Reducing Lambda Terms with Traversals

02/28/2018 ∙ by William Blum, et al. ∙ 0

We introduce a method to evaluate untyped lambda terms by combining the theory of traversals, a term-tree traversing technique inspired from Game Semantics, with judicious use of the eta-conversion rule of the lambda calculus. The traversal theory of the simply-typed lambda calculus relies on the eta-long transform to ensure that when traversing an application, there is a subterm representing every possible operator's argument. In the untyped setting, we instead exhibit the missing operand via ad-hoc instantiation of the eta-expansion rule, which allows the traversal to proceed as if the operand existed in the original term. This gives rise to a more generic concept of traversals for lambda terms. A notable improvement, in addition to handling untyped terms, is that no preliminary transformation is required: the original unaltered lambda term is traversed. We show that by bounding the non-determinism of the traversal rule for free variables, one can effectively 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. We prove correctness by showing that traversals implement leftmost linear reduction, a generalization of the head linear reduction of Danos et. al.



There are no comments yet.


page 13

page 24

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.