Functions as Processes: The Case of Collapsing Non-determinism

This paper studies typed translations of $\lambda$-calculi into $\pi$-calculi, both with non-determinism, informed by the Curry-Howard isomorphism between linear logic and session types (propositions-as-sessions). Prior work considered calculi with non-collapsing non-determinism, a non-committal form of choice in which all alternatives are preserved, ensuring confluence. A question left open is whether there is a correct translation for calculi with the more traditional (and non-confluent) collapsing non-determinism, which commits to one single alternative and discards the rest. A session-typed $\pi$-calculi with collapsing non-determinism is proposed. Next, (i) the key meta-theoretical properties of typed processes (type preservation and deadlock-freedom) are proven following propositions-as-sessions, and (ii) a correct translation of a resource $\lambda$-calculus with non-determinism is given. An alternative semantics for non-determinism is then shown to unlock stronger correctness results for the translation.