Formal synthesis of neural Craig interpolant via counterexample guided deep learning

Craig interpolation is a significant and efficient application to formal verification and synthesis. However, there still remains a challenge in the synthesis of Craig interpolation for nonlinear theory. For quantifier-free theories of nonlinear arithmetic, this paper proposes a new approach to generate nonlinear Craig interpolants represented as deep neural networks. The approach exploits a CEGIS framework where a learner yields a neural candidate interpolant satisfying the interpolant conditions against training data sets, and a verifier adopts computer algebra methods to confirm the correctness of the candidate or to generate counterexamples for further refining the candidate. We implement the tool SyntheNI based on our CEGIS procedure, and assess the performance against a collection of benchmark examples. The tool SyntheNI performs better than existing methods in the aspect of the iteration number and the computational time. As an application, the tool SyntheNI is used to synthesize loop invariants. The SyntheNI can generate nonlinear Craig interpolants for quantifier free nonlinear real arithmetic. The experimental evaluation confirms the high performance of our synthesis method.