Computations with finite closure systems and implications

The lattice of strict completions of a finite poset

“…This possibly accounts for the small number of lattices M P and can help in their study. On the other hand there exist several algorithms to compute lattices defined by implications (see for instance Wild [20]). Using such an algorithm, Nourine [12] computed the cardinality of M P for P an antichain of size n ≤ 6 (i.e.…”

Attribute-incremental construction of the canonical implication basis

“…Of special interest is the canonical implication basis of the context (usually called the Duquenne-Guigues basis), which is a minimal cover of the set of implications that hold in the context. The fact that all bases (minimal covers) have the same cardinality was already known in the database literature [16], but the specific role of the canonical basis in that was made explicit in [23] and [24] (see also [10] A quasi-closed set A is called pseudo-closed if A = A and B ⊂ A for any quasiclosed B ⊂ A. A pseudo-intent of the context is a pseudo-closed (with respect to ) attribute set.…”

“…To make the description of the algorithm self-contained, we describe a "naïve" procedure to calculate the saturation. There are other suitable algorithms for this purpose: LinClosure, which is well known in the database theory [16]; the one proposed in [24], which has nonlinear time complexity but is claimed to be an enhanced version of LinClosure; etc.…”

“…It is a modification of his Next Closure algorithm for computing the concept set of a context. There are also algorithms constructing other implication bases: direct basis [17,24], proper basis [20], etc. The output of these algorithms can then be reduced to the canonical basis [6].…”

Consequences of the Brylawski–Lucas Theorem for Binary Matroids

“…Details of most of the above , and of several numerical tables , ‡ are given in [14] . See also [16] . 3 …”