Resourceful Lenses for Ordered Data

A lens is a bi-directional program. When read from left to right, it denotes an ordinary function. When read from right to left, it denotes an “update translator” that takes an input to this function together with an updated output and produces a new input reflecting the update. Several variants of the idea of lenses have been explored. But none of them deal fully with ordered data. If an update involves a change in the order of a list (for example) in the output, the correspondence between the items in the output list and the items in the input list can be lost, leading to loss or corruption of data. We propose a refined semantic space of resourceful lenses, enriching the well-behaved lenses of Foster, Greenwald, Moore, Pierce, and Schmitt (2007) with an equivalence relation embodying a notion of “reordering,” with respect to which the translation of updates must be invariant. We then present a regular-expressionlike syntax for building resourceful lenses over string data. The key technical challenge is devising an intermediate semantic space that supports global matching of corresponding chunks of input and outputs structures while providing suitable denotations for a compositional language of lens combinators with local static checks guaranteeing that the semantic laws are satisfied. We give examples showing how our language can be used to build realworld string transformations.