Online verification of commutativity

Systems of transformations arise in many programming systems, such as in graphs of implicit type conversion functions. It is important to ensure that these diagrams commute: that composing any path of transformations from the same source to the same destination yields the same result. However, a straightforward approach to verifying commutativity must contend with cycles, and even so it runs in exponential time. Previous work has shown how to verify commutativity in the special case of acyclic diagrams in O(|V|4|E|2) time, but this is a batch algorithm: the entire diagram must be known ahead of time. We present an online algorithm that efficiently verifies that a commutative diagram remains commutative when adding a new edge. The new incremental algorithm runs in O(|V|2(|E| + |V|)) time. For the case when checking the equality of paths is expensive, we also present an optimization that runs in O(|V|4) time but reduces to the minimum possible number of equality checks. We implement the algorithms and compare them to batch baselines, and we demonstrate their practical application in the compiler of a domain-specific language for geometry types. To study the algorithms’ scalability to large diagrams, we apply them to discover discrepancies in currency conversion graphs.