A Language for Bi-Directional Tree Transformations (Working Draft of Full Version)

We develop a semantic foundation and a core programming language for bi-directional transformations on tree-stuctured 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. The challenge of understanding and designing these transformations 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” whose two components are constrained to fit together in a sensible way. We study definedness and continuity in this setting, show that well-behaved lenses form a complete partial order, and state a precise connection with the classical theory of “update translation under a constant complement” from databases. We then instantiate our semantic framework as a small programming language, called Hocus Focus, whose expressions denote well-behaved lenses operating on tree-structured data. The primitives 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 Hocus Focus can be used to define a lens that translates between a native HTML representation of browser bookmarks and a generic form of abstract bookmark structures.