On parametric timed automata and one-counter machines

Two decades ago, Alur, Henzinger, and Vardi introduced the reachability problem for parametric timed automata. Their main results are that reachability is decidable for timed automata with a single parametric clock, and undecidable for timed automata with three or more parametric clocks.In the case of two parametric clocks, decidability was left open, with hardly any progress that we are aware of in the intervening period.In this manuscript, we establish a correspondence between reachability in parametric timed automata with at most two parametric clocks and reachability for a certain class of parametric one-counter machines. We leverage this connection (i) to improve decision procedure for one parametric clock from nonelementary to 2NEXP; (ii) to show decidability for two parametric clocks and a single parameter; (iii) to show lower bounds for reachability problem for one and two parametric clocks; (iv) to show decidability for various classes of parametric one-counter machines.