A general theory of concept lattice with tractable implication exploration

Abstract

This work develops the general concept lattice for the problem concerning categorisation of objects according to their properties. Unlike the conventional approaches, such as the formal concept lattice and the rough set lattice, the general concept lattice is designed to adhere to the general principle that the information content should be invariant regardless how the variables/parameters are presented. Here, one will explicitly demonstrate the existence of such a construction by a sequence of fulfilments compatible with the conventional lattice structure. The general concept lattice promises to be a comprehensive categorisation for all the distinctive object classes according to whatever properties they are equipped with. It will be shown that one can always regain the formal concept lattice and rough set lattice from the general concept lattice.One also speaks of the tractability of the general concept lattice for both its lattice structure and logical content. The general concept lattice permits a feasible construction that can be completed in a single scan of the formal context, though the conventional formal-concept lattice and rough-set lattice can be regained from the general concept lattice. The logic implication deducible from the general concept lattice takes the form of μ1→μ2 where μ1,μ2∈M⁎ are composite attributes out of the concerned formal attributes M. Remarkable is that with a single formula based on the contextual truth 1η one can deduce all the implication relations extractable from the formal context. For concreteness, it can be shown that any implication A→B (A,B being subsets of the formal attributes M) discussed in the formal-concept lattice corresponds to a special case of μ1→μ2 by means of μ1=∏A and μ2=∏B. Thus, one may elude the intractability due to searching for the Guigues-Duquenne basis appropriate for the implication relations deducible from the formal-concept lattice. Likewise, one may identify those μ1→μ2 where μ1=∑A and μ2=∑B with the implications that can be acquired from the rough-set lattice. (Here, the product ∏ stands for the conjunction and the summation ∑ the disjunction.)