A note on the complexity of the causal ordering problem

In this note we provide a concise report on the complexity of the causal ordering problem, originally introduced by Simon to reason about causal dependencies implicit in systems of mathematical equations. We show that Simon's classical algorithm to infer causal ordering is NP-Hard---an intractability previously guessed but never proven. We present then a detailed account based on Nayak's suggested algorithmic solution (the best available), which is dominated by computing transitive closure---bounded in time by $O(|\mathcal V|\cdot |\mathcal S|)$, where $\mathcal S(\mathcal E, \mathcal V)$ is the input system structure composed of a set $\mathcal E$ of equations over a set $\mathcal V$ of variables with number of variable appearances (density) $|\mathcal S|$. We also comment on the potential of causal ordering for emerging applications in large-scale hypothesis management and analytics.