We consider the problem of verifying the safety of well-structured transition systems (WSTS) with auxiliary storage. WSTSs with storage are automata that have ( possibly) infinitely many control states along with an auxiliary store, but which have a well-quasi-ordering on the set of control states. The set of reachable configurations of the automaton may themselves not be well-quasi-ordered because of the presence of the extra store. We consider the coverability problem for such systems, which asks if it is possible to reach a control state (with some store value) that covers some given control state. Our main result shows that if control state reachability is decidable for automata with some store and {it finitely} many control states then the coverability problem can be decided for WSTSs (with infinitely many control states) and the same store, provided the ordering on the control states has some special property. The special property we require is defined in terms of the existence of a ranking function compatible with the transition relation. We then show that there are several classes of infinite state systems that can be viewed as WSTSs with an auxiliary storage. These observations can then be used to both reestablish old decidability results, as well as discover new ones.
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