Decidability and Complexity for Quiescent Consistency and its Variations

Abstract

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. The condition enables greater exibility in object design by allowing more behaviours to be admitted, which in turn allows the algorithms implementing quiescent consistent objects to become more e cient (when executed in a multithreaded environment). Quiescent consistency of an implementation object is de ned in terms of a corresponding abstract speci- cation. This gives rise to two important veri cation questions: membership (checking whether a behaviour of the implementation is allowed by the speci cation) and correctness (checking whether all behaviours of the implementation are allowed by the speci cation). In this paper, we show that the membership problem for quiescent consistency is NP-complete and that the correctness problem is decidable, but coNEXPTIME- complete. 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 PSPACE-complete. 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 speci cation. 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 placing a limit on the number of events between two quiescent points, membership is in PTIME, while correctness is PSPACE-complete. 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.