of a Normalized Spanning Interval
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.