In recent years, the notion of local robustness (or robustness for short) has emerged as a desirable property of deep neural networks. Intuitively, robustness means that small perturbations to an input do not cause the network to perform misclassifications. In this paper, we present a novel algorithm for verifying robustness properties of neural networks. Our method synergistically combines gradient-based optimization methods for counterexample search with abstraction-based proof search to obtain a sound and (δ -)complete decision procedure. Our method also employs a data-driven approach to learn a verification policy that guides abstract interpretation during proof search. We have implemented the proposed approach in a tool called Charon and experimentally evaluated it on hundreds of benchmarks. Our experiments show that the proposed approach significantly outperforms three state-of-the-art tools, namely AI 2 , Reluplex, and Reluval.

Many example-guided program synthesis techniques use abstractions to prune the search space. While abstraction-based synthesis has proven to be very powerful, a domain expert needs to provide a suitable abstract domain, together with the abstract transformers of each DSL construct. However, coming up with useful abstractions can be non-trivial, as it requires both domain expertise and knowledge about the synthesizer. In this paper, we propose a new technique for learning abstractions that are useful for instantiating a general synthesis framework in a new domain. Given a DSL and a small set of training problems, our method uses tree interpolation to infer reusable predicate templates that speed up synthesis in a given domain. Our method also learns suitable abstract transformers by solving a certain kind of second-order constraint solving problem in a data-driven way. We have implemented the proposed method in a tool called Atlas and evaluate it in the context of the Blaze meta-synthesizer. Our evaluation shows that (a) Atlas can learn useful abstract domains and transformers from few training problems, and (b) the abstractions learned by Atlas allow Blaze to achieve significantly better results compared to manually-crafted abstractions. Abstraction learner Synthesizer ATLAS approach Current abstractionSynthesized programs DSL + training problemsAbstraction ( predicates + transformers )

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Scientists given the jitters for the DoD to issue corrected satellite positions from which fully accurate positions can be calculated. But a wait of several weeks is too long for most earthquake forecasting, says Yukio Fujinawa of the National Research Centre for Disaster Prevention in Tsukuba, near Tokyo. Washington & Tokyo THE US Department of Defense (DoD), whose network of navigation satellites has recently allowed scientists to determine geographical positions with unprecedented accuracy, has begun intentionally degrading the signals so that potential military adversaries are unable to take advantage of the system. But researchers in Japan and elsewhere say that the real victims of this misinformation could be scientists trying to forecast earthquakes by using the satellites to detect minute precursor ground motions. The Global Positioning System (GPS), a constellation of two dozen satellites that broadcast signals to ground-based receivers, has been in place since the late 1970s. Each satellite broadcasts a signal that specifies its position and the relative positions of receivers on the ground. Using the signals from several satellites, aircraft, ships and other vehicles can find their position by triangulation to within 40 metres. But in recent years, researchers have begun averaging many readings over time to find locations with errors of as little as one centimetre (Nature 343, 590 & 631, 15 February 1990). This accuracy makes GPS measurements useful for seismology: movements in the Earth's crust in the weeks before an earthquake can translate into positional changes of several centimetres, and by monitoring the relative positions of GPS receivers that have been placed around likely trouble spots, researchers may be able to predict impending earthquakes by watching for unexpected changes in positions. When the GPS system was first launched in 1978, these sorts of precise scientific applications were completely unanticipated. But although the usefulness of the first 'block' of GPS satellites proved to be an unexpected windfall for scientists and other users, the second block of satellites now going up has even greater potential. Indeed, from the beginning, the DoD considered the second generation of GPS satellites too accurate for 'unauthorized' use. Built into the second-generation GPS satellites are electronics that slightly alter the transmitted signal to cloak the true position of the satellite. 'Authorized users' have been given equipment that can unscramble the distortion to reveal a satellite's exact position at any time. Known as 'selective availability' (SA), this method has been in use for about a month. For unauthorized users-a group that includes most scientists-the effect of SA is that signals that could be trusted to provide 1 cm accuracy a month ago are now only good to about 6 cm. Although

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