Dualization of a monotone Boolean function on a finite lattice can be represented by transforming the set of its minimal 1 values to the set of its maximal 0 values. In this paper we consider finite lattices given by ordered sets of their meet and join irreducibles (i.e., as a concept lattice of a formal context). We show that in this case dualization is equivalent to the enumeration of so-called minimal hypotheses. In contrast to usual dualization setting, where a lattice is given by the ordered set of its elements, dualization in this case is shown to be impossible in output polynomial time unless P = NP. However, if the lattice is distributive, dualization is shown to be possible in subexponential time.
Abstract. Concept stability was used in numerous applications for selecting concepts as biclusters of similar objects. However, scalability remains a challenge for computing stability. The best algorithms known so far have algorithmic complexity quadratic in the size of the lattice. In this paper the problem of approximate stability computation is analyzed. An approximate algorithm for computing stability is proposed. Its computational complexity and results of computer experiments in comparing stability index and its approximations are discussed.
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