Inverse inference based on interpretable constrained solutions of fuzzy relational equations with extended max–min composition

Abstract

In this paper, the method of inverse inference based on interpretable constrained solutions of the system of fuzzy relation equations (SFRE) with extended max-min composition is developed. The properties of interval and constrained solutions of the extended maxmin SFRE with granular and relational structure of the solution set are investigated. The extended max-min SFRE can be represented in the form of the max-min subsystems aggregated using the min operator or dual min-max subsystems aggregated using the max operator. When decomposing the max-min and dual min-max subsystems, the set of interval solutions can be decomposed into the lower and upper subsets bounded by the same aggregating solutions.Each lower (upper) subset is defined by the unique greatest (least) or aggregating solution and the set of minimal (maximal) solutions. Following (Bartl et al. 2012), to avoid excessive granularity and ensure interpretability of the interval solutions when restoring causes through observed effects, the constraints in the form of linguistic modifiers are imposed on the measures of causes significances. Given constraints, solving the SFRE is reduced to finding unknown relations which take the values 1(0) if the modifier is present (absent) in the linguistic description of the interval solution. To completely cover the interval solution, the notion of the complete crisp solution is introduced. Then, the reduced set of constrained solutions can be decomposed into the lower and upper subsets of complete crisp solutions bounded by the constrained aggregating solutions. The search for the set of constrained solutions amounts to solving the discrete optimization problem. The properties of the set of interval (constrained) solutions allow parallelizing the genetic search for the lower and upper subsets for each aggregating solution. The developed method makes it possible to simplify the search for the solution set based on the constraints on accuracy (interpretability) of the applied problem.