  Artificial Neural Networks (ANNs) are being deployed for an increasing number
of safety-critical applications, including autonomous cars and medical
diagnosis. However, concerns about their reliability have been raised due to
their black-box nature and apparent fragility to adversarial attacks. These
concerns are amplified when ANNs are deployed on restricted system, which limit
the precision of mathematical operations and thus introduce additional
quantization errors. Here, we develop and evaluate a novel symbolic
verification framework using software model checking (SMC) and satisfiability
modulo theories (SMT) to check for vulnerabilities in ANNs. More specifically,
we propose several ANN-related optimizations for SMC, including invariant
inference via interval analysis, slicing, expression simplifications, and
discretization of non-linear activation functions. With this verification
framework, we can provide formal guarantees on the safe behavior of ANNs
implemented both in floating- and fixed-point arithmetic. In this regard, our
verification approach was able to verify and produce adversarial examples for
$52$ test cases spanning image classification and general machine learning
applications. Furthermore, for small- to medium-sized ANN, our approach
completes most of its verification runs in minutes. Moreover, in contrast to
most state-of-the-art methods, our approach is not restricted to specific
choices regarding activation functions and non-quantized representations. Our
experiments show that our approach can analyze larger ANN implementations and
substantially reduce the verification time compared to state-of-the-art
techniques that use SMT solving.
