Decidability and Complexity Results for Timed Automata and Semi-linear Hybrid Automata

We define a new class of hybrid automata for which reachability is decidable—a proper superclass of the initialized rectangular hybrid automata—by taking parallel compositions of simple components. Attempting to generalize, we encounter timed automata with algebraic constants. We show that reachability is undecidable for these algebraic timed automata by simulating two-counter Minsky machines. Modifying the construction to apply to parametric timed automata, we reprove the undecidability of the emptiness problem, and then distinguish the dense and discrete-time cases with a new result. The algorithmic complexity— both classical and parametric—of one-clock parametric timed automata is also examined. We finish with a table of computability-theoretic complexity results, including that the existence of a Zeno run is Σ11-complete for semi-linear hybrid automata; it is too complex to be expressed in first-order arithmetic.