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| | Alyson is the name people use to contact me and I think it sounds fairly great when you say it. Distributing manufacturing has been his profession for some time. For a while I've been in Alaska but I will have to move in a year or two. To play lacross is one of the things she enjoys most.<br><br>My webpage [http://m-card.co.kr/xe/mcard_2013_promote01/29877 real psychic] |
| ''For the type of boolean algebra called interval algebra, see [[Boolean_algebra_(structure)#Examples|Boolean algebra (structure)]]''
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| '''Allen's interval algebra''' is a calculus for [[spatial-temporal reasoning|temporal reasoning]] that was introduced by [[James F. Allen]] in 1983.
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| The calculus defines possible relations between time intervals and provides a composition table that can be used as a basis
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| for reasoning about temporal descriptions of events.
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| ==Formal description==
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| === Relations ===
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| The following 13 base relations capture the possible relations between two intervals.
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| {| class="wikitable"
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| !Relation
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| !Illustration
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| !Interpretation
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| |-
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| |<math>X \mathbf{\operatorname{<}} Y</math>
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| <math>Y \mathbf{\operatorname{>}} X</math>
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| |[[Image:Allen calculus before.png|X takes place before Y]]
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| |X takes place before Y
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| |-
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| |<math>X \mathbf{\operatorname{m}} Y</math>
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| <math>Y \mathbf{\operatorname{mi}} X</math>
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| |[[Image:Allen calculus meet.png|X meets Y]]
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| |X meets Y (''i'' stands for '''''i'''nverse'')
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| |-
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| |<math>X \mathbf{\operatorname{o}} Y</math>
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| <math>Y \mathbf{\operatorname{oi}} X</math>
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| |[[Image:Allen calculus overlap.png|X overlaps with Y]]
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| |X overlaps with Y
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| |-
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| |<math>X \mathbf{\operatorname{s}} Y</math>
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| <math>Y \mathbf{\operatorname{si}} X</math>
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| |[[Image:Allen calculus start.png|X starts with Y]]
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| |X starts Y
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| |-
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| |<math>X \mathbf{\operatorname{d}} Y</math>
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| <math>Y \mathbf{\operatorname{di}} X</math>
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| |[[Image:Allen calculus during.png|X during Y]]
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| |X during Y
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| |-
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| |<math>X \mathbf{\operatorname{f}} Y</math>
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| <math>Y \mathbf{\operatorname{fi}} X</math>
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| |[[Image:Allen calculus finish.png|X finishes with Y]]
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| |X finishes Y
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| |-
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| |<math>X \mathbf{\operatorname{=}} Y</math>
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| |[[Image:Allen calculus equal.png|X is equal to Y]]
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| |X is equal to Y
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| |}
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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.
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| The sentence
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| : ''During dinner, Peter reads newspaper. Afterwards, he goes to bed.''
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| is formalized in Allen's Interval Algebra as follows:
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| <math>\mbox{newspaper } \mathbf{\{ \operatorname{d}, \operatorname{s}, \operatorname{f} \}} \mbox{ dinner}</math>
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| <math>\mbox{dinner } \mathbf{\{ \operatorname{<}, \operatorname{m} \}} \mbox{ bed}</math>
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| In general, the number of different relations between n intervals is 1, 1, 13, 409, 23917, 2244361... [http://oeis.org/A055203 OEIS A055203]. The special case shown above is for n=2.
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| ===Composition of relations between intervals===
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| 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 [[Inverse relation|converse]] operation, this turns Allen's Interval Algebra into a [[relation algebra]].
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| For the example, one can infer <math>\mbox{newspaper } \mathbf{\{ \operatorname{<}, \operatorname{m} \}} \mbox{ bed}</math>.
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| ==Extensions==
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| 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.
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| ==Implementation==
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| * [http://code.google.com/p/allenintervalrelationships/ A simple java library implementing the concept of Allen's temporal relations and the path consistency algorithm]
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| ==See also==
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| * [[Temporal logic]]
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| * [[Logic]]
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| * [[Region Connection Calculus]].
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| ==References==
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| * James F. Allen: ''Maintaining knowledge about temporal intervals''. In: ''Communications of the ACM''. 26 November 1983. ACM Press. pp. 832–843, ISSN 0001-0782
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| * Bernhard Nebel, Hans-Jürgen Bürckert: ''Reasoning about Temporal Relations: A Maximal Tractable Subclass of Allen's Interval Algebra.'' In: ''Journal of the ACM'' 42, pp. 43–66. 1995.
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| * Peter van Beek, Dennis W. Manchak: ''The design and experimental analysis of algorithms for temporal reasoning.'' In: ''Journal of Artificial Intelligence Research'' 4, pp. 1–18, 1996.
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| [[Category:Knowledge representation]]
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| [[Category:Constraint programming]]
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