Hybrid Automata with Finite Mutual Simulations

Many decidability results for hybrid automata rely upon the finite region bisimulation of timed automata [AD94]. Rectangular automata do not have finite bisimulations [Hen95], yet have many decidable verification problems [PV94,HKPV95]. We prove that every two-dimensional rectangular automaton A with positive-slope variables has a finite mutual simulation relation, which is the intersection of the region bisimulations defined by the extremal slopes of the variables of A. While the mutual simulation is infinite for two-dimensional automata with one variable taking both positive and negative slopes, it forms a regular tesselation of the plane, and therefore can be encoded by one counter. As a corollary, we obtain the decidability of model checking linear temporal logic on these automata.