Spatial–temporal reasoning is used in both the fields of psychology and computer science.
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Spatial-temporal reasoning is the ability to visualize spatial patterns and mentally manipulate them over a time-ordered sequence of spatial transformations.
This ability is important for generating and conceptualizing solutions to multi-step problems that arise in areas such as architecture, engineering, science, mathematics, art, games, and everyday life.
Spatial-temporal reasoning is also studied in computer science. An emphasis has been on qualitative spatial-temporal reasoning which is based on qualitative abstractions of temporal and spatial aspects of the common-sense background knowledge on which our human perspective on the physical reality is based. Methodologically, qualitative constraint calculi restrict the vocabulary of rich mathematical theories dealing with temporal or spatial entities such that specific aspects of these theories can be treated within decidable fragments with simple qualitative (non-metric) languages. Contrary to mathematical or physical theories about space and time, qualitative constraint calculi allow for rather inexpensive reasoning about entities located in space and time. For this reason, the limited expressiveness of qualitative representation formalism calculi is a benefit if such reasoning tasks need to be integrated in applications. For example, some of these calculi may be implemented for handling spatial GIS queries efficiently and some may be used for navigating, and communicating with, a mobile robot.
Examples of temporal calculi include Allen's interval algebra, and Vilain's & Kautz's point algebra. The most prominent spatial calculi are mereotopological calculi, Frank's cardinal direction calculus, Freksa's double cross calculus, Egenhofer and Franzosa's 4- and 9-intersection calculi, Ligozat's flip-flop calculus, and various region connection calculi (RCC). Recently, spatio-temporal calculi have been designed that combine spatial and temporal information. For example, the spatiotemporal constraint calculus (STCC) by Gerevini and Nebel combines Allen's interval algebra with RCC-8. Moreover, the qualitative trajectory calculus (QTC) allows for reasoning about moving objects.
Most of these calculi can be formalized as abstract relation algebras, such that reasoning can be carried out at a symbolic level. For computing solutions of a constraint network, the path-consistency algorithm is an important tool.
| Calculus | Type | Domain | Arity of base relations | № of base relations | Permutations of base relations are base relations | Composition table has been proven | Calculus is a relation algebra | Calculus is a semi associative algebra | Calculus is a weakly associative algebra | Calculus is a non-associative algebra | Complexity of consistency problem for atomic networks | Complexity of consistency problem for arbitrary networks | a-closure decides consistency for atomic networks | Type of composition | Tractable Subsets |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Interval Algebra | Temporal | 1D Line Segments | 2 | 13 | yes | yes | yes | yes | yes | yes | P | NP-hard | yes | strong | ORDHorn |
| INDU | Temporal | 1D Line Segments × Size | 2 | 25 | no | NP-complete | no | weak | |||||||
| Calculus | Type | Domain | Arity of base relations | № of base relations | Permutations of base relations are base relations | Composition table has been proven | Calculus is a relation algebra | Calculus is a semi associative algebra | Calculus is a weakly associative algebra | Calculus is a non-associative algebra | Complexity of consistency problem for atomic networks | Complexity of consistency problem for arbitrary networks | a-closure decides consistency for atomic networks | Type of composition | Tractable Subsets |