Polynomial Template Generation using Sum-of-Squares Programming

19 pages, 3 figures, 1 tableTemplate abstract domains allow to express more interesting properties than classical abstract domains. However, template generation is a challenging problem when one uses template abstract domains for program analysis. In this paper, we relate template computation with the program properties that we want to prove. We focus on one-loop programs with a conditional branch and assume that all the functions involved in these programs are polynomials. We formally define the notion of well-representative template basis with respect to a given property. The definition relies on the fact that template abstract domains produce inductive invariants. We show that these invariants can be obtained by solving certain systems of functional inequalities. Then, such systems can be strengthened using a hierarchy of sum-of-squares feasibility problems. Each step of the SOS hierarchy can possibly provide a solution which in turn yields feasible invariant bound together with a certificate that the desired property holds. The interest of this approach is illustrated on nontrivial program examples in polynomial arithmetic

Polynomial Template Generation using Sum-of-Squares Programming

19 pages, 3 figures, 1 tableTemplate abstract domains allow to express more interesting properties than classical abstract domains. However, template generation is a challenging problem when one uses template abstract domains for program analysis. In this paper, we relate template computation with the program properties that we want to prove. We focus on one-loop programs with a conditional branch and assume that all the functions involved in these programs are polynomials. We formally define the notion of well-representative template basis with respect to a given property. The definition relies on the fact that template abstract domains produce inductive invariants. We show that these invariants can be obtained by solving certain systems of functional inequalities. Then, such systems can be strengthened using a hierarchy of sum-of-squares feasibility problems. Each step of the SOS hierarchy can possibly provide a solution which in turn yields feasible invariant bound together with a certificate that the desired property holds. The interest of this approach is illustrated on nontrivial program examples in polynomial arithmetic