Representations of Space and Time
The representation of time and space plays a crucial role in Artificial Intelligence and Computational Linguistics, but also in practical areas such as business intelligence or geographical information systems. Representing time and space properly eases querying and reasoning thereof.
Geographic information systems use traditional techniques from image processing and CAD, like a pixel/raster or a vector representation of spatial data. Temporal data bases extend tradional relational data models by the notion of valid time and transaction time.
When we speak of time, we mean a linear, dense, and one-dimensional time. With space, we usually refer to a two- or three-dimensional space, often using a latitude/longitude measurement in degrees in case of a spherical 2D geometry or a system of coordinates/trajectories for a Euclidean 2D/3D space. Classical observable spacetime then refers to a 3+1-dimensional continuum in which physical objects move through time and space (Tegmark 1997). Even abstract events or processes can be seen to take place in 4D spacetime. Physical and abstract 4D entities are usually composed of simpler entities -- here, both time and space are the glue to achieve a decomposition. For instance, Allen (1984) has defined a natural system of 13 temporal topological relations. Randell et al. (1992) came up with a logic (later called RCC) to support qualitative reasoning about space.
Human natural language usually comes up with means to refer to time and space, for instance through the use of temporal and spatial prepositions, adverbs, or verb tense and aspect (Vendler 1967, Moens & Steedman 1988, Herskovits 1986). Several algorithms exist which compute the temporal structure of a discourse (e.g., Hitzeman et al. 1995). When constructing mental models of space derived from text, it has been shown that the representation is more topological than Euclidean (Langston et al. 1998).
Within theoretical and computational linguistics, type-logical semantics and categorial grammar have given a rigorous account to tense, aspect, and temporal modification through the use of a possible-worlds semantics (see, e.g., Carpenter 1997).
Studying time from a formal perspective, focussing on the inherent properties of a theory, has a long tradition in logic (e.g., Hayes 1995), artificial intelligence (e.g., McDermott 1982), description logics (e.g., Bry & Spranger 2003, Lutz 2004), or data base theory (Date et al. 2002).
Looking from a more practical viewpoint on time, a number of frameworks have been proposed within the World Wide Web Consortium W3C: (i) ISO 8601 is an international standard for date and time representations issued by the International Organization for Standardization (Wolf & Wicksteed 1998); (ii) XSD (XML Schema Datatypes) comes up with a number of built-in primitive datatype for time and date (Peterson et al. 2008); (iii) OWL-Time (formerly DAML-Time; see Hobbs & Pan 2004) is an OWL ontology of temporal concepts and properties for describing the temporal content of Web pages and the temporal properties of Web services.
Representing changing relationships over time and space is related to the problem of diachronic identity which describes the identification of individuals that look different at different times, but still refer to the same entity. The four-dimensional or perdurantist view assumes that all entities (the perdurants) only exist for some period of time. Entities under this view are sometimes refered to as spacetime worms (Sider 2001), since a 4D trajectory is all one needs to identify/follow a perdurant through time and space. Parts of such a worm are called time slices, encoding cooccurent information that stay constant over the specified period of time. There exist several well-known techniques of extending a relation with time and space: (i) equip the relation with further arguments as is done in temporal data bases; (ii) apply a meta-logical predicate (McCarthy & Hayes 1969); (iii) reifiy the original relation as used in RDF (Manola & Miller 2004). Unfortunately, (i) and (ii) are not applicable to OWL (Smith et al. 2004), whereas (iii) requires ontology rewriting. Welty & Fikes (2006) and Krieger et al. (2008) present alternative approaches compatible with OWL.