A study on shuffle, stopwatches and independently evolving clocks

We show that stopwatch automata are equivalent with timed shuffle expressions, an extension of timed regular expressions with the shuffle operation. Since the emptiness problem is undecidable for stopwatch automata, and hence also for timed shuffle expressions, we introduce a decidable subclass of stopwatch automata called partitioned stopwatch automata. We give for this class an equivalent subclass of timed shuffle expressions and investigate closure properties by showing that partitioned stopwatch automata are closed under union, concatenation, star, shuffle and renaming, but not under intersection. We also show that partitioned stopwatch automata are equivalent with distributed time-asynchronous automata, which are asynchronous compositions of timed automata in which time may evolve independently.