I'm currently investigating a family of temporal relations called 'Ordered Horn' ($OH$ for short). This class was introduced in 'Reasoning about Temporal Relations: A Maximal Tractable Subclass of Allen's Interval Algebra' by B. Nebel and H.J. Bürckert. Following this paper, we define an OH clause as a formula over $\mathbb{N}^k$ of the form:
$F_{H,R}(x_1,\ldots,x_k) = \wedge_{(i,j) \in E(H)} (x_i = x_j) \Rightarrow (x_p R x_q)$,
for a given graph $H$ on $[k]$ and a relation $R \in \{=,\leq,\neq\}$.
An OH relation is then a conjunction of OH clauses. The authors state in Theorem 5 of the paper that the satisfiability of an OH relation can be decided in polynomial time, but the algorithm doesn't seem 'self-contained' as it relies on a generic algorithm for propositional Horn theories.
In relation to a problem I'm currently studying, such a self-contained algorithm seems desirable but it's unclear to me whether it is possible. To tell the truth, I have an algorithm for a subclass called Restricted Ordered Horn (containing the relations expressible using clauses without $\neq$ in the rhs) but unfortunately it can't be adapted to the full $OH$ class.