A Language for Bi-Directional Tree Transformations

We present a semantic foundation and a collection of program combinators for bi-directional transformations on tree-structured data. In one direction, these transformations map a “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. The challenge of understanding and designing these transformations—called lenses—arises from their asymmetric nature: information is discarded when mapping from concrete to abstract, and must be restored on the way back. We identify a natural mathematical space of well-behaved lenses, in which the two components are constrained to fit together in a sensible way. We 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 our semantic framework in the form of a collection of lens combinators that can be assembled to describe powerful transformations on trees and whose well-behavedness follows from simple, local checks. These combinators include familiar constructs from functional programming (composition, mapping, projection, recursion) together with some novel primitives for manipulating trees (splitting, pruning, pivoting, etc.). 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.