One might try to implement event logic using only closed intervals of the
form [q,r], where 
.
Such a closed interval would represent the set 
of
real numbers.
With such closed intervals, one would define Allen's relations as follows:
One difficulty with doing so is that it would be possible for more than one
Allen relation to hold between two intervals when one or both of them are
instantaneous intervals, such as [q,q].
Both 
and 
would hold between [q,q] and [q,r],
both 
and 
would hold between [q,r]
and [q,q], both and 
would hold
between [q,r] and [r,r], both and 
would
hold between [r,r] and [q,r], and =, , and
would all hold between [q,q] and itself.
To create a domain where exactly one Allen relation holds between any pair of
intervals, let us consider both open and closed intervals.
Closed intervals contain their endpoints while open intervals do not.
The intervals (q,r], [q,r), and (q,r), where q<r, represent the
sets 
, 
, and 
of real numbers respectively.
The various kinds of open and closed intervals can be unified into a single
representation 
, where 
and 
are true
or false to indicate the interval being closed or open on the left or right
respectively.
More specifically, 
denotes [q,r],
denotes (q,r],
denotes [q,r), and
denotes (q,r).
To do this, let us use 
to mean when is
true and q<r when is false.
Similarly, let us use 
to mean 
when is
true and q>r when is false.
More precisely, 
and 
.
With these, represents the set
of real numbers.
One can extend the definition of Allen's relations to both open and closed
intervals as follows.
The relation 
holds if the corresponding endpoints of 
and 
are equal and have the same openness.
The relation 
holds if the right endpoint of precedes the
left endpoint of or if they are equal and both open.
For example, [1,3]<[4,5] and [1,3)<(3,5], but
, 
, and 
.
The relation 
holds if the right endpoint of
equals the left endpoint of and one of those endpoints is open while
the other is closed.
For example, 
and 
but 
and 
.
The relation 
holds if
precedes the left endpoint of
or they are equal while the former is closed and the latter is open, precedes the right endpoint of
or they are equal while both endpoints are closed, and precedes the right endpoint
of or they are equal while the former is open and the latter is closed.
, 
,
, and 
, but
, 
, and
.
The relation 
holds if
and are equal and have the same
openness and precedes the right endpoint
of or they are equal while the former is open and the latter is closed.
, 
, and
, but 
,
, 
, and
.
The relation 
holds if
and are equal and have the same
openness and follows the left endpoint of
or they are equal while the former is open and the latter is closed.
, 
, and
, but 
,
, 
, and
.
The relation 
holds if
follows the left endpoint of
or they are equal while the former is open and the latter is closed and precedes the right endpoint
of or they are equal while the former is open and the latter is closed.
and 
, but
, 
,
, and 
.
The inverse Allen relations >, , 
, ,
, and 
are defined analogously to the <,
, 
, , , and 
relations
respectively with the arguments reversed.
The above definitions can be stated more precisely as follows:
With the above definitions, exactly one Allen relation holds between any pair of intervals.
I refer to the set of real numbers represented by an interval as its
extension.
Given the above definition of interval, any interval, such as [5,4],
(5,4], [5,4), or (5,4), where the upper endpoint is less than the lower
endpoint represents the empty set.
Furthermore, any open interval, such as [5,5), (5,5], or (5,5), where the
upper endpoint equals the lower endpoint also represents the empty set.
To create a situation where the extension of each interval has a unique
representation, let us represent all such empty sets of real numbers
as 
.
Thus whenever we represent an interval explicitly,
it will have a nonempty extension and will satisfy the following normalization
criterion: 
.
of a Normalized Spanning Interval
of two Normalized Spanning Intervals
