Widening Operators for Weakly-Relational Numeric Abstractions

In recent years there has been a lot of interest in the definition of so-called weakly-relational numeric domains, whose complexity and precision are in between the (non-relational) abstract domain of intervals [9] and the (relational) abstract domain of convex polyhedra [10]. The first weakly-relational domain proposed in the literature is based on systems of constraints of the form x−y ≤ c and ±x ≤ c, typically represented by Difference-Bound Matrices (DBMs). Even though DBMs have a long tradition in Computer Science, their use in the Abstract Interpretation field is quite recent. The idea of defining an abstract domain of DBMs was put forward in [1], where these constraints were called bounded differences. An independent application can be found in [19], where an abstract domain of transitively closed DBMs is defined. In this case, the transitive closure requirement was meant as a simple and well understood way to obtain a canonical form for the domain elements, so as to abstract away from merely syntactic differences. In [19] the specification of all the required abstract semantics operators is provided, including an operator that is meant to match the standard widening operator defined on the domain of convex polyhedra [10]. Unfortunately, as pointed out in [14, 15], this operator is not a widening since it does not provide a convergence guarantee for the abstract iteration sequence. The abstract domain of (not necessarily transitively closed) DBMs is considered in [14]. In this more concrete, syntactic domain the transitive closure operator behaves as a kernel operator (monotonic, idempotent and reductive) mapping each DBM into the smallest DBM (with respect to the componentwise ordering) encoding the same geometric shape. As done in [19], a widening operator is also defined in [14] and it is observed that this widening “has some intriguing interactions” with transitive closure, therefore identifying the divergence issue faced in [19]. This observation has led to the conclusion that