Complexity Issues in Automated Addition of Time-Bounded Liveness Properties

In this paper, we concentrate on synthesis of real-time programs modeled by Alur and Dill timed automata for automatic addition of different types of time-bounded liveness properties. Time-bounded liveness (also called time-bounded response) – that something good will happen soon, in a certain amount of time – captures a wide range of requirements for specifying real-time and embedded systems. We show that the problem of automatic addition of a time-bounded liveness property to a given timed automaton while maintaining maximal nondeterminism is NP-hard in the size of locations of the input automaton. Furthermore, we show that by relaxing the maximality requirement we can devise a sound and complete algorithm that adds a time-bounded liveness property to a given timed automaton, while preserving its existing MTL specification. This synthesis method is useful in adding properties that are later discovered as a crucial part of a program. Moreover, we show that addition of interval time-bounded liveness, where the good thing should not happen sooner than a certain amount of time, is also NP-hard in the size of locations even without maximal nondeterminism. Finally, we show that adding time-bounded and interval time-bounded as well as unbounded liveness properties are all PSPACE-complete in the size of the input timed