The objective of this article is to investigate the problem of generating both positive and negative exact association rules when a formal context K of (positive) attributes is provided. A straightforward solution to this problem consists of conducting an apposition of the initial context K with its complementary context K, construct the concept lattice B(K|K) of apposed contexts and then extract rules. A more challenging problem consists of exploiting rules generated from each one of the contexts K and K to get the whole set of rules for the context K|K. In this paper, we analyze a set of identified situations based on distinct types of input, and come out with a set of properties. Obviously, the global set of (positive and negative) rules is a superset of purely positive rules (i.e., rules with positive attributes only) and purely negative ones since it generally contains mixed rules (i.e., rules in which at least a positive attribute and a negative attribute coexist). The paper presents also a set of inference rules to generate a subset of all mixed rules from positive, negative and mixed ones. Finally, two key conclusions can be drawn from our analysis: (i) the generic basis containing negative rules, Σ_K, cannot be completely and directly inferred from the set Σ_K of positive rules or from the concept lattice B(K), and (ii) the whole set of mixed rules may not be completely generated from Σ_K alone, Σ_K ∪Σ_K alone, or B(K) alone.
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