A Consistency in Non-Transactional Distributed Storage Systems

executions. We model system executions using the concept of abstract execution, following Burckhardt [2014]. An abstract execution is a multi-graphA = (H, vis, ar) built on a given history H, which it complements with two relations on elements of H, i.e., vis and ar. Whereas histories describe the observable outcomes of executions, vis and ar, intuitively, capture the non-determinism of the asynchronous environment (e.g., message delivery order), as well as implementation-specific constraints (e.g., conflictresolution policies). In other words, vis and ar determine the relations between pairs of operations in a history that explain and justify its outcomes. More specifically: — vis (visibility) is an acyclic natural relation that accounts for the propagation of write operations. Intuitively, a be visible to b (i.e., a vis −→ b) means that the effects of a are visible to the process invoking b (e.g., b may read a value written by a). Two write operations are invisible to each other if they are not ordered by vis. — ar (arbitration) is a total order on operations of the history that specifies how the system resolves conflicts due to concurrent and invisible operations. In practice, such total order can be achieved in various ways: through the adoption of a distributed timestamping [Lamport 1978] or consensus protocol [Birman et al. 1991; Hadzilacos and Toueg 1994; Lamport 2001], using a centralized serializer, or using a deterministic conflict resolution policy. Depending on constraints expressed by vis, during an execution processes may observe different orderings of write operations, which we call serializations. We further define the happens-before order (hb) as the transitive closure of the union of so and vis, denoted by: