Observational Equivalence and Full Abstraction in the Symmetric Interaction Combinators

The symmetric interaction combinators are an equally expressive variant of Lafont’s interaction combinators. They are a graph-rewriting model of deterministic computation. We define two notions of observational equivalence for them, analogous to normal form and head normal form equivalence in the lambda-calculus. Then, we prove a full abstraction result for each of the two equivalences. This is obtained by interpreting nets as certain subsets of the Cantor space, called edifices, which play the same role as Boehm trees in the theory of the lambda-calculus.

💡 Research Summary

The paper investigates the theory of symmetric interaction combinators (SIC), a variant of Lafont’s interaction combinators that serves as a graph‑rewriting model of deterministic computation. The authors address two fundamental questions: (1) how to define observational equivalence for nets in this system, and (2) how to give a denotational semantics that is fully abstract with respect to the chosen equivalence.

First, the authors recall the syntax of SIC nets. A net consists of cells of three kinds—δ, ζ (both binary, each with two auxiliary ports numbered 1 and 2) and ε (nullary)—connected by wires and possibly loops. Each cell has a distinguished principal port; wires attach to ports, and free ports constitute the interface of the net. Reduction proceeds by applying a small set of local, constant‑time rewriting rules that either create or eliminate a special connection called an observable axiom (or simply “axiom”). These axioms are the only observable events that can be generated during reduction.

Inspired by the λ‑calculus notion of head‑normal‑form (hnf) and normal‑form (nf) equivalence, the authors introduce two observational equivalences for SIC nets. A net is observable if some reduction sequence yields at least one observable axiom; it is finitely observable if only finitely many axioms ever appear. Using these notions they define:

• Axiom‑equivalence (AE) – for every context C, C