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Abstract The present paper defines four new kinds of measures of roughness of covering rough sets induced by locally finite covering approximation (LFC -, for brevity) spaces which are generalizations of finite covering approximation spaces. More precisely, consider an LFC -space (U , C) and a nonempty set X (⊆ U). Using a reduction of a given LFC -space (U , C) , the present paper firstly establishes two types of rough membership functions of the set X with respect to LFC -spaces (U , C) , where each of the cardinalities of the sets U , C , and X need not be finite. Next, it also develops another two kinds of notions of roughness of digital topological rough sets. Indeed, these notions are based on the concepts of accuracy of rough sets derived from LFC -spaces. Furthermore, we use them to the estimation of roughness of a covering rough set of X (⊆ U). Besides, we estimate roughness of a digital topological rough set, such as measures of roughness of the Khalimsky and Marcus–Wyse topological rough sets. Moreover, we compare between the measures of the Khalimsky topological and the Marcus–Wyse topological roughness. [ABSTRACT FROM AUTHOR]

Abstract In Massanet et al. (2017) [1] a new property of fuzzy implication functions, called the invariance property with respect to powers of a continuous t-norm, was introduced and its application in approximate reasoning was studied. In the same paper, the novel family of power based implications was presented as a family of fuzzy implication functions that satisfy such invariance property. Unfortunately, this fact is not entirely true since as we will prove in this paper, there exist some power based implications generated from specific ordinal sum t-norms which do not fulfil the invariance property. Thus, a characterization of which continuous t-norms can be used to generate power based implications satisfying the power based invariance property is presented. Additionally, as an alternative solution, we introduce in this paper a slight modification of this property in such a way that all power based implications satisfy it. [ABSTRACT FROM AUTHOR]

Abstract This paper introduces an innovative approach for handling 2D compound hypotheses within the Belief Function framework. We propose a polygon-based generic representation which relies on polygon clipping operators, as well as on a topological ordering of the focal elements within a directed acyclic graph encoding their interconnections. This approach allows us to make the computational cost for the hypothesis representation independent of the cardinality of the discernment frame. For belief combination, canonical decomposition and decision making, we propose efficient algorithms which rely on hashes for fast lookup, and which benefit from the proposed graph representation. An implementation of the functionalities proposed in this paper is provided as an open source library. In addition to an illustrative synthetic example, quantitative experimental results on a pedestrian localization problem are reported. The experiments show that the solution is accurate and that it fully benefits from the scalability of the 2D search space granularity provided by our representation. [ABSTRACT FROM AUTHOR]

Abstract An evenly convex credal set is a set of probability measures that is evenly convex; that is, a set that is an arbitrary intersection of open halfspaces. An evenly convex credal set can for instance encode preference judgments through strict and non-strict inequalities such as P (A) > 1 / 2 and P (A) ≤ 2 / 3. This paper presents an axiomatization of evenly convex sets from preferences, where we introduce a new (and very weak) Archimedean condition. We examine the duality between preference orderings and credal sets; we also consider assessments of almost preference and natural extensions. We then discuss regular conditioning, a concept that is closely related to evenly convex sets. Highlights • A set of axioms is presented for evenly convex credal sets. • An evenly convex set is an intersection of arbitrarily many open halfspaces; a credal set is a set of probability measures. • Axioms show how preference orderings can be represented by evenly convex credal sets. • Axioms include irreflexivity, transitivity, cancellation, monotonicity, and a new Archimedean condition. • Paper discusses duality, natural extensions, regular conditioning. [ABSTRACT FROM AUTHOR]

Abstract How to manage conflict is still an open issue in Dempster–Shafer (D-S) evidence theory. The conflict coefficient k in D-S evidence theory cannot represent conflict reasonably, especially sometimes two Basic Probability Assignments (BPAs) are identical but k is not zero. Jousselme distance can well measure the similarity or conflict between two BPAs but it becomes invalid in some cases. Some scholars introduced the concept of correlation coefficient to measure the similarity of two BPAs. However, existing correlation coefficients of BPAs are unstable or insensitive to quantify the conflict and sometimes deduce wrong results. In this paper, a novel correlation coefficient is proposed which takes into consideration both the non-intersection and the difference among the focal elements. It satisfies all the requirements for a metric. Some numerical examples and comparisons of this paper demonstrate the effectiveness of the correlation coefficient. Highlights • A novel correlation coefficient is proposed to measure the similarity of two BPAs. • Both the non-intersection and the difference among the focal elements are taken into consideration in the new method. • The new method satisfies all the requirements for a metric. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an analytical type-reduction method called concave analytical type-reduction method with spikes (CATRS) of two kinds of spiked concave type-2 fuzzy sets is proposed. The concave type-2 fuzzy sets which are considered in the paper include triangular type-2 fuzzy set, trapezoidal type-2 fuzzy set and its spiked versions. The analytical type-reduction method is free of the following steps, such as discourse partition and discourse refinement for algorithms' discrete implementation, thus its calculation complexity is reduced. To obtain the formulae of concave triangular type-2 fuzzy set, the method proposed by Starczewski is extended from the convex case to the concave one, secondary membership function with spikes is considered as a special form of triangular fuzzy set. Centroid set is formed by all the type-1 fuzzy sets that is derived from the convex part of the primary membership with or without spikes at a given discourse partition point. Moreover, the method proposed for concave triangular type-2 fuzzy set is extended to concave trapezoidal type-2 fuzzy set with a trapezoidal fuzzy number approximation operator. The proposed analytical type-reduction methods can address a more general class of type-2 fuzzy sets, it is more convenient for type-2 fuzzy modeling and inference, and the application of type-reduction method to variable universe of discourse controller design shows that the approximation method is applicable for the adaptive fuzzy controller design. Highlights • An analytical type-reduction method for spiked concave type-2 fuzzy sets has been proposed. • The analytical type-reduction method is free of much of the discourse partition and discourse refinement. • The proposed analytical type-reduction methods can address a more general class of type-2 fuzzy sets' type reduction problem. [ABSTRACT FROM AUTHOR]

A game-theoretical concept of an exact (cooperative) game corresponds to the notion of a discrete coherent lower probability, used in the context of imprecise probabilities. The collection of (suitably standardized) exact games forms a pointed polyhedral cone and the paper is devoted to the recognition of extreme rays of that cone, whose generators are called extreme exact games . We give a necessary and sufficient condition for an exact game to be extreme. Our criterion leads to solving a simple linear equation system determined by a certain min-representation of the game. It has been implemented on a computer and a web-based platform for testing the extremity of an exact game is available, which works with a modest number of variables. The paper also deals with different min-representations of a fixed exact game μ , which can be compared with the help of the concept of a tightness structure (of a min-representation) introduced in the paper. The collection of tightness structures (of min-representations of μ ) is shown to be a finite lattice with respect to a refinement relation. We give a method to obtain a min-representation with the finest tightness structure, which construction comes from the coarsest standard min-representation of μ given by the (complete) list of vertices of the core (polytope) of μ . The newly introduced criterion for exact extremity is based on the finest tightness structure. [ABSTRACT FROM AUTHOR]

Analogical proportions are statements of the form a is to b as c is to d . Such expressions compare the pair ( a , b ) with the pair ( c , d ) . Previous papers have developed logical modelings of such proportions both in Boolean and in multiple-valued settings. They emphasize a reading of the proportion as “the way a and b differ is the same as c and d differ”. The ambition of this paper is twofold. The paper first provides a deeper understanding and further justifications of the Boolean modeling, before introducing analogical inequalities, where “as” is replaced by “as much as” in the comparison of two pairs. From an abstract viewpoint, analogical proportions are supposed to obey at least three postulates expressing reflexivity, symmetry, and stability under central permutation. Nevertheless these postulates are not enough to determine a single model and a minimality condition has to be added as shown in this paper. These models are organized in a complete lattice based on set inclusion. This leads us to discuss lower and upper approximations of the minimal model. Apart from being minimal, this model can also be evaluated in terms of Kolmogorov complexity via an expression reflecting the intended meaning of analogy. We show that the six Boolean patterns of the minimal model that make Boolean analogy true minimize this expression. Besides, analogical proportions extend to 4-tuples of Boolean vectors. This enables us to explain why analogical proportions also reads in terms of similarity (rather than difference, i.e., dissimilarity): a and d share the same presence or absence of features as b and c . Moreover, we establish a link between analogical proportion and Hamming distances between components of the proportion. We also emphasize that analogical proportions are pervasive in any comparison of two vectors a and d that implicitly induce the existence of “intermediary” vectors b and c forming together such a proportion. The similarity reading and the dissimilarity reading of a Boolean analogical proportion are no longer equivalent in the multiple-valued setting, where they give birth to two distinct options that are recalled. These options are also discussed with respect to their capability to handle so-called “continuous” logical proportions of the form a is to b as b is to c involving some idea of “betweenness”. In all the previously investigated issues, the pairs involved in the 4-tuples were compared via equalities of similarities or equalities of dissimilarities. This observation suggests to also consider statements of the form “ a is to b at least as much as c is to d ”, leading to the concept of “analogical inequalities”. Thus, instead of expressing equality between differences or similarities, as it is the case for analogical proportions, it is also interesting to express inequalities between such differences or similarities. Starting from the modeling of analogical proportions, we investigate the logical modeling of analogical inequalities, both in the Boolean and in the multiple-valued cases, and discuss their potential use in relation with some recent related work in computer vision. [ABSTRACT FROM AUTHOR]

This paper clarifies the relation between case-based prediction and analogical transfer. Case-based prediction consists in predicting the outcome associated with a new case directly from its comparison with a set of cases retrieved from a case base, by relying solely on a structured memory and some similarity measures. Analogical transfer is a cognitive process that allows to derive some new information about a target situation by applying a plausible inference principle, according to which if two situations are similar with respect to some criteria, then it is plausible that they are also similar with respect to other criteria. Case-based prediction algorithms are known to apply analogical transfer to make predictions, but the existing approaches are diverse, and developing a unified theory of case-based prediction remains a challenge. In this paper, we show that a common principle underlying case-based prediction methods is that they interpret the plausible inference as a transfer of similarity knowledge from a situation space to an outcome space. Among all potential outcomes, the predicted outcome is the one that optimizes this transfer, i.e. , that makes the similarities in the outcome space most compatible with the observed similarities in the situation space. Based on this observation, a systematic analysis of the different theories of case-based prediction is presented, where the approaches are distinguished according to the type of knowledge used to measure the compatibility between the two sets of similarity relations. [ABSTRACT FROM AUTHOR]