A Hierarchy for Delimited Continuations in Call-by-Name

Abstract

Abstract. Λμ-calculus was introduced as a Böhm-complete extension of Parigot's λμ-calculus. Λμ-calculus, contrarily to Parigot's calculus, is a calculus of CBN delimited control as evidenced by Herbelin and Ghilezan. In their seminal paper on (CBV) delimited control, Danvy and Filinski introduced the CPS Hierarchy of control operators (shifti/reseti)i∈ω.In a similar way, we introduce in the present paper the Stream Hierarchy, a hierarchy of calculi extending and generalizing Λμ-calculus. The (Λ n )n∈ω-calculi have Church-Rosser and Böhm theorems. We then present sound and complete CPS translations for the hierarchy. Next, we investigate the operational content of the hierarchy through its abstract machines, the (Λ n )n∈ω-KAM. Finally, we establish that the Stream hierarchy is indeed a CBN analogue to the CPS hierarchy.

Standardization and Böhm Trees for Λμ-Calculus

“…This is, in our opinion, the deep reason for the failure of separation in λµ-calculus. Below, we show a classi cation of several calculi (or logical formalisms) depending on whether they have the arity matching constraint, whether they are linear or not (single vs multiple occurrences of variables) and whether they have separation: ABT denotes Curien's Abstract Böhm Trees, CP S ∞ denotes an in nitary languages studies by Streicher and Loew [21] and Λ n the languages of the Stream hierarchy [29] (see appendix). Interestingly one observes that Separation is obtained only when either arities don't match or when there is linearity.…”

“…λ-calculus no yes yes [5] λµ-calculus yes yes no [7] Λµ-calculus no yes yes [30] ABT yes yes no [22] CPS ∞ yes yes no [21] Ludics yes no yes [11] (Λ n ) n∈ω no yes yes [29] 5 Conclusion, Perspectives and Future Works. Perspectives and Future Works.…”

“…We shall characterize those Λµ-Böhm trees which are the image of a Λµ-term; Λµ-BT can be generalized to Böhm trees for the Λ n -calculi of the Stream hierarchy [29]:…”

“…While the emphasis was traditionally given to the delimited-control languages in call-by-value, recent works [13,17] have advocated the study CBN delimited control. In a recent work [29], we introduced a hierarchy of calculi generalizing both λ-calculus and Λµ-calculus, the Stream Hierarchy, that we proved to be a call-by-name analogous to the CPS hierarchy.…”

“…In this appendix, we brie y describe Böhm trees for the stream hierarchy which has been introduced in [29] as a generalization of Λµ-calculus which is a call-byname version of the CPS hierarchy.…”