Strong Normalization as Safe Interaction

When enriching the λ-calculus with rewriting, union types may be needed to type all strongly normalizing terms. However, with rewriting, the elimination rule (∨ E) of union types may also allow to type non normalizing terms (in which case we say that (∨ E) is unsafe). This occurs in particular with non-determinism, but also with some confluent systems. It appears that studying the safety of (∨ E) amounts to the characterization, in a term, of safe interactions between some of its subterms. In this paper, we study the safety of (∨ E) for an extension of the λ-calculus with simple rewrite rules. We prove that the union and intersection type discipline without (∨ E) is complete w.r.t. strong normalization. This allows to show that (∨ E) is safe if and only if an interpretation of types based on biorthogonals is sound for it. We also discuss two sufficient conditions for the safety of (∨ E), and study an alternative biorthogonality relation, based on the observation of t...

• Type: Conference
• Year: 2007
• Where: LICS
• Authors: Colin Riba
• Keywords: Computer Science, LICS 2007, Non Normalizing Terms, Strongly Normalizing, Union Types