Specificity shift in solving fuzzy relational equations

Studied is a system of "N" fuzzy relational equations with a max-t composition x(k)□R = y(k), where the input-output fuzzy sets (x(k) and y(k)) are available while the fuzzy relation (R) needs to be determined. The solution to these equations is derived through a new paradigm of specificity shift. The main objective is to modify a level of specificity of the fuzzy sets (relational constraint) so that the modified constraints allow for the use of some standard theoretical results of the theory of fuzzy relational equations that otherwise would have been found totally unjustifiable. In more detail, the specificity of the available input fuzzy sets becomes gradually increased while an opposite tendency is observed for the output fuzzy sets. The optimization of the specificity levels is discussed in detail. Numerical studies are also included. Finally, the use of the specificity shift is discussed in a conjunction with the exploitation of standard gradient-based techniques.