Optimal 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 defining and using mechanisms that modify 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 described by a deterministic automaton 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 should satisfy important properties, namely soundness, compliance and optimality - meaning that enforcement mechanisms should output as soon as possible correct executions that are as close as possible to the input execution. We define the conditions for a property to be enforceable with uncontrollable events. Moreover, we synthesise sound, compliant and optimal descriptions of runtime enforcement mechanisms at two levels of abstraction to facilitate their design and implementation.