A key problem in the adoption of artificial neural networks in safety- related applications is that misbehaviors can be hardly ruled out with traditional analytical or probabilistic techniques. In this paper we focus on specific networks known as Multi-Layer Perceptrons (MLPs), and we propose a solution to ver- ify their safety using abstractions to Boolean combinations of linear arithmetic constraints. We show that our abstractions are consistent, i.e., whenever the ab- stract MLP is declared to be safe, the same holds for the concrete one. Spurious counterexamples, on the other hand, trigger refinements and can be leveraged to automate the correction of misbehaviors. We describe an implementation of our approach based on the H Y SAT solver, detailing the abstraction-refinement process and the automated correction strategy. Finally, we present experimental results confirming the feasibility of our approach on a realistic case study.