An Interval-Based Representation of Temporal Knowledge

I. Introduction

This paper describes a method for representing and reasoning about temporal knowledge. While most work in artificial intelligence would appear to require some reasoning about time, this problem is typically avoided and delegated to "future research." In order to make further progress in the areas of problem solving and natural language understanding, these problems cannot be ignored any longer.

The techniques for modeling time that have been developed in problem-solving systems seem too weak for our purposes. The predominant temporal models in these systems are those using state variables and state spaces. In this world, time is represented as a sequence of instantaneous time slices. Each time slice is described by the set of facts that hold at that time. This has been a successful model in problem solving systems that deal with a single agent operating in a simple discrete world (such as the blocks world). Unfortunately, current problem solving work is attempting to deal with multiple agents, each acting not only on what is true now, but on what each expects to be true in the future. To treat these issues adequately, one needs a more flexible model of time. For further discussion of these issues, see [McDermott, 1978] and [Hayes, 1979).

The techniques for modeling time developed in natural language understanding systems are typically cruder than those in problem solving systems. In part, this is because many efforts have been directed at question answering systems in a static world (with no conception of time), or in domains small enough so that simple before­ after chains of relations are computationally feasible.

Let us consider what general characteristics such a temporal model should have. The following seem particularly relevant:

• --The model should allow significant imprecision of scale. Many temporal relations are stricOy relative and have little relation to some absolute sense of Lime.
• --The model should allow uncertainty. On many occasions, the exact relationship between two time intervals is not known, but some constraints are known about how they could be related.
• --The model should have a strong sense of now. This must be done in a manner that allows the present moment (now) to change without requiring major changes to the knowledge base.
• -- The model should allow a sense of persisrence. It should facilitate default inferences of the form, "if P is true now, it will remain true until I notice otherwise."

With a few notable exceptions, there has been little work done on computer representations of time. The work that there is, however, contains many interesting ideas which will be freely used in this paper. Kahn and Gorry [1977) suggest that the temporal aspects of knowledge can he separated out for intlividual treatment. They describe a system that is a Lime specialist; it maintains temporal relations and provides the rest of the system with the tools to test, retrieve, add, and delete temporal information. The time specialist knows nothing else about the knowledge representation. A similar approach is taken here: the model knows only about temporal relations, and provides the tools for manipulating this knowledge. This is one part of a larger effort to design and implement a temporal logic· which can describe and reason about temporally qualifietl propositions, events, actions, and plans [Allen, in progress]. Thus. the long-term goal of this work is similar to that in Hayes [1979) and McDermott [1981).

The model described here differs from the model of Kahn and Gorry, as well as that of Bruce [1972], in that it emphasizes the aspects of temporal reasoning that are not concerned with date lines. In particular, it allows relative temporal relationships to be maintained in a highly structured fash ion. l:urthermore. it includes a notion of the present time (i .e., "now"), which is maintained in a manner that does not require knowledge of the exact present time, and that does not involve extensive modifications to the knowledge base every time "now" is updated.

The system described here has been implemented in LISP on a VAX ll/780. In particular, the constraint propagation with reference intervals and the manipulations of the "now" are completed and fully tested. All examples in Sections II through V inclusive have actually been run on the system.

II. Time Points vs. Time Intervals

The references to temporal relations in English are often both implicit and vague. In particular, the majority of temporal references are implicitly introduced by tense and by the description of how events are related to other events. Thus we have

"We found the letter while John was away." "We found the letter after we made the decision."

These sentences are introducing temporal relations between the times (intervals) at which the events occurred. For example, the temporal connective "while" indicates that the time when the find event occurred is during the time when John was away. The tense indicates that John being away occurred in the past (i .e., before now).

Al though some events appear to be instantaneous (e.g. one might argue that the event "finding the letter" is instantaneous), the majority of events described take place over a time interval. Thus, our representation is designed to maintain these interval relations conveniently and concisely.

The most apparent scheme for representing intervals seems to be modeling the endpoints. Thus, if we assume a model consisting of a fully ordered set of points of time, then an interval is an ordered pair of points with the first point less than the second. We then can define the following relations between intervals (for any interval t, the lesser endpoint is denoted by t-; the greater by t+ ):

Thus we can now map our intervals and interval relations onto the simpler time point representation .

I am not going to implement intervals using this scheme. The main reason is that the representation is too uniform and does not facilitate structuring the knowledge in a way which is convenient for typical temporal reasoning tasks. The central issue here is the importance of the during relation. A major part of our temporal knowledge is of the form

"event E' occurred during event E"

and that our knowledge of the during relation allows a highly structured representation of time. In particular, a key fact used in testing whether some condition P holds during an interval t is that if t is during an interval T, and P holds during T, then P holds during t. Thus the during relation can be used to define a hierarchy of intervals in which properties can be inherited.

Furtherrno-,e, such a during hierarchy allows reasoning processes to be localized so that irrelevant facts are never considered. For instance, if one is concerned with what is true "today," one need consider only those intervals that are during "today," or above "today" in the during hierarchy. If a fact is indexed by an interval wholly contained by an interval representing "yesterday," then it cannot affect what is true now.

Whenever we need to refer to points, they will be explicitly related to an interval. Thus, we can talk about the beginning of an interval t (t- ). the end of t (t + ). or any arbitrary point during t. The only difference between such points and a standard interval is that points cannot participate in the overlaps relation, and the meet~· relatior: indicates that the point is an endpoint of the interval.

Ill. Maintaining Temporal Relations: The Simple View

We saw above five relations that can hold between intervals. Considering the inverses of these relations, there are a total of nine ways in which an ordered pair of intervals can be related. These are shown in Figure 2.

These relationships between intervals are maintained in a network where the nodes represent individual Intervals. Each arc is labelled to indicate the relationship between the two intervals represented by its nodes. In cases where there is uncertainty about the relationship, all possible cases are entered on the arc. Note that since the nine possible relationships are mutually exclusive, there is no ambiguity in this notation. Figure 3 contains some examples. Throughout, let Ni be the node representing interval i. Notice that the third set of conditions captures the one sense of the notion of disjoint intervals.

Relation

1. i during j

2. i during j or i before j or j during i

3. (i < j) or (i > j)

Network Representation

Ni -- (< d di) --> Nj

Ni -- (< >) --) Nj

Figure 3: Representing Knowledge of Temporal Relations in a Network

Throughout this paper, both the above notations will be used for the sake of readability. In general, if the arc asserts more than one possible relationship, the network form will be used, and in the case where only one relationship is possible, the relation form will be used.

In. this simple view, it is assumed that the network always maintains complete information about how the intervals in it could be related. Thus when a new fact is entered, its full consequences must be computed. This can be done by computing the transitive closure on the temporal relations as follows: the new fact adds a constraint about how the two intervals could be related, which may in turn introduce new constraints between other intervals because of the transitivity behavior of the temporal relationships. For instance, if the fact that i is during j is added, and j is before k, then it is inferred that i must be before k. This new fact is then added to the network in an identical fashion, possibly introducing further constraints on the relationship between other intervals. A few of the transitivity relations are summarized in Figure 4. The full transitivity table can be found in [Allen, 1981].

As an example of using this table: if A ajrer B. and n ~~rr.ng C, then follow the 2n.d row (">") to the 3rd column ( d ). to obtain A - (> 01 d mi) --> C.

A nice · prop~rty of this transitive closure algorithm is that It on.ly continues t? operate as long as it is inferring new relations between Intervals. If the intervals involved we~e p_re~iously related, then the algorithm only continues while It IS. further constraining the old values. The bad proper.ty is that it requires vast amounts of memory, since ~~ery Interval IS explicitly related to every other interval. I his problem IS addressed in the next section.

IV. Maintaining Temporal Relations Using Reference Intervals

We need some method in which to restrict the propagation of constraints throughout the network but to allow the inferences locally. Furthermore we should later be able to. i~fer the relati.onships that we~e not produced by the ongmal propagation of constraints. This latter conditio~ eliminates the possibility of restricting the propagation by only allowing propagation paths of some fixed length, for there seems to be no method available to recover the information not inferred except by a brute force search. A more structured view of the network is needed.

The solution described here allows the user of the system to determine what inferences are important to perform "automatically," and what ones can be deferred. It entails introducing the concept of reference intervals. Each time a relation is added, it is added with respect to a reference interval. The user of this system may set up the re_ference intervals as desired. In general, they probably will often reflect part of the during hierarchy, but also may reflect the semantic clustering of events.

The constraint propagation is then restricted to inferring new relationships between points that share a common reference interval. We shall allow time intervals to have more than one reference interval. In the graphical notation. reference intervals will be indicated in parentheses following each node. Thus the situation described above would be captured by the nodes

Throughout the paper, nodes will indicate their reference intervals only when it is relevant to the discussion. Since there is a fixed set of reference intervals for any one node, no confusion should arise.

Reference intervals are simply other intervals themsel ves, and so have their own reference intervals as well. Whi le it is up to the user of this system to choose what reference intervals are present, there are some particularly useful techniques worth mentioning.

The first type of interval useful as a reference is an interval that more or less divides a large set of time intervals into those before and those after. Kahn and Gorry [ 1977] use these types of reference intervals, and give examples such as "the birthdate of a person," "my graduation from school." These events divide the set of intervals relating to a person's life into those before and those after. In the representation here we get this behavior us ing two reference intervals, one for the time before the event, and one for the time afler.

As a simple example, let G be the time of my graduation from school, from which Gb (before G) ami Ga (after G) are defined as reference intervals. Let L be the time of my learning to play chess, and W be the time of my winn ing a state lottery. Assume further that L is before G, so is stored with respect to the reference interval Gb, and that W is afler G, so is stored with respect to the reference interval Ga. Since L and W do not share a co111mon reference interval, the explicit relationship between them is not computed or stored. This information is summarized below, using X as a dummy reference interval for Ga and Gb.

Gb(X) beforeGa(X) L(Gb) duringGb(X) W(Ga) during Ga(X)

To find the relation between L and W, we follow up the reference interval chains until we find an explicit relationship. Thus, from L we first find Gb, and from W we find Ga. It is explicitly asserted that Gb is before Ga. The trace of this search path indicates the facts relevant to in fer that L is before W using the transitivity table (Figure 4). Of course, the real benefit of this approach is only realized when there are many intervals that either reference Ga or that reference Gb.

Reference intervals can also be used to control inference that is more semantically or pragmatically based. For instance, in a dialogue system, one needs to keep track of the focus and topic of discussion. The discussion of one topic may introduce many facts that should be treated as a un it. Expl icit signals are often given to indicate changes in topic. In general, we do not want to examine the re la tionships between time intervals introduced across topics as closely as those introduced within a single topic. Thus we should introduce a new reference interval for each topic. ·

V. Maintainin~ the Present

For this representation to be truly useful in a dialogue system it r:nust be able to capture the notions of past, present, and future . Furthermore, since the "the present" is conti nually moving into "the future," this updating must not involve a large-scale reorganization of the data base each time it is done.

The method used here is simply to map "now" onto an Interval that is maintained in a similar manner to al l other intervals in the system. If we are clever about using Its reference mterval, so that most relations are inferred via the reference interval rather than directly, then updating "now" to a new interval that has the same reference interval should not be too expensive.

For example, let "now" be interval Nl, which is during its reference interval Rl. An example state of the data base would be

NJ(Rl) during Rl

Rl before Il, Rl after 12, Rl during 13

From this we can infer easily that the present (i.e., Nl) is during 13, before 11, and after 12. If "now" then is updated (slightly), N2 can be defined as the new "now" using the same reference interval by adding the facts

N2(Rl) during Rl, N2(Rl) after Nl(Rl)

Thus, "now" has been updated but most of the relations in the data base have been unaffected, for the effects of N2 will only propagate to intervals referenced by Rl. The reference interval R 1 has "protected" the rest of the data base from minor- change in the present moment. Of course, eventually "now" will cease to be during Rl and a new reference interval will be needed. This will involve a more major update to the data base, but the amount of work can be reduced if Rl itself has a reference interval that "protects" much of the data base from it.

Thus we need a hierarchy of reference intervals, each containing the present moment. This hierarchy should be designed to mirror the set of English terms that can be used to refer to the present. For example, in English we can refer to the exact moment of an utterance, as well as to larger intervals such as "this morning," "today," "during this lecture," and "while at this bar." These intervals typically have well defined starting and termination points. Thus it is reasonable to assume that the temporal data base will receive explicit notification when one of them ceases to contain the present. This allows the following important assumption:

when updatin~ the "now" interval, unless otherwise stated, its relationship to its reference interval(s) remains constant.

When one of the reference intervals in the hierarchy ceases to con lain the present moment, a new reference interval is selected. (This new interval should usually be provided by the user.) This update is done in the identical fashion as describeq above with "now." In particular, the relationship with the higher level reference interval remains constant. A new "now" interval, below the new reference interval in the hierarchy, must be introduced. For example, the beginning of a new day would make much of the old hierarchy part of the past (i.e., "yesterday").

While many intervals will be generated by this succession of "nows," a large number of them will have been used only to index the time of an utterance. Thus, they generally can be deleted without harm.

Consider the simple example in Figure S, which describes a possible hierarchy for the presen~ where my uttering the present sentence is during interval now2. Let indentation indicate a during relationship and the ordering from top to bottom indicate the before relation.

Thus, day3, which is today, contains the intervals am3 and pm3, and is also their reference interval. Interval am3 is before pm3. In turn, am3 contains 11 and amtalks, and is their reference interval. The intervals of the form nowi are the intervals that at one time or another were/will be part of the present. The underlined intervals give the structure that is currently present. Thus "now" refers to interval now2.

Given this state, we can infer that interval 13 is part of the past by tracing up the hierarchy from 13 until we find an explicit relationship with the present (the underlined nodes). Since I3 is during am3, which is before pm3, we can infer that 13 is before the present.

To update "now" without changing its reference point, we simply add a new interval now3, assert that now3 during talk and now2 before now3. The fact that 13 is still in the past after this update is calculated in the identical manner as before. The result is shown in Figure 6 (omitting the reference interval information).

To update a reference interval, say to update the interval talk when I finish my talk, the same operation as with a minor update is performed, and then a new interval to represent the new "now" is added. In many cases, however, the user will specify an already known interval to be the new reference (e.g. updating "today" to "tomorrow"). Assuming that aftertalk is the newly specified reference interval, we need only create a new "now" node, say now4.

In addition, when the reference node talk is updated, some of the old nodes that represented "now" could be deleted. In particular, now! , now2, and nowJ might not be needed any longer. The new hierarchy is shown in Figure 7.

VI. Persistence of Intervals

The last requirement described in the introduction was that the representation should facilitate plausible inferences of the form "if fact P is true now, it will remain true until noticed . otherwise." Most of the issues concerning this fall outside the range of this paper, as this system only knows about time intervals. However, a simple trick using this representation makes inferences of the above form easy to implement.

Typically, when a new fact is learned, its exact extent in time is not known. For instance, when I parked my car in the parking lot this morning I knew its location. Sitting at my desk now, I assume it is still there, though I have no proof of that fact. In general, I assume it will remain where it is until I pick it up. Thus, although I don't know the extent of the interval in which my car is parked, I want to be able to assume that this fact holds later in the day.

The temporal representation is already based on the observation that most time intervals do not have precisely defined limits. If we allow the user to specify that some intervals should be assumed to extend as far as possible given the constraints, then we can use such intervals to index facts that are assumed to persist until discovered otherwise.

Thus, if we let a fact P be indexed by a persistent interval Tp, then testing P later during an interval t will succeed (by assumption) if it is possible that t is during Tp. Che~,;king whether relationships between intervals are possible is easy, since the representation explicitly maintains this information.

For example, let Tp represent the interval in which my car is in the parking lot. I know that Tp is met by Tarrive, where Tarrive is the time that I arrived at school today. Then, let "now" be represented by the interval Tnow, where Tnow after Tarrive. We can conclude that my car is in the parking lot as follows. Since it is there during Tp, we are interested in whether it is possible that Tnow is during Tp. The known constraints allow us to infer the following:

Tp met by Tarrive, Tarrive before Tnow = > Tp--(< o di m)-> Tnow Thus it is possible that Tnow is during Tp, since it is possible that Tp contains ("di") Tnow. So the test succeeds.

Of course, if it is later learned that the car was found to be missing during some time interval Tmiss, then Tp is constrained to be before Tmiss (even though it is still persistent). If Tnow is then after or during Tmiss, then it is not possible any longer that Tnow is during Tp.

McDermott (1981] introduces a similar notion that he also c<J lls persistence (must be a good term!). His scheme involves specifying a (numeric) duration over which the proposition will be assumed to hold after it becomes true. In my scheme described here, limits are introduced by constraining the persistent interval to be before (or some other relation) some other interval. Thus our approaches differ on how central a role explicit duration information plays. I have explicitly avoided all duration scales here. but deal with them briefly in [Allen, 1981].

Managing a system such as this is a difficult problem that requires some form of truth maintenance (e.g. see [Doyle, 1979]). These issues, however, are independent of the temporal representation. All that is shown here is that the necessary temporal calculations are easily done within this framework .

VII. Summary

This paper argues that a model for representing temporal information such as that acquired in dialogues is more naturally interval-based rather than point-based. It describes a method of representing the relationships between temporal intervals in a hierarchical manner using constraint propagation techniques. lly using a notion of reference intervals, the amount of computation involved when adding a fact can be controlled in a predictable manner.

Evidence for the viability of this representation is presented by discussing a few important problems that arise in dialogues involving temporal reference. The first problem involved the representation of the present moment, which must be able to be updated frequently and thus must be done efficiently. The use of reference intervals to insulate the present moment from the bulk of the facts in the knowledge base was suggested as a promising technique.

The other problem discussed involved the ability to assume various conditions would hold until explicitly discovered otherwise. Since the techniques suggested already computed the set of possible relationships that could hold between two intervals, it was easy to implement this assumption mechanism. If an interval was labelled as persistent, then it was assumed to extend as far as possible given its constraints.

Ack nowlcdgerncnts

Many thanks to Jerry Feldman, Alan Frisch, Don Perlis, Ken Sloan. and the AI Study Group for their comments on a draft version of this paper, and for other general discussions of issues while the research was in progress.

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