Modern distributed systems often rely on so called weakly-consistent databases, which achieve scalability by sacrificing the consistency guarantee of distributed transaction processing. Such databases have been formalised in two different styles, one based on abstract executions and the other based on dependency graphs. The choice between these styles has been made according to intended applications: the former has been used to specify and verify the implementation of these databases, and the latter to prove properties of programs running on top of the databases. In this paper, we present a set of novel algebraic laws (i.e. inequations) that connect these two styles of specifications; the laws relate binary relations used in a specification based on abstract executions, to those used in a specification based on dependency graphs. We then show that this algebraic connection gives rise to so called robustness criteria, conditions which ensures that a program running on top of a weakly-consistent database does not exhibit anomalous behaviours due to this weak consistency. These criteria make it easy to reason about programs running on top of these databases, and may become a basis for dynamic or static program analyses. For a certain class of consistency models specifications, we prove a full abstraction result that connects the two styles of specifications.
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