Allen's Interval Based Temporal Logic

Allen's theory of time and action [All83,All84] has intervals as primitive temporal elements and assumes a linear model of time. A set of thirteen mutually exclusive binary relations - $\{ Equals$, $Before$, $After$, $Meets$, $MetBy$, $Overlaps$, $OverlappedBy$, $Starts$, $StartedBy$, $During$, $Contains$, $Finishes$, $FinishedBy \}$ - describe different ways of relating two convex intervals. In [AF94] all these thirteen relations are formally defined using the single primitive relation Meets. The logic has three types of temporal entities - properties, processes and events. These entities differ in the subintervals over which they hold. Thus, properties are homogeneous, events are discrete and processes are either homogeneous or heterogeneous. The 2-place predicate - $HOLDS(p, i)$ - specifies that property $p$ holds over the interval $i$. Another predicate - $OCCUR(e, i)$ - denotes that an event $e$ occurs over an interval $i$. Finally, $OCCURING(p, i)$ denotes that a process $p$ occurs over an interval $i$.