Normalizable Horn Clauses, Strongly Recognizable Relations, and Spi

Abstract

We exhibit a rich class of Horn clauses, which we call $\mathcal{H}_{\text{1}}$, whose least models, though possibly infinite, can be computed effectively. We show that the least model of an $\mathcal{H}_{\text{1}}$ clause consists of so-called strongly recognizable relations and present an exponential normalization procedure to compute it. In order to obtain a practical tool for program analysis, we identify a restriction of $\mathcal{H}_{\text{1}}$ clauses, which we call $\mathcal{H}_{\text{2}}$, where the least models can be computed in polynomial time. This fragment still allows to express, e.g., Cartesian product and transitive closure of relations. Inside $\mathcal{H}_{\text{2}}$, we exhibit a fragment $\mathcal{H}_{\text{3}}$ where normalization is even cubic. We demonstrate the usefulness of our approach by deriving a cubic control-flow analysis for the Spi calculus [1] as presented in [14].