Stepwise functional refoundation of relational concept analysis

Relational concept analysis (RCA) is an extension of formal concept analysisallowing to deal with several related contexts simultaneously. It has beendesigned for learning description logic theories from data and used withinvarious applications. A puzzling observation about RCA is that it returns asingle family of concept lattices although, when the data feature circulardependencies, other solutions may be considered acceptable. The semantics ofRCA, provided in an operational way, does not shed light on this issue. In thisreport, we define these acceptable solutions as those families of conceptlattices which belong to the space determined by the initial contexts(well-formed), cannot scale new attributes (saturated), and refer only toconcepts of the family (self-supported). We adopt a functional view on the RCAprocess by defining the space of well-formed solutions and two functions onthat space: one expansive and the other contractive. We show that theacceptable solutions are the common fixed points of both functions. This isachieved step-by-step by starting from a minimal version of RCA that considersonly one single context defined on a space of contexts and a space of lattices.These spaces are then joined into a single space of context-lattice pairs,which is further extended to a space of indexed families of context-latticepairs representing the objects manippulated by RCA. We show that RCA returnsthe least element of the set of acceptable solutions. In addition, it ispossible to build dually an operation that generates its greatest element. Theset of acceptable solutions is a complete sublattice of the interval betweenthese two elements. Its structure and how the defined functions traverse it arestudied in detail.