Analogical proportions and the factorization of information in distributive lattices

Nelly Barbot (1) , Laurent Miclet (2) , Henri Prade (3, 4)

1 CORDIAL - Human-machine spoken dialogue
2 Dyliss - Dynamics, Logics and Inference for biological Systems and Sequences
3 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
4 CNRS - Centre National de la Recherche Scientifique

Analogical proportions are statements involving four enti- ties, of the form 'A is to B as C is to D'. They play an important role in analogical reasoning. Their formalization has received much attention from different researchers in the last decade, in particular in a proposi- tional logic setting. Analogical proportions have also been algebraically defined in terms of factorization, as a generalization of geometric nu- merical proportions (that equate ratios). In this paper, we define and study analogical proportions in the general setting of lattices, and more particularly of distributive lattices. The decomposition of analogical pro- portions in canonical proportions is discussed in details, as well as the resolution of analogical proportion equations, which plays a crucial role in reasoning. The case of Boolean lattices, which reflects the logical mod- eling, and the case corresponding to entities described in terms of gradual properties, are especially considered for illustration purposes.

Mots clés: analogical proportion; lattice; factorization

Domaines: Intelligence artificielle [cs.AI]

Nelly Barbot, Laurent Miclet, Henri Prade. Analogical proportions and the factorization of information in distributive lattices. 10th International Conference on Concept Lattices and Their Applications (CLA), Oct 2013, La Rochelle, France. ⟨hal-00908005⟩