  Nonlinear interpolants have been shown useful for the verification of
programs and hybrid systems in contexts of theorem proving, model checking,
abstract interpretation, etc. The underlying synthesis problem, however, is
challenging and existing methods have limitations on the form of formulae to be
interpolated. We leverage classification techniques with space transformations
and kernel tricks as established in the realm of machine learning, and present
a counterexample-guided method named NIL for synthesizing polynomial
interpolants, thereby yielding a unified framework tackling the interpolation
problem for the general quantifier-free theory of nonlinear arithmetic,
possibly involving transcendental functions. We prove the soundness of NIL and
propose sufficient conditions under which NIL is guaranteed to converge, i.e.,
the derived sequence of candidate interpolants converges to an actual
interpolant, and is complete, namely the algorithm terminates by producing an
interpolant if there exists one. The applicability and effectiveness of our
technique are demonstrated experimentally on a collection of representative
benchmarks from the literature, where in particular, our method suffices to
address more interpolation tasks, including those with perturbations in
parameters, and in many cases synthesizes simpler interpolants compared with
existing approaches.
