A Logic for the Imprecision of Abstract Interpretations

In numerical analysis, error propagation refers to how small inaccuracies in input data or intermediate computations accumulate and affect the final result, typically governed by the stability and sensitivity of the algorithm with respect to some perturbations. The definition of a similar concept in approximated program analysis is still a challenge. In abstract interpretation, inaccuracy arises from the abstraction itself, and the propagation of this error is dictated by the abstract interpreter. In most cases, such imprecision is inevitable. In this paper we introduce a logic for deriving (upper) bounds on the inaccuracy of an abstract interpretation. We are able to derive a function that bounds the imprecision of the result of an abstract interpreter from the imprecision of its input data. When this holds we have what we call partial local completeness of the abstract interpreter, a weaker form of completeness known in the literature. To this end, we introduce the notion of a generator for a property represented in the abstract domain. Generators allow us to restrict the search space when verifying whether the bounding function holds for a given program and input. We then introduce a program logic, called Error Propagation Logic (EPL), for propagating the error bounds produced by an abstract interpretation. This logic is a combination of correctness and incorrectness logics and a logic for program omega-continuity that is also introduced in this paper.