On parametric timed automata and one-counter machines

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.