Strong Normalization as Safe Interaction

Abstract

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 the least reducibility candidate.

On the Values of Reducibility Candidates

“…and (⊔ E), we can type non-strongly normalizing terms, while the type system with (Fun + ) but without (⊔ E) is strongly normalizing [14].…”

“…⇒ U causes strong normalization problems similar to those of (⊔ E) [14]. Our solution does not handle it because it would imply to have a call-by-value arrow, while we want to force call-by-value only locally in (⊔ E let ).…”

“…Write ρ for CR ρ . The correctness of the subtyping relation is standard [14], excepted that distributivity is ensured by Cor. 3.8.…”

“…The straightforward elimination of union types is known to break subject reduction [1], and for some extensions of the λ-calculus, to break strong normalization as well [14]. The counter example for subject reduction given in [1] has been adapted to implicit existential types in [18].…”

“…when closure preserves unions). But results of [14] on failures of strong normalization with elimination of union types show that some rewrite systems are not compatible with type interpretations which are closed under unions.…”