Although Deep Neural Networks (DNNs) have great generalization and prediction capabilities, their functioning does not allow a detailed explanation of their behavior. Opaque deep learning models are increasingly used to make important predictions in critical environments, and the danger is that they make and use predictions that cannot be justified or legitimized. Several eXplainable Artificial Intelligence (XAI) methods that separate explanations from machine learning models have emerged, but have shortcomings in faithfulness to the model actual functioning and robustness. As a result, there is a widespread agreement on the importance of endowing Deep Learning models with explanatory capabilities so that they can themselves provide an answer to why a particular prediction was made. First, we address the problem of the lack of universal criteria for XAI by formalizing what an explanation is. We also introduced a set of axioms and definitions to clarify XAI from a mathematical perspective. Finally, we present the Greybox XAI, a framework that composes a DNN and a transparent model thanks to the use of a symbolic Knowledge Base (KB). We extract a KB from the dataset and use it to train a transparent model (i.e., a logistic regression). An encoder-decoder architecture is trained on RGB images to produce an output similar to the KB used by the transparent model. Once the two models are trained independently, they are used compositionally to form an explainable predictive model. We show how this new architecture is accurate and explainable in several datasets.

Accepted in Knowledge-Based Systems Journal