Departmental Papers (CIS)

Date of this Version

January 2005

Document Type

Conference Paper


Postprint version. Copyright ACM, 2005. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Proceedings of the 32nd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 233-246.
Publisher URL:


We propose a novel approach to the well-known view update problem for the case of tree-structured data: a domain-specific 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 well-typed lenses.

We identify a natural 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, merging, etc.). We illustrate the expressiveness of these combinators by developing a number of bi-directional list-processing transformations as derived forms.


Bi-directional programming, Harmony, XML, lenses, view update problem



Date Posted: 10 September 2005