Interpolants in Nonlinear Theories Over the Reals

We develop algorithms for computing Craig interpolants for first-order formulas over real numbers with a wide range of nonlinear functions, including transcendental functions and differential equations. We transform proof traces from δ-complete decision procedures into interpolants that consist of Boolean combinations of linear constraints. The algorithms are guaranteed to find the interpolants between two formulas A and B whenever A ∧ B is not δ-satisfiable. At the same time, by exploiting δ-perturbations one can parameterize the algorithm to find interpolants with different positions between A and B. We show applications of the methods in control and robotic design, and hybrid system verification.