  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.
