Yoan Renaud

Uncovering and reducing hidden combinatorics in guigues-duquenne bases

Mannila and Raiha [5] have shown that minimum implicational bases can have an exponential number of implications. Aim of our paper is to understand how and why this combinatorial explosion arises and to propose mechanisms which reduce it.

Computing Implications with Negation from a Formal Context

The objective of this article is to define an approach towards generating implications with (or without) negation when only a formal context K = (G, M, I) is provided. To that end, we define a two-step procedure which first (i) computes implications whose premise is a key in the context K |

Generating positive and negative exact rules using formal concept analysis: problems and solutions

\tilde{\rm K}

Recursive decomposition and bounds of the lattice of Moore co-families

representing the apposition of the context K and its complementary

Join on Closure Systems Using Direct Implicational Basis Representation

\tilde{\rm K}

Recursive decomposition tree of a Moore co-family and closure algorithm

with attributes in

An Inference System for Exhaustive Generation of Mixed and Purely Negative Implications from Purely Positive Ones.

\tilde{\rm M}

A new generic class of Frankl’s families

(negative attributes), and then (ii) uses an inference axiom we have defined to produce the whole set of implications.

About The Recursive Decomposition of the lattice of co-Moore Families

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.

A closure algorithm using a recursive decomposition of the set of Moore co-families

A collection of sets on a ground set Un (Un = {1,2,...,n}) closed under intersection and containing Un is known as a Moore family. The set of Moore families for a fixed n is in bijection with the set of Moore co-families (union-closed families containing the empty set) denoted