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Representations of Space and Time

definition: 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.
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Temporal Reasoning Bibliography

From Discourse to Logic.
Kamp, Hans and Reyle, Uwe.
Dordrecht, Kluwer, 1993.

Maintaining Knowledge about Temporal Intervals.
Allen, James F.
CACM 26 1983, 832-843.

James F. Allen (1984). A General Model of Action and Time. Artificial Intelligence 23(2), 123-154.
Francois Bry & Stephanie Spranger (2003). Temporal Constructs for a Web Language. Proceedings 4th Workshop on Interval Temporal Logics and Duration Calculi. ESSLLI.
Bob Carpenter (1997). Type-Logical Semantics. MIT Press.
Christopher J. Date, Hugh Darwen & Nikos A. Lorentzos (2002). Temporal Data and the Relational Model. Morgan Kaufmann.
Patrick J. Hayes (1995). A Catalog of Temporal Theories. Technical report UIUC-BI-AI-96-01, University of Illinois.
Annette Herskovits (1986). Language and Spatial Cognition: An Interdisciplinary Study of the Prepositions in English. Cambridge University Press.
Janet Hitzeman, Marc Moens & Claire Grover (1995). Algorithms for analysing the temporal structure of discourse. Proceedings 7th EACL, 253-260.
Jerry R. Hobbs & Feng Pan (2004). Time Ontology in OWL. http://www.w3.org/TR/owl-time/
Hans-Ulrich Krieger, Bernd Kiefer & Thierry Declerck (2008). A Framework for Temporal Representation and Reasoning in Business Intelligence Applications. AAAI Spring Symposium on AI Meets Business Rules and Process Management, 59-70.
William Langston, Douglas C. Kramer & Arthur M. Glenberg (1998). The Representation of Space in Mental Models Derived From Text. Memory and Cognition 26(2), 247-262.
Carsten Lutz (2004). Combining interval-based temporal reasoning with general TBoxes. Artificial Intelligence 152(2), 235-274.
Frank Manola & Eric Miller (2004). RDF Primer. http://www.w3.org/TR/rdf-primer/
John McCarthy & Patrick J. Hayes (1969). Some Philosophical Problems from the Standpoint of Artificial Intelligence. Machine Intelligence 4, 463-502. Edinburgh University Press.
Drew V. McDermott (1982). A Temporal Logic for Reasoning about Processes and Plans. Cognitive Science 6, 101-155.
Marc Moens (1987). Tens, Aspect and Temporal Reference. Ph.D. thesis. Centre for Cognitive Science, University of Edinburgh.
Marc Moens & Mark Steedman (1988). Temporal Ontology and Temporal Reference. Computational Linguistics 14(2), 15-28.
David Peterson, Shudi Gao, Ashok Malhotra, C.M. Sperberg-McQueen & Henry S. Thompson (2008). W3C XML Schema Definition Language (XSD) 1.1 Part 2: Datatypes. http://www.w3.org/TR/xmlschema11-2/
David A. Randell, Zhan Cui & Anthony G. Cohn (1992). A Spatial Logic based on Regions and Connection. Proceedings 3rd Interantional Conference on Knowledge Representation and Reasoning, 165-176.
Theodore Sider (2001). Four Dimensionalism: An Ontology of Persistence and Time. Oxford University Press.
Michael K. Smith, Chris Welty & Deborah L. McGuinness (2004). OWL Web Ontology Language Guide. http://www.w3.org/TR/owl-guide/
Max Tegmark (1997). On the dimensionality of spacetime. Classical and Quantum Gravity, 14, L69-L75.
Zeno Vendler (1967). Verbs and Times. Linguistics in Philosophy, 97-121. Cornell University Press.
Christopher Welty & Richard Fikes (2006). A reusable ontology for fluents in OWL. Proceeding 4th FOIS, 226-236.
Misha Wolf & Charles Wicksteed (1998). Date and Time Formats. http://www.w3.org/TR/NOTE-datetime