## Initialization

The data-based synthesis algorithm ensures that in the controlled system, the state requirement invariants hold in all reachable states. It also ensures that in the controlled system, for all transitions from reachable states, the events only occur if the requirement automata and state/event exclusion invariants allow them.

The synthesis algorithm does not restrict any uncontrollable events. Instead, such restrictions are propagated backwards to the source state of the edge with the uncontrollable event, and from there to the transitions that lead to the source state, etc. They are propagated backwards until an edge with a controllable event is encountered (for which the guard can be restricted) or the initial state is reached (and the initialization predicate can be restricted).

If a variable in the uncontrolled system has a single initial value, and the initialization predicate is restricted to not allow this initial value, initialization will be impossible, causing an empty supervisor error. For discrete variables with multiple potential initial values, the synthesis algorithm may restrict initialization to disallow certain initial values, while still leaving possibilities for initialization. For discrete variables declared to initially have an arbitrary initial value, as well as for input variables, the synthesis algorithm essentially determines under which conditions the system can be started, and still exhibits only safe, non-blocking behavior.

If the controlled system requires more strict initialization than the uncontrolled system, an additional initialization predicate is added to the resulting supervisor. The exact predicate may differ, depending on the simplifications that are applied.