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

An important property of normalized spanning intervals is that for any
normalized spanning interval , contains at most 1, 4, 4,
2, 2, 4, 4, 2, 2, 2, 2, 4, or 4 normalized spanning
intervals when *r* is =, <, >, , , ,
, , , , , ,
or respectively.
In practice, however, fewer normalized spanning intervals are needed, often
only one.

The intuition behind the above definition is as follows. Let us handle each of the cases separately.

*r*=<- For any intervals and in the extensions of and respectively we want . From (2) we get . Furthermore, from (14) we get . Combining these we get . In this case, both and are free indicating that either endpoint of can be open or closed.
*r*=>- For any intervals and in the extensions of and respectively we want . From (3) we get . Furthermore, from (14) we get . Combining these we get . In this case, both and are free indicating that either endpoint of can be open or closed.
- For any intervals and in the extensions of and respectively we want . From (4) we get and . Furthermore, from (14) we get . Combining these we get and . In this case, only is free indicating that the upper endpoint of can be open or closed.
- For any intervals and in the extensions of and respectively we want . From (5) we get and . Furthermore, from (14) we get . Combining these we get and . In this case, only is free indicating that the lower endpoint of can be open or closed.
- For any intervals and in the extensions of and respectively we want . From (6) we get . Furthermore, from (14) we get and . Combining these we get and . In this case, both and are free indicating that either endpoint of can be open or closed.
- For any intervals and in the extensions of and respectively we want . From (7) we get . Furthermore, from (14) we get and . Combining these we get and . In this case, both and are free indicating that either endpoint of can be open or closed.
- For any intervals and in the extensions of and respectively we want . From (8) we get , , and . Furthermore, from (14) we get and . Combining these we get , , and . In this case, only is free indicating that the upper endpoint of can be open or closed.
- For any intervals and in the extensions of and respectively we want . From (9) we get , , and . Furthermore, from (14) we get and . Combining these we get , , and . In this case, only is free indicating that the upper endpoint of can be open or closed.
- For any intervals and in the extensions of and respectively we want . From (10) we get , , and . Furthermore, from (14) we get and . Combining these we get , , and . In this case, only is free indicating that the lower endpoint of can be open or closed.
- For any intervals and in the extensions of and respectively we want . From (11) we get , , and . Furthermore, from (14) we get and . Combining these we get , , and . In this case, only is free indicating that the lower endpoint of can be open or closed.
- For any intervals and in the extensions of and respectively we want . From (12) we get and . Furthermore, from (14) we get and . Combining these we get and . In this case, both and are free indicating that either endpoint of can be open or closed.
- For any intervals and in the
extensions of and respectively we want
.
From (13) we get and
.
Furthermore, from (14) we get and
.
Combining these we get
and
.
In this case, both and are free indicating that either
endpoint of can be open or closed.

Wed Aug 1 19:08:09 EDT 2001