Deep neural networks are usually considered black-boxes due to their complex internal architecture, that cannot straightforwardly provide human-understandable explanations on how they behave. Indeed, Deep Learning is still viewed with skepticism in those real-world domains in which incorrect predictions may produce critical effects. This is one of the reasons why in the last few years Explainable Artificial Intelligence (XAI) techniques have gained a lot of attention in the scientific community. In this paper, we focus on the case of multi-label classification, proposing a neural network that learns the relationships among the predictors associated to each class, yielding First-Order Logic (FOL)-based descriptions. Both the explanation-related network and the classification-related network are jointly learned, thus implicitly introducing a latent dependency between the development of the explanation mechanism and the development of the classifiers. Our model can integrate human-driven preferences that guide the learning-to-explain process, and it is presented in a unified framework. Different typologies of explanations are evaluated in distinct experiments, showing that the proposed approach discovers new knowledge and can improve the classifier performance.

Explainable artificial intelligence has rapidly emerged since lawmakers have started requiring interpretable models for safety-critical domains. Concept-based neural networks have arisen as explainable-by-design methods as they leverage human-understandable symbols (i.e. concepts) to predict class memberships. However, most of these approaches focus on the identification of the most relevant concepts but do not provide concise, formal explanations of how such concepts are leveraged by the classifier to make predictions. In this paper, we propose a novel end-to-end differentiable approach enabling the extraction of logic explanations from neural networks using the formalism of First-Order Logic. The method relies on an entropy-based criterion which automatically identifies the most relevant concepts. We consider four different case studies to demonstrate that: (i) this entropy-based criterion enables the distillation of concise logic explanations in safety-critical domains from clinical data to computer vision; (ii) the proposed approach outperforms state-of-the-art white-box models in terms of classification accuracy.