Yoav Shoham, "Consistency enforcing in temporal
reasoning", Section 6.4 in Y. Shoham, <I> Artificial Intelligence
techniques in Prolog</I>, Morgan Kaufmann, San Francisco, CA, 1994, pp.
152--161 (references are on pp. 317--323). 
<P>

This section describes Allen's interval formalism for temporal
reasoning. In this formalism, different events $A,B,\ldots,$ 
are represented by
(time) intervals, and expert knowledge consists of relations $ARB$ between
pairs $(A,B)$ of events, e.g., "$A$ precedes $B$", "$A$ ends $B$", etc.
Initially, we know some relations between some pairs of intervals. 
A natural problem is to describe what relations can be deduced from
the given ones. For that, the author uses Allen's
"propagate-interval-constraints" algorithm that is based on
statements of the type "If $AR_1B$ and $BR_2C$, then $AR_3C$". This
idea is similar to transitive closure, and, similarly to transitive
closure, it leads to a cubic-time
algorithm.
<P>
A Prolog program implementing this algorithm is  described and traced
on an example of robot control.