A Tutorial on Domain Theory in Abstract Interpretation

Abstract

The success of abstract interpretation in the analysis of discrete dynamic systems stems from its simple, but nevertheless rigorously defined, underlying idea that the specification of the behavior of a system, e.g. a program, at different levels of abstraction, is an approximation of its formal semantics. This provides a variety of formal methods for specifying, modifying, and implementing program analysis tools. In particular, one of the most fundamental facts of abstract interpretation is that most interesting properties of the approximated semantics, like its precision, completeness, and compositionality, which may involve complex operators, fixpoints etc., all depend upon the notion of abstraction, which is precisely and uniquely specified by the chosen abstract domain of properties. Because of the key role played by domains in abstract interpretation, any formal method to compare or transform abstract interpretations is therefore inherently based on corresponding methods to compare and transform abstract domains: Modifying domains corresponds to modify abstract interpretations.