Reducing Lambda Terms with Traversals (preprint) February 26, 2018

We introduce a method to evaluate untyped lambda terms by combining the theory of traversals,a term-tree traversing techniques inspired from Game Semantics [12, 8], with judicious use of theeta-conversion rule of the lambda calculus.The traversal theory of the simply-typed lambda calculus relies on the eta-long transform toensure that when traversing an application, there is a subterm representing each possible operator’sargument[8, 12]. In the untyped setting, we instead exhibit the missing operand via ad-hoc instan-tiation of the eta-expansion rule, which allows the traversal to proceed as if the operand existedin the original term. This gives rise to a more generic concept of traversals for lambda terms. Anotable improvement, in addition to handling untyped terms, is that no preliminary transformationis required: the original unaltered lambda term is traversed.We show that by bounding the non-determinism of the traversal rule for free variables, one caneffectively compute a set of traversals characterizing the paths in the tree representation of the beta-normal form, when it exists. This yields an evaluation algorithm for untyped lambda-terms. Weprove correctness by showing that traversals implement leftmost linear reduction, a generalizationof the head linear reduction of Danoset al. [10, 9].