Crisp Relations Research Articles

Automatic Construction of Interval-Valued Fuzzy Hindi WordNet using Lexico-Syntactic Patterns and Word Embeddings

A computational lexicon is the backbone of any language processing system. It helps computers to understand the language complexity as a human does by inculcating words and their semantic associations. Manually constructed famous Hindi WordNet (HWN) consists of various classical semantic relations (crisp relations). To handle uncertainty and represent Hindi WordNet more semantically, Type- 1 fuzzy graphs are applied to relations of Hindi WordNet. But uncertainty in the crisp membership degree is not considered in Type 1 fuzzy set (T1FS). Also collecting billions (5,55,69,51,753 relations in HWN) of membership values from experts (humans) is not feasible. This paper applied the concept of Interval-Valued Fuzzy graphs and proposed Interval- Valued Fuzzy Hindi WordNet (IVFHWN). IVFHWN automatically identifies Interval- Valued Fuzzy relations between words and their degree of membership using word embeddings and lexico-syntactic patterns. The experimental results for the word sense disambiguation problem show better outcomes when IVFHWN is being used in place of Type 1 Fuzzy Hindi WordNet and classical Hindi WordNet.

A quantitative fuzzy-valued intersection matrix for obtaining fuzzy relationships between vague spatial objects

Topological relation between geographical objects is one of the key components of the Geographical Information System (GIS). Considering the objects to be exact with sharp boundaries, four intersection, and nine intersection matrices initiated the study of calculating the topological relations. Due to the extensive existence of vague spatial phenomenons, the study of uncertainty object modeling has grown rapidly. Parallelly, the relationship calculating tools have been upgraded to handle vague concepts. The fuzzy 9-intersection matrix and Egg-Yolk methods are popular tools. Although the geographical objects are uncertain, surprisingly, most of these tools generate certain relations between them. The qualitative nature of these tools overlooks the essence of their uncertainty. We propose a quantitative fuzzy-valued 9-intersection matrix to overcome these drawbacks to obtain a fuzzy relationship between uncertain geographical objects. Then two new similarity computation techniques are introduced to calculate the membership grade between the proposed matrix and the known crisp matrices. These similarity computations allow two spatial objects to have partial membership against the eight established topological relations. The quantitative calculations indicate the strength of the relationship. The superiority of the proposed model is exhibited through comparative studies. Further, certain linguistic variables are linked to the evaluated membership grades to generate an immediate association with the known crisp relations. Examples are provided to demonstrate the applicability and efficacy of this association.

On compositions of (L-fuzzy) automata: A categorical approach

This paper studies different compositions of (L-fuzzy) automata by using the notions from category theory, where L is a complete residuated lattice. Specifically, we introduce four different categories of (L-fuzzy) automata with morphisms as crisp relations and L-fuzzy relations, and show that each category is a symmetric monoidal with a product, coproduct, internal monoid, and comonoid. Such categorical study advances the existing categories because of the choice of morphisms. Also, among these compositions, the monoidal description of these categories further enriches the more established fuzzy automata theory.

Fuzzy Bisimulations for Nondeterministic Fuzzy Transition Systems

Bisimulations are established forms of behavioral equivalences for discrete event systems. Recently, bisimulations have been introduced to fuzzy transition systems (FTSs) and developed quickly. Relational lifting by using weight functions is one of the most used methods in bisimulation research, which lifts a crisp relation on the state spaces to a crisp relation on the distributions of state spaces of FTSs. This paper proposes a method, called fuzzy relational lifting method, via fuzzy similarity measures induced by residua (implications) in complete residuated lattices. In contrast with the existing method, fuzzy relational lifting method relaxes the conditions of lifting relation by using weight functions and is a generalization of the method. Some properties of fuzzy lifting operation are presented. Based on the fuzzy lifting operation, a fuzzy bisimulation for nondeterministic fuzzy transition systems (NFTSs) is introduced, which is much more natural and robust than some behavioral measures, and can measure the degree of similarity between any two states which are deemed to be equivalent intuitively but simply distinguished by other behavioral measures. Then some properties of fuzzy bisimulations are provided. Moreover, a fixedpoint characterization of fuzzy bisimulations is given and the greatest fuzzy bisimulation is computed. Finally, a real-valued logic is introduced to characterize fuzzy bisimulations over Godel algebra.

Picture Fuzzy Relations over Picture Fuzzy Sets

Nowadays, picture fuzzy set theory is a flourishing field in mathematics with uncertainty by incorporating the concept of positive, negative and neutral membership degrees of an object. A traditional crisp relation represents the satisfaction or the dissatisfaction of relationship, connection or correspondence between the objects of two or more sets. However, there are some problems that can’t be solved through classical relationships, such as the relationship between two objects being vague. In those situations, picture fuzzy relation over picture fuzzy sets is an important and powerful concept which is suitable for describing correspondences between two vague objects. It represents the strength of association of the elements of picture fuzzy sets. It plays an important role in picture fuzzy modeling, inference and control system and also has important applications in relational databases, approximate reasoning, preference modeling, medical diagnosis, etc. In this article, we define picture fuzzy relations over picture fuzzy sets, including some other fundamental definitions with illustrations. The max-min and min-max compositions of picture fuzzy relations are defined in the light of picture fuzzy sets and discussed some properties related to them. The reflexivity, symmetry and transitivity of a picture fuzzy relation are described over a picture fuzzy set. Finally, various properties are explored related to the picture fuzzy relations over a picture fuzzy set.

Post-symptomatic detection of COVID-2019 grade based mediative fuzzy projection

The concept of fuzzy set, intuitionistic set, and mediative fuzzy set as a generalization of a crisp set have been introduced in many real-life applications. The concept of crisp relation between elements of sets can be extended to fuzzy relations. Extended relations will be considered as relations on fuzzy sets. In this work, we developed the concept of mediative fuzzy relation and meditative fuzzy projection in the context of fuzzy relation and fuzzy projection. We extended the basic operations of fuzzy projection into intuitionistic fuzzy projection and then in the mediative fuzzy projection. We have shown the credibility and impact of mediative index factor involves in the mediative fuzzy projection in context of prediction work in relation to the proposed model. Further, we applied the mediative fuzzy projection in the medical diagnosis in post-COVID-19 patients. The obtained results have also been discussed with their geometrical representation.

Metaphysics, Meaning, and Morality: A Theological Reflection on AI

Theologians often reflect on the ethical uses and impacts of artificial intelligence, but when it comes to artificial intelligence techniques themselves, some have questioned whether much exists to discuss in the first place. Ancient thinkers—namely Augustine of Hippo— enable us to draw forth the moral and metaphysical significance of current developments like the “deep neural networks” responsible for some of the most remarkable achievements of the past few years. First, Augustine’s theology of the natural world’s rationes seminales makes sense of neural networks’ success by explaining the world as a kaleidoscopic refraction of divine Wisdom rather than as a collection of discrete objects standing in crisp relations. Second, his account of interpretive judgment as a moral act bound up with love reveals how our training and use of AI relates to the Christian’s assimilation to Wisdom. The contingent meaning of the neural network reveals AI’s potential either for standing between us and the world or in some sense facilitating a Christian’s regathering of the created echoes of divine Wisdom throughout his or her journey into the Trinity.

Preference implication-based approach to ranking fuzzy numbers

Dombi and Baczy'{n}ski presented a new approach to the problem of implication operation by introducing the preference implication, which has very advantageous properties. In this paper, it is presented how the preference implication is connected with soft inequalities and with sigmoid functions. Utilizing this connection the preference implication-based preference measure for two fuzzy numbers is introduced and its key properties, including the reciprocity, are described. Then, the exact expression for computing the new preference measure for trapezoidal fuzzy numbers is presented. Here, using the new preference measure, two crisp relations over trapezoidal fuzzy numbers are introduced. It is shown that one of them is a strict (but not a total) order relation, and the other one is an equivalence relation. The strict order relation can be used to rank comparable fuzzy numbers, while the equivalence relation, which we call the indifference relation, expresses that the order of some fuzzy numbers is indifferent. These two crisp relations can be used to rank a collection of trapezoidal fuzzy numbers. Lastly, the proposed ranking method is compared with some well-known existing fuzzy number ranking methods.

Limited approximate bisimulations and the corresponding rough approximations

To measure the similarity of nodes in the neighboring subgraphs, Milner introduced the notion of k-limited bisimilarity. Recently, as a weaker version of k-limited bisimilarity, the notion of k-limited similarity was proposed and applied to graph pattern matching. (Bi)simulations have been widely used in comparing the behavior of fuzzy transition systems. In order to study the (bi)simulation semantics of labeled fuzzy transition systems in the residuated lattice-valued logic setting, we introduce an extension of labeled approximation spaces, called the quantitative fuzzy approximation spaces (QFASs), whose labels are equipped with a residuated lattice-valued equality relation. In a QFAS, we define a new notion of limited approximate similarity, to quantify to what extent one state is simulated by another in the neighboring subgraphs, and provide its properties. Based on the new notion, we give a definition of limited approximate simulation and discuss its properties. Then we introduce an ordered pair of relations, one on the state (vertex) set (limited approximate simulation) and one on the edge (transition) set induced by the relation on state set, called VE limited approximate simulation in this paper. We also present a new notion of limited approximate bisimilarity in a QFAS, to quantify to what extent two states are similar in the neighboring subgraphs, and give its properties. One main contribution of the paper is to give a condition for two states to be limited approximate bisimilar and investigate the degree of similarity between two states in a QFAS. Finally, we discuss the relationships between the rough approximations based on the underlying crisp relation induced by underlying labeled fuzzy relation and the rough approximations based on limited approximate bisimilarity.

Soft multicriteria computing supporting decisions on the Forex market

The paper relates to decisions made by traders on the Forex market. Existing trading systems based on technical analysis use crisp relations–binary activation functions to generate BUY or SELL signals for the trader. A large number of signals generated independently by different indicators and relatively low accuracy of the signals is observed, so that they cannot be effectively used in decision-making processes.A new fuzzy multicriteria approach is proposed in this paper. It includes the application of the theory of fuzzy systems and multicriteria analysis. The crisp relations are replaced by fuzzy ones. The signals generated by different indicators at the same time are analyzed in a multicriteria space. Each criterion relates to different indicator with the value defined by a respective membership function. A multicriteria optimization problem is formulated in which the currency pairs — variants in the analysis, having maximum values of all criteria are looked for. An original dominance-based algorithm is presented. It calculates a selected set of non-dominated variants that can be presented to the trader.A computer-based system is constructed on the basis of the proposed approach and including the proposed algorithm. The system having data from the market derives selected variants for further analysis and also check their prediction accuracy. Numerical experiments on real data from the Forex market show advantages of the proposed approach in comparison to the existing traditional crisp systems.

Further characterizations of possibility-theoretical indices in fuzzy optimization

In this paper, we investigate six types of fuzzy relations based on Zadeh's possibility theory. Such relations were originally proposed by Dubois and Prade as comparison indices for fuzzy numbers and were extended by Inuiguchi, Ichihashi, and Kume to general cases. The fuzzy relations we consider are defined for fuzzy sets in a vector space preordered by a convex cone. We reveal the connection of the fuzzy relations with set relations, order-like crisp relations known in the area of set optimization. Specifically, we prove some equalities and equivalences involving the fuzzy relations and set relations under three different assumptions. We finally consider a fuzzy optimization problem in a general form and define its solution concepts. Applying the above equivalences to the problem, we obtain its relationship to certain set optimization problems.

Ranking trapezoidal fuzzy numbers using a parametric relation pair

Huynh et al. introduced a probability-based fuzzy relation for comparing fuzzy numbers (see V. Huynh, Y. Nakamori and J. Lawry (2008) [40]), but they did not detail how to compute it. Here, we will consider this fuzzy relation as a probability-based preference intensity index and present closed formulas for the integrals needed to compute this index for fuzzy sets that have trapezoidal membership functions. Also, we will propose an algorithm to compute this index and a numerical method to approximate it. The comparison of two fuzzy numbers should also be able to capture the situation where the order of the fuzzy numbers cannot be judged; and so, their order may be considered as being indifferent. Here, using the probability-based preference intensity index, we will introduce two crisp relations, which have a common parameter, over a collection of fuzzy sets that have trapezoidal membership functions. Next, we will show that - depending on the parameter value - one of them is a strict order relation and the other one may be interpreted as a relation that expresses the order indifference of fuzzy numbers. We will call this latter one the order indifference relation. Lastly, we will demonstrate how these two relations can be utilized to rank a collection of fuzzy sets that have trapezoidal membership functions.

T-Equivalences: The Metric Behavior Revisited

Since the notion of T-equivalence, where T is a t-norm, was introduced as a fuzzy generalization of the notion of crisp equivalence relation, many researchers have worked in the study of the metric behavior of such fuzzy relations. Concretely, a few techniques to induce metrics from T-equivalences, and vice versa, have been developed. In several fields of computer science and artificial intelligence, a generalization of pseudo-metric, known as partial pseudo-metrics, have shown to be useful. Recently, Bukatin, Kopperman and Matthews have stated that the notion of partial pseudo-metric and a type of generalized T-equivalence are linked. Inspired by the preceding fact, in this paper, we state a concrete relationship between partial pseudo-metrics and the aforesaid generalized T-equivalences. Specifically, a method for constructing partial pseudo-metrics from the new type of T-equivalences and, reciprocally, for constructing the generalized T-equivalences from partial pseudo-metrics are provided. However, important differences between the new approach and the classical one are established. Special interest is paid to the case in which the minimum, drastic, and Łukasiewicz t-norms are under consideration.

Relational Galois connections between transitive fuzzy digraphs

Fuzzy‐directed graphs are often chosen as the data structure to model and implement solutions to several problems in the applied sciences. Galois connections have also shown to be useful both in theoretical and in practical problems.In this paper, the notion of relational Galois connection is extended to be applied between transitive fuzzy directed graphs. In this framework, the components of the connection are crisp relations satisfying certain reasonable properties given in terms of the so‐called full powering.

Fuzzified Crisp Relative Cure Hierarchical Clustering for Temporal Relational Data

A bitemporal data clustering is a significant solution to the diverse problems for finding the intrinsic structure and compact information over temporal data. The temporal data are collected in the series of particular time periods. The various data mining methods have been developed in the temporal relational data analysis. But the accurate analysis was not performed with minimum time. An efficient technique called Fuzzy Crisp Relative Spherical CURE Hierarchy Clustering (FCRSCHC) is introduced for improving the temporal relational data analysis by partitioning the total dataset into different clusters with minimum time as well as space complexity. The CURE hierarchical structure takes the number of scattered temporal data points in the spherical surface for the clustering. After that, ‘k’ number of clusters and the representative points (i.e. cluster centroid) are initialized. Then the distance between the representative point and the temporal data point are calculated using spherical coordinates. The minimum distance between the data points are grouped into a particular cluster. Then the fuzzy memberships between the two cluster representative points are calculated based on the distance metric. The CURE hierarchical structure merges the two clusters based on the crisp relation between the representative points. Then, the newly obtained clusters are validated using the silhouette coefficient to identify the data points are close to its own cluster or their neighboring clusters. Finally, the optimal numbers of clusters are obtained and minimize the incorrect data clustering which improves the accuracy. The experimental evaluation is performed using a bitmeporal dataset with various parameters such as clustering accuracy, false alarm rate, clustering time and space complexity. The results show that FCRSCHC technique improves the clustering accuracy and minimize the time as well as space complexity as compared to the state-of-the-art- works.

Cutting levels of the winning probability relation of random variables pairwisely coupled by a same Frank copula

We consider the winning probability relation associated with a set of (real-valued) random variables. Its computation requires knowledge of the marginal cumulative distribution functions of the random variables and of the bivariate cumulative distribution function of any couple of these random variables. Here, the copulas underlying these bivariate cumulative distribution functions are assumed to be identical and to belong to the parametric family of Frank copulas, although this most likely does not represent the real dependence and might even not be feasible at all; the Frank copula parameter should thus be regarded as a parameter of the method presented. This winning probability relation is then exploited to establish a strict partial order on the set of random variables. This is realized by computing an appropriate α-cut that yields a cycle-free crisp relation, resulting in a strict partial order upon computing its transitive closure. For any given Frank copula, we focus on finding the lowest possible value of α, called (minimal) cutting level, such that cutting the winning probability relation strictly above that value results in a crisp relation that is free from cycles of length m, with m∈{3,4,…}, irrespective of the marginal cumulative distribution functions of the random variables. We are able to give closed-form expressions for these cutting levels: (1) for all m when the copula is either the independence copula or one of the two Fréchet-Hoeffding bounds; (2) for any Frank copula when m=4 or when m tends to infinity. The relationship between the cutting level, the cycle length and the Frank copula parameter is discussed.

On Neutrosophic Crisp Relations

The aim of this paper is to introduce a new types of neutrosophic crisp relations as a generalization to intuitionistic relations due to Indira et al.[9], and study some of its properties. Finally, the concepts of the star and retract neutrosophic relations are introduces and studied and some properties of these concepts will be investigated.

The preposition at from a spatial language, cognition, and information systems perspective

While both linguistic semantics and geographic information science have struggled to accurately and usefully define the meanings of natural language prepositions, there has been less dialogue between these disciplines than one might expect. In this paper we set out to describe the meaning of the English preposition at using well-established formal models of qualitative spatial relations within geographic information science (GIScience), ultimately proposing an account for its meaning which is novel for both discipline areas, and we relate this description to previous definitions of at from within linguistic semantics. The investigation in this paper illustrates that at cannot be handled using the mathematical formalizations of crisp spatial relations in GIScience. However, the model proposed in this paper based on contrast sets, Voronoi Diagrams and Galton & Hood’s (2005) Anchoring theory, contributes to a better understanding of the meaning of at. The model also enables improved automatic interpretations of the preposition by partitioning space into contrast sets and representing at as indicating that a spatial entity is more closely related to a reference object than to any other in the set. We show how the model can be framed within a formal semantics account and how it can in principle be extended to account for spatial uses of other prepositions as well. The paper demonstrates potential areas of fruitful cross-fertilisation between GIScience and linguistic semantics. EARLY ACCESS

A mixed integer programming-based global optimization framework for analyzing gene expression data

The analysis of high throughput gene expression patients/controls experiments is based on the determination of differentially expressed genes according to standard statistical tests. A typical bioinformatics approach to this problem is composed of two separate steps: first, a subset of genes with altered expression level is identified; then the pathways which are statistically enriched by those genes are selected, assuming they play a relevant role for the biological condition under study. Often, the set of selected pathways contains elements that are not related to the condition. This is due to the fact that the statistical significance is not sufficient for biological relevance. To overcome these problems, we propose a method based on a large mixed integer program that implements a new feature selection model to simultaneously identify the genes whose over- and under-expressions, combined together, discriminate different cancer subtypes, as well as the pathways that are enriched by these genes. The innovation in this model is the solutions are driven towards the enrichment of pathways. That may indeed introduce a bias in the search; such a bias is counter-balanced by a wide exploration of the solution space, varying the involved parameters in their feasible region, and then using a global optimization approach. The conjoint analysis of the pool of solutions obtained by this exploration should indeed provide a robust final set of genes and pathways, overcoming the potential drawbacks of relying solely on statistical significance. Experimental results on transcriptomes for different types of cancer from the Cancer Genome Atlas are presented. The method is able to identify crisp relations between the considered subtypes of cancer and few selected pathways, eventually validated by the biological analysis.

ENHANCEMENT OF GENERIC BUILDING MODELS BY RECOGNITION AND ENFORCEMENT OF GEOMETRIC CONSTRAINTS

Abstract. Many buildings in 3D city models can be represented by generic models, e.g. boundary representations or polyhedrons, without expressing building-specific knowledge explicitly. Without additional constraints, the bounding faces of these building reconstructions do not feature expected structures such as orthogonality or parallelism. The recognition and enforcement of man-made structures within model instances is one way to enhance 3D city models. Since the reconstructions are derived from uncertain and imprecise data, crisp relations such as orthogonality or parallelism are rarely satisfied exactly. Furthermore, the uncertainty of geometric entities is usually not specified in 3D city models. Therefore, we propose a point sampling which simulates the initial point cloud acquisition by airborne laser scanning and provides estimates for the uncertainties. We present a complete workflow for recognition and enforcement of man-made structures in a given boundary representation. The recognition is performed by hypothesis testing and the enforcement of the detected constraints by a global adjustment of all bounding faces. Since the adjustment changes not only the geometry but also the topology of faces, we obtain improved building models which feature regular structures and a potentially reduced complexity. The feasibility and the usability of the approach are demonstrated with a real data set.