Bounding Skeletons, Locally Scoped Terms and Exact Bounds for Linear Head Reduction

Bounding skeletons were recently introduced as a tool to study the length of interactions in Hyland/Ong game semantics. In this paper, we investigate the precise connection between them and execution of typed λ-terms. Our analysis sheds light on a new condition on λ-terms, called local scope. We show that the reduction of locally scoped terms matches closely that of bounding skeletons. Exploiting this connection, we give upper bound to the length of linear head reduction for simply-typed locally scoped terms. General terms lose this connection to bounding skeletons. To compensate for that, we show that λ-lifting allows us to transform any λ-term into a locally scoped one. We deduce from that an upper bound to the length of linear head reduction for arbitrary simply-typed λ-terms. In both cases, we prove the asymptotical optimality of the upper bounds by providing matching lower bounds.