An Investigation into Functions as Processes

In [Mil90] Milner examines the encoding of the λ-calculus into the π-calculus [MPW92]. The former is the universally accepted basis for computations with functions, the latter aims at being its counterpart for computations with processes. The primary goal of this paper is to continue the study of Milner's encodings. We focus mainly on the lazy λ-calculus [Abr87]. We show that its encoding gives rise to a λ-model, in which a weak form of extensionality holds. However the model is not fully abstract: To obtain full abstraction, we examine both the restrictive approach, in which the semantic domain of processes is cut down, and the expansive approach, in which λ-calculus is enriched with constants to obtain a direct characterisation of the equivalence on λ-terms induced, via the encoding, by the behavioural equivalence adopted on the processes. Our results are derived exploiting an intermediate representation of Milner's encodings into the Higher-Order π-calculus, an ω-order extension of π-calculus where also agents may be transmitted. For this, essential use is made of the fully abstract compilation from the Higher-Order π-calculus to the π-calculus studied in [San92a].