Bottom-up β-reduction: uplinks and λ-DAGs

• Authors: Olin Shivers;Mitchell Wand
• Venue: ESOP'05 Proceedings of the 14th European conference on Programming Languages and Systems
• Year: 2005

Representing a λ term as a DAG rather than a tree allows us to represent the sharing that arises from β, thus avoiding combinatorial explosion in space. By adding uplinks from a child to its parents, we can efficiently implement β in a bottom-up manner, thus avoiding combinatorial explosion in time required to search the term in a top-down fashion. We present an algorithm for performing β onλ-terms represented as uplinked DAGs; discuss its relation to alternate techniques such as Lamping graphs, explicit-substitution calculi and director strings; and present some timings of an implementation. Besides being both fast and parsimonious of space, the algorithm is particularly suited to applications such as compilers, theorem provers, and type-manipulation systems that may need to examine terms in-between reductions—i.e., the “readback” problem for our representation is trivial. Like Lamping graphs, and unlike director strings or the suspension λ, the algorithm functions by side-effecting the term containing the redex; the representation is not a “persistent” one. The algorithm additionally has the charm of being quite simple: a complete implementation of the core data structures and algorithms is 180 lines of SML.