A fully-abstract semantics of lambda-mu in the pi-calculus

Abstract

We study the λµ-calculus, extended with explicit substitution, and define a compositional outputbased interpretation into a variant of the π-calculus with pairing that preserves single-step explicit head reduction with respect to weak bisimilarity. We define four notions of weak equivalence for λµ -one based on weak reduction ∼ wβµ , two modelling weak head-reduction and weak explicit head reduction, ∼ wH and ∼ wxH respectively (all considering terms without weak head-normal form equivalent as well), and one based on weak approximation ∼ A -and show they all coincide. We will then show full abstraction results for our interpretation for the weak equivalences with respect to weak bisimilarity on processes.

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].…”

“…But, in fact, this is not the same β (and the named term has changed), as reflected in the fact that its type changes during reduction. Moreover, when making the substitution explicit as in[11], it becomes clear that this other approach in fact is a short-cut, which our definition does without.…”

Fully abstract encodings of λ-calculus in HOcore through abstract machines

“…Related work: Since Milner's seminal work [19], other translations of the λ-calculus or one of its variants into π-calculus have been proposed, e.g., to study connections with logic [2,5,23], termination [9,4,25], sequentiality [6], control [9,12,24], or Continuation-Passing Style (CPS) transforms [21,22,10]. These works use the more expressive first-order π-calculus, except for [21,22], discussed below; full abstraction is proved w.r.t.…”

“…These works use the more expressive first-order π-calculus, except for [21,22], discussed below; full abstraction is proved w.r.t. contextual equivalence in [6,25,12], normal-form bisimilarity in [24], and normalform and applicative bisimilarities in [22]. The definitions of the encodings and the equivalences of [6,25,12] are driven by types, and therefore cannot be compared to our untyped setting.…”

“…The definitions of the encodings and the equivalences of [6,25,12] are driven by types, and therefore cannot be compared to our untyped setting. In [24], van Bakel et al establish a full abstraction result between the λµ-calculus with normal-form bisimilarity and the π-calculus. Their encoding relies on an unbounded number of restricted names to evaluate several translations of λ-terms in parallel, while we rely on flags and on barbed equivalence to know which translated λ-term is being evaluated.…”