Asynchronous games: innocence without alternation

The notion of innocent strategy was introduced by Hyland and Ong in order to capture the interactive behaviour of lambda-terms and PCF programs. An innocent strategy is defined as an alternating strategy with partial memory, in which the strategy plays according to its view. Extending the definition to non-alternating strategies is problematic, because the traditional definition of views is based on the hypothesis that Opponent and Proponent alternate during the interaction. Here, we take advantage of the diagrammatic reformulation of alternating innocence in asynchronous games, in order to provide a tentative definition of innocence in non-alternating games. The task is interesting, and far from easy. It requires the combination of true concurrency and game semantics in a clean and organic way, clarifying the relationship between asynchronous games and concurrent games in the sense of Abramsky and Melli`es. It also requires an interactive reformulation of the usual acyclicity criterion of linear logic, as well as a directed variant, as a scheduling criterion.

💡 Research Summary

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The paper tackles a fundamental limitation of the classic notion of innocence in game semantics, which has traditionally relied on a strict alternation between Opponent and Proponent moves. In Hyland‑Ong’s pointer games, an innocent strategy is defined by the fact that the player reacts only according to its “view” of the play, a view that is extracted by recursively removing “inauthentic” moves using justification pointers. This extraction presupposes that moves strictly alternate; when this hypothesis is dropped, the usual definition collapses.

To overcome this, the authors turn to the framework of asynchronous games, a setting that enriches transition systems with two‑dimensional tiles. A tile expresses the independence of two transitions m and n: the square

x ──m──► y
│          │
n          n
│          │
z ──m──► w

states that performing m then n leads to the same state as performing n then m. Such squares generate a homotopy equivalence on paths, identifying different interleavings that differ only by commuting independent moves. In this picture, justification pointers of pointer games are replaced by the absence of a tile that would allow a forbidden reordering; thus causality is captured diagrammatically.

Using this diagrammatic machinery, the authors propose a new definition of innocence that no longer mentions views. The key property is locality with respect to positions (states) modulo homotopy: if two alternating plays s and t end in the same position x and are homotopic, then for any Opponent move m from x, the Proponent’s response n is forced uniformly:

 s·m·n ∈ σ ⇔ t·m·n ∈ σ.

Consequently, an innocent strategy is completely determined by the set of positions it can reach, where a position is understood as a homotopy class of plays. This set of positions defines a closure operator on the underlying partial order of information states, exactly the representation used in Abramsky‑Melliès’s concurrent games. Hence innocent strategies in the asynchronous setting are simultaneously alternating‑style strategies (as sets of plays) and concurrent‑style strategies (as closure operators).

The paper illustrates the ideas with the Boolean game B and its tensor product B⊗B. In B the interaction is simple (Opponent asks a question q, Proponent answers true or false). In B⊗B two independent Boolean cells are queried. A traditional alternating innocent strategy would contain a single linearisation, e.g. qL·trueL·qR·falseR. A non‑alternating innocent strategy, however, contains both linearisations (left‑first and right‑first) and, more generally, every possible interleaving of the underlying partial order. All these interleavings converge to the same homotopy class (the same final position), and the strategy’s causal structure can be recovered as the partial order on moves that generated that position.

Beyond examples, the authors formalise “ingenious strategies” – strategies equipped with an underlying causality order for every reachable position and satisfying a suite of diagrammatic conditions (tile completeness, cube property, etc.). While ingenious strategies are not yet closed under composition, they provide the first systematic way to talk about non‑alternating innocent strategies in a categorical setting.

Finally, the work positions asynchronous games as a unifying bridge between pointer games (interleaving semantics) and concurrent games (truly concurrent semantics). By redefining innocence without alternation, the authors open the door to fully concurrent denotational models for languages such as Parallel Algol, asynchronous π‑calculus, and other process‑oriented calculi. Future research directions include establishing compositionality for ingenious strategies, exploring full categorical structures, and applying the framework to richer type systems and effectful concurrent languages.