The approximation theorem for the Λμ-calculus

ACM Trans. Comput. Logic

2018

We show characterisation results for normalisation, head-normalisation, and strong normalisation for λ μ using intersection types. We reach these results for a strict notion of type assignment for λ μ that is the natural restriction of the domain-based system of van Bakel et al. (2011) for λ μ by limiting the type inclusion relation to just intersection elimination. We show that this system respects β μ-equality, by showing both soundness and completeness results. We then define a notion of reduction on derivations that corresponds to cut-elimination, and show that this is strongly normalisable. We use this strong normalisation result to show an approximation result, and through that a characterisation of head-normalisation. Using the approximation result, we show that there is a very strong relation between the system of van Bakel et al. (2011) and ours. We then introduce a notion of type assignment that eliminates ω as an assignable type, and show, using the strong normalisation result for derivation reduction, that all terms typeable in this system are strongly normalisable as well, and show that all strongly normalisable terms are typeable. We conclude by adding type variables to our system, and show that system essentially is that of van Bakel (2010b).

Section: Type Assignment For (Strong) Normalisationsupporting

confidence: 68%

Section: Type Assignment For (Strong) Normalisationmentioning

confidence: 83%

See 1 more Smart Citation

Characterisation of Normalisation Properties for λμ using Strict Negated Intersection Types

ACM Trans. Comput. Logic

2018

“…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].…”

Section: Approximation Semantics For λµmentioning

confidence: 99%

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

Electron. Proc. Theor. Comput. Sci.

2017

We study the strict type assignment for λµ that is presented in [7]. We define a notion of approximants of λµ-terms, show that it generates a semantics, and that for each typeable term there is an approximant that has the same type. We show that this leads to a characterisation via assignable types for all terms that have a head normal form, and to one for all terms that have a normal form, as well as to one for all terms that are strongly normalisable.

“…Essentially following [26], we now define a weak approximation semantics for λµ. Approximation for λµ has been studied by others as well [25,12]; however, seen that we are mainly interested in weak reduction here, we will define weak approximants, which are normally not considered.…”

mentioning

confidence: 99%

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

Electron. Proc. Theor. Comput. Sci.

Vigliotti

2014

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.