Non-Hausdorff parallelized manifolds over geometric models of conservative programs

Abstract Every directed graph $G$ induces a locally ordered metric space $\mathcal{X}_{(G)}$ together with a local order $\tilde {\mathcal{X}}_{(G)}$ that is locally dihomeomorphic to the standard pospace $\mathbb{R}$ ; both are related by a morphism ${\beta }_{(G)} G:\tilde {\mathcal{X}}_{(G)}\to {\mathcal{X}}_{(G)}$ satisfying a universal property. The underlying set of $\tilde {\mathcal{X}_{(G)}}$ admits a non-Hausdorff atlas $\mathcal{A}_{G}$ equipped with a non-vanishing vector field ${{f}}_{G}$ ; the latter is associated to $\tilde {\mathcal{X}}_{(G)}$ 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 ${{\beta }_{G_1}} \times \cdots \times {{\beta }}_{G_n}$ to a subset $M$ of the parallelized manifold $\mathcal{A}_{G_1} \times \cdots \times \mathcal{A}_{G_n}$ . By assigning the suitable norm to each tangent space of $\mathcal{A}_{G_1} \times \cdots \times \mathcal{A}_{G_n}$ , the length of every directed smooth path $\gamma$ on $M$ , i.e. $\int {{|\gamma '(t)|}}_{\gamma (t)}dt$ , corresponds to the execution time of the sequence of multi-instructions associated to $\gamma$ . This induces a pseudometric ${{d}}_{\mathcal{A}}$ whose restrictions to sufficiently small open sets of $\mathcal{A}_{G_1} \times \cdots \times \mathcal{A}_{G_n}$ (we refer to the manifold topology, which is strictly finer than the pseudometric topology) are isometric to open subspaces of ${\mathbb{R}}^n$ with the $\alpha$ -norm for some $\alpha \in [{{1}},{{\infty }}]$ . The transition maps of $\mathcal{A}_{G}$ are translations, so the representation of a tangent vector does not depend on the chart of $\mathcal{A}_{G}$ in which it is represented; consequently, differentiable maps between open subsets of $\mathcal{A}_{G_{1}} \times \cdots \times \mathcal{A}_{G_{n}}$ are handled as if they were maps between open subsets of ${\mathbb{R}}^n$ . For every directed path $\gamma$ on $M$ (possibly the representation of a sequence $\sigma$ of multi-instructions),