The Least Must Speak with the Greatest

We extend a propositions-as-types correspondence between linear logic and session types to include recursive sessions. Our extension takes least and greatest fixed points as dual, an idea well-known to theorists, but which has not previously appeared in the treatment of recursive sessions. We preserve the freedom from races, deadlock, and livelock that is a hallmark of the propositions-as-types approach, and avoid a problem with preservation of duality that appears in the standard treatment of recursive session types. We treat two systems, one based on classical linear logic, derived from work of Caires and Pfenning and of Wadler, and one based on a linear functional language, derived from work of Gay and Vasconcelos. Our treatment of recursion is inspired by Baelde.