Complements Witness Consistency

Much of the existing bx literature, especially that from the PL community on lenses, has described extensional, state-based formalisms. More recently, attention has turned to incorporating intensional information about edits (typically based on monoid actions), or more generally, deltas (typically based on categories), describing how models are updated. Pervasive in both the conceptual modelling, and the mathematics, of varieties of such bx, is the role played by the complement, which generalises the ‘constant complement’ case of the view-update problem in databases. Complements typically reify, or correspond to, data which is abstracted away by passing from a source to a view. In this paper, we present an alternative perspective, which has perhaps been implicit in the lens literature, but not, to our knowledge, previously made explicit anywhere: namely that elements of the complement are witnesses to the consistency relation maintained by the transformation. We illustrate this idea with examples drawn from the bx literature, especially that on lenses.