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),

A language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group on the collection of homogeneous languages of length n ∈ ℕ. One recovers the isothetic regions from (Haucourt 2017, to appear (online since October 2017)) by considering the alphabet of connected subsets of the space |G|, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by |G|) are co-unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections |G| satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections |G| to the geometric properties of the space |G|. On the way, the algebraic structure |G| is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure |G|.

The programs, we consider are written in a restricted form of the language introduced by Dijkstra (1968). A program is said to beconservativewhen each of its loops restores all the resources it consumes. We define the geometric model of such a program and prove that the collection of directed paths on it is a reasonable over-approximation of its set of execution traces. In particular, two directed paths that are close enough with respect to theuniform distanceresult in the same action on the memory states of the system. The same holds forweakly dihomotopicdirected paths. As a by-product, we obtain a notion of independence, which is favourably compared to more common ones. The geometric models actually belong to a handy class oflocal pospaceswhose elements are calledisothetic regions. The local pospaces we use differ from the original ones, we carefully explain why the alternative notion should be preferred. The title intentionally echoes the article by Carson and Reynolds (1987).