A New Encoding of Not Necessarily Closed Convex Polyhedra

Many applications of static analysis and verification compute on some abstract domain based on convex polyhedra. Traditionally, most of these applications are restricted to convex polyhedra that are topologically closed. When adopting the Double Description (DD) method [8], a closed convex polyhedron can be specified in two ways, using a constraint system or a generator system: the constraint system contains a finite set of linear non-strict inequality constraints; the generator system contains two finite sets of vectors, collectively called generators, which are rays and points of the polyhedron. Some applications of static analysis and verification, including recent proposal such as [3], need to compute on the domain of not necessarily closed (NNC) convex polyhedra. By definition, any NNC polyhedron can be represented by a so-called mixed constraint system, that is, a constraint system where a further finite set of linear strict inequality constraints is allowed to occur. The usual approach for implementing NNC polyhedra is to embed them into closed polyhedra in a vector space with one extra dimension. While this idea, originally proposed in [6] and also described in [7], proved to be quite effective, its direct application results in a low-level user interface where most of the geometric intuition of the DD method gets lost under the “implementation details”. A much cleaner approach was proposed in [1, 2], where the concept of generator of an NNC polyhedron is extended to also account for the closure points of the polyhedron. In particular, it is shown that any NNC polyhedron can be elegantly and intuitively represented by means of extended generator systems. The combined use of mixed constraint systems and extended generator systems provides a higher level interface to the domain of NNC polyhedra, allowing for simpler definitions of some of the corresponding operators.