Innocent strategies as presheaves and interactive equivalences for CCS (expanded version)

Seeking a general framework for reasoning about and comparing programming languages, we derive a new view of Milner’s CCS. We construct a category E of ‘plays’, and a subcategory V of ‘views’. We argue that presheaves on V adequately represent ‘innocent’ strategies, in the sense of game semantics. We equip innocent strategies with a simple notion of interaction. We then prove decomposition results for innocent strategies, and, restricting to presheaves of finite ordinals, prove that innocent strategies are a final coalgebra for a polynomial functor derived from the game. This leads to a translation of CCS with recursive equations. Finally, we propose a notion of ‘interactive equivalence’ for innocent strategies, which is close in spirit to Beffara’s interpretation of testing equivalences in concurrency theory. In this framework, we consider analogues of fair testing and must testing. We show that must testing is strictly finer in our model than in CCS, since it avoids what we call ‘spatial unfairness’. Still, it differs from fair testing, and we show that it coincides with a relaxed form of fair testing.

💡 Research Summary

The paper presents a novel categorical reconstruction of Milner’s Calculus of Communicating Systems (CCS) by leveraging ideas from game semantics, presheaf theory, and coalgebra. The authors first introduce a category E of “plays”, where a play is not merely a linear trace but a more flexible string‑diagram that captures concurrent, independent moves. Plays are equipped with two kinds of morphisms: prefix inclusion (extending a play) and position enlargement (adding information about additional participants). A subcategory V ⊂ E, called “views”, is then defined; objects of V represent the limited information that a single participant can observe.

The central technical claim is that V‑presheaves (i.e. functors Vᵒᵖ → Set) precisely model innocent strategies in the sense of game semantics: a presheaf F on V is innocent when it is isomorphic to the right Kan extension of its restriction along the inclusion V → E. In other words, the behaviour of a strategy on the whole play category is completely determined by its behaviour on the view category. This yields a clean, compositional description of strategies as presheaves on views, while ordinary strategies (on all plays) are obtained by right Kan extension.

Interaction between strategies is captured by an amalgamation operation. Given a position X that decomposes into two disjoint sub‑positions X₁ and X₂, and innocent strategies F₁ on X₁ and F₂ on X₂, there exists a unique innocent strategy r(F₁,F₂) on X whose restrictions recover F₁ and F₂. This mirrors the standard game‑semantic composition where two two‑player games interact through a shared interface. The paper does not yet develop a full “hiding” step, but notes that amalgamation already suffices to define a notion of testing.

When restricting to presheaves of finite ordinals (i.e. finite sets), the collection of innocent strategies forms a final coalgebra for a polynomial functor P derived from the underlying game. This connects the construction to Kleene coalgebra: the functor encodes possible input, output, and a distinguished “tick” move that signals success. Consequently, CCS terms (including recursive equations) can be translated into elements of this final coalgebra by unfolding recursive definitions indefinitely.

The authors then introduce interactive equivalence, inspired by testing equivalences in concurrency theory. A test is an innocent strategy G on a separate position X₁ that shares the same channel set as the strategy under test F. To decide whether F passes G, one forms the amalgamated strategy r(F,G), extends it to the combined position X ⊎ X₁ via right Kan extension, and then restricts to a “closed‑world” sub‑game that forbids interaction with the outside. F passes G iff the resulting strategy belongs to a pre‑specified class K of “successful” strategies. Two concrete classes are studied:

• Kₘ: strategies whose maximal states (those with no further extensions) are all successful (i.e., contain a tick). This class corresponds to the traditional must‑testing equivalence.
• K_f: strategies where every finite play can be extended to a successful one. This class corresponds to fair‑testing.

The paper shows that, in this framework, must‑testing becomes strictly finer than in standard CCS because the notion of play already eliminates “spatial unfairness”: a process that can eventually output on a channel but is blocked by an infinite silent loop is no longer considered maximal, so the unfair scheduler cannot hide the output. However, must‑testing still differs from fair‑testing because other sources of unfairness (e.g., scheduler bias) remain; consequently Kₘ does not coincide with K_f. The authors prove that Kₘ coincides with the set of strategies whose states all admit a successful extension, and that the restriction to finite plays in K_f is essential to rule out these remaining unfair behaviours.

Beyond the technical results, the paper outlines several future directions: refining the definition of interactive equivalence (especially the hiding step), tightening the connection between the induced equivalence on CCS terms and classical testing equivalences, extending the framework to richer calculi such as the π‑calculus or λ‑calculus, exploring probabilistic or quantitative testing, and relating the construction to graph rewriting, computads, and higher‑dimensional rewriting systems. Ultimately, the authors envision a systematic pipeline that, given any process calculus, produces a category of innocent strategies and an associated interactive equivalence, providing a uniform semantic foundation for comparing languages and reasoning about compilation.