Non-Hausdorff parallelized manifolds over geometric models of conservative programs

Every directed graph G induces a locally ordered metric space X₍G₎ together with a local order X̃₍G₎ that is locally dihomeomorphic to the standard pospace R; both are related by a morphism β₍G₎G: X̃₍G₎ → X₍G) satisfying a universal property. The underlying set of X^̃₍G₎ admits a non-Hausdorff atlas A_G equipped with a non-vanishing vector field f_G; the latter is associated to X̃₍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 β_G1 × · · · × β_Gn to a subset M of the parallelized manifold A_G1 × · · · × A_Gn. By assigning the suitable norm to each tangent space of A_G1 × · · · × A_Gn, the length of every directed smooth path γ on M, i.e. ∫ |γ^'(t)|_γ(t₎dt, corresponds to the execution time of the sequence of multi-instructions associated to γ. This induces a pseudometric d_A whose restrictions to sufficiently small open sets of A_G1 × · · · × A_Gn (we refer to the manifold topology, which is strictly finer than the pseudometric topology) are isometric to open subspaces of Rⁿ with the α-norm for some α ∈ [1, ∞]. The transition maps of A_G are translations, so the representation of a tangent vector does not depend on the chart of A_G in which it is represented; consequently, differentiable maps between open subsets of A_G1 × · · · × A_Gn are handled as if they were maps between open subsets of Rⁿ. For every directed path γ on M (possibly the representation of a sequence σ of multi-instructions), there is a shorter directed smooth path on M that is arbitrarily close to γ, and that can replace γ as a representation of σ.