Membership questions for timed and hybrid automata

Timed and hybrid automata are extensions of finite-state machines for formal modeling of embedded systems with both discrete and continuous components. Reachability problems for these automata are well studied and have been implemented in verification tools. In this paper for the purpose of effective error reporting and testing, we consider the membership problems for such automata. We consider different types of membership problems depending on whether the path (i.e. edge-sequence), or the trace (i.e. event-sequence), or the timed trace (i.e. timestamped event-sequence), is specified We give comprehensive results regarding the complexity of these membership questions for different types of automata, such as timed automata and linear hybrid automata, with and without epsilon-transitions. In particular we give an efficient O(n.m(2)) algorithm for generating timestamps corresponding a path of length n in a timed automaton with m clocks. This algorithm is implemented in the verifier COSPAN to improve its diagnostic feedback during timing verification. Second, we show that for automata without epsilon-transitions, the membership question is NP-complete for different types of automata whether or not the timestamps are specified along with the trace. Third, we show that for automata with epsilon-Transitions, the membership question is as hard as the reachability question even for timed traces: it is PSPACE-complete for rimed automata, and undecidable for slight generalizations.