Formal Concept Analysis (FCA) is a mathematical framework for knowledge representation and discovery. It performs a hierarchical clustering over a set of objects described by attributes, resulting in conceptual structures in which objects are organized depending on the attributes they share. These conceptual structures naturally highlight commonalities and variabilities among similar objects by categorizing them into groups which are then arranged by similarity, making it particularly appropriate for variability extraction and analysis. Despite the potential of FCA, determining which of its properties can be leveraged for variability-related tasks (and how) is not always straightforward, partly due to the mathematical orientation of its foundational literature. This paper attempts to bridge part of this gap by gathering a selection of properties of the framework which are essential to variability analysis, and how they can be used to interpret diverse variability information within the resulting conceptual structures.

Distributivity is a well-established and extensively studied notion in lattice theory. In the context of data analysis, particularly within Formal Concept Analysis (FCA), lattices are often observed to exhibit a high degree of distributivity. However, no standardized measure exists to quantify this property. In this paper, we introduce the notion of rises in (concept) lattices as a means to assess distributivity. Rises capture how the number of attributes or objects in covering concepts change within the concept lattice. We show that a lattice is distributive if and only if no non-unit rises occur. Furthermore, we relate rises to the classical notion of meet- and join distributivity. We observe that concept lattices from real-world data are to a high degree join-distributive, but much less meet-distributive. We additionally study how join-distributivity manifests on the level of ordered sets.

Automating the extraction of concept hierarchies from free text is advantageous because manual generation is frequently labor- and resource-intensive. Free result, the whole procedure for concept hierarchy learning from free text entails several phases, including sentence-level text processing, sentence splitting, and tokenization. Lemmatization is after formal context analysis (FCA) to derive the pairings. Nevertheless, there could be a few uninteresting and incorrect pairings in the formal context. It may take a while to generate formal context; thus, size reduction formal context is necessary to weed out irrelevant and incorrect pairings to extract the concept lattice and hierarchies more quickly. This study aims to propose a framework for reducing formal context in extracting concept hierarchies from free text to reduce the ambiguity of the formal context. We achieve this by reducing the size of the formal context using a hybrid of a WordNet-based method and a frequency-based technique. Using 385 samples from the Wikipedia corpus and the suggested framework, tests are carried out to examine the reduced size of formal context, leading to concept lattice and concept hierarchy. With the help of concept lattice-invariants, the generated formal context lattice is compared to the normal one. In contrast to basic ones, the homomorphic between the resultant lattices retains up to 98% of the quality of the generating concept hierarchies, and the reduced concept lattice receives the structural connection of the standard one. Additionally, the new framework is compared to five baseline techniques to calculate the running time on random datasets with various densities. The findings demonstrate that, in various fill ratios, hybrid approaches of the proposed method outperform other indicated competing strategies in concept lattice performance.

Likewise, the current trend is to produce new resources in a digital format (e.g., in the context of social networks), which entails an in-depth paradigm shift in almost all the humanistic, social, scientific and technological fields. In particular, the field of the humanities is one which is going through a significant transformation as a result of these digitalization efforts and the paradigm shift associated with the digital age. Indeed, we are witnessing the emergence of a whole host of disciplines, those of Digital Humanities (Berry 2012), which are closely dependent on the production and proper organization of digital collections. As a result of the undoubted importance of digital collections in modern society, the search for effective and efficient methods to carry out the production, preservation and enhancement of such digital collections has become a key challenge in modern society (Calhoun, 2013). In particular, the annotation of resources with metadata that enables their proper cataloging, search, retrieval and use in different application scenarios is one of the key elements to ensuring the profitability of these collections of digital objects.

Formal Concept Analysis (FCA) is an approach to creating a conceptual hierarchy in which a \textit{concept lattice} is generated from a \textit{formal context}. That is, a triple consisting of a set of objects, $G$, a set of attributes, $M$, and an incidence relation $I$ on $G \times M$. A \textit{concept} is then modelled as a pair consisting of a set of objects (the \textit{extent}), and a set of shared attributes (the \textit{intent}). Implications in FCA describe how one set of attributes follows from another. The semantics of these implications closely resemble that of logical consequence in classical logic. In that sense, it describes a monotonic conditional. The contributions of this paper are two-fold. First, we introduce a non-monotonic conditional between sets of attributes, which assumes a preference over the set of objects. We show that this conditional gives rise to a consequence relation that is consistent with the postulates for non-monotonicty proposed by Kraus, Lehmann, and Magidor (commonly referred to as the KLM postulates). We argue that our contribution establishes a strong characterisation of non-monotonicity in FCA. Typical concepts represent concepts where the intent aligns with expectations from the extent, allowing for an exception-tolerant view of concepts. To this end, we show that the set of all typical concepts is a meet semi-lattice of the original concept lattice. This notion of typical concepts is a further introduction of KLM-style typicality into FCA, and is foundational towards developing an algebraic structure representing a concept lattice of prototypical concepts.

Categorization of business processes is an important part of auditing. Large amounts of transactional data in auditing can be represented as transactions between financial accounts using weighted bipartite graphs. We view such bipartite graphs as many-valued formal contexts, which we use to obtain explainable categorization of these business processes in terms of financial accounts involved in a business process by using methods in formal concept analysis. We use Dempster-Shafer mass functions to represent agendas showing different interest in different set of financial accounts. We also model some possible deliberation scenarios between agents with different interrogative agendas to reach an aggregated agenda and categorization. The framework developed in this paper provides a formal ground to obtain and study explainable categorizations from the data represented as bipartite graphs according to the agendas of different agents in an organization (e.g. an audit firm), and interaction between these through deliberation. We use this framework to describe a machine-leaning meta algorithm for outlier detection and classification which can provide local and global explanations of its result and demonstrate it through an outlier detection algorithm.

Posets are discrete mathematical structures which are ubiquitous in a broad range of data analysis and machine learning applications. Research connecting posets to the data science domain has been ongoing for many years. In this paper, a comprehensive review of a wide range of studies on data analysis and machine learning using posets are examined in terms of their theory, algorithms and applications. In addition, the applied lattice theory domain of formal concept analysis will also be highlighted in terms of its machine learning applications.

With our work, we contribute towards a qualitative analysis of the discourse on controversies in online news media. For this, we employ Formal Concept Analysis and the economics of conventions to derive conceptual controversy maps. In our experiments, we analyze two maps from different news journals with methods from ordinal data science. We show how these methods can be used to assess the diversity, complexity and potential bias of controversies. In addition to that, we discuss how the diagrams of concept lattices can be used to navigate between news articles.

We propose BERT4FCA, a novel method for link prediction in bipartite networks, using formal concept analysis (FCA) and BERT. Link prediction in bipartite networks is an important task that can solve various practical problems like friend recommendation in social networks and co-authorship prediction in author-paper networks. Recent research has found that in bipartite networks, maximal bi-cliques provide important information for link prediction, and they can be extracted by FCA. Some FCA-based bipartite link prediction methods have achieved good performance. However, we figured out that their performance could be further improved because these methods did not fully capture the rich information of the extracted maximal bi-cliques. To address this limitation, we propose an approach using BERT, which can learn more information from the maximal bi-cliques extracted by FCA and use them to make link prediction. We conduct experiments on three real-world bipartite networks and demonstrate that our method outperforms previous FCA-based methods, and some classic methods such as matrix-factorization and node2vec.

In this paper, we introduce a new method for querying triadic concepts through partial or complete matching of triples using an inverted index, to retrieve already computed triadic concepts that contain a set of terms in their extent, intent, and/or modus. As opposed to the approximation approach described in Ananias, this method (i) does not need to keep the initial triadic context or its three dyadic counterparts, (ii) avoids the application of derivation operators on the triple components through context exploration, and (iii) eliminates the requirement for a factorization phase to get triadic concepts as the answer to one-dimensional queries. Additionally, our solution introduces a novel metric for ranking the retrieved triadic concepts based on their similarity to a given query. Lastly, an empirical study is primarily done to illustrate the effectiveness and scalability of our approach against the approximation one. Our solution not only showcases superior efficiency, but also highlights a better scalability, making it suitable for big data scenarios.