Decoupling the ascending and descending phases in Abstract Interpretation

Abstract Interpretation approximates the semantics of a program by mimicking its concrete fixpoint computation on an abstract domain $\mathbb{A}$. The abstract (post-) fixpoint computation is classically divided into two phases: the ascending phase, using widenings as extrapolation operators to enforce termination, is followed by a descending phase, using narrowings as interpolation operators, so as to mitigate the effect of the precision losses introduced by widenings. In this paper we propose a simple variation of this classical approach where, to more effectively recover precision, we decouple the two phases: in particular, before starting the descending phase, we replace the domain $\mathbb{A}$ with a more precise abstract domain $\mathbb{D}$. The correctness of the approach is justified by casting it as an instance of the A$^2$I framework. After demonstrating the new technique on a simple example, we summarize the results of a preliminary experimental evaluation, showing that it is able to obtain significant precision improvements for several choices of the domains $\mathbb{A}$ and $\mathbb{D}$.