Abstract
Relational concept analysis (RCA) extends formal concept analysis (FCA) by taking into account binary relations between formal contexts. It has been designed for inducing description logic TBoxes from ABoxes, but can be used more generally. It is especially useful when there exist circular dependencies between objects. In this case, it extracts a unique stable concept lattice family grounded on the initial formal contexts. However, other stable families may exist whose structure depends on the same relational context. These may be useful in applications that need to extract a richer structure than the minimal grounded one. This issue is first illustrated in a reduced version of RCA, which only retains the relational structure. We then redefine the semantics of RCA on this reduced version in terms of concept lattice families closed by a fixed-point operation induced by this relational structure. We show that these families admit a least and greatest fixed point and that the well-grounded RCA semantics is characterised by the least fixed point. We then study the structure of other fixed points and characterise the interesting lattices as the self-supported fixed points.
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Notes
- 1.
An anonymous reviewer complements the remarks of Sect. 2.2 noting that RCA\(^0\) is also very related to Graph-FCA as they both have only one context and using existential scaling.
- 2.
Instead of developing both \(\mathscr {K}\) and \(\mathscr {L}\) independently and maintaining an equivalence between them, it would have been possible to use a more FCA-like structure associating the corresponding contexts and lattices.
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Acknowledgements
This work has been partially funded by the ANR Elker project (ANR-17-CE23-0007-01). The author thanks Philippe Besnard for pointing to the Knaster-Tarski theorem.
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Euzenat, J. (2021). Fixed-Point Semantics for Barebone Relational Concept Analysis. In: Braud, A., Buzmakov, A., Hanika, T., Le Ber, F. (eds) Formal Concept Analysis. ICFCA 2021. Lecture Notes in Computer Science(), vol 12733. Springer, Cham. https://doi.org/10.1007/978-3-030-77867-5_2
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