Non-Hausdorff parallelized manifolds over geometric models of conservative programs
Tobias Marauli, Hubert Gattringer, Andreas MΓΌller Β· HAL (Le Centre pour la Communication Scientifique Directe) Β· 2023
Every directed graph πΊ induces a locally ordered metric space X πΊ together with a local order XπΊ that is locally dihomeomorphic to the standard pospace R; both are related by a morphism π½ πΊ : XπΊ β X πΊ satisfying a universal property.The underlying set of XπΊ admits a non-Hausdorff atlas A πΊ equipped with a non-vanishing vector field π πΊ ; the latter is associated to XπΊ through the correspondence between local orders and cone fields on manifolds.The above constructions are compatible with cartesian products, so the geometric model of a conservative program is lifted through π½ πΊ1 Γ β’ β’ β’ Γ π½ πΊπ to a subset π of the parallelized manifold A πΊ 1 Γ β’ β’ β’ Γ A πΊπ .By assigning the suitable norm to each tangent space of A πΊ 1 Γ β’ β’ β’ Γ A πΊπ the length of every directed smooth path πΎ on π, i.e. β« |πΎ β² (π‘)| πΎ(π‘) ππ‘, corresponds to the execution time of the sequence of multi-instructions associated to πΎ.This induces a pseudometric π A whose restrictions to sufficiently small open sets of A πΊ 1 Γ β’ β’ β’ Γ A πΊπ (we refer to the manifold topology, which is strictly finer than the pseudometric topology) are isometric to open subspaces of R π with the πΌ-norm for some πΌ β [1, β].The transition maps of A πΊ are translations so the representation of a tangent vector does not depend on the chart of A πΊ in which it is represented; consequently, differentiable maps between open subsets of A πΊ 1 Γ β’ β’ β’ Γ A πΊπ are handled as if they were maps between open subsets of R π .For every directed path πΎ on π (possibly the representation of a sequence π of multi-instructions) there is a shorter directed smooth path on π that is arbitrarily close to πΎ, and that can replace πΎ as a representation of π.