China 
Africa 
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Epilogue
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
Another very interesting branch of lattice theory, that has not been discussed in this treatise, is Formal Concept Analysis (FCA). The main reason for this is that FCA evolved around the same time as the lattice algebra described in this book. Formal concept analysis is based on concept lattices and has proven to be an excellent tool for certain types of data analysis. It is a method which takes binary relations and produces a complete lattice. The current FCA theory is based on the pioneering efforts in the early 1980's by a research group led by Rudolf Wille, Bernhard Ganter, and Peter Baumeister at the Technical University of Darmstadt [313, 88, 89]. The motivation and goals of FCA and lattice algebra are the same, namely to expand lattice theory for utilization of real-world applications based on a solid mathematical foundation. There already exist examples where these two branches of lattice theory overlap [41]. We can foresee that in the future FCA could be added to an expanded version of lattice algebra by additional chapters devoted to FCA. This would be similar as in adding numerical linear algebra to linear algebra. It would also expand this book from a one-semester to a two-semester study.
Using Boolean factors for the construction of an artificial neural networks
Published in International Journal of General Systems, 2018
Lauraine Tiogning Kueti, Norbert Tsopze, Cezar Mbiethieu, Engelbert Mephu-Nguifo, Laure Pauline Fotso
The optimal factors belong to the set of formal concepts that comes from FCA (Wille 1982). Formal concept analysis is a field of mathematics widely used in data analysis, knowledge representation and information management. Its aim is the representation of data using the notion of concept as consisting of intention and extension and on the organization of the concepts through a conceptual hierarchy (Ganter and Wille 1999). FCA can be understood as conceptual clustering method, which clusters simultaneously objects and their descriptions. There is a growing interest in applications of FCA: data mining, neural networks, software engineering, linguistics, psychology, information retrieval and so on. The initial data to be used are represented in the form of a binary relation on a set of objects and attributes called formal context.
Cyber-Physical Systems, a new formal paradigm to model redundancy and resiliency
Published in Enterprise Information Systems, 2020
Formal Concept Analysis has been applied in many domains as a knowledge representation and discovery tool. The current paper adapts this approach and its evaluation for the needs of CPS modelling and analysis. The proposed result highlights the FCA bottom-up approach focusing with the particularities of the domain and building upon them a structure to allow to discover the general dependencies. Some research studying relations between the FCA and the graph modelling methods (Carbonnel et al. 2016), (Morozov, Lezoche, and Panetto 2017) specifies the need of the use of extra filtering after the lattice building process. As a further development of the approach, we plan to study the use of the resulting lattice-models to response some of the pressing questions of Industry 4.0 such as the identification of redundancies in functionalities, the improvement of the systems plasticity and their auto-adaptation to environment changes. The future research would be to implement an approach of reinforcement learning algorithms in the CPS so that, by themselves, following several iterations, the systems could learn the specificities of each other and build, autonomously and in a collaborative way, their own global model. Another feature to add to the presented approach would be the interpretation of the models through some extra knowledge derived from formal representations (e.g. Ontologies). This property would improve the automatic modelling and discovery steps. The formal knowledge representation would be complementary in relation to the informal (expert experiences) source of knowledge. The definition of correspondences between the lattices will be used for a much more optimised improvement of systems’ resilience.