Synchronous Games, Simulations and lambda-calculus

We refine a model for linear logic based on two well-known ingredients: games and simulations. We have already shown that usual simulation relations form a sound notion of morphism between games; and that we can interpret all linear logic in this way. One particularly interesting point is that we interpret multiplicative connectives by synchronous operations on games. We refine this work by giving computational contents to our simulation relations. To achieve that, we need to restrict to intuitionistic linear logic. This allows to work in a constructive setting, thus keeping a computational content to the proofs. We then extend it by showing how to interpret some of the additional structure of the exponentials. To be more precise, we first give a denotational model for the typed lambda-calculus; and then give a denotational model for the differential lambda-calculus of Ehrhard and Regnier. Both this models are proved correct constructively.

💡 Research Summary

The paper presents a constructive denotational model for intuitionistic linear logic (ILL) that combines two well‑established ideas: game semantics and simulation relations. In earlier work the authors showed that simulation relations between games form a sound notion of morphism and that the multiplicative connectives (⊗ and ⅋) can be interpreted as synchronous operations on games. However, that work did not attach computational content to the proofs. By restricting to ILL, the authors obtain a setting where every proof step can be interpreted constructively, allowing simulations to be realized as actual program transformations rather than mere existence statements.

The core technical development proceeds as follows. A game is defined as a set of positions together with a set of moves; each move leads deterministically to a successor position. A simulation relation R ⊆ P₁ × P₂ between games G₁ and G₂ satisfies the usual back‑and‑forth condition: whenever the player of G₁ makes a move from a position p₁, there exists a matching move in G₂ from the related position p₂ such that the resulting pair of successor positions remains in R. Crucially, the authors require R to be constructively realizable: R itself is represented by a computable function (or program) that, given a pair of positions, produces the matching move in the second game. This makes the simulation a concrete strategy transformer.

Next, the logical connectives of ILL are interpreted as operations on games:

• Tensor (⊗) is modeled by the synchronous parallel composition of two games; a player must make a move in both components simultaneously.
• Par (⅋) is the dual synchronous composition, where the player may interleave moves but must respect the linear usage constraints.
• Linear implication (⊸) is interpreted as a simulation from the antecedent game to the consequent game; a term of type A ⊸ B corresponds to a strategy that, given a strategy for A, produces a strategy for B.

The authors prove that these interpretations preserve the simulation relation: if R₁ : G₁ → H₁ and R₂ : G₂ → H₂ are simulations, then R₁ ⊗ R₂ : G₁ ⊗ G₂ → H₁ ⊗ H₂ is again a simulation, and similarly for the other connectives. This closure under the logical operations is essential for building a categorical model of ILL.

With the game‑simulation framework in place, the paper moves to a denotational semantics for the simply‑typed λ‑calculus enriched with linear types. Types are identified with ILL formulas, and λ‑terms are mapped to strategies on the corresponding games. Function abstraction λx.t becomes a simulation that, given a strategy for the argument type, yields the strategy for the body; application corresponds to the synchronous interaction of the argument and function strategies. The authors establish soundness with respect to β‑reduction and η‑expansion, showing that the semantic interpretation respects the operational calculus. Moreover, they provide a constructive normalization algorithm derived directly from the game dynamics, thereby giving computational content to proofs.

The final contribution is an extension to the differential λ‑calculus of Ehrhard and Regnier. The differential operator D, which captures linear approximation of terms, is modeled by a game operation that duplicates a move (reflecting the linear‑resource sensitivity) and then distributes it across two synchronized sub‑games. The simulation relations are shown to be stable under this duplication, satisfying the axioms of the differential λ‑calculus (linearity, Leibniz rule, etc.). Consequently, the authors obtain a constructive denotational model for the differential λ‑calculus, proving its correctness within the same game‑simulation setting.

In summary, the paper achieves three major results: (1) a constructive interpretation of ILL where simulations are realized as computable strategies; (2) a sound and complete denotational model for the typed λ‑calculus based on this interpretation; and (3) an extension of the model to accommodate differential operators, yielding a constructive semantics for the differential λ‑calculus. By grounding linear logic in synchronous games and explicit simulations, the work bridges the gap between logical proof theory and concrete computational processes, offering a robust framework for future investigations into resource‑aware programming languages and their semantics.