Interval linear systems as a necessary step in fuzzy linear systems

Weldon Lodwick (1) , Didier Dubois (2, 3)

1 University of Colorado [Denver]
2 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
3 CNRS - Centre National de la Recherche Scientifique

This article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization of scalar equality to intervals. Each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals. We also consider the smallest multidimensional intervals containing the solution sets. We propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals. This unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies.

Mots clés: Interval linear systems; Fuzzy interval analysis; Constraint intervals

Domaines: Intelligence artificielle [cs.AI]; Apprentissage [cs.LG]; Logique en informatique [cs.LO]; Informatique et langage [cs.CL]

Weldon Lodwick, Didier Dubois. Interval linear systems as a necessary step in fuzzy linear systems. Fuzzy Sets and Systems, 2015, 281, pp.227-251. ⟨10.1016/j.fss.2015.03.018⟩. ⟨hal-01285665⟩