Elsevier

Information Sciences

Volume 530, August 2020, Pages 180-200
Information Sciences

General framework for consistencies in decision contexts

https://doi.org/10.1016/j.ins.2020.02.045Get rights and content

Abstract

We present a general framework which provides a unified approach to attribute reduction in decision contexts across various generalizations of Formal Concept Analysis. The framework is demonstrated on an attribute reduction method which has been studied multiple times separately for each of the generalizations. Specifically, we use the framework to elevate the discernibility matrix based method to a more general setting.

Introduction

Formal Concept Analysis (FCA) [2], [5] is a method of information management and data analysis invented by Rudolf Wille. Solid mathematical and computational foundations of FCA were developed in the 1980s. In the past two decades or thereabouts, FCA has seen considerable interest in various communities. Many papers on applications of FCA in various domains have been published, including those in premier journals and conferences. The method is based on a formalization of a certain philosophical view of conceptual knowledge which goes back to Port-Royal logic [1], [9].

The basic input for FCA is a flat table, called a formal context, in which rows represent objects and columns represent attributes. Each entry of the formal context contains a cross if the corresponding object has the corresponding attribute. FCA provides two outputs. The first one is a concept lattice—a hierarchy of particular clusters present in the formal context. The second one is a system of attribute implications—if-then rules describing dependencies among attributes in the formal context.

Recently, many papers considering a particular extension of FCA for purposes of decision making have been published (see Appendix A). In this extension, the input is a decision context – a formal context where the attributes are divided into conditions and decisions.1 The desired output is a collection of decision rules – attribute implications where conditions are only in antecedents and decisions are only in consequents.

A majority of the papers considers the problem of attribute reduction in decision contexts. The attribute reduction seeks to find a minimal subset of conditions which satisfies a given consistency property.

Many generalizations of FCA appeared to enable us to analyze graded data, real valued data, data with missing values, etc [29]. With almost each such generalization of FCA, a study of analogously generalized decision contexts and decision rules was made. The results within these studies usually utilize very little of the properties of the specific generalizations. We observed that the only essential property shared is the fact that the generalized decision context induces two closure systems. This enables us to propose a general framework which covers all the individual approaches described in the literature (see Appendix A). The key idea of the general framework is based on the fact that all settings of FCA considered in all papers on decision contexts share a particular property. Namely, in all the settings extents of both the condition part and decision part of decision context are crisp sets which form two closure systems. The consistency conditions then can be expressed as a relationship between the two closure systems. We use the already existing framework of pattern structures [7].

This study enables us to extend results on relationships between consistency properties from Li et al. [16] to more general settings of FCA, like those handling real-valued data [21], incomplete data [22], or fuzzy data [28]. We also provide a description of discernibility matrix-based reduction in decision contexts in our setting and provide an illustrative example.

The structure of this paper is as follows: First, we recall preliminary notions used in the rest of the paper (Section 2). Next, we list consistency conditions in decision contexts utilized in the relevant papers and express them as properties of pairs of closures systems (Section 3). Then, we describe our general framework based on pattern structures (Section 4) and provide examples of recently studied generalized decision contexts as particular instances in the framework (Section 5). Then, we comment on an additional notion of consistency (Section 6). Finally, we summarize our conclusions (Section 7). Additionally, this paper has an appendix, which serves as an overview of the papers which are covered by the proposed framework.

Section snippets

Formal concept analysis

An input to FCA is a triplet ⟨X, Y, I⟩, called a formal context, where X, Y are finite non-empty sets of objects and attributes, respectively, and I is a binary relation between X and Y; ⟨x, y⟩ ∈ I means that the object x has the attribute y. Finite formal contexts are usually depicted as tables, in which rows represent objects, columns represent attributes, and each entry contains a cross if the corresponding object has the corresponding attribute.

The binary relation in the formal context

Consistency conditions in decision contexts

In this section, we recall consistency conditions used in decision contexts. We use the terminology of [16] where the various consistency conditions found in literature are called I-, II-,..., V-consistencies.

Pattern structures and decision pattern structures

In this section, we briefly recall the pattern structures [7] and state an extension analogous to decision contexts in FCA.

Discernibility matrix method in the general framework

Most papers on decision formal context describe a method of enumerating all reducts – the discernibility matrix method inspired by similar methods in Rough Set Theory [34]. It is a purely theoretical method as it is unfeasible even for what is considered medium sized data [10].

Definition 9

Let ⟨X, C, D, δC, δD⟩ be a decision pattern structure and let ⟨A, c⟩, 〈A′,c′〉∈B(X,C̲,δC). A discernibility set of ⟨A, c⟩, ⟨A′, c′⟩ is a set of condition attributes in which the intents c, c′ differ, i.e.D(〈A,c〉,〈A′,c′〉)={C

Remarks on IV-consistency

There is one more consistency type considered in [16], called IV-consistency (or graded consistency). We treat the IV-consistency separately, because it leads to a different form of discernibility matrix which is not covered by Definition 11. Still, it possible include it to the proposed framework of pattern decision structures as we explain below.

The IV-consistency was proposed by Wu et al. [43]. It is only based on the relationship between concepts generated by some object. A decision context

Conclusions

Many results introduced for specific settings of decision contexts exploit only the fact that extents form a closure system and that distinguishable condition attributes occur in the settings. We have used the pattern structures to propose a unifying framework which covers all the specific settings. The framework brings the following:

  • (i)

    No additional settings, where extents form closure systems, need to be separately studied. For instance, in Examples 6 and 7, we considered the attribute C11 (file

CRediT authorship contribution statement

Radek Janostik: Conceptualization, Methodology, Validation, Investigation, Writing - review & editing. Jan Konecny: Conceptualization, Methodology, Validation, Investigation, Writing - original draft, Writing - review & editing, Project administration, Funding acquisition.

Declaration of Competing Interest

None.

Acknowledgment

The authors acknowledge support by the grants:• IGA 2018 of Palacký University Olomouc, No.IGA_PrF_2018_030, • JG 2019 of Palacký University Olomouc, No. JG_2019_008.

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