Translating Asynchronous Games for Distributed Synthesis (Full Version)

In distributed synthesis, we generate a set of process implementations that, together, accomplish an objective against all possible behaviors of the environment. A lot of recent work has focussed on systems with causal memory, i.e., sets of asynchronous processes that exchange their causal histories upon synchronization. Decidability results for this problem have been stated either in terms of control games, which extend Zielonka’s asynchronous automata by partitioning the actions into controllable and uncontrollable, or in terms of Petri games, which extend Petri nets by partitioning the tokens into system and environment players. The precise connection between these two models was so far, however, an open question. In this paper, we provide the first formal connection between control games and Petri games. We establish the equivalence of the two game models based on weak bisimulations between their strategies. For both directions, we show that a game of one type can be translated into an equivalent game of the other type. We provide exponential upper and lower bounds for the translations. Our translations make it possible to transfer and combine decidability results between the two types of games. Exemplarily, we translate decidability in acyclic communication architectures, originally obtained for control games, to Petri games, and decidability in single-process systems, originally obtained for Petri games, to control games.

💡 Research Summary

The paper addresses a fundamental gap in the theory of distributed synthesis for asynchronous systems with causal memory. Two prominent formal models have been used independently to study this problem: control games, which extend Zielonka’s asynchronous automata by distinguishing controllable and uncontrollable actions, and Petri games, which extend Petri nets by partitioning tokens into system and environment players. Although both models capture the same intuition—processes make decisions based on their causal past—their syntactic treatment of control differs, making it unclear whether results proved for one model can be transferred to the other.

The authors first formalize both models. In a control game, each process observes a local view (the shortest trace that contains all actions the process has participated in or learned about via synchronisation). A controller is a family of local functions that, based on these views, may enable or block controllable actions; uncontrollable actions are always possible. Winning conditions are expressed as sets of local states that must be avoided (safety) or reached (reachability). In a Petri game, the underlying net defines a global unfolding; tokens placed on system places can restrict outgoing transitions, while tokens on environment places cannot. Strategies are defined as restrictions of the unfolding that guarantee the desired property for all possible environment moves.

The core technical contribution is a bidirectional translation between the two models that preserves winning strategies up to weak bisimulation. The main obstacle is the different granularity of control: a single action in a control game is either fully controllable or fully uncontrollable for all participating processes, whereas a transition in a Petri game may be controllable by some tokens and uncontrollable by others. To bridge this, the authors introduce commitment sets. Before executing an action, a process (or token) moves to an intermediate state that explicitly records which actions it intends to enable. This intermediate state carries the same causal information as the original view, allowing the translation to encode the control decision at the level required by the target model.

Control‑to‑Petri translation: For each process p, its local automaton is turned into a set of places and transitions. Every controllable action a is split into a “commit‑a” transition (which records the intention to enable a) followed by the actual a transition that synchronises with the other participants. Uncontrollable actions become ordinary transitions that are always fireable. The resulting Petri game’s system places correspond to the original processes, and the added commit transitions ensure that the system’s ability to block a is represented by the presence or absence of a token in a specific place.

Petri‑to‑Control translation: Each system place becomes a dedicated process in a control game. Transitions outgoing from system places are mapped to controllable actions, while those from environment places become uncontrollable. Again, a commit action is inserted before each real action to capture the token’s decision in the local view of the corresponding process. The resulting control game reproduces the causal information of the original unfolding.

Both translations are shown to be strategy‑preserving: a winning controller in the source game can be transformed into a winning controller in the target game, and vice versa. The authors prove that the two resulting transition systems are weakly bisimilar, meaning that any observable behaviour (up to the independence relation of asynchronous actions) is matched in the other model.

Complexity analysis reveals that the translations are exponential in the worst case. The upper bound stems from the need to introduce a commit transition for each original action and to encode the domain of each action (the set of processes that must synchronise). The authors also provide matching lower bounds by constructing families of games where any correct translation necessarily incurs an exponential blow‑up, showing that the exponential cost is inherent.

Having established the equivalence, the paper demonstrates transfer of decidability results. Known decidability for control games on acyclic communication architectures (EXPTIME‑complete) can now be applied to Petri games with the same architecture, because the translation preserves the structure of the communication graph. Conversely, decidability results for single‑process Petri games (or Petri games with a bounded number of system tokens) can be lifted to control games with a single process, yielding new algorithmic insights for that class.

The authors conclude that the two formalisms, previously thought to be distinct, are in fact two views of the same underlying problem. This unification opens the door to cross‑fertilisation of techniques: algorithms, tool support, and complexity results can be shared across the communities. Future work may focus on reducing the exponential overhead (e.g., via symbolic representations), extending the translation to richer communication topologies (including cyclic graphs), and implementing the translations in practical synthesis tools to evaluate their impact on real‑world distributed systems.