Acyclic Solos and Differential Interaction Nets

We present a restriction of the solos calculus which is stable under reduction and expressive enough to contain an encoding of the pi-calculus. As a consequence, it is shown that equalizing names that are already equal is not required by the encoding of the pi-calculus. In particular, the induced solo diagrams bear an acyclicity property that induces a faithful encoding into differential interaction nets. This gives a (new) proof that differential interaction nets are expressive enough to contain an encoding of the pi-calculus. All this is worked out in the case of finitary (replication free) systems without sum, match nor mismatch.

💡 Research Summary

The paper investigates a refined fragment of the solos calculus—called the acyclic solos calculus—and shows how it can serve as a robust intermediate language for encoding the π‑calculus into differential interaction nets (DINs). The motivation stems from a mismatch between the unification‑based communication of the solos calculus and the substitution‑based communication of the π‑calculus. In the original solos calculus, when two occurrences of the same name are unified, a “self‑identification” may arise: a name can be identified with itself, which in the graphical representation of DINs manifests as a self‑loop link that cannot be eliminated by reduction. This phenomenon does not occur in the π‑calculus, where communication is always realized by substituting the received name for the transmitted one.

To eliminate self‑identification, the authors introduce two technical devices. First, a simple typing discipline assigns each occurrence of a name a protocol label: either a send (S) or a receive (R). This breaks the symmetry of unification: an S‑occurrence can only be unified with an R‑occurrence, which can be interpreted as a directed flow from the sender to the receiver, mimicking substitution. Second, they define five acyclicity conditions (AC1–AC5) that constrain how bound names may appear in a solos term. Roughly, these conditions ensure that:

• No name has two R‑occurrences (AC1);
• A bound name never appears in an R‑position before its binding (AC2);
• All other occurrences of a bound name are S‑occurrences that appear “after” the binding in the graph (AC3);
• The dependency graph induced by bindings is a forest (AC4);
• No cycles are created when following the flow from S‑to‑R occurrences (AC5).

These constraints guarantee that any reduction step of an acyclic solos term preserves the acyclicity property, i.e., self‑identification never arises during reduction. Consequently, the reduction semantics of acyclic solos is stable under the typing and acyclicity discipline.

The paper then revisits the known translation from the π‑calculus to the solos calculus (as presented in