Implicative Logic based encodings of the λ-calculus into the π-calculus

We study output-based encodings of the λ-calculus into the asynchronous π-calculus – enriched with pairing – that have its origin in mathematical logic, and show that these encodings respect reduction. We will also show that, for closed terms, the encoding fully encodes explicit spine reduction -which encompasses lazy reduction-, in that term-substitution as well as each reduction step are modelled up to contextual 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. We conclude by optimising our encoding, essentially disabling reduction inside a substitution.