{"id":469512,"date":"2018-02-25T02:47:51","date_gmt":"2018-02-25T10:47:51","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=469512"},"modified":"2018-10-16T22:34:07","modified_gmt":"2018-10-17T05:34:07","slug":"reducing-lambda-terms-with-traversals-preprint","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/reducing-lambda-terms-with-traversals-preprint\/","title":{"rendered":"Reducing Lambda Terms with Traversals (preprint)"},"content":{"rendered":"<div id=\"pageContainer1\" class=\"page\">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.<\/div>\n<div>\n<div>\n<div><\/div>\n<div>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&#8217;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.<\/div>\n<div><\/div>\n<div>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 <em>leftmost linear reduction<\/em>, a generalization of the <em>head linear reduction<\/em>\u00a0of Danos et. al.<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>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 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