Allen's Interval Algebra

Allen's Interval Algebra is a calculus for temporal reasoning that has been introduced by James F. Allen in 1983.

The calculus defines possible relations between time intervals and provides a composition table that can be used as a basis for reasoning about temporal descriptions of events.

Relations
The following 13 base relations capture the possible relations between two intervals.

Using this calculus, given facts can be formalized and then used for automatic reasoning. Relations between intervals are formalized as sets of base relations.

The sentence
 * During dinner, Peter reads newspaper. Afterwards, he goes to bed.

is formalized in Allen's Interval Algebra as follows:

$$\mbox{newspaper } \mathbf{\{ \operatorname{d}, \operatorname{s}, \operatorname{f} \}} \mbox{ dinner}$$

$$\mbox{dinner } \mathbf{\{ \operatorname{<}, \operatorname{mi} \}} \mbox{ bed}$$

Composition of relations between intervals
For reasoning about the relations between temporal intervals, Allen's Interval Algebra provides a composition table. Given the relation between $$X$$ and $$Y$$ and the relation between $$Y$$ and $$Z$$, the composition table allows for concluding about the relation between $$X$$ and $$Z$$. Together with a converse operation, this turns Allen's Interval Algebra into a relation algebra.

For the example, one can infer $$\mbox{newspaper } \mathbf{\{ \operatorname{m}, \operatorname{<} \}} \mbox{ bed}$$.

Extensions
Allen's Interval Algebra can be used for the description of both temporal intervals and spatial configurations. For the latter use, the relations are interpreted as describing the relative position of spatial objects. This also works for three-dimensional objects by listing the relation for each coordinate separately.