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Fundamenta Informaticae - Concept Lattices and Their Applications
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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, ΣK, cannot be completely and directly inferred from the set ΣK of positive rules or from the concept lattice B(K), and (ii) the whole set of mixed rules may not be completely generated from ΣK alone, ΣK ∪ ΣK alone, or B(K) alone.