Strong Normalization as Safe Interaction
Résumé
The elimination rule of union types (denoted by (UE)) can break strong normalization (i.e. is unsafe) in presence of non-determinism. This is quite disturbing, since union types are often required in presence of non-deterministic computations. Surprisingly, there are also confluent systems for which this rule is unsafe. It appears that studying the safety of (UE) amounts to the characterization in a term of safe interactions between some of its subterms. In this paper, we study the safety of the elimination rule of union types for an extension of the lambda calculus with simple rewrite rules. We prove that the union and intersection types discipline is complete w.r.t. strong normalization. This allows to show that (UE) is safe iff an interpretation of types based on biorthogonals is sound for it. We then discuss two sufficient conditions for the safety of (UE). The first one is issued from the closure by union of reducibility candidates while the second is the closure by union of a biorthogonality operator. We show that closure by union of reducibility candidates implies completeness of intersection type assignment. Finally, we discuss an alternative biorthogonality relation, based on the observation of the least reducibility candidate.
Domaines
Logique en informatique [cs.LO]
Origine : Fichiers produits par l'(les) auteur(s)