Maintaining knowledge about temporal intervals

“An extensive study of the relations between time intervals is present in the pioneering work of Allen [115].”

“User behavior patterns can be expressed in a number of different ways [2, 4, 34].”

“6 An ontology of time has been published by Hobbs and Pan [20], which includes Allen’s set of time relations [1].”

“Keywords: Spatial reasoning, modal logic, topological interpretation, topo-bisimulation, serial set.”

“The most successful example for an import from the temporal to the spatial domain is probably Allen's (1983) system of interval relations. Explicit use of the interval relations is made for example by Guesgen (1989) and by Hernfindez (1992) who both devise spatial reasoning formalisms that are more general than Allen'…”

“For other event formalisms, see also [28,29,49,76,85,105].”

“Allen [4, 5] discusses the different representations of timestamps adopted in literature, including one where timestamps are logical timestamps only, merely a sequential numbering of events on a system.”

“Temporal objects include points (instants as defined in Freska 1992) and lines (intervals as defined in Allen 1983).”

“Allen (1983) gives an O( n 3 ) approximation algorithm for the all-to-all minimal labels problem that is a special case of path consistency algorithms (Montanari, 1974; Mack-worth, 1977).”

“In general, any of the 13 temporal relations defined in Allen’s interval algebra for temporal reasoning [3] can be the described relation, that is, the equal relation as well as the six relations before , meets , overlaps , during , starts , and finishes with respective inverses (see Fig.”

“…of a constraint can be expressed without explicitly storing the set of forbidden tuples, path consistency can be used (e.g., temporal networks Allen, 1983, constraint networks Smith, 1992). tightnesses lower than .63 since it does not delete any value, while for SRPC the limit is .57 of…”

“It can be observed that representing each task by its execution duration (as an interval), the interdependencies between these tasks can be modeled [6].”

“Others have shown the general solution to temporal uncertainty to be an NP-hard problem [1,10].”

“4, have been originally proposed for temporal reasoning [5].”

“[EF91], any of the 13n (n = 1, 2) possible directional operators between two n-dimensional objects [ All83 ,”

“Allen's temporal operators [1] can be used to describe relationships between situations.”

“weak composition = composition Interval Algebra [1] STAR calculus [21] rectangle algebra [9] containment algebra [10] block algebra [3] cyclic algebra [2]”

“Allen’s interval algebra for temporal reasoning [1] enumerates all possible relationships between pairs of time intervals: A before B, A meets B, A overlaps B, A starts B, A finishes B, and A equals B.”

“In particular, Allen (1983) brie(cid:13)y describes an extension of his temporal interval rea-soner to handle duration, the temporal equivalent of size.”

“T e m po r a l r e l a ti on s t og e t h e r w it h t e m po r a l r ea s on i ng r u l e s f o r m t e m po r a l e v e n t ca l c u l u s ( A ll e n , 1983 ) .”

“A commonly used approach is Allen’s Interval Alge-bra [2], which defines several relations between time intervals, with BTR = { <, >, m, mi, o, oi, s, si, d, di, f, fi, = } .”

“Examples of such approaches are: 2D-String (Chang et al ., 1987) as shown in Figure 10, its extension 2D-CString (Lee & Hsu, 1990) and 2D-GString (Chang et al ., 1996), the two-dimensional extension of Interval Algebra (Allen, 1983), as shown in Figure 11, the Rectangle Algebra (Balbiani et al ., 1998, 1999a), as show…”

“While research in the area has focused mainly on logics involving constraint systems that have as domains intervals [9,1] and regions [10–12], with a variety of decidability and complexity results, there seems to have been little attention paid to commonly used constraint systems of the form ( D, …”

“We could also equivalently use Allen’s algebra operators [All83].”

“Let n be the set of interpretations for this language.”

“Since an event is represented by an interval or by one point in time, Point-interval temporal logic [23], which is an extension to Allen’s interval logic [1], is used to handle different possibilities, which are • both events are intervals • both events are points • one event is an interval and the other is a point.”

“However, while there is a wide consensus in the NLP community over the modeling of temporal relations between events, mainly based on Allen’s interval algebra (Allen, 1983), the question on how to model other types of event relations is still open.”

“The set of relation types employed in TimeML is based on the thirteen relations proposed by Allen in his interval algebra (Allen, 1983; Allen, 1984) and includes the six binary relations, before, immediately before, includes, begins, ends, during, the six inverses of these, simultaneous, and in addition identity, whic…”

“We have incorporated in our language and includes Allen’s 13 binary interval relations [2] and several other basic useful functions on intervals.”

“The classes of TCSPs surveyed are qualitative point Vilainetal89, qualitative interval Allen83, metric point Dechteretal91 and some of their combinations KautzLadkin91, Meiri96.”

“The constraints we currently use for the temporal dimen­sion are based on the temporal relations de.ned by Allen [1]: equals, before, meets, overlaps, during, starts and .nishes, with similar relations for the spatial dimensions X and Y.”

“Temporal Prolog is not directly based on temporal logic: it uses Allen's temporal constraint model [8] for reasoning about time intervals and their relationships, but with some modi cations for e ciency reasons [54, section 3].”

“Examples for such systems include spatial constraint networks based on the RCC8 relations [6, 9, 30] or on cardinal direction relations [10], and temporal constraint networks based on Allen’s relations of time intervals [1, 28, 34] or on relations between time points [33, 34].”

“Considerable effort has gone into constructing pieces of such a theory, e.g., the aspects of time [1], causality [30], activity [2], authority [27], constraint representation [12], and ownership [21].”

“Allen Newell and Herbert A. Simon (1976)”

“In particular, we consider bidimensional extensions of Allen’s interval relations [4], that is, rectangular relations.”

“Then we can provide composition operators that allow for interval relations [1].”

“The most interesting property from a representational standpoint is Time-Constraints, the syntax for which is given in the Appendix and allows speci cation of any of the 13 Allen relations [1].”

“The temporal relations between them was then clarified, based on [1].”

“XXX YYY Figure 7: Selection of temporal operators for composition (based on Allen [4]).”

“To be specific, the relations are:precedes , meets, overlaps, starts, during, finishes, equals, and their inverses, identified by addingi to the relation name [17].”

“2 ALLEN’S CALCULUS Allen’s [1] approach to reasoning about time(3)is based on the notion of time intervals and binary relations on them.”

“Future work will look at the articulation details of bimanual gestures, i.e., the way the dominant and nondominant hands move to perform a ring gesture, such as by employing temporal algebra [1].”

“The relations are self-explanatory, the interested reader is referred to [15] and [16] for further details.”

“, Allen’s interval constraints [1] and qualitative constraints on their duration.”

“Allen distinguishes 13 cases to express qualitative relations between intervals in 1D [39].”

“Such tables are also used by other relational calculi, notably James Allen's calculus of temporal intervals (Allen 1983) and are often referred to by the alternative name of transitivity tables.”

“9 Interval logics and granularities There are a range of interval temporal logics that have been developed for use in Computer Science and Artificial Intelligence; although we cover one variety in the next section and another in Chapter 7, the interested reader might begin by consulting [14, 241, 382, 449].”

“on representations of time [2, 10, 32, 52, 71] generally involves complex constraint propagation mechanisms to maintain partial ordering relations between time intervals.”

“, “before” [Allen, 83]) constraints were considered.”

“For the former, we base the ordering of assertions on the nature of their dimension and we use the relations already defined in Allen’s Interval Algebra [38] for relatively positioning intervals (here value ranges) on a dimension.”

“The possible types of spatial and temporal relations are considered in the literature on temporal and spatial reasoning (e.g. Allen 1983, Egenhofer 1991, Frank 1992) and on geographic information systems (e.g. Jones 1997, Longley et al. 1999).”

“Naldurg et al. (2004), Ning et al. (2001), and Kohout et al. (2002) describe attack patterns or normal behavior patterns based on Allen’s temporal relations (Allen, 1983) such as ‘‘before”, ‘‘after” and ‘‘meet” between events (e.g., a login by user u should be followed by a logout).”

“Considerable research examines the dimensions of time and how the notion of time can be incor­porated into information system [Allen 1983].”

“Extending a combination of Tables 1 and 2, Vernier & Nigay [39] define a comprehensive design space for multimodal fusion, based Allen’s relationships [1] applied to the levels of Nielsen’s model of interaction.”

“This intuition is in similar flavor with that of Allen’s Algebra, which proposes that a small number of temporal relations are sufficient to encode the structure of temporal sequences [2].”

“Temporal and causal relations follow a certain transitivity rules (Allen, 1983; Gerevini and Schubert, 1995), e.g., if there exists “A BEFORE B” and “B BEFORE C”, “A BEFORE C” also holds.”

“If ei temporally precedes event ej , τ(ei) and τ(ej) must possess the following temporal relationships as defined by Allen [7]: τ(ei) before τ(ej), τ(ei) meets τ(ej), τ(ei) overlaps τ(ej), τ(ei) starts τ(ej), τ(ei) equal to τ(ej), τ(ei) contains τ(ej), τ(ei) started by τ(ej), and τ(ei) finished by τ(ej).”

“Qualitative temporal constraints, e.g., (Al (before) A2), are represented in an “Allen-style” fashion (cf. [2]) while quantitative constraints appear as metric (inequalities.”

“Our proposed semantics is also expressive enough to capture all thirteen Allen’s temporal relationships [3].”

“This region-based approach to spatial reasoning closely mirrors Allen’s interval-based approach to temporal reasoning [2]—they both take extended entities, rather than points, as primitives.”

“Several suggestions have emerged from generic fields of computer science, such as artificial intelligence, or the knowledge and data management areas [63] [64] [65] [66] [3] [1].”