Past Pushdown Timed Automata

We consider past pushdown timed automata that are discrete pushdown timed automata [15] with past-formulas as enabling conditions. Using past formulas allows a past pushdown timed automaton to access the past values of the finite state variables in the automaton. We prove that the reachability (i.e., the set of reachable configurations from an initial configuration) of a past pushdown timed automaton can be accepted by a nondeterministic reversal-bounded multicounter machine augmented with a pushdown stack (i.e., a reversal-bounded NPCM). Using the known fact that the emptiness problem for reversal-bounded NPCMs is decidable, we show that model-checking past pushdown timed automata against Presburger safety properties on discrete clocks and stack word counts is decidable. An example ASTRAL specification is presented to demonstrate the usefulness of the results.