Abstract
The ability to represent and manage temporal knowledge about the world is fundamental in humans as well as in artificial agents. Some examples of important “intelligent” activities that require time representation and reasoning include the following:
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Given that certain conditions, actions, and events occurred in the world prior to a time t,predict whether certain conditions will hold at t or whether certain actions can be executed or certain events will occur at or after t;
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Hypothesize conditions, actions, or events prior to a certain time t to explain why certain conditions, actions, or events occurred at or after t;
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Given a description of the world at a certain time t, plan a set of actions that can be executed in a certain order starting at t, to achieve a desired goal;
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Schedule a set of given activities to meet some constraints imposed on the order, duration, and temporal position or separation of the activities, such as a stipulated deadline;
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Given (possibly incomplete or uncertain) information about the temporal relations holding between events or facts in the represented domain, answer temporal queries about other implicit (entailed) relations; for example, queries about the possibility or the necessity that two particular future events will temporally overlap or about the shortest temporal distance separating two events that have occurred.
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Notes
We choose to deepen this subfield, partly because the book does not contain a specific contribution on this subject and partly because this is one of the areas in which the author has conducted active research for several years.
This section gives only quite a general introduction. For a more detailed treatment the interested reader is referred to Ladkin (1987b), Allen and Hayes (1989), van Benthem (1991), Galton (1995b), Galton’s chapter in this book (Chapter 10).
Other less commonly used primitive units are semintervals (Freksa, 1992a) and nonconvex intervals (Ladkin, 1986; Ligozat, 1990, citeyearLig91; Morris et al., 1993 ). Vilain (1982), Mein (1996), and more recently Jonsson, Drakengren, and Bäcström (1996) studied frameworks where both points and intervals are primitive units.
Suppose,for example, that the travel times of a train vary according to the kind and quantity of load transported.
As observed in Sandewall and Shoham (1995) the predicate Holds is commonly used in the context of SC, though it was not used in the original formulation of SC.
This example and the one that we will use to illustrate the qualification problem are borrowed from Shoham and Goyal (1988).
Other methodologies are briefly reviewed in Sandewall’s chapter of this book, Chapter 9.
For a critical survey of these and other methods see Sandewall and Shoham (1995) and Sandewall (1994a).
The actions that can be handled by uCPOP are a subset of those that can be expressed by Pednault’s action description language (ADL) (1989), which is more expressive than the STRIPS’S language. In particular, uCPOP operates with actions that have conditional effects, universally quantified goals, preconditions and effects, and disjunctive goals and preconditions (Penberthy and Weld, 1992).
From a computational point of view, when the events (actions) are partially ordered, and their descriptions correspond to instances of STRIPS operators, temporal projection is NP-hard (Nebel and Bäckström, 1994; Dean and Boddy, 1988b).
This is likely to be the case for instances of NP-complete reasoning problems that lie around critical values (a “phase transition”, Cheeseman et al., 1991) of certain parameters of the problem space. Ladkin and Reinefeld (1992; ming), and more recently Nebel (1996) identified phase transitions for the problem of determining the consistency of a set of assertions of relations in the interval algebra.
Allen and Hayes (1989) showed that one of these basic relations, meets, can be used to express all the others.
van Beek calls the problem of computing the deductive closure computing the “minimal labels” (between all pairs of intervals or points) in van Beek (1992) and computing the “feasible relations” in (van Beek, 1992).
IA contains 213 = 8192 relations, SIA contains 188 relations (Ladkin and Maddux, 1988a; van Beek and Cohen, 1990), and the ORD-Horn subclass is a strict superset of the relations in SIA containing 868 relations (Nebel and Biirckert, 1995).
On parallel machines the complexity of iterative local path-consistency algorithms lies asymptotically between n 2 and n 2 log n (Ladkin and Maddux, 1988a, 1994).
Two networks are equivalent when the variables represented admit the same set of consistent interpretations in both the representations. A relation R1 is stronger than a relation R2 if R1 implies R2. Also recall that the relations of IA and PA form algebras, and hence they are closed under the operation of composition, as well as under the operations of converse and intersection. For more details the interested reader may consult (Tarski, 1941; Ladkin and Maddux, 1994; Nebel and Bürckert, 1995; Hirsh, 1996 ).
Another algorithm for accomplishing this task has been proposed by Ligozat (1996a), without a worst-case complexity analysis.
Note that two such relations used in conjunction can also express disjointness of two intervals.
As discussed in (Dechter et al., 1991; Kautz and Ladkin, 1991), the TCSP model can easily be extended to the case where the intervals of the constraints are (semi)open, or have -/+ infinity as bounds.
Two temporal constraint networks are equivalent when the corresponding set of constraints admit exactly the same solutions.
Actually, the tasks of consistency checking and of finding a solution can be accomplished using O(e) space, where e is the number of the input constraints.
Another related study is Isli (1994).
Combined-metric-Allen is not able to infer certain strict inequalities entailed by the input set of constraints.
See Gerevini and Cristani (1995) for an example illustrating this source of incompleteness.
Recently Delgrande and Gupta proposed an interesting extension to the chain partitioning of a timegraph that is based on “serial-parallel graphs” (Delgrande and Gupta, 1996).
Some related studies on “fuzzy” temporal reasoning are 6Dubois and Prade 1989, Console et al. (1991), Godo and Vila (1995).
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Gerevini, A. (1997). Reasoning about Time and Actions in Artificial Intelligence: Major Issues. In: Stock, O. (eds) Spatial and Temporal Reasoning. Springer, Dordrecht. https://doi.org/10.1007/978-0-585-28322-7_2
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