Enforcement of (Timed) Properties with Uncontrollable Events

This paper deals with runtime enforcement of untimed and timed properties with uncontrollable events. Runtime enforcement consists in modifying the executions of a running system to ensure their correctness with respect to a desired property. We introduce a framework that takes as input any regular (timed) property over an alphabet of events, with some of these events being uncontrollable. An uncontrollable event cannot be delayed nor intercepted by an enforcement mechanism. Enforcement mechanisms satisfy important properties, namely soundness and compliance meaning that enforcement mechanisms output correct executions that are close to the input execution. We discuss the conditions for a property to be enforceable with uncontrollable events, and we define enforcement mechanisms that modify executions to obtain a correct output, as soon as possible. Moreover, we synthesize sound and compliant descriptions of runtime enforcement mechanisms at two levels of abstraction to facilitate their design and implementation.