We propose a novel approach to the view update problem for tree-structured data: a domainspecific programming language in which all expressions denote bi-directional transformations on trees. In one direction, these transformations—dubbed lenses—map a “concrete ” tree into a simplified “abstract view”; in the other, they map a modified abstract view, together with the original concrete tree, to a correspondingly modified concrete tree. Our design emphasizes both robustness and ease of use, guaranteeing strong well-behavedness and totality properties for welltyped lenses. We identify a natural mathematical space of well-behaved bi-directional transformations over arbitrary structures, study definedness and continuity in this setting, and state a precise connection with the classical theory of “update translation under a constant complement ” from databases. We then instantiate this semantic framework in the form of a collection of lens combinators that can be assembled to describe transformations on trees. These combinators include familiar constructs from functional programming (composition, mapping, projection, conditionals, recursion) together with some novel primitives for manipulating trees (splitting, pruning, copying, merging, etc.). We illustrate the expressiveness of these combinators by developing a number of bi-directional listprocessing transformations as derived forms. An extended example shows how our combinators can be used to define a lens that translates between a native HTML representation of browser bookmarks and a generic abstract bookmark format.

We propose a novel approach to the classical view update problem. The view update problem arises from the fact that modifications to a database view may not correspond uniquely to modifications on the underlying database; we need a means of determining an “update policy ” that guides how view updates are reflected in the database. Our approach is to define a bi-directional query language, in which every expression can be read both (from left to right) as a view definition and (from right to left) as an update policy. The primitives of this language are based on standard relational operators. Its type system, which includes recordlevel predicates and functional dependencies, plays a crucial role in guaranteeing that update policies are well-behaved, in a precise sense, and that they are total—i.e., able to handle arbitrary changes to the view.

We develop a semantic foundation and a core programming language for bi-directional transformations on treestuctured data. In one direction, these transformations, called lenses, map a complex "concrete" tree into a simplified "abstract" one; in the other, they map a modified abstract tree, together with the original concrete tree, to a correspondingly modified concrete tree.