An Implicative Logic based encoding of the λ-calculus into the π-calculus

We study an output-based encoding 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 encoding respects reduction. We will define the notion of (explicit) head reduction -which encompasses (explicit) lazy reductionand show that the encoding fully represents this reduction in that (explicit) term-substitution as well as individual single reduction steps are modelled. We show that all the main properties (soundness, completeness, and semantics) hold for our encoding, as well as that termination of reduction is preserved. We also show that our encoding gives a semantics for the equivalence on terms generated by explicit reduction as well as for normal β-equivalence. 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 encoding. keywords: λ-calculus, π-calculus, intuitionistic logic, classical logic, encoding, type assignment