Timed recursive state machines: Expressiveness and complexity

The paper proposes a temporal extension of Recursive State Machines (RSMs), called Timed RSMs (TRSMs), which consists of an indexed collection of Timed Automata, called components. Each component can invoke other components in a potentially recursive manner. Besides being able to model procedure calls and recursion, TRSMs are equipped with the ability to suspend the evolution of time within a component when another component is invoked and to recover it when control is given back at return time. This mechanism is realized by storing clock valuations into an implicit stack at invocation time and restoring them upon return. Indeed, TRSMs can be related to an extension of Pushdown Timed Automata, called EPTAs, where an additional stack, coupled with the standard control stack, is used to store temporal valuations of clocks. The expressiveness and computational properties of the resulting model are analyzed, showing that it can be used to recognize timed languages exhibiting context-free properties not only in the untimed “control” part, but also in the associated temporal dimension. The reachability problem for both TRSMs and EPTAs is investigated, showing that the problem is undecidable in the general case. However, the problem becomes decidable for two meaningful subclasses, called I-TRSM and L-TRSM, obtained by suitably constraining the set of clocks to reset at invocation time and to restore at return time. The considered subclasses are compared with the corresponding EPTAs subclasses through bisimulation of their timed LTSs. The complexity of the reachability problem for L-TRSM and I-TRSM is proved to be EXPTIME-complete and PSPACE-complete, respectively. Moreover, we prove that such classes strictly enhance the expressive power of Timed Automata and of Pushdown Timed Automata, forming a proper hierarchy.