the one hand and with paid work in the other hand. Clerical work womens current alternative to the 19th century factory has agreeable social elements combined with tolerable and limited duties. Staying home with children can be lonely 7 days a week; it lacks crisp challenges and interpersonal relations. If parents do not spend their money and time producing children they can apply both money and time to the purchase and use of dazzling array of other goods. Children are no longer investments in the traditional sense because 1) children are in large part no longer formed by parents but by television schools and peer groups; and 2) parents rely on their own savings and the state to provide for their old age. A feature of earlier high fertility was the inculcation of differentiated gender roles starting long before marriage. Women has few choices beyond raising children. The spread of high-fertility cultures did not need to be planned by anyone; sheer aithmetic worked at 2nd remove to make male dominance universal. This article argues that under modern conditions there will be few children.

Structure of fuzzy binary relations

The structure of fuzzy binary relations of indifference and preference is studied. The full description of fuzzy equivalence relations in terms of fuzzy partitions is given. All possible logical relations between various transitivity properties of fuzzy preferences are established. These relations permit to choose the notion of a fuzzy quasi-order relation which is an analog of a corresponding crisp relation. The possible approaches to the fuzzy choice are outlined. Finally, the general notion of a relative ‘degree of fuzziness’ is defined in connection with a study of the quantitative description of preservation properties under fuzzy mappings.