Martin Hofmann, Edit Lenses

Edit Lenses (Martin Hofmann)

A lens is a bidirectional transformation between a pair of connected data structures, capable of translating an edit on one structure into an appropriate edit on the other. Many varieties of lenses have been studied, but none, to date, has offered a satisfactory treatment of how edits are represented. Many foundational accounts only consider edits of the form "overwrite the whole structure," leading to poor behavior in many situations by failing to track the associa- tions between corresponding parts of the structures when elements are inserted and deleted in ordered lists, for example. Other theo- ries of lenses do maintain these associations, either by annotating the structures themselves with change information or with auxiliary data structures, but every extant theory assumes that the entire original source structure is part of the information passed to the lens--a rather special (and perhaps inefficient) choice. We offer here a general theory of edit lenses, which work with descriptions of changes to structures, rather than with the structures themselves. We identify a simple notion of "editable structure"-- a set of states plus a monoid of edits with a partial action on the states--and construct a semantic space of lenses between such structures, with natural laws governing their behavior. We show how a range of constructions from earlier papers on "state-based" lenses--composition, products, sums, lists operations, etc.--can be carried out in this space. Further, we show how to construct edit lenses for arbitrary containers in the sense of Abbott, Altenkirch, and Ghani. Finally, we show that edit lenses refine a well-known formulation of state-based lenses, in the sense that every state- based lens gives rise to an edit lens over structures with a simple overwrite-only edit language, and conversely every edit lens on such structures gives rise to a state-based lens.

This is joint work with Benjamin Pierce and Daniel Wagner.