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Although Deep Neural Networks (DNNs) have great generalization and predictioncapabilities, their functioning does not allow a detailed explanation of theirbehavior. Opaque deep learning models are increasingly used to make importantpredictions in critical environments, and the danger is that they make and usepredictions that cannot be justified or legitimized. Several eXplainableArtificial Intelligence (XAI) methods that separate explanations from machinelearning models have emerged, but have shortcomings in faithfulness to themodel actual functioning and robustness. As a result, there is a widespreadagreement on the importance of endowing Deep Learning models with explanatorycapabilities so that they can themselves provide an answer to why a particularprediction was made. First, we address the problem of the lack of universalcriteria for XAI by formalizing what an explanation is. We also introduced aset of axioms and definitions to clarify XAI from a mathematical perspective.Finally, we present the Greybox XAI, a framework that composes a DNN and atransparent model thanks to the use of a symbolic Knowledge Base (KB). Weextract a KB from the dataset and use it to train a transparent model (i.e., alogistic regression). An encoder-decoder architecture is trained on RGB imagesto produce an output similar to the KB used by the transparent model. Once thetwo models are trained independently, they are used compositionally to form anexplainable predictive model. We show how this new architecture is accurate andexplainable in several datasets.

Greybox XAI: a Neural-Symbolic learning framework to produce interpretable predictions for image classification