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
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: .