Given an Allen relation *r* and two sets *I* and *J* of intervals, let
denote the set *K* of all intervals such that
for some and , where .
Given an Allen relation *r* and two normalized spanning intervals
and , let denote a set of normalized spanning intervals
whose extension is , where *I* and *J* are the extensions
of and respectively.
One can compute as follows:

Here, denotes the inverse relation corresponding to *r*, i.e. the
same relation as *r* but with the arguments reversed.
It is easy to see that .
Thus an important property of normalized spanning intervals is that for any
two normalized spanning intervals and , contains at
most 4, 64, 64, 16, 16, 64, 64, 16, 16, 16, 16, 64,
or 64 normalized spanning intervals, when *r* is =, <, >, ,
, , , , , ,
, , or respectively.
While simple combinatorial enumeration yields the above weak bounds on the
number of normalized spanning intervals needed to represent ,
in practice, far fewer normalized spanning intervals are needed, in most cases
only one.

The intuition behind the above definition is as follows.
Let *I* and *J* be the extensions of and respectively.
The extension of the set of all is the set of all intervals such
that for some in *J*.
Furthermore, the extension of the set of all is the set of all
intervals in *I* such that for some in *J*.
Similarly, the extension of the set of all is the set of all
intervals such that for some in *I*.
Analogously, the extension of the set of all is the set of all
intervals in *J* such that for some in *I*.
Thus the extension of the set of all is the set of all
intervals such that where is in *I*, is
in *J*, and .

Wed Aug 1 19:08:09 EDT 2001