Hybrid automata model systems with both digital and analog components, such as embedded control programs. Many verification tasks for such programs can be expressed as reachability problems for hybrid automata. By improving on previous decidability and undecidability results, we identify a boundary between decidability and undecidability for the reachability problem of 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 independent trajectories within piecewise-linear envelopes and are reinitialized whenever the envelope changes. Our algorithm is based on the construction of a timed automaton that contains all reachability information about a given initialized rectangular automaton. The translation has practical significance for verification, because it guarantees the termination of symbolic procedures for the reachability analysis of initialized rectangular automata. The translation also preserves theω-languages of initialized rectangular automata with bounded nondeterminism. 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 automata augmented with a single stopwatch.

We present algorithms for computing similarity relations of labeled graphs. Similarity relations have applications for the refinement and verification of reactive systems. For finite graphs, we present an O(mn) algorithm for computing the similarity relation of a graph with n vertices and m edges (assuming m/spl ges/n). For effectively presented infinite graphs, we present a symbolic similarity-checking procedure that terminates if a finite similarity relation exists. We show that 2D rectangular automata, which model discrete reactive systems with continuous environments, define effectively presented infinite graphs with finite similarity relations. It follows that the refinement problem and the /spl forall/CTL* model-checking problem are decidable for 2D rectangular automata.

Hybrid automata model systems with both digital and analog components, such az embedded control programs.Many verification tasks for such programs can be expressed as reachability problems for hybrid automata.By improving on previous decidability and undecidability 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 trajectories 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 defines the same timed language.The translation has practical significance for verification, 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 automata with a single stopwatch.