Standardization and Böhm Trees for Λμ-Calculus

Abstract

Abstract. Λµ-calculus is an extension of Parigot's λµ-calculus which (i) satis es Separation theorem: it is Böhm-complete, (ii) corresponds to CBN delimited control and (iii) is provided with a stream interpretation. In the present paper, we study solvability and investigate Böhm trees for Λµ-calculus. Moreover, we make clear the connections between Λµ-calculus and in nitary λ-calculi. After establishing a standardization theorem for Λµ-calculus, we characterize sovalbility in Λµ-calculus. Then, we study in nite Λµ-Böhm trees, which are Böhm-like trees for Λµ-calculus; this allows to strengthen the separation results that we established previously for Λµ-calculus and to shed a new light on the failure of separation in Parigot's original λµ-calculus. Our construction clari es Λµ-calculus both as an in nitary calculus and as a core language for dealing with streams as primitive objects.

The approximation theorem for the Λμ-calculus

“…The Λμ-calculus is an extension of Parigot's λμ-calculus (Parigot 1992), proposed by de Groote (1994a,b) and developed by Saurin (2005Saurin ( , 2008bSaurin ( , 2010a. The interest of Λμ lies in the fact that it preserves the separability property, namely the Böhm theorem of the λcalculus with βη-conversion (Saurin 2005), which is not the case of λμ (David et al 2001).…”

“…The interest of Λμ lies in the fact that it preserves the separability property, namely the Böhm theorem of the λcalculus with βη-conversion (Saurin 2005), which is not the case of λμ (David et al 2001). Indeed, many basic concepts and properties from ordinary λ-calculus in the classic book (Barendregt 1984) extend to Λμ: confluence of the reduction relation even in presence of the η-rule (Py 1998); Böhm-out technique and separability (Saurin 2005); standardisation, head-normal forms and solvability, Böhm trees (Saurin 2012).…”

Characterisation of Approximation and (Head) Normalisation for λμ using Strict Intersection Types

“…Approximation for Λµ (a variant of λµ where naming and µ-binding are separated [17]) has been studied by others as well [22,18]; weak approximants for λµ are studied in [11].…”

Call-by-Value Solvability, Revisited

“…Since λ vsub is essentially a refinement of λ CBV , we compare them explicitly in Section 2.1, while we refer to the introduction of [11] for more relations with the literature. Solvability has also been recently studied for some extensions of λ-calculus in [18,25], but both works consider a call-by-name calculus.…”