Criteria for the Simple Path Property in Timed Automata

Timed automata have been studied in the past and have been found to have a complexity dependent on the relative scale of the time constants involved in the timing constraints imposed, even if the timing constraints are restricted to the form x<k where x is a clock variable and k is a constant. We have previously shown that this complexity dependence on the time constants can be eliminated if the timed automaton has the simple path property (state A is reachable from state B if and only if it is reachable along a path with no cycles), and gave a set of conditions on the placement of clock queries and resets which imply this simple path property. These automata were called alternating RQ timed automata. We gave a technique for using this properly to iteratively constrain an untimed automaton to rule out simple paths which cannot meet their timing constraints. The simple path property means that only simple paths need be constrained. In this paper, we give conditions for a timed automaton with arbitrary constraint equations to have the simple path property. As far as we know all practical examples in the literature meet these criteria. For example, this includes all automata with constraints of the form for each state s, a trace must remain in s for a time t where \(t_{s_{min} } < t < t_{s_{max} }\). We are currently working on an efficient implementation for timed automata where arbitrary linear inequalities among the clock values are allowed. Using linear programming, we iteratively detect simple paths which are not traversable and construct untimed automata which disallow these paths. The present paper serves to extend this approach to a wide class of applications. In addition, we define extended RQ timed automata which include all the examples in the literature and are easily tested for this property.