Decidability and complexity for quiescent consistency and its variations

Quiescent consistency is a notion of correctness for a concurrent object that gives meaning to the object's behaviours in quiescent states, i.e., states in which none of the object's operations are being executed. Correctness of an implementation object is defined in terms of a corresponding abstract specification. This gives rise to two important verification questions: membership (checking whether a behaviour of the implementation is allowed by the specification) and correctness (checking whether all behaviours of the implementation are allowed by the specification). In this paper, we show that the membership problem for quiescent consistency is NP-complete and that the correctness problem is decidable, but coNP-hard and in EXPSPACE. For both problems, we consider restricted versions of quiescent consistency by assuming an upper limit on the number of events between two quiescent points. Here, we show that the membership problem is in PTIME, whereas correctness is in PSPACE. Quiescent consistency does not guarantee sequential consistency, i.e., it allows operation calls by the same process to be reordered when mapping to an abstract specification. Therefore, we also consider quiescent sequential consistency, which strengthens quiescent consistency with an additional sequential consistency condition. We show that the unrestricted versions of membership and correctness are NP-complete and undecidable, respectively. When by placing a limit on the number of events between two quiescent points, membership is in PTIME, while correctness is in PSPACE. Finally, we consider a version of quiescent sequential consistency that places an upper limit on the number of processes for every run of the implementation, and show that the membership problem for quiescent sequential consistency with this restriction is in PTIME.