Polyhedral Approximation of Multivariate Polynomials using Handelman's Theorem

Alexandre Maréchal (1) , Alexis Fouilhé (1) , Tim King (1) , David Monniaux (1, 2) , Michaël Périn (1)

1 VERIMAG - IMAG - VERIMAG
2 CNRS - Centre National de la Recherche Scientifique

Convex polyhedra are commonly used in the static analysis of programs to represent over-approximations of sets of reachable states of numerical program variables. When the analyzed programs contain nonlinear instructions, they do not directly map to standard polyhedral operations: some kind of linearization is needed. Convex polyhe-dra are also used in satisfiability modulo theory solvers which combine a propositional satisfiability solver with a fast emptiness check for polyhedra. Existing decision procedures become expensive when nonlinear constraints are involved: a fast procedure to ensure emptiness of systems of nonlinear constraints is needed. We present a new linearization algorithm based on Handelman's representation of positive polynomials. Given a polyhedron and a polynomial (in)equality, we compute a polyhedron enclosing their intersection as the solution of a parametric linear programming problem. To get a scalable algorithm, we provide several heuristics that guide the construction of the Handelman's representation. To ensure the correctness of our polyhedral approximation , our Ocaml implementation generates certificates verified by a checker certified in Coq.

Mots clés: abstract interpretation; polyhedra; SMT-solvers; linearization; Handelman's theorem

Domaines: Informatique [cs]

Alexandre Maréchal, Alexis Fouilhé, Tim King, David Monniaux, Michaël Périn. Polyhedral Approximation of Multivariate Polynomials using Handelman's Theorem. International Conference on Verification, Model Checking, and Abstract Interpretation 2016, Barbara Jobstmann; Rustan Leino, Jan 2016, St. Petersburg, United States. ⟨hal-01223362⟩