In 1979, Cousot and Cousot gave a specification of the best (most-precise) abstract transformer possible for a given concrete transformer and a given abstract domain. Unfortunately, their specification does not lead to an algorithm for obtaining the best transformer. In fact, algorithms are known for only a few abstract domains.

This paper presents a parametric framework that, for a given abstract domain A and logic L, computes successively better A values that over-approximate the set of states defined by an arbitrary formula in L. Because the method approaches the most-precise A value from ``above'', if it is taking too much time, a safe answer can be returned at any time. For certain combinations of A and L, the framework is complete -- i.e., with enough resources, it computes the most-precise abstract value possible.

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