A Logical Interpretation of the λ-Calculus into the π-Calculus, Preserving Spine Reduction and Types

Abstract

We define a new, output-based encoding of the λ-calculus into the asynchronous π-calculusenriched with pairing -that has its origin in mathematical logic, and show that this encoding respects one-step spine-reduction up to substitution, and that normal substitution is respected up to similarity. We will also show that it fully encodes lazy reduction of closed terms, in that term-substitution as well as each reduction step are modelled up to similarity. We then define a notion of type assignment for the π-calculus that uses the type constructor →, and show that all Curry-assignable types are preserved by the encoding.

Duality and i/o-Types in the π-Calculus

“…Note that · B is written in [vBV09] using asynchronous free output and restriction instead of delayed bound output. We can adopt this more concise notation since (νx)(ax | P ) and a(x):P are strongly bisimilar processes, and similarly for x(p ′ ):p ′ p and x(p ′ ).p ′ p. (Another difference is that the replication in the encoding of the application is guarded, as in [vBV10], to force a tighter operational correspondence between reductions in λ and in the encodings.…”

“…The first one is the ordinary encoding by Milner [Mil92], the second one is by van Bakel and Vigliotti [vBV09]. The two encodings are syntactically quite different.…”

Linear Lambda Calculus and Deep Inference

“…It is worth remarking that the above embedding · · strongly recalls output-based embedding of standard λ-calculus into π-calculus [17]. The reason why we think this is relevant is simple.…”

“…Remark 5.1 We keep stressing that · · strongly recalls output-based embedding of standard λ-calculus with explicit substitutions into π-calculus [17]. In principle, this means that extending SBVr with the right logical operators able to duplicate atoms, and consequently upgrading · · , we could model full β-reduction as proof-search.…”

“…the base clause of the, so called, output-based embedding of the standard λ-calculus with explicit substitutions into π-calculus [17]. In (3), "."…”