Coupling Policy Iteration with Semi-definite Relaxation to Compute Accurate Numerical Invariants in Static Analysis

We introduce a new domain for finding precise numerical invariants of programs by abstract interpretation. This domain, which consists of sub-level sets of nonlinear functions, generalizes the domain of linear templates introduced by Manna, Sankaranarayanan, and Sipma. In the case of quadratic templates, we use Shor’s semi-definite relaxation to derive safe and computable abstractions of semantic functionals, and we show that the abstract fixpoint can be over-approximated by coupling policy iteration and semi-definite programming. We demonstrate the interest of our approach on a series of examples (filters, integration schemes) including a degenerate one (symplectic scheme).