Worst-case groundness analysis using positive Boolean functions

This note illustrates a theoretical worst-case scenario for groundness analyses obtained through abstract interpretation over the abstract domain of positive Boolean functions. A sequence of programs is given for which any Pos-based abstract interpretation for groundness analysis follows an exponential chain. Another sequence of programs is given for which a state-of-the-art implementation based on ROBDDs gives a result of exponential size in only three iterations. The moral of the story is that a serious Pos analyser must incorporate some form of widening to protect itself from the inherent complexity of the underlying domain.