  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.
