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|X is equal to Y
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|}Using this calculus, given facts can be formalized and then used for automatic reasoning. Relations between intervals are formalized as sets of base relations.▼
To see that the 13 relations are exhaustive, note that each point of <math>X</math> can be at 5 possible locations relative to <math>Y</math>: before, at the start, within, at the end, after. These give <math>5 + 4 + 3 + 2 + 1 = 15</math> possible relative positions for the start and the end of <math>X</math>. Of these, we cannot have <math>X_0 = X_1 = Y_0</math> since <math>X_0 < X_1</math>, and similarly we cannot have <math>X_0 = X_1 = Y_1</math>, thus giving us 13 possible relations.
In general, the number of different relations between ''n'' intervals, starting with ''n'' = 0, is 1, 1, 13, 409, 23917, 2244361... [
===Composition of relations between intervals===▼
For reasoning about the relations between temporal intervals, Allen's interval algebra provides a [[Relation composition|composition]] table. Given the relation between <math>X</math> and <math>Y</math> and the relation between <math>Y</math> and <math>Z</math>, the composition table allows for concluding about the relation between <math>X</math> and <math>Z</math>. Together with a [[converse relation|converse]] operation, this turns Allen's interval algebra into a [[relation algebra]].▼
▲
The sentences
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<math>\mbox{dinner } \mathbf{\{ \operatorname{<} \}} \mbox{ bed}</math>
▲In general, the number of different relations between ''n'' intervals, starting with ''n'' = 0, is 1, 1, 13, 409, 23917, 2244361... [http://oeis.org/A055203 OEIS A055203]. The special case shown above is for ''n'' = 2.
▲===Composition of relations between intervals===
▲For reasoning about the relations between temporal intervals, Allen's interval algebra provides a [[Relation composition|composition]] table. Given the relation between <math>X</math> and <math>Y</math> and the relation between <math>Y</math> and <math>Z</math>, the composition table allows for concluding about the relation between <math>X</math> and <math>Z</math>. Together with a [[converse relation|converse]] operation, this turns Allen's interval algebra into a [[relation algebra]].
For the example, one can infer <math>\mbox{newspaper } \mathbf{\{ \operatorname{<} \}} \mbox{ bed}</math>.
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The study of [[overlapping markup]] uses a similar algebra (see <ref>Steven DeRose. Markup Overlap: A Review and a Horse. In Proceedings of Extreme Markup Languages 2004, Montréal, Québec, August 2-6, 2004.
http://xml.coverpages.org/DeRoseEML2004.pdf</ref>). Its models have more variations depending on whether endpoints of document structures are permitted to be truly co-located, or merely [tangent].
== Temporal primitives ==
In the cultural heritage ontology [[CIDOC CRM]], Allen relations are replaced by so-called ''temporal primitives'', which facilitate the formulation of attestable statements as well as reasoning about these statements.<ref>CIDOC CRM Version 7.3: https://cidoc-crm.org/versions-of-the-cidoc-crm, section ''Temporal Relation Primitives based on fuzzy boundaries''</ref>
Temporal primitives split up the Allen relations into individual statements about the start or end of the intervals. For example, ''X overlaps with Y'' (<math>X \mathbf{\operatorname{o}} Y</math>) can be split as follows:
* <math>X \mathbf{\operatorname{o}} Y</math> ⇔ ''starts before the start of'' (<math>X</math>,<math>Y</math>) ∧ ''ends after the start of'' (<math>X</math>,<math>Y</math>) ∧ ''ends before the end of'' (<math>X</math>,<math>Y</math>)
In addition, the ''equal to'' of the Allen relations is replaced by ''before or with'' and ''after or with''. A simple example:
* The reign of King [[Harold Godwinson|Harold II]] ''starts before the start of'' the [[Battle of Hastings]]
* The reign/life of Harold II ''ends after or with the start of'' the Battle of Hastings
* The reign/life of Harold II ''ends before or with the end of'' the Battle of Hastings
In the example, it is not necessary to specify whether Harold II was killed at the beginning or during or at the end of the battle, i.e. whether <math>X \mathbf{\operatorname{m}} Y</math>, <math>X \mathbf{\operatorname{o}} Y</math> or <math>X \mathbf{\operatorname{fi}} Y</math> applies (disjunctions such as <math>\mathbf{\{ \operatorname{m}, \operatorname{o}, \operatorname{fi} \}}</math> cannot be expressed in CIDOC CRM, except in queries). If it is relevant for a particular historical question, it can be specified later by adding e.g. ''ends after the start of''.
CIDOC CRM distinguishes between events and their corresponding time intervals. Allen relations and temporal primitives are statements between events and only as a consequence between their time intervals. Another difference is that temporal, spatial and spatiotemporal entities in CIDOC CRM are seen as having fuzzy borders. Especially statements about exact simultaneity are otherwise extremely rare.
== Implementations ==
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