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In the domain of rough set theory, super-reducts represent subsets of attributes possessing the same discriminative power as the complete set of attributes when it comes to distinguishing objects across distinct classes in supervised classification problems. Within the realm of super-reducts, the concept of reducts holds significance, denoting subsets that are irreducible. Contrastingly, constructs, while serving the purpose of distinguishing objects across different classes, also exhibit the capability to preserve certain shared characteristics among objects within the same class. In essence, constructs represent a subtype of super-reducts that integrates information both inter-classes and intra-classes. Despite their potential, constructs have garnered comparatively less attention than reducts. Both reducts and constructs find application in the reduction of data dimensionality. This paper exposes key concepts related to constructs and reducts, providing insights into their roles. Additionally, it conducts an experimental comparative study between optimal reducts and constructs, considering specific criteria such as shortest length and coarsest granularity, and evaluates their performance using classical classifiers. The outcomes derived from employing seven classifiers on sixteen datasets lead us to propose that both coarsest granularity reducts and constructs prove to be effective choices for dimensionality reduction in supervised classification problems. Notably, when considering the optimality criterion of the shortest length, constructs exhibit clear superiority over reducts, which are found to be less favorable. Moreover, a comparative analysis was conducted between the results obtained using the coarsest granularity constructs and a technique from outside of rough set theory, specifically correlation-based feature selection. The former demonstrated statistically superior performance, providing further evidence of its efficacy in comparison. [ABSTRACT FROM AUTHOR]

Few-shot learning, which aims to identify new classes with few samples, is an increasingly popular and crucial research topic in the machine learning. Recently, the development of deep learning has deepened the network structure of a few-shot model, thereby obtaining deeper features from the samples. This trend led to an increasing number of few-shot learning models pursuing more complex structures and deeper features. However, discarding shallow features and blindly pursuing the depth of sample feature levels is not reasonable. The features at different levels of the sample have different information and characteristics. In this paper, we propose a few-shot image classification model based on deep and shallow feature fusion and a coarse-grained relationship score network (HFFCR). First, we utilize networks with different depth structures as feature extractors and then fuse the two kinds of sample features. The fused sample features collect sample information at different levels. Second, we condense the fused features into a coarse-grained prototype point. Prototype points can better represent the information in this class and improve classification efficiency. Finally, we construct a relationship score network, concatenating the prototype points and query samples into a feature map and sending it into the network to calculate the relationship score. The classification criteria for learnable relationship scores reflect the information difference between the two samples. Experiments on three datasets show that HFFCR has advanced performance. [ABSTRACT FROM AUTHOR]

Multi-label learning (MLL) suffers from the high-dimensional feature space teeming with irrelevant and redundant features. To tackle this, several multi-label feature selection (MLFS) algorithms have emerged as vital preprocessing steps. Nonetheless, existing MLFS methods have their shortcomings. Primarily, while they excel at harnessing label-feature relationships, they often struggle to leverage inter-feature information effectively. Secondly, numerous MLFS approaches overlook the uncertainty in the boundary domain, despite its critical role in identifying high-quality features. To address these issues, this paper introduces a novel MLFS algorithm, named VMFS. It innovatively integrates multi-granulation rough sets with three-way decision, leveraging multi-granularity decision-theoretic rough sets (MGDRS) with variable degrees for optimal performance. Initially, we construct coarse decision (RDC), fine decision (RDF), and uncertainty decision (RDU) functions for each object based on MGDRS with variable degrees. These decision functions then quantify the dependence of attribute subsets, considering both deterministic and uncertain aspects. Finally, we employ the dependency to assess attribute importance and rank them accordingly. Our proposed method has undergone rigorous evaluation on various standard multi-label datasets, demonstrating its superiority. Experimental results consistently show that VMFS significantly outperforms other algorithms on most datasets, underscoring its effectiveness and reliability in multi-label learning tasks. [ABSTRACT FROM AUTHOR]

Trust is an extremely helpful construct when reasoning under uncertainty. Thus, being able to logically formalize the concept in a suitable language is important. However, doing so is problematic for three reasons. First, in order to keep track of the contextual nature of trust, situation trackers are required inside the language. Second, in order to produce trust estimations, agents rely on evidence personally gathered or reported by other agents; this requires elements in the language that can track which agents are used as referrals and how much weight is placed on their opinions. Finally, trust is subjective in nature, thus, personal thresholds are needed to track the trust-propensity of different evaluators. In this paper we propose an interpretation of a probabilistic modal language à la Hennessy-Milner in order to capture a context-aware quantitative notion of trust based on evidence. We also provide an axiomatization for the language and prove soundness, completeness, and decidability results. [ABSTRACT FROM AUTHOR]

Among the multi-source data clustering tasks, there is a kind of frequently encountered tasks where only one of the multi-source datasets is available for sake of privacy and other reasons. The only available dataset is called local dataset, and the other are called external datasets. The horizontal collaborative fuzzy clustering (HCFC) model is a typical one that can deal with such clustering tasks. In HCFC, each external dataset is used through the knowledge mined from it rather than itself. The knowledge expressed as a knowledge partition matrix is fused into the clustering process of the local dataset. Reviewing the existing HCFC models, we can find three issues that need improvement. Firstly, the existing HCFC models quantify the collaboration contribution of each external knowledge by a hyperparameter at dataset-level, and moreover, do not distinguish the collaboration contributions of objects in the same external dataset. This may lead to counterintuitive clustering results. Focused on this issue, this paper proposes an enhanced HCFC (EHCFC) algorithm that extends the collaboration from dataset-level to object-level, and assigns different weights to objects based on the information amount provided by objects. Through EHCFC, a more flexible collaboration and a more intuitive clustering result can be reached. Secondly, the collaboration mechanisms of the existing HCFC models require that the dimensionalities of the partition matrices of external datasets and local dataset are the same, which makes the HCFC algorithms unable to work in many real situations. Focused on this limitation, a knowledge adaption mechanism based on relative entropy and spectral clustering is proposed resulting in a further refined EHCFC-KA algorithm, i.e., EHCFC with knowledge adaption. The proposed knowledge adaption mechanism makes both the HCFC algorithms and the EHCFC algorithm effective and successful in more application scenarios. Finally, we define two indexes in terms of consistency (the consistency of the clustering result with external knowledge) to evaluate the performance of collaborative clustering. Experiments on synthetic datasets and UCI public datasets demonstrate that the proposed EHCFC and EHCFC-KA algorithms outperform the existing HCFC algorithms and achieve significantly better intuitive collaborative clustering performance. [ABSTRACT FROM AUTHOR]

Evidential likelihood-based inference is a new approach to statistical inference in which the relative likelihood function is interpreted as a possibility distribution. By expressing new data as a function of the parameter and a random variable with known probability distribution, one then defines a random fuzzy set and an associated predictive belief function representing uncertain knowledge about future observations. In this paper, this approach is applied to binomial and multinomial regression. In the binomial case, the predictive belief function can be computed by numerically integrating the possibility distribution of the posterior probability. In the multinomial case, the solution is obtained by a combination of constrained nonlinear optimization and Monte Carlo simulation. In both cases, computations can be considerably simplified using a normal approximation to the relative likelihood. Numerical experiments show that decision rules based on predictive belief functions make it possible to reach lower error rates for different rejection rates, as compared to decisions based on posterior probabilities. [ABSTRACT FROM AUTHOR]

Considering L to be a complete residuated lattice, by introducing the notion of L -valued covering on an L -set (as a universe), and then an L -valued neighborhood based on it, we present the concept of L -valued covering-based rough sets. We mainly address the following issues in this paper: Firstly, we present four types of L -valued neighborhood operators and study some of their respective properties. Secondly, we construct L -valued lower (resp., upper) approximation operators and then discuss some of their properties. Finally, we try to propose a kind of MADM problem based on an L -valued covering-based rough set model (for L = [ 0 , 1 ]). [ABSTRACT FROM AUTHOR]

Multi-label feature selection based on fuzzy rough sets, as a key step of multi-label data preprocessing, has been widely concerned by scholars in recent years. Most of the existing multi-label feature selection algorithms directly treat labels as logical labels and use a single distance metric to describe similarity. However, the variability of label descriptions and the limitations of a single distance measure should not be overlooked. In this paper, we propose a fuzzy rough set model with metric learning and label enhancement. Specifically, we use a kernel membership label enhancement algorithm based on JS divergence to convert logical labels into numerical labels, which not only reflects the importance of different labels, but also takes into account the differences in label distribution. In addition, a multi-metric learning algorithm is proposed for multi-label learning, in which the metric distance function under the label space and feature space can be learned autonomously. Then, based on the proposed model, we propose a novel multi-label feature selection algorithm based on metric learning and fuzzy rough sets. On this basis, a fast multi-label feature selection algorithm is further designed to improve the computational efficiency. In the experiments, compared with other nine algorithms on real-world datasets, the results show the superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The criterion fuzzy concept refers to a fuzzy set that represents the decision-maker's subjective preference for each criterion within the universe of criteria. Addressing the challenge of ranking all alternatives based on a given criterion fuzzy concept is a novel research direction in the field of fuzzy multi-criteria ranking issues. This paper proposes a three-phase approach for multi-criteria ranking in fuzzy environments, which combines the criterion fuzzy concept and three-way decision thinking. The proposed approach not only analyzes the decision-making characteristics of all alternatives but also facilitates their ranking. During the first phase, a qualitative classification method based on the criterion fuzzy concept and ideal solutions is defined, which divides all alternatives into three independent decision sub-regions. During the second phase, by analyzing the priority relationships among the alternatives within every sub-region, three local ranking rules for alternatives are proposed to determine the ranking of alternatives in each classification region. During the third phase, the semantic relations among three classification regions are considered to give an overall ranking of all alternatives. Finally, combined with two existing quantitative ranking indicators, multiple data sets are employed to verify the feasibility and superiority of the proposed three-phase multi-criteria ranking approach. [ABSTRACT FROM AUTHOR]

Distributivity of fuzzy implications over t-norms and t-conorms provides an effective way to solve the combinatorial rule explosion problems caused by addition of inputs in fuzzy inference systems, and hence has received considerable attention in the literature. In this paper, we explore the distributivity of a newly-born class of ordinal sum fuzzy implications with respect to t-norms and t-conorms, where the involving fuzzy implications are constructed by following the structure of (S , N) -implications generated from ordinal sum t-conorms and fuzzy negations, and are called N -ordinal sum fuzzy implications with the typical feature that the summand domains of linear transformations of given fuzzy implications are rectangles displayed almost along the minor diagonal of [ 0 , 1 ] 2. The main results of this paper are characterizations of the t-norm and t-conorm solutions to the four usual distributive equations of N -ordinal sum fuzzy implications, where, of particular interest is that the t-norm and t-conorm solutions have the ordinal sum representations with only one nontrivial summand, and then the necessary and sufficient conditions under which N -ordinal sum fuzzy implications distribute over t-norms and t-conorms in different ways are obtained. [ABSTRACT FROM AUTHOR]

Reasoning about uncertainty is one of the main cornerstones of Knowledge Representation. Formal representations of uncertainty are numerous and highly varied due to different types of uncertainty intended to be modeled such as vagueness, imprecision and incompleteness. There is a rich body of theoretical results that has been generated for many of these approaches. It is often the case though, that pragmatic tools for reasoning with uncertainty lag behind this rich body of theoretical results. Rough set theory is one such approach for modeling incompleteness and imprecision based on indiscernibility and its generalizations. In this paper, we provide a pragmatic tool for constructively reasoning with generalized rough set approximations that is based on the use of Answer Set Programming (Asp). We provide an interpretation of answer sets as (generalized) approximations of crisp sets (when possible) and show how to use Asp solvers as a tool for reasoning about (generalized) rough set approximations situated in realistic knowledge bases. The paper includes generic Asp templates for doing this and also provides a case study showing how these techniques can be used to generate reducts for incomplete information systems. Complete, ready to run clingo Asp code is provided in the Appendix, for all programs considered. These can be executed for validation purposes in the clingo Asp solver. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to solve the problem proposed by Çaylı about uninorms on bounded lattices and build close relationships among uninorms constructed in this paper. Based on the known construction methods and researchers' work, we obtain new uninorms on L with the given t-norm and t-conorm by using closure (interior) operators. The new construction methods provide answers to the problem presented by Çaylı. All classes of uninorms constructed via closure (interior) operators in this paper can be closely connected in a quadruple from the views of the logics of finite observations. [ABSTRACT FROM AUTHOR]

Automata theory based on quantum logic has been established by Ying. Nondeterministic fuzzy finite automata theory has been proposed by Cao, and further generalized by Pan et al. In this paper, we propose the notion of nondeterministic finite automaton based on quantum logic whose underlying structure is a complete orthomodular lattice l , called nondeterministic l -valued finite automaton (N l -FA, for short). We show that nondeterministic l -valued finite automata, deterministic l -valued finite automata, and nondeterministic l -valued finite automata with ε -moves are equivalent if and only if l satisfies the distributive law. Next, two l -valued relations in terms of language equivalence are given. Some properties of the two l -valued relations under various operations such as reversal, union, product, and concatenation, are investigated. Finally, we give the degree of the closeness between two N l -FAs with the same set of states to reflect the robustness, and obtain some interesting results. The theory developed in this paper enriches the automata theory based on quantum logic. [ABSTRACT FROM AUTHOR]

In the context of multicriteria decision analysis, Pairwise Comparison Matrices (PCMs) represent a widely-used mathematical structure to capture comparisons about criteria or alternatives. When a confidence (or uncertainty) level is associated to the provided comparison measures, PCMs are naturally extended to the so-called Interval Pairwise Comparison Matrices (IPCM). Classical approaches, based on the eigenvector associated to the largest eigenvalue of the matrix (Analytic Hierarchy Process) or optimization problems (e.g. the Logarithmic Least Squares approach), are frequently adopted in order to estimate absolute utilities from relative judgements in absence of expert's uncertainty. Conversely, in the presence of uncertain measures, computationally intensive approaches based on Monte Carlo analysis have been adopted. In this paper, starting from a given PCM, we provide an approach able to find the smallest multiplicative perturbations of its elements able to perturb the obtained ranking from the ordinal perspective. On the basis of such approach, we provide theoretical results which can be adopted also in the case of uncertain measures (e.g., IPCMs) with the aim to characterize the stability of the obtained ranking on the basis of the given uncertainty. As testified by numerical examples that conclude the paper, the proposed framework can greatly simplify the classical approaches by providing the possibility to make direct assumptions on the final outcome of Monte Carlo analysis and reducing or avoiding its required computational effort. [ABSTRACT FROM AUTHOR]

Three-way decision is about thinking, problem-solving, and computing in threes or through triads. Its philosophy and practice of triadic thinking appear everywhere. In particular, there are a number of three-world or triworld models in different fields and disciplines, where a complex system, a complicated issue, or an intricate concept is explained and understood in terms of three interrelated worlds, with each world enclosing a group of elements or representing a particular view. This paper examines various triadic metaphors and structures in general and three-world metaphors and conceptions in particular for building and explaining basic concepts of three-way decision. The three major objectives are (a) to formulate a TAO (Triading-Acting-Optimizing) framework of three-way decision, (b) to discuss the Dao, the Way of three-way decision, and (c) to articulate the Dao, the Way of three-world thinking. Three-world conceptions offer more insights into three-way decision with new viewpoints, methods, and modes. An emphasis of the paper is on the philosophy and the power of three-way decision for building high-level conceptual models. By using three-part structures, by focusing on only three things, or by considering three basic ingredients, we may build a theory, a model, or a method that is simple-to-understand, easy-to-remember, and practical-to-use. [ABSTRACT FROM AUTHOR]

Fuzzy rough set theory can be used as a tool for dealing with inconsistent data when there is a gradual notion of indiscernibility between objects. It does this by providing lower and upper approximations of concepts. In classical fuzzy rough sets, the lower and upper approximations are determined using the minimum and maximum operators, respectively. This is undesirable for machine learning applications, since it makes these approximations sensitive to outlying samples. To mitigate this problem, ordered weighted average (OWA) based fuzzy rough sets were introduced. In this paper, we show how the OWA-based approach can be interpreted intuitively in terms of vague quantification, and then generalize it to Choquet-based fuzzy rough sets (CFRS). This generalization maintains desirable theoretical properties, such as duality and monotonicity. Furthermore, it provides more flexibility for machine learning applications. In particular, we show that it enables the seamless integration of outlier detection algorithms, to enhance the robustness of machine learning algorithms based on fuzzy rough sets. [ABSTRACT FROM AUTHOR]

As a hot topic in machine learning, prediction has attracted a lot of attention nowadays. Scientific prediction can provide a guide for reducing decision-making losses and making reasonable decisions. However, most of existing prediction models still suffer from limited performance, which cannot reasonably handle complex prediction problems. In addition, there are certain limitations in the scope of different prediction models. In light of the above limitations, the paper proposes a novel error correction prediction model based on the idea of three-way decision (TWD), which is titled an ECP-TWD model. First, the back propagation algorithm optimized neural network (BPNN) model is used to achieve the pre-prediction and obtain initial prediction error series. Second, we further combine the strengths of TWD with ensemble learning, tri-divide all alternatives according to the magnitude of the prediction error of the BPNN model, and apply different strategies to re-predict the prediction error sequence in each region, so as to achieve the correction of predicted values of the BPNN model. Finally, the validity, stability and superiority of the presented model are verified based on the case analysis and experimental analysis. The results show that the ECP-TWD model has the better prediction performance compared to other state-of-the-art prediction models. [ABSTRACT FROM AUTHOR]

This paper investigates the modularity equation including semi-t-operators and T-uninorms. We discuss the modularity equation in two cases, which are semi-t-operators over T-uninorms, and T-uninorms over semi-t-operators, respectively. For the case of semi-t-operators over T-uninorms, all solutions to the equation are given. Moreover, by considering the underlying uninorm of the T-uninorm is locally internal on the boundary, necessary and sufficient conditions of the case for T-uninorms over semi-t-operators are established. [ABSTRACT FROM AUTHOR]

It is common for people to remark that a particular argument is a strong (or weak) argument. Having a handle on the relative strengths of arguments can help in deciding on which arguments to consider, which arguments to regard as acceptable, and on which arguments to present to others in a discussion. In computational models of argument, there is a need for a deeper understanding of argument strength. It is a multidimensional problem, and in this paper, we focus on one aspect of argument strength for deductive argumentation based on a defeasible logic. We assume a probability distribution over models of the language and consider how there are various ways to calculate argument strength based on the probabilistic necessity and sufficiency of the premises for the claim, the probabilistic sufficiency of competing premises the claim, and the probabilistic necessity of the premises for competing claims. We provide axioms for characterizing probability-based measures of argument strength, and we investigate four specific probability-based measures. [ABSTRACT FROM AUTHOR]

In this paper, we investigate capacity preserving Z + d -actions and their ergodicity. We give the concepts of weak ergodicity and strong ergodicity for capacity preserving Z + d -actions. In particular, we establish a weak ergodic theorem and a strong ergodic theorem for capacity preserving Z + d -actions. [ABSTRACT FROM AUTHOR]

Formal concept analysis was first used in collaborative filtering for over one decade. Popular approaches are based on superconcept-subconcept relationship or boolean matrix factorization. In this paper, we design a heuristic approach to construct a set of approximately strong concepts for recommendation. Here strong refers to not only big intent to ensure the similarity among users, but also big extent to ensure the stability of user groups. First, we use the intent threshold as the constraint and the area as the optimization objective to obtain approximately strong concepts. Second, we generate pre-recommendations based on the local popularity of the items implied by each concept. Finally, we determine the actual recommendation according to the number of times the item is pre-recommended to the user. Experiments have been undertaken on five popular datasets with different versions. Results show that our algorithm has lower runtime and/or higher recommendation quality compared with approaches based on concept lattice, matrix factorization, k -nearest neighbors, item-based collaborative filtering and boolean matrix factorization. [ABSTRACT FROM AUTHOR]

The existing three-way conflict analysis models focus on complete situation tables and rarely involve incomplete situation tables. In this paper, we give a preliminary approach to three-way conflict analysis models based on incomplete situation tables. Firstly, we construct three-way conflict analysis models based on an incomplete three-valued situation table (ITVST). We use two different methods to establish the models. One is to establish straightly the model based on the original ITVST. The other is to convert the ITVST into a complete three-valued situation table (TVST) by supplementing the missing values with the maximum probability value method, which is called a complete TVST induced by the maximum probability value method. Then we construct the three-way conflict analysis model based on the induced complete TVST. Secondly, to deal with more uncertainty and complexity in the actual decision-making process, we generalize an ITVST to an incomplete many-valued situation table (IMVST) and establish the corresponding three-way conflict analysis models. Finally, we further extend an IMVST to an incomplete interval-valued situation table (IIVST) and construct three-way conflict analysis models based on an IIVST. We use examples to demonstrate the process of constructing conflict, neutrality, and alliance relations of agents on incomplete situation tables. Some algorithms and basic properties are also addressed. [ABSTRACT FROM AUTHOR]

When dealing with sets of probabilities, distortion or neighbourhood models are convenient practical tools, as they rely on very little parameters. In this paper, we study their behaviour when such models are combined and processed through some reasoning tools. More specifically, we study their behaviour when merging different distortion models quantifying uncertainty on the same quantity, and when manipulating distortion models defined over multiple variables. [ABSTRACT FROM AUTHOR]

One of the challenges in Dempster-Shafer theory is that the data fusion calculation resulting from the popular Dempster's rule of combination is #P-complete. This imposes a computational constraint on the number of belief functions and the number of focal sets that can be combined using Dempster's rule. In this paper we develop Monte Carlo algorithms to approximate Dempster's rule of combination. The algorithms incorporate importance sampling and low-discrepancy sequences. Numerical results suggest the algorithms make it possible to apply Dempster's rule to a much larger number of belief functions and focal sets, and consequently widen the scope of applications of Dempster-Shafer theory. [ABSTRACT FROM AUTHOR]

In this paper, we propose Minimax Probability Machine for Multi-label data classification and is termed as Multi-Label Minimax Probability Machine (MLMPM). Based on data mean and covariance information, MLMPM builds a classifier that minimizes an upper bound on the mis-classification probability of unseen future data. For capturing label correlation we have considered asymmetric co-occurrency matrix into the model. The proposed model has also been extended to non-linear settings using the Mercer Kernel trick. To accelerate the training procedure, iterative weighted least squares is used to train the underlying optimization model efficiently. Extensive experimental comparisons of our proposed method with related multi-label algorithms on synthetic as well as real world multi-label datasets, along with Amazon rainforest satellite images dataset, prove its efficacy. [ABSTRACT FROM AUTHOR]

Pattern mining is one of the most studied fields in data mining. Being mostly motivated by practitioners, pattern mining algorithms are often based on heuristics and are lacking suitable formalization. In this paper, we are revisiting pattern mining, and especially itemset mining, which allows one to analyze binary datasets in searching for interesting and meaningful itemsets and respective association rules. We introduce a concise representation –the closure structure– based on closed itemsets and their minimum generators (called "passkeys") for capturing the intrinsic content of a dataset. The closure structure allows one to understand the content of the dataset in terms of closed sets and equivalence classes of itemsets. We discuss theoretical properties of passkeys which are concise representatives of closed itemsets. We propose a formalization of the closure structure and passkeys in terms of Formal Concept Analysis, which is well adapted to studying such elements. Besides theoretical results, we present the GDPM algorithm for enumerating passkeys and discovering the closure structure. GDPM is rather unique as it returns a characterization of a dataset content in terms of complexity levels, highlighting the diversity and the distribution of the itemsets. Finally, some experiments show how the GDPM algorithm and the closure structure can be practically used. [ABSTRACT FROM AUTHOR]

Fuzzy probabilistic rough set is an important mathematical model in rough set theory and has been widely used in applications. Although many studies have been taken on this topic, the existing fuzzy probabilistic rough set models are insufficient for dealing with data noise and data discretization. This motivates us to develop a better method to handle complex data. In this paper, we propose a novel fuzzy probabilistic rough set model on two universes by using λ -fuzzy sets. To further extend application field, we define upper and lower approximation operators of fuzzy probabilistic rough set in a multigranulation context and propose a novel multigranulation fuzzy probabilistic rough set model on two universes. Moreover, some interesting properties of the models are studied. The effectiveness and superiority of them are illustrated by numerical examples and experiments compared with existing models. Experimental results show that the proposed models have better performance in classification and have potential applications in decision making problems. [ABSTRACT FROM AUTHOR]

Attribute reduction is an important application of rough set theory. Most existing rough set models do not consider the weight information of attributes in information systems. In this paper, we first study the weight of each attribute in information systems by using data binning method and information entropy theory. Then, we propose a new rough set model of weighted neighborhood probabilistic rough sets (WNPRSs) and investigate its basic properties. Meanwhile, the dependency degree formula of an attribute relative to an attribute subset is defined based on WNPRSs. Subsequently, we design a novel attribute reduction method by using WNPRSs and the corresponding algorithm is also given. Finally, to evaluate the performance of the proposed algorithm, we conduct a data experiment and compare it with other existing attribute reduction algorithms on eight public datasets. Experimental result demonstrates that the proposed attribute reduction algorithm is effective and performs better than some of the existing algorithms. [ABSTRACT FROM AUTHOR]

The theory of three-way decision was originally introduced as three regions to explain rough sets. Nowadays we can apply this theory to many important fields by defining reasonable trisecting rules. In this paper, we reconsider the two-universe model of three-way decision proposed by Xu et al. and extend it to a more general level. Specifically, we introduce a pair of thresholds to the model and propose a generalized model of three-way decision with ranking and reference tuple. We prove that, although the pairs of thresholds are infinite, we only need to consider a finite number of pairs of thresholds and their corresponding trisections. Based on this theoretical foundation, we further define a unique measure to assess the trisections, compare trisections through this measure, and propose an algorithm to compute the optimal trisection in finite steps. By comparison with other models and an example of application, we demonstrate that the generalized model is more expressive and more practical than the previous one, and has some advantages over the compared models in accuracy of trisecting and validity of explaining. [ABSTRACT FROM AUTHOR]

We study the representation of sets of desirable gambles by sets of probability mass functions. Sets of desirable gambles are a very general uncertainty model, that may be non-Archimedean, and therefore not representable by a set of probability mass functions. Recently, Cozman (2018) has shown that imposing the additional requirement of even convexity on sets of desirable gambles guarantees that they are representable by a set of probability mass functions. Already more that 20 years earlier, Seidenfeld et al. (1995) gave an axiomatisation of binary preferences—on horse lotteries, rather than on gambles—that also leads to a unique representation in terms of sets of probability mass functions. To reach this goal, they use two devices, which we will call 'SSK–Archimedeanity' and 'SSK–extension'. In this paper, we will make the arguments of Seidenfeld et al. (1995) explicit in the language of gambles, and show how their ideas imply even convexity and allow for conservative reasoning with evenly convex sets of desirable gambles, by deriving an equivalence between the SSK–Archimedean natural extension, the SSK–extension, and the evenly convex natural extension. [ABSTRACT FROM AUTHOR]

By applying the principles of three-way decision as thinking in threes, in this paper I introduce a conceptual model of data science in three steps. First, I examine examples of triadic thinking in general and trilevel thinking in specific in data science. Then, based on Weaver's trilevel categorization of communications problems, I propose the concept of the symbols-meaning-value (SMV) space and discuss three perspectives on the SMV space from the viewpoints of information science and management science, cognitive science, and computer science. I label the operations on the SMV three levels metaphorically as seeing, knowing, and doing. Finally, I put forward a SMV-space-based conceptual model of data science, in which data are a resource, the power of data is the knowledge embedded in data, and the value of data is the wise decision and the best course of action supported by data. The goals and functions of data science at the SMV three levels are, respectively, making data available, making data meaningful, and making data valuable. To demonstrate the potential contributions of the conceptual model, I comment on some of its practical values and implications. [ABSTRACT FROM AUTHOR]

In this paper we propose efficient methods for elicitation of complexly structured preferences and utilize these in problems of decision making under (severe) uncertainty. Based on the general framework introduced in Jansen et al. (2018) [37] , we now design elicitation procedures and algorithms that enable decision makers to reveal their underlying preference system (i.e. two relations, one encoding the ordinal, the other the cardinal part of the preferences) while having to answer as few as possible simple ranking questions. Here, two different approaches are followed. The first approach directly utilizes the collected ranking data for obtaining the ordinal part of the preferences, while their cardinal part is constructed implicitly by measuring the decision maker's consideration times. In contrast, the second approach explicitly elicits also the cardinal part of the decision maker's preference system, however, only an approximate version of it. This approximation is obtained by additionally collecting labels of preference strength during the elicitation procedure. For both approaches, we give conditions under which they produce the decision maker's true preference system and investigate how their efficiency can be improved. For the latter purpose, besides data-free approaches, we also discuss ways for statistically guiding the elicitation procedure if data from elicitations of previous decision makers is available. Finally, we demonstrate how the proposed elicitation methods can be utilized in problems of decision under (severe) uncertainty. Precisely, we show that under certain conditions optimal decisions can be found without fully specifying the preference system. [ABSTRACT FROM AUTHOR]

As two important methods used to deal with uncertainty, the rough set theory and the evidence theory have close connections with each other. The purpose of this paper is to examine relationships between the relation-based rough set theory and the evidence theory, and to present interpretations of belief structures in relation-based rough set algebras. The probabilities of relation lower and upper approximations from a serial relation yield a pair of belief and plausibility functions and its belief structure. Properties of the belief structures induced by different relation-based rough set algebras are explored in this paper. The belief structure induced from a reflexive (serial and transitive, serial and symmetric, serial and Euclidean, respectively) relation is reflexive (transitive, symmetric, Euclidean, respectively). Conversely, for a reflexive (transitive, symmetric, Euclidean, respectively) belief structure, there exist a probability and a reflexive (serial and transitive, serial and symmetric, serial and Euclidean, respectively) relation such that the belief and plausibility functions defined by the known belief structure are, respectively, the belief and plausibility functions induced by the relation approximation operators. Then, necessary and sufficient conditions for a belief structure to be the belief structure induced by the relation approximation operators from different binary relations are presented. [ABSTRACT FROM AUTHOR]

We make gradual generalizations in this paper, from the concepts of twin approximation operators in covering rough set theory to the concepts of couple fuzzy covering rough set models in fuzzy rough set theory, and further to the concepts of couple L -fuzzy covering rough set models in L -fuzzy rough set theory. Given a fuzzy covering approximation space (U , C ˜) and β ∈ (0 , 1 ] , for each x ∈ U , we divide C ˜ into two parts ⊤ x = { C ˜ i ∈ C ˜ : C ˜ i (x) ≥ β } and ⊥ x = { C ˜ i ∈ C ˜ : C ˜ i (x) < β }. To fully describe x from positive aspect and negative aspect, both the two parts ⊤ x and ⊥ x are important, especially the combination of them. So, in this paper, based on the two parts, we define 1 st couple β -fuzzy covering rough set models [ ( P − ˜ , P + ˜) , ( Q − ˜ , Q + ˜) ] and 2 nd couple fuzzy β -covering rough set models [ ( P − ‾ , P + ‾) , ( Q − ‾ , Q + ‾) ]. Both the two types of couple fuzzy β -covering rough set models are generalizations of the twin approximations defined in covering rough set theory. Since each pair of operators in these models are not only closely related but complementary, they can be used to analyze and solve practical problems from positive and negative aspects so as to make a crucial decision. So we then give some examples to show their practical value. The relationships between our models and some other models introduced in previous literature are investigated, and the matrix methods are given to calculate the related approximations and to describe the relationships between every couple models. To generalize the couple fuzzy β -covering rough set models to the CCD lattice are of a bit complicated, because any two elements in the lattice cannot be compared with each other generally. After some effort we successfully construct the ideal couple L -fuzzy β -covering rough set models in L -fuzzy rough set theory which are just the generalizations of the two types of couple fuzzy β -covering rough set models. We also obtain the lattice matrix representations to calculate the related approximations. [ABSTRACT FROM AUTHOR]

To solve hard problems, AI relies on a variety of disciplines such as logic, probabilistic reasoning, machine learning and mathematical programming. Although it is widely accepted that solving real-world problems requires an integration amongst these, contemporary representation methodologies offer little support for this. In an attempt to alleviate this situation, we position and motivate a new declarative programming framework in this paper. We focus on the semantical foundations in service of providing abstractions of well-known problems such as SAT, Bayesian inference, generative models, learning and convex optimization. Programs are understood in terms of first-order logic structures with semiring labels, which allows us to freely combine and integrate problems from different AI disciplines and represent non-standard problems over unbounded domains. Thus, the main thrust of this paper is to view such well-known problems through a unified lens in the hope that appropriate solver strategies (exact, approximate, portfolio or hybrid) may emerge that tackle real-world problems in a principled way. [ABSTRACT FROM AUTHOR]

Granular computing is becoming a very hot research field, which has received extensive attention in recent years. It helps us to analyze and solve problems better by dividing complex problems into several simpler ones. Three-way granular concept is an important concept proposed by combining granular computing, formal concept analysis and three-way decision. Using traditional updating methods of three-way granular concepts, a lot of time and space resources are needed when multiple attributes or objects are deleted in formal context. In order to improve the efficiency and flexibility of obtaining three-way concepts, this paper discusses a novel dynamic update method of three-way granular concepts. In this paper, we firstly introduce the related knowledge of three-way granular concepts. Secondly, the update rules of the extension and connotation of attribute-induced three-way granular concepts are discussed in the dynamic formal context to construct three-way granular concepts. Moreover, we develop a method for establishing attribute-induced three-way granular concept by dynamic changes in the case of deleting multiple objects and attributes in the formal context. Furthermore, we design four algorithms to compare between the proposed approaches and traditional updating ways of three-way granular concepts. Finally, the validity of dynamic update method of attribute-induced three-way granular concept is verified through the experimental evaluation using six datasets coming from the University of California-Irvine (UCI) repository. [ABSTRACT FROM AUTHOR]

This paper proposes a sequential three-way decision-making approach based on BWM (best-worst method) and MULTIMOORA for multiple levels of granularity to deal with the multi-attribute group decision-making problems under uncertainty. First, using BWM to preprocess the attribute indicators of the decision problem, the relationship between the criteria important for decision problem is determined. Then, according to the importance of the determined level of granularity, from rough to clear, three-way decisions are made in sequence at each level. In this way, we obtain a lattice based three-way decision-making process. Also, in the process at each granularity, we consider the costs of both the decision process and the decision result. Thus, this paper builds a multigranulation sequential three-way decisions model with cost-sensitive. Next, based on the above-mentioned model, we propose a decision procedure and an algorithm for multi-attribute group decision making. It is worth noting that according to the research goals of the decision problem, MULTIMOORA is used to sort the classified alternatives. And a ranking result of the decision problem is given. Finally, the effectiveness and validity of the proposed model is verified by a multi-attribute group decision problem. The problem is the case of emergency diagnosis and treatment after a highway accident. And put forward reasonable suggestions for further development direction. [ABSTRACT FROM AUTHOR]

This paper considers the issue of aggregation of experts' opinions expressed through not necessarily independent and even conflicting probability distributions or multiple priors. The paper addresses the elicitation of the consensus distribution by introducing the Steiner point of the set of all common distributions among experts and derives its conditional probabilities on future events. [ABSTRACT FROM AUTHOR]

Bayesian games offer a suitable framework for games where the utility degrees are additive in essence. This approach does nevertheless not apply to ordinal games, where the utility degrees do not capture more than a ranking, nor to situations of decision under qualitative uncertainty. The present paper proposes a representation framework for ordinal games under possibilistic incomplete information and extends the fundamental notions of pure and mixed Nash equilibrium to this framework. We show that deciding whether a pure Nash equilibrium exists is a difficult task (NP-hard) and propose a Mixed Integer Linear Programming (MILP) encoding of the problem; as to the problem of computing a possibilistic mixed equilibrium, we show that it can be solved in polynomial time. An experimental study based on the GAMUT game generator confirms the feasibility of the approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a parametrised factor that enables inference on Gaussian networks where linear dependencies exist among the random variables. Our factor representation is effectively a generalisation of traditional Gaussian parametrisations where the positive-definite constraint of the covariance matrix has been relaxed. For this purpose, we derive various statistical operations and results (such as marginalisation, multiplication and affine transformations of random variables) that extend the capabilities of Gaussian factors to these degenerate settings. By using this principled factor definition, degeneracies can be accommodated accurately and automatically at little additional computational cost. As illustration, we apply our methodology to a representative example involving recursive state estimation of cooperative mobile robots. [ABSTRACT FROM AUTHOR]

The concept of q-rung orthopair fuzzy sets (q-ROFSs) severs as an extended form of orthopair fuzzy sets, which excels in flexibly depicting imprecise information existed in multiple attribute group decision making (MAGDM) via allowing a greater space in terms of membership degrees (MD) and non-membership degrees (ND). Moreover, incomplete information systems play a significant role in depicting incomplete information existed in MAGDM. In this paper, for the sake of exploring MAGDM with imprecise and incomplete information, a new MAGDM method based on multigranulation probabilistic models, MULTIMOORA (Multi-Objective Optimization by Ratio Analysis plus the full MULTIplicative form) and the technique of precise order preference (TPOP) method in incomplete q-rung orthopair fuzzy (q-ROF) information systems is systematically investigated. First, the concept of multigranulation (MG) incomplete q-ROF information systems is established. Then, a completion technique is put forward to obtain MG q-ROF information systems via patching lost values. Moreover, three versions of MG q-ROF probabilistic rough sets (PRSs) are put forward in light of MG q-ROF information systems. Afterwards, a new MAGDM method is constructed by means of MG q-ROF PRSs, MULTIMOORA and the TPOP method. At last, two experimental studies from open datasets are performed to present the applicability of the constructed MAGDM method, and corresponding sensitivity analysis and comparative analysis are further conducted to demonstrate the validity of the presented methodology. [ABSTRACT FROM AUTHOR]

The full characterization of different pairs of aggregation operators that fulfill the distributivity law is crucial for many different areas such as the utility theory and the integration theory. The problem of distributivity between uni-nullnorms and Mayor's aggregation operators is being addressed through this paper. Here presented study is the next step in research of this topic and it upgrades some previously obtained results. [ABSTRACT FROM AUTHOR]

Subject-specific causal discovery methods could help to make better decisions in domains where there are variations on causal relations across subjects, such as neuroscience and genetics. However, discovering subject-specific causal models from limited data sets could be challenging. Although there are several causal discovery methods, most of them are focused on finding the common causal relations of a population from data sets with enough samples. The issue of discovering subject-specific causal relations from limited data sets has not been sufficiently explored. In this paper, we propose a knowledge transfer method for discovering subject-specific causal models. Our method discovers causal probabilistic graphical models, up to Markov equivalence classes, from discrete and continuous data sets. We hypothesized that transferring weighted instances of additional sources according to their discrepancy with target instances helps to compensate for the lack of target data and improves the performance of a scored-based causal discovery method. Experimental results on synthetic data sets and benchmark causal Bayesian networks show that our proposal, under few target samples, outperforms significantly in adjacency and arrowhead recovery the baseline and alternative methods. [ABSTRACT FROM AUTHOR]

In this paper, the Ordered Weighted Averaging (OWA) operators will be considered to propose general fuzzy regression modeling technique for crisp/fuzzy-input fuzzy-output data. Here, by considering various kinds of a special form of OWA operators we will be able to 1. investigate different approaches for aggregation of residuals/errors, 2. construct a multi-objective optimization problem, 3. provide high-breakdown estimators, 4. identify poorly fitted data points, which might be considered as outliers while limiting their effects on the model, 5. show how robust alternatives to other fuzzy regression models can be obtained as a result. Finally, a special case of the OWA-based fuzzy regression model known as OWA-Least Trimmed Absolutes Deviations (OWA-LTAD) fuzzy regression model is studied in details for crisp-input fuzzy-output data. An algorithm is investigated in the present study for estimation process in which 1. the highest breakdown value will always be achieved (up to the maximum of 0.5) for problems with multiple outliers, 2. the poorly fitted dataset can be labeled as outliers in an elemental set. In the presence of multiple outliers, the new method is particularly useful, and the performance of the proposed approach is illustrated for modeling several simulated datasets and a real valued dataset all contaminated with multiple outliers as well. We will look at an application of the proposed approach, in particular, to detect outliers in such models. [ABSTRACT FROM AUTHOR]

In this paper we create a bridge between desirability and information algebras: we show how coherent sets of gambles, as well as coherent lower previsions, induce such structures. This allows us to enforce the view of such imprecise-probability objects as algebraic and logical structures; moreover, it enforces the interpretation of probability as information, and gives tools to manipulate them as such. [ABSTRACT FROM AUTHOR]

Planning in stochastic and partially observable environments is a central problem in artificial intelligence. To address this issue, an accurate model or a black-box simulator of the environment is usually needed in the literature. Although some recent approaches have been proposed for learning optimal behaviors under model uncertainty, prior knowledge about the environment is still required to guarantee the performance of the proposed algorithms. With the benefits of the Predictive State Representations (PSRs) approach for state representation and model prediction, in this paper, we introduce an approach for planning under partial observability with no prior domain knowledge, where an offline PSR model is firstly learned and then combined with online Monte-Carlo tree search for planning under model uncertainty. Furthermore, we also showed that with the proposed framework, the PSR models learned via other techniques, e.g., the online PSR model learning approach, can be integrated straightforwardly. By comparing with the state-of-the-art approach of planning under model uncertainty, we demonstrated the effectiveness of the proposed approaches along with the proof of their convergence. [ABSTRACT FROM AUTHOR]

Efficiently discovering causal relations from data and representing them in a way that facilitates their use is an important problem in science that has received much attention. In this paper, we propose an adaptation of the Formal Concept Analysis formalism to the problem of discovering and representing causal relations. We show that Formal Concept Analysis structures and algorithms are well-suited to this problem. [ABSTRACT FROM AUTHOR]

Few-shot learning is one of the significant areas of machine learning, which aims to recognize novel visual classes from few labeled examples. Many existing models make full use of the similarity of inner-class features and achieve satisfactory results. However, these models assume that classes are independent of each other, ignoring the inter-class relationship. In this paper, we propose a few-shot hierarchical classification model via multi-granularity relation networks (HMRN) considering both the inner-class similarity and inter-class relationship. The multi-granularity relationship among coarse- and fine-grained classes is an important auxiliary information in the class hierarchical structure originated from data. Thus, we first extract hierarchical features of different granularity classes according to the membership relationship among the classes. Second, we build multi-granularity relation networks to obtain the inner-class similarity relation of different granularity classes using the hierarchical features. Finally, we consider the tradeoff among the inner-class similarity relation of different granularity classes for hierarchical few-shot learning, which takes the information of coarse-grained classes to assist the learning of fine-grained classes. Experimental results show that our model outperforms several state-of-the-art flat (without hierarchical structure) models and hierarchical models. For example, the accuracy of HMRN is about 3.00% better than that of flat models on the tieredImageNet dataset. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of lattice-valued tree pushdown automata that extends the concepts of lattice-valued pushdown automata, lattice-valued tree automata and fuzzy tree pushdown automata. Also, we consider fuzzy tree languages recognized by lattice-valued (deterministic) tree pushdown automata (called lattice-valued context-free tree languages). Then, we prove the pumping lemma that can be used in proving that certain sets of tree languages are not recognizable. Also, we study closure properties of lattice-valued context-free tree languages with respect to some operations. We show that lattice-valued context-free tree languages recognized by lattice-valued deterministic tree pushdown automata are closed under union, concatenation, star, linear and alphabetic tree homomorphism and inverse tree homomorphism. By applying the pumping lemma on an example, we show that lattice-valued context-free tree languages are not closed under intersection. However, we prove that the intersection of lattice-valued context-free tree language and lattice-valued regular tree language is recognizable by a lattice-valued tree pushdown automaton. Also, we show that lattice-valued non-deterministic tree pushdown automata are only closed under union. However, if L satisfies distributivity, then non-deterministic tree pushdown automata are closed under all of these operations. [ABSTRACT FROM AUTHOR]