A note on the complexity of the causal ordering problem

In this paper we provide a concise report on the complexity of the causal ordering problem, originally introduced by Simon to reason about causal relations implicit in mathematical models. We show that Simon's classical algorithm to infer causal ordering is NP-Hard---an intractability previously guessed by Nayak but not yet 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 $O(|\mathcal S|^3)$, 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 density $|\mathcal S|$. We also comment on the potential of causal ordering for emerging applications in large-scale hypothesis management and predictive analytics.