A General Theory of Concept Lattice (II): Tractable Lattice Construction and Implication Extraction

As the second part of the treatise 'A General Theory of Concept Lattice', this paper speaks of the tractability of the general concept lattice for both its lattice structure and logic 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 {\mu}1 {\rightarrow} {\mu}2 where {\mu}1,{\mu}2 {\in} M^{\ast} are composite attributes out of the concerned formal attributes M. Remarkable is that with a single formula based on the contextual truth 1{\eta} one can deduce all the implication relations extractable from the formal context. For concreteness, it can be shown that any implication A {\rightarrow} B (A, B being subsets of the formal attributes M) discussed in the formal-concept lattice corresponds to a special case of {\mu}1 {\rightarrow} {\mu}2 by means of {\mu}1 = {\prod} A and {\mu}2 = {\prod} 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 {\mu}1 {\rightarrow} {\mu}2 where {\mu}1 = {\sum} A and {\mu}_2 = {\sum} B with the implications that can be acquired from the rough-set lattice. (Here, the product {\prod} stands for the conjunction and the summation {\sum} the disjunction.)}