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

We study an output-based translation of the λ-calculus with explicit substitution into the synchronous π-calculus – enriched with pairing – that has its origin in mathematical logic, and show that this translation respects reduction. We will define the notion of (explicit) head reduction -which encompasses (explicit) lazy reductionand show that the translation fully represents this reduction in that term-substitution as well as each single reduction step are modelled. We show that all the main properties (soundness, completeness, and adequacy) hold for these notions of reduction, as well as that termination is preserved with respect to a notion of call by need reduction for the π-calculus. We then define a notion of type assignment for the π-calculus that uses the type constructor →, and show that all Curry types assignable to λ-terms are preserved by the translation. We will also show that the π-calculus gives a semantics for the (standard) λ-calculus by defining an encoding that will fully represent reduction with explicit substitution, β-reduction, and equality, mapping equivalent term to equivalent processes. keywords: λ-calculus, π-calculus, intuitionistic logic, classical logic, translation, type assignment