One of the main problems of knowledge representation is
checking consistency of the expert's knowledge.
In particular, we may want to check consistency of
the knowledge about time intervals ${\bf t}_i=[t^-_i,t^+_i]$
of different events. In many cases, the experts only have the
information about the * ordering*of different events, e.g.,
"${\bf t}_i$ starts ${\bf t}_j$",
or "${\bf t}_i$ precedes ${\bf t}_j$".
This knowledge can be formulated as follows:
\begin{itemize}
\item we start with * basic* ordering relations between numbers
$a$, $b$: $a**b$;
\item by applying these relations to endpoints of the intervals
${\bf t}_i$ and ${\bf t}_j$, we get basic ordering relations $R_1,R_2,
\ldots,$ between
intervals; e.g., "${\bf t}_i$ starts ${\bf t}_j$" can be described
as "$t^-_i=t^-_j\,\&\, t^+_i intractable).
Several **

The authors propose a new tractable class that not only
contains all previously known tractable classes, but
is also * maximal* in the sense that if we add one more of 8192
possible statements to
the list of 868 statements allowed for this class,
the class becomes intractable.

This class contains more than 10\% of 8192 possible statements about pairs of intervals, as opposed to previously known subclasses which allow only 1--2\% of statements.