Maintaining knowledge about temporal intervals

“Allen’s paper Allen (1983) in 1983 listed 13 possible cases for the temporal relationships of two time intervals.”

“As shown at right in Figure 4, R EKALL provides a standard library of common spatial and temporal predicates such as the Allen interval operations [5] and their 2D spatial analogues [9].”

“Temporal planning has been extensively studied in the operations research and artificial intelligence communities, with early works such as (Baker and Baker 1974; Allen 1983; Valdes-Perez 1986).”

“We therefore chose Allen’s Interval Algebra [2] as our basis.”

“In the TempEval Challenge, only a restricted set of Allen-style [2] temporal relations are used.”

“Comparing to the solution proposed in [28], we use the " exists " operator that influences our inference algorithm.”

“Siskind (2001) represented human actions in a form similar to a first order logic with Allen’s temporal predicates. Their event logic focused on a particular class of activities called ‘liquid’ events, and it was able to represent activities having multiple levels of hierarchy by limiting a sub-event to be used only o…”

“As observed by Allen [1983], there are 13 possible relationships between two intervals such as, for example, the meets relation: two intervals i1 and i2 are related by meets iff the right endpoint of i1 is identical to the left endpoint of i2 —see [Allen,1983] for an exact definition of the other…”

“…(cid:133) wet (cid:133) p ; s 0 (cid:134) ; s (cid:134) Allen (1984) presents a further development of the situation calculus where events, processes, activities and causality are expressed as predicates occurring over time intervals along with binary operators over temporal intervals (Allen 1983).”

“They are mostly influenced by Allen’s calculus for time relations (Allen 1983) and follow a mathematical tradition, to give as a result the conjunction of all possible results.”

“PhD thesis, University of Groningen, Netherlands, May 2010.”

“Reference [ 1 ] argues that all events have duration and considers intervals to be the basic timing concept.”

“In [10] the representation of temporal knowledge in a case is based on Allen ’s thirteen temporal relations [ 1 ].”

“Meanwhile there exist a number of qualitative temporal and spatial calculi, such as Allen’s [1] interval calculus for reasoning about time, a calculus for reasoning about qualitative directions [4], and a calculus for reasoning about topological relations [2, 21].”

“For further references to interval temporal logics and their applications in such diverse areas and AI and Hardware Verification, see Allen (1983), Schwartz et al. (1983), Moszkowski and Manna (1984) and Allen and Hayes (1985).”

“The elements of PA are the set of binary relations {<, > , = , -<,>_,5, -<>,0}, through which the "pointizable" subset of relations of Allen's Interval Algebra (IA) [2] (sometimes called PIA) can be represented [14].”

“Let the core and the support of 1 be [2,7] and [1,8] ; that of , 2 be [4,6] and [3,7] ; that of 3 be [9,13] and [7.”

“part of** represent all the possible relations between time interval such as Allen’s interval algebra[22].”

“Now the FTCM is converted into a rule based fuzzy inference system by integrating neuro fuzzy rules with Allen’s temporal relations [3] to get neuro fuzzy temporal rules.”

“An event model M of an event E is composed of five parts: ”physical objects” (a set of variables whose values correspond to the physical objects involved in E), ”components” (a set of variables whose values correspond to the event instances composing E), ”forbidden components” (a set of variables corresponding to all…”

“These properties are the well-know Allen Interval Relations [15].”

“• a set of constraints (symbolic, logical, spatial and temporal constraints including Allen’s interval algebra operators [2]) involving these variables.”

“INPUT: [1] An access request (u; o; p), [2] The authorization rules set AR”

“The interface implicitly includes the 13 temporal interval relations proposed in [ 1 ], although it only defines seven.”

“can be any of the possible relationships between time interval, firstly identified in [ 2 ] and later extended in [8].”

“For further references to interval temporal logics and their applications in such diverse areas and AI and Hardware Verification, see [ 5 , 64, 58, 7] 10 Michael Fisher & Richard Owens”

“Major work in AI has focused on de ning relationships between di ering time intervals and creating a calculus for the manipulation of these relations [1, 2, 3, 14, 15], the use of constraint propagation techniques to narrow ambiguous bounds on temporal statements [13, 17].”

“In particular, interval-based temporal logics built over Allen’s Interval Algebra [17] or Moszkowski’s Interval Temporal Logic[18] appear as adequate choices to support practical temporal reasoning for the class of CPS systems with above-mentioned characteristics.”

“Most timeinterval mining methods rely on Allen’s temporal relations [1] to represent the temporal logical relations among the time intervals.”

“…can be used for semantic Q/A , inference etc. to support decision-making. et al. , 1999] suffice; RA uses the relations of Interval Algebra (IA) [Allen, 1983] • Σ dyn : The set of fluents Φ = { φ 1 , ..., φ n } and events Θ = { θ 1 , ..., θ n } respectively characterise the dynamic properties…”

“This calculus has 218 basic relations, which is quite large when compared with RCC8 and Allen’s Interval Algebra [1].”

“These axioms were derived from Allen’s interval algebra (Allen, 1983).”

“The main principle of the composition tables, initially introduced by Allen [ 2 ]a nd applied to temporal intervals, is to support the derivation of the relationship between A and C given the relationships between A and B ,a ndB and C. Composition tables can be used for deriving transitive relationships and reasoning…”

“For our work, Allen’s taxonomy of temporal relationships is adopted to describe the basic relationships between intervals [4].”

“J.F. Allen (1983).”

“We use the relations between time intervals in Allen’s interval algebra [2].”

“Many existing models for temporal composition of multimedia data are based on the Allen relations [1].”

“Arbitrarily complex multimedia synchronization scenarios are built using a structuring minimalist approach based on the seven basic synchronization constructs of Allen’s logic of intervals [All83], [Ham72] depicted by Fig.”

“Further, to ensure tractable and complete reasoning, one must restrict the expressiveness of a knowledge representation language almost to the point of unuseability (except within restricted domains [1, 6]).”

“Arguments put forward by computer scientists for making the move towards intervals or periods, rather than points, have been partly similar to classical ones ('greater ease and naturalness of expression': Halpem & Shoham 1986, Kowalski & Sergot 1985), partly new, stressing computational advantages of period-based infe…”

“Draft part of Allen’s theory is based on his seminal interval temporal logic [Allen 1983, Allen and Hayes 1985]. Bennett and Galton [2004] propose Versatile Event Logic (VEL), which consists of a general temporal ontology and semantics encompassing many other representations such as the situation calculus and event ca…”

“CASEˆE allows for modeling temporal relations between events based on the Allen’s time calculus [1] as well as subevent relationships along the temporal dimension.”

“The most common approach to extract TIRPs is using Allen’s temporal relations [27], with seven relations to capture the state of two alphabetic time intervals against each other (e.g., overlap , equals , meets ).”

“In the case of overlapping, more relations are considered between these temporal events (Allen relationships) [31].”

“According to Allen’s interval algebra [41], there are 13 temporal relations between a1 and a2 .”

“2 Topological relations: when boundaries are known, Allen for 1D [2], Egenhofer for 2D [15] relations can be used (Fig.”

“:3) t Þj ð e ; t Þð e 0 ; h end for 11: end for 15: return D the Allen’s 13 qualitative temporal relations (Allen, 1983).”

“The optimized query in Section 9.2.2 shows the effect of these ranslation rules.”

“We base our temporal model on [1] in which logical events or propositions can be ordered consistently along a timeline and durations of events or propositions holding true can be derived from their relationships with others.”

“The algorithm never compares more than three arcs at a time and there are temporal networks where each subgraph of three arcs is consistent but where there is no consistent labeling for the whole graph, an example is given in (Allen, 1983).”

“Reasoning about temporal intervals addresses the problem that much of our temporal knowledge is relative and methods are needed that allow for significant imprecision in reasoning [16].”

“Generalizing Allen’s [1] well-known relations on temporal intervals to the spatial case, these describe the containment relations between two regionsand their boundaries.”

“As a result, the following qualitative angles can be inferred: [ 6 BCD]= Acute (22) [ 6 DBC]= SlightlyAcute Obtuse (23) (iii) Add qualitative angles [ 6 ABD] and [ 6 DBC] with the QADD inference rules, namely, entries QADD[1; 2], QADD[1; 3], QADD[1; 4], and QADD[1; 5] of Table 2.”

“This is the main difference between our approach and approaches taken by various interval logics [2], where intervals represent points in a two-dimensional (half-)space.”

“To solve this relation confusion problem, inspired by Allen interval algebra (Allen, 1983) and the neural-symbolic system in (Zhou et al., 2021), we decompose the decision of a temporal relation into questions about start time, end time, and duration.”

“two approaches: (1) Reasoning directly with Allen’s interval relations (shown in Table 1), by constructing rules like: If event X occurs before Y, and event Y before Z then X should happen before Z (Allen, 1990).”

“Depending on the nature of the time variables and the constraints used, BTR can be represented using different types of algebra, such as point algebra (PA), interval algebra (IA) or, the most known, Allen’s interval algebra (Allen 1983), in which the BTR is composed of the relations presented in Table 1.”

“Primitive of time Point Ideal for specifying accurate positions in time Scheduling, planning, temporal constraints and temporal relations are barely supported [7,8] Interval Ideal for representing coarse and incomplete temporal knowledge [9] Used by Allen’s temporal logic [10]”

“Allen algebra [7, 8], for instance, provides 13 relations corresponding to all the possible order relations between the beginnings and the ends of two intervals.”

“Essentially, the approach enhances ontology-based activity models with qualitative temporal information based on Allen’s temporal logic [7].”

“However, for the majority of models, time is seen as a continuous and linear line, consisting of points called ‘instants’, and identified by dates (Allen 1983).”

“Both ontologies adopted Allen’s Interval Based Temporal Logic [1], which provides a foundation of temporal logic for many temporal models.”

“algebraic intervals with a hierarchical relationship like Allen’s [22] interval, such a structure clearly appears in the expression ‘‘the first part of the XIXth Century’’; metric as in the expression ‘‘for 80 years’’; operation of sequencing as ‘‘in the next period’’.”

“A contrasting approach is the presumption of event boundaries, for example, on the basis of formal characterisations in artificial intelligence (Allen 1983) and qualitative spatial reasoning.”

“An extension of temporal logic [10] to two and three dimensions is then used to describe the inter-object relationships, as described in [11].”

“Note that these predicates are conceptually similar to interval arithmetic [1], [24].”

“Its advantage is being easily interfaced with a declarative symbolic representation based on a set of quantified Allen operators [ 3 ] together with a set of causal operators.”

“Similar to Allen's calculus for temporal reasoning [1], which is based on 13 elementary relations that can hold between time intervals, in RCC8, there are eight elementary topological relations that can hold between regions.”

“Allen [2] gives a temporal interval algebra (Table 1) for representing and reasoning about temporal relations between events represented as intervals.”

“1 Mapping Topological Relations onto Allen's Intervals Topological relations between objects are mapped onto topological relations between MBRs, as was done by both Clementini et al. (1994) and Papadias et al.”

“Allen (1983) introduced an algebra of temporal relations on intervals.”

“Reference [18] shows that the Face-Centered Cubic Lattice (F CC) model is a well-suited, realistic model for 3D conformations of proteins (see also References [10] [12] [13] [19]).”

“For the running and demo examples of this paper, positional relations on axis-aligned rectangles based on the Rectangle Algebra (RA) [5] suffice; RA uses the relations of Interval Algebra (IA) [2] RIA ≡ {before, after, during, contains, starts, started by, finishes, finished by, overlaps, overlapped by, meets, met by,…”

“5.2 Example I: Allen's Interval Algebra Recall the standard representation of Allen's Interval algebra over the rationals that we presented in Section 2.5.”

“Table 1 shows the detailed Allen’s temporal relations of activities [1], which position any pair of intervals by comparing their starting and ending time points to support qualitative temporal knowledge representation.”