Allen’s Interval Algebra


Allen’s Interval Algebra [1] (AIA) is a system for reasoning about temporal relations. 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 [2].

AIA is used in a variety of QSRs and other topics in QSRlib, e.g. in the Regional Algebra QSR, in the QSR graphs, etc.


Allen’s Interval Algebra defines 13 possible qualitative temporal base relations between two intervals X and Y:

Relation Inverse Illustration Interpretation
X < Y Y > X allen_before X takes place before Y.
X = Y Y = X allen_equal X is equal to Y.
X m Y Y mi X allen_meets X meets Y.
X o Y Y oi X allen_overlaps X overlaps with Y.
X d Y Y di X allen_during X takes place during Y.
X s Y Y si X allen_starts X starts Y.
X f Y Y fi X allen_finishes X finishes Y.


[1]James F. Allen: Maintaining knowledge about temporal intervals. Communications of the ACM, 1983.
[2]Wikipedia: Allen’s interval algebra. Wikipedia, accessed October 2015,