Allen's interval algebra

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

Relation	Illustration	Interpretation
${\displaystyle X\,{\mathrel {\mathbf {<} }}\,Y}$ ${\displaystyle Y\,{\mathrel {\mathbf {>} }}\,X}$		X precedes Y Y is preceded by X
${\displaystyle X\,{\mathrel {\mathbf {m} }}\,Y}$ ${\displaystyle Y\,{\mathrel {\mathbf {mi} }}\,X}$		X meets Y Y is met by X (i stands for inverse)
${\displaystyle X\,{\mathrel {\mathbf {o} }}\,Y}$ ${\displaystyle Y\,{\mathrel {\mathbf {oi} }}\,X}$		X overlaps with Y Y is overlapped by X
${\displaystyle X\,{\mathrel {\mathbf {s} }}\,Y}$ ${\displaystyle Y\,{\mathrel {\mathbf {si} }}\,X}$		X starts Y Y is started by X
${\displaystyle X\,{\mathrel {\mathbf {d} }}\,Y}$ ${\displaystyle Y\,{\mathrel {\mathbf {di} }}\,X}$		X during Y Y contains X
${\displaystyle X\,{\mathrel {\mathbf {f} }}\,Y}$ ${\displaystyle Y\,{\mathrel {\mathbf {fi} }}\,X}$		X finishes Y Y is finished by X
${\displaystyle X\,{\mathrel {\mathbf {=} }}\,Y}$		X is equal to Y

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 the newspaper. Afterwards, he goes to bed.

is formalized in Allen's Interval Algebra as follows:

${\displaystyle {\mbox{newspaper }}\mathbf {\{\operatorname {d} \}} {\mbox{ dinner}}}$

${\displaystyle {\mbox{dinner }}\mathbf {\{\operatorname {<} \}} {\mbox{ bed}}}$

In general, the number of different relations between n intervals, starting with n = 0, is 1, 1, 13, 409, 23917, 2244361... OEIS A055203. The special case shown above is for n = 2.

For reasoning about the relations between temporal intervals, Allen's interval algebra provides a composition table. Given the relation between ${\displaystyle X}$ and ${\displaystyle Y}$ and the relation between ${\displaystyle Y}$ and ${\displaystyle Z}$, the composition table allows for concluding about the relation between ${\displaystyle X}$ and ${\displaystyle Z}$. Together with a converse operation, this turns Allen's interval algebra into a relation algebra.

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