What ' s Decidable about Hybrid Automata ? 1

Hybrid automata model systems with both digital and analog components, such as embedded control programs. Many veriication tasks for such programs can be expressed as reachability problems for hybrid au-tomata. By improving on previous decidability and un-decidability results, we identify the precise boundary between decidability and undecidability of the reachability problem for hybrid automata. On the positive side, we give an (optimal) PSPACE reachability algorithm for the case of initialized rectangular automata, where all analog variables follow trajecto-ries within piecewise-linear envelopes and are reinitialized whenever the envelope changes. Our algorithm is based on a translation of an initialized rectangular automaton into a timed automaton that deenes the same timed language. The translation has practical signiicance for verii-cation, because it guarantees the termination of symbolic procedures for the reachability analysis of initialized rectangular automata. On the negative side, we show that several slight generalizations of initialized rectangular automata lead to an undecidable reachability problem. In particular, we prove that the reachability problem is undecidable for timed au-tomata with a single stopwatch.