Brief Announcement: Semantics of Eventually Consistent Replicated Sets

This paper studies the semantics of sets under eventual consistency. The set is a pervasive data type, used either directly or as a component of more complex data types, such as maps or graphs. Eventual consistency of replicated data supports concurrent updates, reduces latency and improves fault tolerance, but forgoes strong consistency (e.g., linearisability). Accordingly, several cloud computing platforms implement eventually-consistent replicated sets [2,4]. The sequential semantics of a set are well known, and are defined by individual updates, e.g., {true}add(e){e ∈ S} (in “{pre-condition} computation {post-condition}” notation), where S denotes its abstract state. However, the semantics of concurrent modifications is left underspecified or implementationdriven. We propose the following Principle of Permutation Equivalence to express that concurrent behaviour conforms to the sequential specification: “If all sequential permutations of updates lead to equivalent states, then it should also hold that concurrent executions of the updates lead to equivalent states.” It implies the following behavior, for some updates u and u′: {P}u;u′{Q} ∧ {P}u′;u{Q′} ∧Q ⇔ Q′ ⇒ {P}u ‖ u′{Q}