Reduction Of Concept Lattices Research Articles

Stream Convolution for Attribute Reduction of Concept Lattices

Attribute reduction is a crucial research area within concept lattices. However, the existing works are mostly limited to either increment or decrement algorithms, rather than considering both. Therefore, dealing with large-scale streaming attributes in both cases may be inefficient. Convolution calculation in deep learning involves a dynamic data processing method in the form of sliding windows. Inspired by this, we adopt slide-in and slide-out windows in convolution calculation to update attribute reduction. Specifically, we study the attribute changing mechanism in the sliding window mode of convolution and investigate five attribute variation cases. These cases consider the respective intersection of slide-in and slide-out attributes, i.e., equal to, disjoint with, partially joint with, containing, and contained by. Then, we propose an updated solution of the reduction set for simultaneous sliding in and out of attributes. Meanwhile, we propose the CLARA-DC algorithm, which aims to solve the problem of inefficient attribute reduction for large-scale streaming data. Finally, through the experimental comparison on four UCI datasets, CLARA-DC achieves higher efficiency and scalability in dealing with large-scale datasets. It can adapt to varying types and sizes of datasets, boosting efficiency by an average of 25%.

Tri-granularity attribute reduction of three-way concept lattices

A real-world dataset often generates a huge concept lattice that may be difficult to understand and impractical to use. The principle of granular computing is that we can understand and use a complex whole at multiple levels of granularity. In light of this, we propose a unified tri-granularity model of various types of concept lattices, which allows us to examine a huge concept lattice at three levels of granularity (i.e., the elementary granularity, local granularity and global granularity). Attribute reduction plays a fundamental role in three-way concept analysis, it simplifies the expression of three-way concepts and thus contributes to a better perception of the knowledge in three-way concept lattices. The tri-granularity model suggests an opportunity to investigate the tri-granularity attribute reduction of three-way concept lattices. The existing research on attribute reduction works at the global granularity but pays little attention to the local granularity, or, even less, to the elementary granularity. Driven by these issues, we supply definitions and methods of local granularity and elementary granularity attribute reduction of three-way concept lattices. These newly proposed two levels of attribute reduction with the existing global granularity attribute reduction together provide a framework for the tri-granularity attribute reduction of three-way concept lattices. We further analyze the relationships among the three levels of attribute reduction. Moreover, the efficacy of our suggested approach is illustrated by an example via the trisections induced by three-way concepts. Finally, we design two tri-granularity attribute reduction algorithms whose effectiveness is further examined by numerical experiments.

Survey on attribute and concept reduction methods in formal concept analysis

Formal concept analysis (FCA) is now widely recognized as a useful approach for extracting, representing, and analyzing knowledge in various domains. The high computational cost of knowledge processing and the difficulty of visualizing the lattice are two key challenges in practical FCA implementations. Moreover, assessing the finalized built-up lattice may be problematic due to the enormous number of formal concepts and the complexity of their connections. The challenge of constructing concept lattices of adequate size and structure to convey high-importance context features remains a significant FCA aim. In the literature, various strategies for concept lattice reduction have been presented. In this work, we suggest a categorization of reduction methods for concept lattice based on three main categories: context pre-processing, non-essential distinctions elimination, and concept filtration, whereby using FCA-based analysis, the most important methods in the literature are analyzed and compared based on six pillars: the preliminary step of the reduction process, domain expert, changing the original data structure, final concept lattice, quality of reduction, and category of reduction method.

A user-guided reduction concept lattice and its algebraic structure

Concept lattice has been widely studied and it is still attracting researchers in many fields by virtue of its well structured knowledge representation. However, it is more obvious that the number of concept lattice nodes generated from the initial formal context is enormous and results in more storage and lower efficiency of building, especially with the increasing of data to deal with. Meanwhile, users are not interested in the whole intents composed of all the attribute sets and it is a time consuming and troublesome job for them to acquire interesting knowledge. In order to deal with these deficiencies, we propose a constrained concept lattice structure referred to as a reduction concept lattice (RCL) guided by user preferences. This type of lattice structure not only can effectively reduce storage space and construction time, but also help to extract the knowledge that users are interested in. In this study, the definition of RCL is first given based on users’ three types of interesting background knowledge described by predicate logic, together with some RCL’s examples to illustrate this type of structure. Next, an algebraic structure of RCL is defined and its completeness is proved, and accordingly a new way is provided for reduction of concept lattice based on user preferences.

Exploring invariance of concept stability for attribute reduction in three-way concept lattice

Exploring invariance of concept stability for attribute reduction in three-way concept lattice

Research on the multi-granularity method of role engineering.

The role-based access control model (RBAC) is an access control approach oriented to enterprise security policy, which plays a leading role in the field of access control due to its good applicability and flexibility. However, as the scale of access subjects and access objects expands, it becomes more difficult to develop a role engineering system that meets security requirements. Particle size calculation can reduce or improve the particle size of the problem and improve efficiency on the premise of obtaining satisfactory solutions. In this study, the granularity calculation method was introduced into the role formulation process of RBAC, which can effectively reduce the complexity of the problem and improve the efficiency of the RBAC role engineer. At the same time, the concept lattice reduction method was used to reduce the role concept lattice and reduce the workload of the system security administrator.

Analysis of Mine Safety Performance Evaluation Law Based on Matter-Element Analysis and Rough Set of Concept Lattice Reduction

The mine safety performance evaluation and its law were taken as research objects; the human factor, mechanical equipment, management system, environment and technology were taken as mine safety state indexes; accident loss and disabling incident frequency rate as mine safety performance evaluation indexes. The classification standard of safety performance was determined according to the range of influence index and evaluation index, and then the influence index value of safety performance was discretized. According to the classification standard, the classical domain and node domain of safety performance evaluation index classification were determined. Combined with the weight of each safety performance evaluation index, the correlation degree of each mine safety performance evaluation was calculated, and then the mine safety performance was evaluated, and the mine safety performance evaluation decision system was constructed. Based on the dominance-based rough set, the evaluation law of mine safety performance evaluation decision system was analyzed. To generate more reliable laws, an efficient concept lattice reduction method was introduced to reduce the attributes of decision system. The example shows that when the concept lattice is used for reduction, there are 84 concepts in the Hasse diagram, and 24 of them can be identified. The decision system has 8 conditional attributes. And 255 power sets can be obtained under 8 conditional attributes. After eliminating the defect attribute, there are 148 reducible sets, among which there are 4 optimal reducible sets with two conditional attributes. The results show that the proposed method can be used to analyze the law of mine safety performance, so as to improve mine safety work.

Reduction of fuzzy-crisp concept lattice based on order-class matrix

Knowledge reduction is one of critical problems in data mining and information processing. It can simplify the structure of the lattice during the construction of fuzzy-crisp concept lattice. In terms of fuzzy-crisp concept, we develop an order-class matrix to represent extents and intents of concepts, respectively. In order to improve the computing efficiency, it is necessary to reduce the size of lattices as much as possible. Therefore the judgement theorem of meet-irreducible elements is proposed. To deal with attribute reductions, we develop a discernibility Boolean matrix in formal fuzzy contexts by preserving extents of meet-irreducible elements via order-class matrix. A heuristic attribute-reduction algorithm is proposed. Then we extend the proposed model to consistent formal fuzzy decision contexts. Our methods present a new framework for knowledge reduction in formal fuzzy contexts.

Reducing concept lattices by means of a weaker notion of congruence

Attribute and size reductions are key issues in formal concept analysis. In this paper, we consider a special kind of equivalence relation to reduce concept lattices, which will be called local congruence. This equivalence relation is based on the notion of congruence on lattices, with the goal of losing as less information as possible and being suitable for the reduction of concept lattices. We analyze how the equivalence classes obtained from a local congruence can be ordered. Moreover, different properties related to the algebraic structure of the whole set of local congruences are also presented. Finally, a procedure to reduce concept lattices by the new weaker notion of congruence is introduced. This procedure can be applied to the classical and fuzzy formal concept analysis frameworks.

Attribute reduction of SE-ISI concept lattices for incomplete contexts.

Three-way concept analysis in incomplete contexts lays the theory dealing with the data in incomplete contexts, especially three kinds of partially known formal concepts including SE-ISI formal concept, ISE-SI formal concept and ISE-ISI formal concept. Generally speaking, not every attribute is essential in an incomplete context since the purpose of research is different. Thus, we propose four kinds of attribute reduction of SE-ISI concept lattices based on different criteria. Then, we discuss the relationships among the four kinds of attribute reduction, including the relationships among the consistent sets and relationships among the reducts. Finally, based on discernibility matrices and discernibility functions, the approaches to obtaining these attribute reduction are presented.

Semantifying formal concept analysis using description logics

Formal Concept Analysis (FCA) is a field of applied mathematics with its roots in order theory, in particular the theory of complete lattices. Over the past 20 years, FCA has been widely studied. Description Logics (DLs) are a family of knowledge representation languages which can be used to represent the terminological knowledge of an application domain in a structured and formally well-understood way. Nowadays, properties and semantics of ontology constructs mainly are determined by DLs. The current research progress and the existing problems of FCA are analyzed. In this paper, we semantify FCA with DLs, in other words, we present an extended FCA (i.e., semantic FCA) by using the concepts of DLs to act as the attributes of formal contexts. Furthermore, we semantify the three components (i.e., formal concepts, attribute implications, and concept lattices) of traditional FCA. In addition, we also study the attribute reduction of formal contexts, formal concepts, and concept lattices from a semantics point of view.

Note on representing attribute reduction and concepts in concept lattice using graphs

Mao (Soft Computing 21(24):7293–7311, 2017) claims to make contributions to the study of reduction in attributes in concept lattices by using graph theory. Some of her results could affect development of formal concept analysis; in particular, her three new algorithms for formal concept enumeration promise a lesser time complexity. Unfortunately, as we show, all the three algorithms are incorrect.

On attribute reduction in concept lattices: The polynomial time discernibility matrix-based method becomes the CR-method

We eliminate unnecessary computations of attribute discernibility sets in discernibility matrix-based methods (DM-methods) of attribute reduction in concept lattices. We obtain a polynomial time algorithm we call a Skim DM-method. The Skim DM-method however is merely slightly different from the standard clarification and reduction (CR-method). This result enables us to express a relationship of the CR-method and the DM-methods: the DM-methods are basically the CR-method with up to an exponential amount of unnecessary computations. Additionally, we experimentally evaluate the Skim DM-method in comparison with the CR-method.

Vague concept lattice reduction using granular computing and vague entropy

Recently, the calculus of Formal Concept Analysis (FCA) in the fuzzy setting is prolonged in bipolar fuzzy space for precise analysis of fuzzy attributes. In this process it is addressed that the attributes like bald (or tadpole) or not bald (not tadpole) cannot be defined through a precise or sharp boundaries. To deal with them, some evidence to support (i.e. true tA membership-values) or reject (i.e. false fA membership-values) the attributes is required in the given boundary 0≤tA+fA≤1. Hence, the proposed method tried to provide the mathematical algebra of vague concept lattice and its navigation at user defined granules with an illustrative example. In addition, the vague entropy measurement is also computed to validate the results.

Stability Prediction Model of Roadway Surrounding Rock Based on Concept Lattice Reduction and a Symmetric Alpha Stable Distribution Probability Neural Network

To combat the uncertainty of the multiple factors affecting roadway surrounding rock stability, five initial indexes are selected for reduction according to concept lattice theory: rock quality designation (RQD), uniaxial compressive strength (Rc), the integrity coefficient of rock mass, groundwater seepage, and joint condition. The aim of this study is to compute correlation coefficients among various indexes and verify the effectiveness of lattice reduction. Alpha stable distribution is used to replace the commonly used Gauss distribution in probabilistic neural networks. A prediction model for the stability of roadway surrounding rock is then established based on a concept lattice and improved probabilistic neural network. 100 groups of training sample data are plugged into this model one by one to examine its rationality. The established model is employed for engineering application prediction with ten indiscriminate sample groups from the Jianlinshan mining area of the Daye iron mine, revealing accuracy of up to 90%. This demonstrates that our prediction model based on a concept lattice and improved probabilistic neural network has high reliability and applicability.

On attribute reduction in concept lattices: Experimental evaluation shows discernibility matrix based methods inefficient

In recent years, discernibility matrix based methods of attribute reduction in concept lattices (DM-methods) enjoyed an increase in attention and were applied in many extensions of formal concept analysis. In our previous paper, we pointed out that there exists an older method (CR-method) with theoretically lesser time complexity and we proposed a wrapping procedure to use the CR-method in any extension where the DM-methods are used. Now we evaluate the methods experimentally. Results of the evaluation assert our previous theoretical findings that the CR-method is strictly superior as it outperforms the DM-methods by several order of magnitude. To emphasize the poor performance of the DM-methods we introduce a new naïve and deliberately slow algorithm called SIMPEL. Subsequently, we show that even its performance is not so bad in comparison with the DM-methods. Our conclusions are that it is inefficient to use the DM-methods for attribute reduction in concept lattices and that the CR-method should be used instead in practice.

0–1 linear integer programming method for granule knowledge reduction and attribute reduction in concept lattices

Knowledge reduction is one of the key issues in formal concept analysis, and there have been many studies on this topic. Granule knowledge reduction and attribute reduction are two of the most important knowledge reduction in formal concept analysis. Firstly, theorem to character granule knowledge reduction is given, and granule knowledge reduction method in concept lattices based on 0–1 linear integer programming is proposed in this paper. Then, characterization theorems of three types attributes are obtained in attribute reduction, and attribute reduction method in concept lattices based on 0–1 linear integer programming is proposed.

Concept Lattice Simplification in Formal Concept Analysis Using Attribute Clustering

In Formal Concept Analysis (FCA), a concept lattice graphically portrays the underlying relationships between the objects and attributes of an information system. One of the key complexity problems of concept lattices lies in extracting the useful information. The unorganized nature of attributes in huge contexts often does not yield an informative lattice in FCA. Moreover, understanding the collective relationships between attributes and objects in a larger many valued context is more complicated. In this paper, we introduce a novel approach for deducing a smaller and meaningful concept lattice from which excerpts of concepts can be inferred. In existing attribute-based concept lattice reduction methods for FCA, mostly either the attribute size or the context size is reduced. Our approach involves in organizing the attributes into clusters using their structural similarities and dissimilarities, which is commonly known as attribute clustering, to produce a derived context. We have observed that the deduced concept lattice inherits the structural relationship of the original one. Furthermore, we have mathematically proved that a unique surjective inclusion mapping from the original lattice to the deduced one exists.

On attribute reduction in concept lattices: Methods based on discernibility matrix are outperformed by basic clarification and reduction

Several recent papers have studied the problem of attribute reduction in concept lattices by using a discernibility matrix in various extensions of formal concept analysis. We recall the clarification and reduction method by Ganter & Wille and show that it is strictly superior to the methods based on discernibility matrix. Consequently, we show how to adapt the clarification and reduction method to the extensions of formal concept analysis considered in the recent papers.

A methodology for analysis of concept lattice reduction

Formal concept analysis (FCA) is a mathematical theory of data analysis with applications in many areas. The problem of obtaining a concept lattice of an appropriate size was identified in several applications as one of the most important problems of FCA. In order to deal with this problem several techniques with different characteristics were proposed for concept lattice reduction. However, there are currently no adequate methods to assess what types of knowledge transformations can result from a reduction. A methodology for analysis of concept lattice reduction is presented here. It is based on the use of sets of proper implications holding in the original and reduced formal contexts or concept lattices. Working with both sets of implications, the methodology is able to show what is preserved, eliminated, inserted or transformed by a reduction technique. Three classes of reduction techniques are analyzed from the standpoint of the methodology in order to highlight techniques of each class have in common with respect to the transformations performed. Such analysis is followed by specific examples in each class.