Explainability of Deep Neural Networks using the Constructive Axiom of Choice Model

The explainability of deep neural networks is a crucial challenge in artificial intelligence, where systems make complex, often opaque decisions that are difficult to interpret. While many explainability approaches are regularly emerging, they generally rely on post hoc methods, which provide explanations after training, without ensuring intrinsic traceability of decisions. This paper proposes a method to improve the intrinsic explainability of deep neural networks by integrating the mathematical model of the constructive axiom of choice. Unlike post hoc explanatory approaches such as LIME (Local Interpretable Model-agnostic Explanations) and SHAP (SHapley Additive exPlanations), which provide explanations after the model has made a decision, our method justifies the decisions made by the network during the training phase, ensuring direct traceability of the neural activations that lead to the final output. The mathematical model of the axiom of choice, commonly used in existence proofs in constructive analysis, guarantees a formal and rigorous selection of significant activations across the network layers, as the selection is made through a choice function that identifies activations meeting a predefined property, thus ensuring that only the activations relevant to the final decision are retained. After implementing our approach, we obtained encouraging results. A graphical visualization of the activation paths in the network shows both the global set of activations and the significant subset selected by the function, relative to the decision made by the model.