The 4-Octahedron Abstract Domain

In the code static analysis, the choice of an adequate abstract domain is an interesting issue. In this paper, we provide a new numerical abstract domain: 4-Octahedron. It is an Octahedra subclass that infers relations of the form : { x ∼ α, x−y ∼ β, (x−y)−(z−t) ∼ λ}, such that: x, y, z and t are real variables, α, β and λ are real constants and ∼ ∈ {≤,≥}. The 4-Octahedron extends the conjunction of octagonal invariants, called Unit Two Variable Per Inequality (UTVPI) constraints, with inequalities of the form: {(x−y)−(z−t) ≤ λ}. Its precision lies between the octagons and polyhedra. We construct a suitable structure for its representation, we provide normalization algorithms for computing its canonical representation and we give methods to compute its transfer functions (Union, Intersection, Assignment, Projection, ...). Complexity of the implementation algorithms is proved to be cubic in the number of system variables.