## On the intractability of computing the Duquenne-Guigues base.(English)Zbl 1277.68258

Summary: Implications of a formal context $$(G,M,I)$$ obey Armstrong rules, which allows for definition of a minimal (in the number of implications) implication base, called Duquenne-Guigues or stem base in the literature. A long-standing problem was that of an upper bound for the size of a stem base in the size of the relation $$I$$. In this paper we give a simple example of a relation where this boundary is exponential. We also prove $$\#$$P-hardness of the problem of determining the size of the stem base (i.e., the number of pseudo-intents).

### MSC:

 68T30 Knowledge representation 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 06A15 Galois correspondences, closure operators (in relation to ordered sets)

### Keywords:

computational complexity; implication base
Full Text: