Counterfactuals in Fuzzy Relational Models

Abstract Given rapidly growing requirements for explainability, counterfactual explanations have gained interest in Machine Learning systems. This study investigates this timely problem in fuzzy relational systems described by fuzzy relational equations and develops a detailed solution to the counterfactual problems encountered in this setting. An underlying optimization problem is formulated, and its gradient-based solution is constructed. It is also demonstrated that the non-uniqueness of the derived solution is conveniently formalized and quantified by admitting a result coming in the form of information granules of a higher type, namely type-2 or interval-valued fuzzy set. The construction of the solution in this format is realized by invoking the principle of justifiable granularity. We also discuss ways of designing fuzzy relations and elaborate on methods of carrying out counterfactual explanation in rule-based models. Illustrative examples are included to present the performance of the method and interpret the obtained results.