Incompleteness, counterexamples and refinements in abstract model-checking

In this paper we study the relation between the lack of completeness in abstract interpretation of model-checking and the structure of the counterexamples produced by a model-checker. We consider two dual forms of completeness of an abstract interpretation: Forward and backward completeness. They correspond respectively to the standard/ completeness of an abstract interpretation and can be related with each other by adjunction. We give a constructive characterization of Clarke et al.'s spurious counterexamples in terms of both forward and backward completeness of the underlying abstract interpretation. This result allows us to understand the structure of the counterexamples that can be removed by systematically refining abstract domains to achieve completeness with respect to a given operation. We apply our result to improve static program analysis by refining the model-checking of an abstract interpretation.